Constant Boundary Condition Research Articles

Experimental Analysis of Additively Manufactured Latticework Coupons

Abstract The capabilities and the precision of additive manufacturing processes have been tremendously increased in the last years, and this trend is not expected to be close to an end. Researchers, in the gas turbine industry, are focusing on the technology implementation on cooling systems. The aim of the present work is to investigate the performances of eight different geometries in order to exploit the cooling potential of some challenging latticework schemes with respect to traditional ones such as smooth channel, dimples, and pin fins. Test coupons consisting of those cooling structures embedded inside a rectangular cooling channel with dimensions of 56 mm width, 5 mm height, and 68 mm length were fabricated by means of direct metal laser melting technique. Since an aim of the present work is to evaluate the shape deformation associated with this additive process, the results of the CT scan imaging technique are presented. The performances are characterized both in terms of averaged heat transfer and friction factor values inside the test coupons by means of steady-state technique with a constant wall temperature boundary condition. The investigated flow Reynolds numbers range from 18, 000 to 40, 000. These values allow the authors to provide a deeper understanding of the latticework structures performances at Reynolds numbers beyond the range typically investigated in the literature. An efficiency index is employed to compare the performances of each geometry. Although the obtained results reveal that none of the tested geometries outperforms the other ones, the latticework arrangements show a heat transfer enhancement with respect to a porosity decrease. In addition, the good capability in terms of heat transfer of the lattice structures measured in the present work suggests the use of such technologies for twofold optimization process based on mechanical resistance and heat transfer characteristics.

Heat transfer enhancement of nanofluids in a four-blade LPD static mixer

The heat transfer performance of nanofluids within a four-bladed LPD static mixer was investigated using a combination of experiments and simulations under constant heat flow boundary conditions with Re ranging from 3519 to 16719. Al2O3-H2O nanofluid with volume fraction φ = 0.3 % was prepared experimentally and the stability of the nanofluid was analyzed by settling observation method and Dynamic light scattering (DLS). Numerical simulations were adopted to analyze the effects of nanoparticle types, volume fractions and types of mixing nanofluids on the heat transfer enhancement inside the four-bladed LPD static mixer. The results show that the maximum error of Nusselt number (Nu) between the predicted values of Mixture model and experimental values was 7.39 %. Because of the combined heat transfer performance coefficient PEC more than 1 with the increasing of Re, the mixers with α = 30° and β = 45° instead of KSM could be recommended when Re > 5279. The PEC of Al2O3-H2O nanofluids gradually increases with the increasing φ. At the same Re and φ, the flied synergy number Fc and PEC of the mixed nanofluids are 1.01–1.47 times and 1.15–3.35 times higher than those of the single nanofluids.

On Dirichlet Problem of Time-Fractional Advection-Diffusion Equation

The significant motivation behind this research article is to utilize a technique depending upon acertain variant of the integral transform (Fourier and Laplace) to investigate the basic solution for theDirichlet problem with constant boundary conditions. The time-fractional derivative one-dimensional,the equation of advection-diffusion and the Liouville-Caputo fractional derivative in a line fragment areintroduced. We also illustrate the results using graphical representations

Investigating heat transmission in a wellbore for Low-Temperature, Open-Loop geothermal systems

Heat transfer processes in geothermal wellbores are an essential component to overall system performance, but can be overlooked in subsurface modelling studies that tend to focus on reservoir response. This study addresses heat transfer in the wellbore through a comprehensive modelling study of 10 different parameters on OpenGeoSys software designed to evaluate heat losses during open-loop production conditions for typical low-temperature (<100 °C) single phase geothermal systems. Models were set-up to focus on wellbore effects only, using a single production well with constant fluid inlet temperature boundary condition. Results indicate that under base case conditions for a 2-km deep well surrounded by rock formations with a thermal conductivity of 2.5 W/(m.K) and bottom-hole temperature of 60 °C, the difference in inlet (reservoir) and production (wellhead) temperature at the end of a 40-year production period is 2.06 °C. This corresponds to minimum heat losses into the surrounding formations of −63.3 W/m, which is ∼ 7 % of the thermal power recorded at the wellhead (assuming a rejection temperature of 30 °C) and a 3.4 % difference to bottom-hole temperature. These losses in heat can be significant, particularly when in combination with surface losses through the heat exchanger or in the reservoir through thermal breakthrough. Wellbore insulation can reduce losses, but it would appear this only impacts the short-term. Wellbore performance can also be improved, with heat losses minimized, when developing wells above the reservoir interval in low thermal conductivity rock with high geothermal gradients.

