Flux Boundary Condition Research Articles

Three weak solutions for a class of fourth order p(x)-Kirchhoff type problem with Leray-Lions operators

In this work, we study the multiplicity of a weak solution for a fourth order p(x)-Kirchhoff type problem involving the Leray-Lions type operators with no flux boundary condition. By using variational approach and critical point theory, we determine an open interval of parameters for which our problem admits at least three distinct weak solutions.

Two-Fluid and Discrete Element Modeling of a Parallel Plate Fluidized Bed Heat Exchanger for Concentrating Solar Power

Abstract A novel high-temperature particle solar receiver is developed using a light trapping planar cavity configuration. As particles fall through the cavity, the concentrated solar radiation warms the boundaries of the receiver and in turn heats the particles. Particles flow through the system, forming a fluidized bed at the lower section, leaving the system from the bottom at a constant flowrate. Air is introduced to the system as the fluidizing medium to improve particle heat transfer and mixing. A laboratory scale cavity receiver is built by collaborators at the Colorado School of Mines and their data are used for model validation. In this experimental setup, near IR quartz lamp is used to provide flux to the vertical wall of the heat exchanger. The system is modeled using the discrete element method and a continuum two-fluid method. The computational model matches the experimental system size and the particle size distribution is assumed monodisperse. A new continuum conduction model that accounts for the effects of solid concentration is implemented, and the heat flux boundary condition matches the experimental setup. Radiative heat transfer is estimated using a widely used correlation during the post-processing step to determine an overall heat transfer coefficient. The model is validated against testing data and achieves less than 30% discrepancy and a heat transfer coefficient greater than 1000 W/m2 K.

Simulation Study on Acetylene Gas Diffusion Process in Large Transformers

To discover the influence of the diffusion process of acetylene gas in large oil filled power transformers on the dissolved gases analysis results, the diffusion process of acetylene gas over long distances and the influencing factors have been investigated by simulation. In this paper, based on the convection-diffusion equation of acetylene gas in oil, a one-dimensional model for the diffusion of acetylene gas in a 10 m long oil duct is established and numerically solved by the backward difference method. The simulation found that in right-angle coordinates, the concentration of acetylene gas in the oil is uniformly diffused outward, and the arrival time of the gas is related to the flow rate; in polar coordinates, the acetylene gas is mainly concentrated in the area near the defects and the tank wall, and the distribution of the gas content is extremely inhomogeneous, and the homogeneous state can not be reached for a long time. The results show that the simulation of gas production from local defects in large transformers is closer to reality by using polar coordinate form and boundary flux conditions; because the diffusion coefficient is very small in magnitude, the slow flow of transformer oil leads to the predominant convective transfer of acetylene gas.

Heat transfer inside a horizontal circular enclosure with a heated semi-circular cylinder at different orientations

Natural convection inside a cylindrical enclosure is important in many industrial applications. Laminar natural convective heat transfer in the annular space between heated semi-circular cylinder in a horizontal concentric cylindrical enclosure, which has relevance to cooling of electronic components, is studied in this work. Constant wall temperature (CWT) and constant wall heat flux (CWHF) boundary conditions are considered on the inner cylinder, whereas the outer cylindrical enclosure is maintained at a fixed lower temperature. The Finite Volume Method (FVM) is used for the computations. The effect of the Rayleigh number ( Ra = 10 2 to 10 5 ) and the angular orientation of the semi-circular cylinder on the convective heat transfer is investigated. The fluid flow, temperature profile, and heat flow are visualized via streamlines, isotherms, and heatlines. Further, the overall heat transfer is explained based on the local and average Nusselt number on the inner cylinder wall. The effect of the angular orientation of the inner cylinder is noticeable for Ra ≥ 10 3 . The angular orientation ϕ = 90 ° greatly enhances the heat transfer as compared to all other orientations for all Ra ≥ 10 3 for both CWT and CWHF boundary conditions. The correlation between the average Nusselt and Rayleigh numbers for a particular orientation is proposed.