Numerical simulation of fluid flow and heat transfer in steady state laminar natural convection in horizontal circular cylindrical enclosures

Heat transfer by natural convection often occurs in a closed enclosure in various applications. Different geometries of the cross section of a two-dimensional enclosure, such as triangle, rectangle, and trapezoid have already been investigated. In this research, a regular polygon with twelve sides is considered. The study focuses on the rate of heat transfer through the enclosure, using one side of the polygon as the hot surface and another side as the cold surface. The dodecagon can serve as a model for circular cylindrical enclosures under constant temperature boundary conditions over a specified length. The Nusselt number of the enclosure is obtained for each side as the hot surface. Both the lowest and highest Nusselt numbers are evaluated based on the position of the cold side. The findings reveal that the highest Nusselt number does not necessarily correspond to configurations where the hot and cold sides are directly opposite each other. In situations where convection heat transfer through the enclosure is the predominant mode of heat transfer, the Nusselt number is a power function of the Rayleigh number. Conversely, a logarithmic relationship is observed in cases of conduction heat transfer within the enclosure.

Effects of buoyancy in a laminar flow of supercritical carbon dioxide in a horizontal microchannel at different gravities

The effects of buoyancy were generally neglected in convective heat transfer inside horizontal microchannels because gravitational effects diminish compared to inertial and viscous forces with diminishing length scale. However, buoyancy effects can be significant for supercritical carbon dioxide (sCO2) slightly above the critical condition in microchannels, and the heat transfer coefficient can peak because of abrupt changes in the fluid's thermophysical properties. Previous studies considered buoyancy in channels with hydraulic diameter above 500 μm at a constant surface heat flux boundary condition. These effects were not well understood for microchannels with a hydraulic diameter as small as 100 μm, particularly with constant wall temperature boundary condition, but they hold great potential in the design of micro heat exchangers.This study numerically investigated the effects of buoyancy inside a 100-μm hydraulic diameter square microchannel with a total length of 27 mm and a heated length of 25 mm for laminar flow of sCO2. A total of 180 cases were analyzed to predict the effects of surface temperature ranging from 310 to 350 K, pressure ranging from 8 to 10 MPa, Reynolds numbers from 100 to 500, and gravitational acceleration from 0 g to 3 g, on the heat transfer. It was revealed that gravity has a significant effect on the heat transfer coefficient and on the thermal entrance length. The average heat transfer coefficient was increased by up to 138% on the bottom surface of the channel for 3 g compared to 0 g. The thermal entrance length showed a mixed behavior, such as it was increased for certain conditions while decreased for other conditions. For some cases, it was also observed that the pressure drop increased slightly with increasing gravity, and it was directly proportional to the wall shear stress. Finally, it was concluded that the gravity had a significant effect on the flow and heat transfer characteristics but was negligible for Richardson number below 10−4.

Thermal-hydraulic and load following performance analysis of a heat pipe cooled reactor

Heat pipe cooled reactors have gained attention as a potential solution for nuclear power generation in space and deep sea applications because of their simple design, scalability, safety and reliability. However, under complex operating conditions, a control strategy for variable load operation is necessary. This paper presents a two-dimensional transient characteristics analysis program for a heat pipe cooled reactor and proposes a variable load control strategy using the recuperator bypass (CSURB). The program was verified against previous studies, and steady-state and step-load operating conditions were calculated. For normal operating condition, the predicted temperature distribution with constant heat pipe temperature boundary conditions agrees well with the literature, with a maximum temperature difference of 0.4 K. With the implementation of the control strategy using the recuperator bypass (CSURB) proposed in this paper, it becomes feasible to achieve variable load operation and return the system to a steady state solely through the self-regulation of the reactor, without the need to operate the control drum. The average temperature difference of the fuel does not exceed 1 % at the four power levels of 70 %,80 %, 90 % and 100 % Full power. The output power of the turbine can match the load change process, and the temperature difference between the inlet and outlet of the turbine increases as the power decreases.

Cyclic Heat Transfer Solver for OpenFOAM

Channels with periodically repeating geometries are often simulated using periodic or cyclic boundary conditions. By calculating the temperature and flow field in one periodic module, the resulting distributions can be generalized to multiple modules. This reduces the computational load by simulating a single module versus the whole structure. This is a particularly useful approach when performing large optimisation studies of periodic geometries, such as compact heat exchangers. Currently, OpenFOAM only supports cyclic boundary conditions for pressure and momentum, but not heat transfer. The present work introduces a steady and an unsteady solver for cyclic heat transfer with constant wall temperature boundary conditions. The solver is validated against analytical Hagen-Poiseuille flow and two configurations of periodic wavy channels. In the latter case, the results are compared to existing literature.

Аспекты численного моделирования процессов разрушения упруго-хрупких тел

Understanding the nucleation and evolution of microdefects in solid bodies is important to ensure the reliability and safety of critical structures and to identify their strength and deformation resources. In numerical modeling, failure zones can be represented as areas with significantly underestimated rigid characteristics by analogy with the method of variable elastic parameters used in solving the boundary-value problems of the theory of plasticity. However, the formal application of numerical algorithms of plasticity does not always lead to an adequate description of failure processes especially in elastic-brittle bodies. This paper considers some aspects of the numerical simulation of failure processes, such as the calculation of a stress-strain state after reducing the rigidity of finite elements under constant boundary conditions by organizing an appropriate iteration procedure, and the selection of the maximum number of finite elements fractured per iteration, the value of a loading step, and the discretization degree of the computational domain. The influence of the above aspects on the results of failure simulation is illustrated by comparing the numerical solutions to the problem of deformation of the strip made of elastic-brittle material with the edge stress concentrator, which were obtained by different algorithms. The loading diagrams were plotted, and the implementation of the post-critical stage at the macro level was demonstrated. Failure kinetics was analyzed for different variants of implementation of the iterative procedure and at a variable number of elements fractured per iteration. It has been found that, in order to get an accurate description of deformation and failure processes, the automatic selection of a loading step seems to be more reasonable. Analysis has indicated that the discretization degree of the computational domain has a strong impact on the modeling results. This suggests that the finite element size should correspond to a certain strength constant of a material having the dimensions of length.

Modeling the effect of grain boundary diffusivity and trapping on hydrogen transport using a phase-field compatible formulation

Hydrogen grain boundary (GB) trapping is widely accepted as the main cause for hydrogen induced intergranular failure. Several studies were conducted to unveil the role of GBs on hydrogen transport; however, a clear understanding is yet to be attained. This is due to the limitations of the state-of-the-art experimental procedures for such highly kinetic processes. In this study, we aim at providing a deeper understanding of hydrogen-GB interactions using full-field representative volume element (RVE). The phase-field method is chosen for generating RVEs, since it is an appropriate numerical tool to represent GBs. A novel fully-kinetic formulation for hydrogen diffusion and GB trapping is presented, which is compatible with the phase-field based RVEs. GB diffusivity (Dgb) and trap-binding energy (Egb) were used as parameters to understand the interactions between diffusion and GB trapping. Uptake and permeation simulations were performed with constant and gradient occupancy boundary conditions respectively. In both cases, increasing Egb, increased the hydrogen GB occupancy. The permeation simulations showed that the hydrogen flux along the GBs increased with increasing both, Dgb and, surprisingly, Egb. Since trapping increases the hydrogen occupancy along GBs, it also increases the occupancy gradients, resulting in a higher flux. This led to the conclusion that, in the case of an external occupancy gradient, GB trapping and diffusion cooperate, rather than compete, to increase the hydrogen flux. On the other hand, the decisive factor for the retention of hydrogen at the GBs in permeation simulations was Dgb rather than Egb.

General wetting energy boundary condition in a fully explicit nonideal fluids solver.