DNS vs. LES: Turbulent flow in square duct with heated foil boundary

Turbulent heat transfer in a square duct, where a finite section of a single duct boundary is heated with an infinitely thin foil, was analyzed with LES and DNS approaches. Both computations were performed with OpenFOAM code with passive scalar approximation, using second order accurate schemes. Results are obtained for the key velocity and temperature statistics in the duct, with an emphasis on the temperature profiles and temperature fluctuations at the foil boundary, which appear due to the imposed constant heat flux boundary condition at the foil. Comparison of DNS and standard LES-WALE model show 20% higher foil temperatures and temperature fluctuations in LES results. These differences are explained with detailed analysis of the LES-WALE model near-wall behavior. It has been demonstrated that LES-WALE yields highly accurate results only when employed with a finely resolved mesh. However, the computational costs associated with such refinement may not be necessarily justified especially with respect to DNS.

Numerical modelling of thermal jet assisted rock cutting with double PDC cutters

AbstractPreconditioning of rock for drilling operations is a potential method to facilitate the mechanical breakage and mitigate the tool wear. This paper numerically investigates one such preconditioning technique, namely, the thermal jet assisted rock cutting. For this end, a numerical method for solving the governing thermo‐mechanical problem is developed and validated. The continuum approach is chosen to describe the rock failure, being based on a damage‐viscoplasticity model with the Drucker–Prager yield surface and the modified Rankine surface as the tensile cut‐off. In the damage part, separate scalar damage variables are employed for tension and compression. The cutters are modelled as rigid bodies and their interaction with the rock is modelled by imposing contact constraints by the forward increment Lagrange multiplier method, which is compatible with the chosen explicit time integration scheme. A damage‐based erosion criterion is applied to remove the contact nodes surrounded by heavily damaged elements. The thermal jets are modelled with a moving external heat flux boundary condition. The global thermo‐mechanical problem is solved with a staggered approach explicitly in time while applying mass scaling to increase the critical time step. The novel 3D numerical simulations involving two cutters demonstrate the capabilities of the method with a special emphasis on cutter‐thermal jet configurations. Therefore, the present method provides a potential tool for the bit design in thermal jet assisted drilling.

Thermal entropy generation and exergy efficiency analysis of rGO/water nanofluid in a tube under turbulent regime using experimental and fully connected neural network

Analyzing entropy generation is one of the most effective methods for examining how well thermal systems operate. To determine the ideal operating conditions, numerous researchers have examined the entropy generation of thermal systems. In this work, the effects of concentrations of reduced graphene oxide (rGO) nanoparticles on the formation of entropy in water-rGO nanofluid flow through a circular pipe with a constant wall heat flux boundary condition in turbulent regimes are investigated experimentally. rGOs were prepared from Hummer's technique. Experiments are conducted at particle volumes from 0.5 % to 2.0 %, mass flow rates from 0.0333 kg/s to 0.25 kg/s, and Reynolds numbers from 1800 to 21,000, respectively. Later, estimated data points were anticipated with a convolutional neural network (CNN). The model was analyzed through several performances like root-mean-square-error (RMSE), mean-square-error (MSE), mean-absolute-percentage-error (MAE), and the coefficient of determination (R2). Results indicated, with used by 2.0 vol% of nanofluid, entropy generation-thermal is reduced to 39.24 %, entropy generation-frictional is higher by 16.92 %, entropy generation-total is decreased by 37.82 %, and second law efficiency is augmented to 34.17 % at Reynolds number of 13,705, against base fluid data. The correlation coefficient R2 evaluated from CNN for thermal entropy generation, frictional entropy generation, total entropy generation, and exergy efficiency are 0.99297, 0.99924, 0.9989, and 0.99273, respectively.

On a diffusive epidemic model with the tendency to move away from the infectious diseases

This paper is purported to investigate an epidemic system with the tendency of the susceptible to move away from the infectious diseases in a bounded domain with no flux boundary condition. The local and global stabilities of positive equilibrium are investigated to this system without cross-diffusion. The sufficient conditions to nonexistence and existence of non-constant positive solution are considered for this epidemic system. The results show that there exists spatiotemporal pattern formation when the tendency of susceptible individuals is strong to move away from the infectious diseases with 1<R0<1+min⁡{dSd,βσ1Aσ2}, while there does not exist spatiotemporal pattern formation when the self pressure is big enough with a weak tendency of the susceptible individuals to keep away from the infectious individual. Moreover, there exists positive solution bifurcation from the semi-trivial solution when the basic reproduction number is equal to one, which implies that endemic disease exists locally.

Saturated Porous Ferroconvection in a Ferrofluid Layer with Viscosity as a Function of Magnetic Field: Focus on Convective Boundary Condition

The present work aims to examine the influence of magnetic field dependent (MFD) viscosity on the onset of ferroconvection (FC) in a horizontal porous layer saturated with a quiescent ferrofluid (FF) and subjected to a uniform vertical magnetic field. It is assumed that the porous boundaries at the bottom and top are rigid-paramagnetic. The thermal conditions consist of a constant heat flux at the lower surface and a convective boundary condition at the upper surface, encompassing fixed temperature and uniform heat flux cases. The application of the Galerkin technique to the resulting eigenvalue problem reveals that the stability region expands as the porous parameter, Biot number, MFD viscosity parameter and magnetic susceptibility increase in magnitude. Conversely, the stability region contracts as the magnetic number and non-linearity of magnetization increase. Furthermore, it is noted that under uniform heat flux boundary conditions, the criterion for the initiation of ferroconvection remains unaffected by the non-linearity of fluid magnetization.

Experimentally determining the thermophysical properties, heat transfer and friction factor Fe3O4-TiO2 magnetic hybrid nanofluids in a mini-heat sink under magnetic field: Proposing new correlations

The heat transfer and friction factor for the Fe3O4-TiO2/water nanofluids circulating in a mini-heat sink of comparable magnitude to personal computer or electronic gadgets with and without a magnetic field have been estimated experimentally. The Fe3O4-TiO2 nanomaterial was prepared using chemical coprecipitation and sol–gel technique. Synthesized Fe3O4-TiO2 nanoparticles were characterized by X-ray diffraction, and vibrating sample magnetometer. The water-based hybrid nanofluids were then used in a series of experiments for assessing heat transfer, and friction factor of a mini-heat sink assembly of five circular channels with 4 mm in diameter and 50 mm in length. The experiments were performed in the operating range: 238.66 < Re < 1873.36; 0 <ϕ < 2 %; and 0 < Gauss < 1000, respectively. Constant heat flux boundary condition was maintained at the bottom surface of the heat sink and an external magnetic field was provided with an electromagnet. Results indicate that, by using the Fe3O4-TiO2 hybrid nanofluid of 2 % vol. the Nusselt number, with no magnetic field, is enhanced over that of water by 38.16 %, and by 88.93 % when the magnetic field of 1000 gauss is applied. However, the hybrid nanofluid of 2 vol% imposed a penalty in increase in friction factor of 25.87 % without magnetic field, and 67.64 % when magnetic field of 1000 G is applied at a Reynolds number of 1873.33 over water data. The experimental data were used to develop generalized Nusselt number and friction factor correlations for a mini-heat sink subjected to an external magnetic field.

Chemically reactive aspects of stagnation-point boundary layer flow of second-grade nanofluid over an exponentially stretching surface

Assessing the stagnation-point flow of a second-grade nanofluid with an induced magnetic field and the effects of Joule heating toward an exponentially extending surface is the leading goal of the current effort. The modeling of the thermal and solute energy equations has taken into account nonlinear thermal radiation as well as the impact of activation energy. The boundary of the sheet is subjected to the zero-mass flux and thermal slip boundary conditions. The simulated flow equations are converted into coupled nonlinear ODEs (ordinary differential equations) using similarity variables. The BVP4C MATLAB technique is used to numerically resolve these ODEs. Graphs and tabular data are used to perform the physical debate. The importance of a greater estimation of the magnetic Prandtl number and magnetic parameter is emphasized in order to obtain a better induced magnetic field profile. Additionally, the greater estimations of the second-order fluid parameter increase the rates of friction and heat transmission.