We present an explicit finite-difference method to simulate the nonideal multiphase fluid flow. The local density and momentum transport are modeled by the Navier-Stokes equationsand the pressure is computed by the van der Waals equationof the state. The static droplet and the dynamics of liquid-vapor separation simulations are performed as validations of this numerical scheme. In particular, to maintain the thermodynamic consistency, we propose a general wetting energy boundary condition at the contact line between fluids and the solid boundary. We conduct a series of comparisons between the current boundary condition and the constant contact angle boundary condition as well as the stress-balanced boundary condition. This boundary condition alleviates the instability induced by the constant contact angle boundary condition at θ≈0 and θ≈π. Using this boundary condition, the equilibrium contact angle is correctly recovered and the contact line dynamics are consistent with the simulation by applying a stress-balanced boundary condition. Nevertheless, unlike the stress-balanced boundary condition for which we need to further introduce the interface thickness parameter, the current boundary condition implicitly incorporates the interface thickness information into the wetting energy.

COMPUTING DEFLECTIONS OF TOPOLOGICALLY OPTIMIZED BEAMS

In civil engineering, computing deflections is a fundamental step in the structural design process. However, most structural optimization codes do not directly compute or evaluate displacements in the resulting optimized structures. In that context, this work presents the basis of an approach to compute and analyze deflections (vertical displacements) in optimized beams. Using Matlab®, we implemented an extension to the FEM-based 99 Line Topology Optimization Code (Sigmund, 2001), which is able to compute deflections and plot deformed shapes of optimized structures, allowing users to analyze deformation during the design process. We also compared maximum deflections of optimized versus nonoptimized beams. According to results, for constant boundary conditions, optimized beams present smaller maximum deflections.

MATHEMATICAL MODELING OF DIFFUSION BOUNDARY PROBLEMS WITH PERIODIC BOUNDARY CONDITIONS USING MATLAB AND C++ FOR NUMERICAL AND ANALYTICAL SOLUTIONS

The article examines a second-order parabolic partial differential equation of a three-dimensional (3D) non-stationary boundary problem with constant diffusion coefficients and periodic boundary conditions in the x and y directions. The method for reducing the (3D) non-stationary boundary problem to the corresponding one-dimensional (1D) non-stationary boundary problem using periodic boundary conditions in the x and y directions is discussed. The stationary (analytical) solution of the obtained (1D) stationary boundary problem is also obtained. The numerical solutions of the 1D boundary problem are obtained using the Matlab package "pdepe" and the C++ programming language. As a practical application of the developed mathematical model, the article discusses calculating the concentration of heavy metal Ca in a peat layer based on the obtained experimental data (measurements).

Analytical model for organic contaminant transport in a cut-off wall and aquifer dual-domain system considering barrier arrangements

This paper presents an analytical study of organic contaminants transport in a cut-off wall and aquifer dual-domain system, considering the effects of the inlet boundary conditions and cut-off structural arrangements. The comprehensive sensitivity analysis of parameters focusing on the breakthrough time, attenuation time and cumulative concentration are presented using the Monte Carlo simulation and Sobol global method. The simplified constant inlet boundary condition can lead to an excessively conservative prediction of the contaminant breakthrough compared to the ‘finite mass’ and ‘decaying source’ boundary conditions. The cut-off wall hydraulic performance can be enhanced by reducing the contaminant's head loss, shape factor, half-life and cut-off wall hydraulic conductivity while increasing the retardation factor. The contaminant's half-life can largely influence the maximum contaminant concentration, attenuation time and breakthrough time. For example, the maximum contaminant concentrations for T1/2 = 1.4 years and T1/2 = 100 years are 13 and 123 times greater than that for T1/2 = 0.1 year, respectively. The influence of the variation of shape factor on the breakthrough curve should be taken into consideration. Altering the structural arrangement of the cut-off wall to accommodate a smaller shape factor increases the contaminant breakthrough time. The proposed solution allows the analysis of a cut-off wall and aquifer system with different inlet boundary conditions and structural arrangements of the cut-off wall.