Lattice Boltzmann Method Simulation of Nanofluid Natural Convection Heat Transfer in a Square Cavity with Constant Heat Flux at Walls

In this study, for the first time, a nanofluid's natural convection heat transfer in a two-dimensional square cavity has been numerically investigated by use of the lattice Boltzmann method with the constant heat flux boundary condition. The horizontal walls of the cavity are insulated, and the vertical walls are kept at a constant heat flux. The diameters of the nanoparticles inside the cavity are the same and have a homogeneous distribution, and there is no chemical reaction between the particles. The flow is also assumed to be the steady state and two-dimensional. Constant temperature, streamlines, velocity, and average Nusselt have been investigated for different nanoparticle volume fractions and Rayleigh numbers. The results showed that the lattice Boltzmann method efficiently analyzes the natural heat transfer of nanofluids; moreover, by use of nanofluid in the cavity increases the heat transfer rate. With the increase in the nanoparticle volume fraction, the average Nusselt number on the right wall of the cavity increased. For a volume fraction of 20% with Grashof number 105, the average Nusselt number increased by almost 50% compared to the base fluid at the same Grashof number. It has been observed that as the volume fraction of nanoparticles in the fluid increases, the fluid’s viscosity also increases; consequently, the velocity of the fluid is found to decrease.

Practical considerations for the adoption of Anderson acceleration in nonlinear diffusion accelerated transport

The use of the Anderson Acceleration (AA) method within the context of the coarse mesh finite difference (CMFD)-accelerated neutron transport k-eigenvalue calculation is explored for practical problems. Current application of CMFD only has a linear convergence rate. The AA method is similar to a quasi-Newton method and can improve the convergence rate of fixed-point iterations. We therefore implement AA for the 2D calculation in MPACT and study its performance in pressurized water reactor (PWR) problems. The initial results show that the traditional AA method can improve the convergence rate in a trivial problem and improve the stability of the CMFD method for some problems. However, the success of AA in the practical problems, where simulations with symmetry or domain decomposition, relies on one essential factor–the treatment of iterative angular flux boundary conditions in the AA. Additionally, the efficiency of AA is also affected by the computational cost for solving the optimization problem, and the computational cost of the transport solver and the CMFD solver. To address these issues, we propose a new implementation of AA that considers only the mutligroup scalar flux in the optimization problem, but the angular flux and the scalar flux in the mixing. The new approach achieves better stability and efficiency in practical simulations, and can be explained by the theory of Randomized Numerical Linear Algebra (RandNLA).

Time-periodic natural convection in a pentagonal house with pitched roof during one day

A numerical investigation of unsteady natural convective flow and heat transfer in a pentagonal house is performed. We set a sinusoidal periodic heat flux boundary condition for the pitched roof. Local radial basis function method and Fourier spectral method are adopted to solve the governing equations. The Rayleigh number varies from 103 to 105, the period of the roof heat flux ranges from 10 to 1,000, and the amplitude is set as 0.1, 0.3, and 0.5, respectively. It is found that all the control parameters significantly impact the fluid flow and heat transfer. The temperature and flow fields exhibit periodic behavior due to the sinusoidal heat source. The increasing Rayleigh number leads to a larger periodic average Nusselt number of the top and right walls and a smaller value of the bottom wall. Moreover, an obvious hysteresis phenomenon of heat transfer occurs especially for high Rayleigh number and large period.

Effects of different cross-sections of Body Centered Cubic cells on pressure drop and heat transfer of additively manufactured heat sinks

In many industrial applications, heat loads management requires the design and production of compact heat exchangers which are expected to handle high thermal loads with acceptable pressure losses, while assuring good mechanical performances. These challenging targets can be achieved by filling the cavities where the cool/hot fluid circulates with lattice structures promoting the heat exchange between the fluid and the cavity boundaries. Such lattice structures can be only produced through Additive Manufacturing due to their high geometric complexity. Recent experimental investigations proved the effectiveness of some kinds of lattice structures having a circular cross section. Here the aerothermal behaviour of Body-Centred Cubic (BCC) lattice stagger arrays in a rectangular channel was experimentally investigated by considerably extending the previous studies to higher Reynolds numbers (up to 30′000) and to new types of lattice structures. Specifically, three new BCC structures having a cam-like, drop-like and elliptical cross section were explored in this work and compared against those having circular cross section. All the samples were manufactured by means of Laser Powder Bed Fusion and made from AlSi10Mg. At first, the heat exchangers were comprehensively characterized by means of optical non-destructive methods. Successively they were tested in a dedicated rig by imposing constant heat flux boundary conditions. The characteristics of the transitional or fully turbulent approaching flow to the test section are also reported thanks to dedicated flow field measurements performed by Particle Image Velocimetry. According to the obtained results, the BCC structure with the circular cross section of larger diameter is the most effective in terms of heat transfer, although it is largely penalized by the pressure losses. Similar heat transfer performances were achieved by the tapered cross section of elliptical shape with the advantage of a considerably lower friction factor. Pressure losses resulted almost identical for all the tapered cross sections but lower than those of the circular one having an equal frontal dimension. When considering the thermal performance factor the circular shape becomes unfavourable for Re>20′000, while the elliptical cross section is the best choice to efficiently promote heat transfer up to Re=30′000.

CONVECTIVE HEAT TRANSFER INSIDE A ROTATING HELICAL PIPE FILLED WITH SATURATED POROUS MEDIA

In this paper, we study the hydrothermal characteristics of flow inside a rotating helical pipe filled with saturated homogeneous porous medium. The analysis is being carried out for the case of small curvature and torsion. Using the perturbation approach, velocity and temperature fields are solved for both uniform wall heat flux and uniform wall temperature boundary conditions. Perturbation expansion up to the third order is carried out to investigate the effect of rotation on the flow. The influence of rotation on velocity is noticed as early as the first order, and on temperature solution, it has an effect in the third order. The influence of rotation on the Nusselt number does not appear till third order, and it is discovered that the Nusselt number grows as dimensionless curvature increases. Moreover, the theoretical results have been verified against experimental data from existing literature for the special case of zero rotation and curvature of the pipe. The available experimental data align well with and support the theoretical results in this limiting case.

Geotechnical Properties of Residual Diyarbakir Basaltic Clay in Turkey

Diyarbakir province mean rock formation is basaltic due to its location. Basaltic parent rocks release much silt or clay when imposed to chemical or physical weathering. The soils are residual in origin. In other words, they formed in situ where the basaltic parent rocks exist. They were not transported by means of wind or water. In this research, firstly, the properties of Diyarbakir basaltic clays were investigated from literature. After that, one soil sample from Kayapinar vicinity has been taken to laboratory to find index properties. The values from two different locations which are obtained from literature and values which are obtained from laboratory were compared. The results indicated that the three samples from different locations showed roughly the same index properties. As per the results, the clay is high plastic (CH), and it has swelling potential. In this thesis, design parameters of posttensioned slabs on ground under different climatic scenarios were found. The widely used method for slabs on ground under swelling soils is the Post‐Tensioning Institute (PTI) method. In order to use this method, Thornthwaite moisture index is required. This index is used in geotechnical engineering to identify moisture flux boundary conditions for unsaturated soils and depth of active zone for expansive soils. This index has been found using water balance sheet tables for Diyarbakir. So as to find the future moisture index, water balance sheet tables have been formed finding the climate projection data for Turkey in literature. In this way, five moisture index values of Diyarbakir have been found for the current situation and with regard to representative concentration pathway (RCP4.5 and RCP8.5) scenarios for 2050. The parameters for the maximum differential swell (em) and the angular moisture change index (ym) for design procedure according to the PTI method were found for each moisture index. Results showed that the ym and em values increased when the Thornthwaite moisture index is reduced. Also results showed that the slab thickness is increased when moisture index is reduced. Considering the moisture index parameters for future values, Diyarbakir is likely to face arid climates in the future.