Theoretical analysis about model simplifications and boundary conditions on two-phase flow instability of parallel heated tube systems

Flow instability may happen in both subcritical two-phase flow boiling systems (such as U tube natural circulation steam generator of Pressurized Water Reactor, once through steam generator of High-Temperature Gas-Cooled Reactor, etc) and supercritical evaporation systems. In previous theoretical researches, the model of a single heated tube (SHT) with constant pressure drop boundary condition (ConPD) was always used to investigate the system stability. However, in real applications, there are usually multiple parallel heated tubes (MHT) and unheated connection regions before the branch heated tubes. The system is usually driven by a pump, and there is always a long unheated main pipe and throttling between the heated tubes (boiler or steam generator) and the pump. However, there lacks rigorous analysis about the effects of these model simplifications and boundary conditions. Take superheated two-phase flow boiling systems as examples, the theoretical models with different flow-driven boundary conditions (ConPD, constant mass flow rate (ConMF), and pump driven (PumpD)), different numbers of heat tubes (SHT, MHT), with or ignoring the unheated region (UHR), with or ignoring the bypass of a pump, with identical or un-identical geometry parameters and operation parameters are established. Linear ordinary differential equations describing the instability are obtained. Eigenvalues of the coefficient matrix of linear ordinary differential equations, and the property of the block matrix and the similarity matrix are used to analyze the effects of the above boundary conditions and model simplifications on the system stability. Two new dimensionless numbers, dimensionless pump number (Npump) and dimensionless bypass number (Nbypass), are proposed to describe the effect of pump and its bypass on stability quantitatively. The effect of the pump bypass on the stability can be attributed to the decrease of Npump. No matter ConPD or PumpD is adopted, SHT considering UHR is proved to be more stable than SHT ignoring UHR. No matter UHR is considered or ignored, SHT with PumpD is proved to be more stable than SHT with ConPD. No matter UHR of MHT at the main pipe is considered or ignored, the stability of MHT with ConPD or ConMF or PumpD is proved to be the same as that of SHT with ConPD ignoring UHR. When ConPD ignoring UHR is adopted, the stability of MHT with un-identical geometry and operation parameters is proved to be the same as that of its most unstable heated tube. The above conclusions about the effect of model simplification and boundary condition are obtained from superheated two-phase flow boiling systems, they also hold for saturated two-phase flow boiling systems and supercritical evaporation systems.

Frictional and poromechanical properties of the Delaware Mountain Group: Insights into induced seismicity in the Delaware Basin

The Delaware Basin in West Texas and Southeast New Mexico has experienced a proliferation in seismic activity since 2016. The seismic activity is primarily due to subsurface injection of wastewater into shallow and deep reservoirs. However, the precise mechanisms connecting fluid injection to seismic activity are not well understood. To shed light on these processes, we measured rate-state friction and poromechanical properties of rocks sampled from the Delaware Mountain Group (DMG) at pressures and stresses representative of in-situ conditions. Experiments were conducted inside a pressure vessel and loaded at a true-triaxial stress state. The samples exhibit velocity-strengthening behavior and transition to velocity-neutral behavior with increasing slip. We also measure frictional healing and demonstrate that the healing rates are consistent with those measured from quartz-feldspathic-rich rocks. Experiments were conducted under a constant pore-pressure boundary condition where transient changes in pore-volume were compensated by modulating the fluid volume within the fault. Our data show that a step increase in sliding velocity leads to dilatancy and causes an influx of fluid into the fault. In contrast, a step decrease in sliding velocity causes the fault zone to compact and reduces fluid flow into the fault. Data show that changes in flow rate scale systematically with fault slip velocity. Broadly speaking, the frictional and poromechanical data indicate that shallow faults within the DMG should favor aseismic creep as opposed to unstable slip. Our data are consistent with elastic dislocation models of fault slip and InSAR data, indicating that faulting in the DMG is predominantly aseismic. The lack of velocity weakening behavior observed in our samples indicates that heterogenous frictional properties are needed to explain the co-existence of both aseismic and seismic slip in the DMG.

Modeling study on the geometric optimization of thermoelectric modules

Abstract The performance of thermoelectric power generators (TEG) primarily depends on the properties of the thermoelectric materials employed. For conventional thermoelectric modules (TEM) utilizing the same material, the geometric parameters also play a significant role in determining TEM performance. As such, optimizing the geometry of TEM can lead to improved performance. In this study, TEM were modeled, designed, fabricated, and tested to investigate the effects of different geometric parameters on their performance. Numerical simulations were conducted under both constant temperature and constant flow boundary conditions, and the results were validated through experimental testing. The simulation results under constant flow boundary conditions exhibited good agreement with the experimental results. The effects of thickness, cross-sectional area, and filling ratio of thermoelectric legs on TEM performance were investigated through numerical simulations and compared with findings from previous studies. It was observed that increasing the cross-sectional area of the thermoelectric legs led to a decrease in the power output of TEM. Conversely, increasing the filling ratio effectively enhanced the TEM's performance. Furthermore, an optimal thermoelectric leg thickness was identified through the numerical simulations that could yield the maximum power output of TEM. The underlying mechanism behind this observation was explained, shedding light on why different reports have identified different optimal thicknesses. Optimizing the thermoelectric leg thickness can help maintain a high effective temperature difference and low internal resistance, which can vary based on the specific type of TEM and the thickness and thermal conductivity of the insulating substrates and copper sheets.

On the equivocal nature of the mass absorption curves

The idea behind the research presented is based upon apparently contradictory experimental results obtained here by means of photoacoustics modalities for the same drug donor/acceptor membrane system, serving as a surrogate for a transdermal delivery system. The first modality allowed for the monitoring of the total amount of mass uptake (m(t)-type data), while the second technique allowed for the quantification of time-dependent concentration distribution within the acceptor membrane (c(x,t)-type data). Despite of a very good agreement between the mt data and the 1st-order uptake fitting model (standard Fickian diffusion with constant source boundary condition), the standard approach failed during the c(x,t) data analysis. The results led to the analysis of the interfacial transfer contribution to the overall mass transfer efficiency, which eventually allowed to question reliability of the mt data analysis for the determination and quantification of the mass transport parameters. A more detailed analysis of the c(x,t) by means of the newly introduced transport rate number parameter revealed, that the mass uptake by the acceptor is almost equally influenced by interfacial and bulk transport processes. The analyses performed were translated into a model-free characteristic times, i.e. parameters independent of the model scheme used.

A predictive method for pipe in pipe, cylindrical modules and spherical packed bed latent thermal energy storage systems

Due to the transient operation of latent thermal energy storage (LTES) heat exchangers, methods from classical heat exchanger theory such as the effectiveness number of transfer unit method are not applicable. As a result, the study of these systems lacks a general theoretical framework. Recently, an approximate analytical solution method was developed which integrates phase fraction under a constant boundary condition to an LTES heat exchanger with a varying heat transfer fluid temperature. In this paper, the method is applied to three geometries: pipe in pipe, cylindrical modules and spherical packed bed geometries. The resulting model predicts the phase change fraction and effectiveness as a function of time, based on one-dimensional solutions for the movement of the phase change front. The predictions of the analytical model are compared to results of a simplified numerical method of an LTES heat exchanger. The average deviation of the predicted phase change fraction is 3.3 10−2 %, 4.6 10−1 % and 3.7 10−1 % for the pipe in pipe, cylindrical and spherical module respectively. The difference in model performance stems from the description of the fundamental heat transfer behavior. In a second validation step, the outlet temperature predicted by the model is compared to an experimental dataset for a LTES shell and tube heat exchanger. The predictions of the model can be improved further by including sensible heat, heat losses and allowing time-varying inlet conditions.

Parameter study for oil spray cooling on endwindings of electric machines via Eulerian–Lagrangian simulation

The demand for larger power density and torque for the power traction motors used in electrified transportation puts forward a requirement for better thermal management methods. Spray cooling is a promising direct liquid cooling technique that has been proved to possess high heat removal capability in previous research. This paper investigates the heat transfer characteristics of spray cooling on endwindings of electric machines via numerical simulation through an Eulerian–Lagrangian approach. The utilized numerical models and calculated results are validated with experimentally measured data. The influence of different parameters and options involved in the simulation settings on the final results, like the stream numbers for the spray injector, the constant heat flux versus constant temperature thermal boundary condition, the influence of splashing, the effect of heat conduction in the endwindings and the Saffman lift force, only solving the energy equation for the air after its flow field reaches a steady-state, are evaluated. Parameter analyses are also conducted for operation conditions, configuration of the spray nozzles, and material properties of the coolant liquid. It is found that larger flow rate, smaller droplet size, lower spray height, more nozzle numbers, larger thermal conductivity and smaller viscosity of the coolant liquid tend to increase the overall heat transfer performance.