IMPACT OF MEASUREMENTS TECHNIQUES ON HEAT TRANSFER CHARACTERISTICS IN AIR JET ARRAYS

Air impact processes have diverse applications in engineering, including backflow welding, textile drying, and gas turbine blade and combustion liner cooling. This research examines the influence of experimental methodologies and measurement tools on convective heat transfer in adjustable air jet assemblies. The experiment involves the use of heated targets made of thin stainless steel foil with constant heat flux boundary conditions. Thermography measures target surface temperatures by analyzing how internal passage cross-flow affects convective heat transfer via outflow adjustments. The experiments involve two arrays of jet nozzles: inline and staggering, each comprising 44 impingement jet nozzles arranged in 4 rows with 11 jet holes in each row. The study presents unsteady time average local and spatial Nusselt numbers as functions of jet Reynolds number (4630-14000) and explores their dependence on the jet nozzle diameter. Cross-flow levels significantly affect spatial and local Nusselt numbers in both local and span-wise averaged values, regardless of the Reynolds number. Strong cross-flow (single configuration) distributes flow causing turbulence and uneven heat distribution, reducing Nusselt numbers. In contrast, moderate cross-flow (double configuration) improves heat transfer and increases Nusselt numbers. The study emphasizes the crucial role of experimental techniques in heat transfer evaluation and demonstrates agreement with prior studies within a standard error below 5%.

Heat Transfer Augmentation for Impingement of Steady Air Jet under Exponential Heat Flux Boundary Condition

Heat Transfer Augmentation for Impingement of Steady Air Jet under Exponential Heat Flux Boundary Condition

PARAMETRIC STUDY AND DEVELOPMENT OF A CORRELATION FOR NUSSELT NUMBER IN NANOFLUID JET IMPINGEMENT

Numerical simulations for nanofluid ( water with Al&lt;sub&gt;2&lt;/sub&gt;O&lt;sub&gt;3&lt;/sub&gt; nanoparticles) jet impinging perpendicularly on a flat circular heated p late have been performed. A heated p late is subjected to constant heat flux boundary condition. A two-phase modified mixture mo del was used for the prediction of heat transfer coefficient, and comparisons are made with standard mixture model. Present results for average Nusselt number are validated with experimental data available in the literature. Though a standard mixture model predicted heat transfer with accepted accuracy, it was found that accuracy of modified mixture model is better (around 5&amp;#37; improvement) compared to standard mixture model. Thereafter, parametric study was performed considering nozzle exit Reynolds number (Re), spacing ratio (&lt;i&gt;H/D&lt;/i&gt;), nanoparticle volume fraction (&amp;phi;), and nanoparticle diameter (dp) on heat transfer prediction. The results reveal that particle diameter in the range 10-100 nm has no effect on the Nusselt number, Furthermore, heat transfer increased with increasing Reynolds number and volume fraction. However, spacing ratio shows first increasing and then, decreasing trend (similar to a log-normal distribution curve) in the prediction of heat transfer. Finally, a new correlation was developed for Nusselt number using nonlinear regression analysis. In the correlation, a two-phase multiplier was used, which is the ratio between two-phase Nusselt number (Nu&lt;sub&gt;nf&lt;/sub&gt; ) and single-phase Nusselt number (Nu&lt;sub&gt;sp&lt;/sub&gt;). The simplified correlation is found to predict data with maximum error of 8.9&amp;#37;, average error of 2.74&amp;#37; and &lt;i&gt;R&lt;/i&gt;&lt;sup&gt;2&lt;/sup&gt; &amp;#61; 0.955.