Conservation Equations Research Articles

Development of HAWT wake under thermal stratification conditions

The evolution analysis of the wind wake downstream of a rotor submerged in a stratified atmospheric boundary layer is discussed in this work. The hybrid model AD-CFD is used, to obtain the velocity field. The changes in thermal stratification are taken into account when solving the energy conservation equation. Results from the unstable stratification case indicate a 41% velocity deficit and 30% is found for stable thermal stratification, at z = 1D. this work show that in a stratified atmospheric boundary layer, an isolated wind turbine’s ability to produce electricity is more impacted by stable thermal stratification than by unstable ones.

Electro-hyperelasticity with phase field for electric breakdown modeling

In this paper, we systematically derive the phase field model for electric breakdown channel propagation. The model includes electric charge, mass, momentum and energy conservation equations as well as Allen–Cahn type equation which governs the phase field evolution. The medium motion is described by the Godunov–Romenski type hyperelastic model. Electromagnetic part of the model consists of Maxwell’s equations in quasi(electro)stationary approximation including electric charge conservation law. Derivation of the model is based on the Coleman–Noll procedure and Gurtin’s microforces and microstress framework.

Generalized Effective Stress of Dual-Porosity Geomaterials Considering Thermochemical Effects

The accurate calculating of effective stress is crucial to predicting soil deformation and ensuring structural safety under complex mechanical, thermal, and chemical loads. In this study, the thermodynamic pressures of the solid phase, and liquid phases in macropores and micropores of dual-porosity geomaterials are determined on the basis of thermodynamics and conservation equations with considering for the impact of temperature and solute concentration on soil-water interactions. In particular, thermal pressurization related to temperature, and generalized osmotic pressure related to solute concentration were incorporated into the liquid-phase thermodynamic pressure equations. Thereafter, an equation for the generalized effective stress that could be applied specifically to dual-porosity geotechnical materials with consideration for environmental loads was derived from the definition of the stress tensor. The deformation behavior of clay under mechanical, thermal, and chemical coupling loading conditions was predicted using the newly proposed generalized effective stress. Predicted results revealed a stronger correlation between the generalized effective stress and void ratio, thereby confirming the accuracy and effectiveness of the proposed approach. The derived equation establishes a solid theoretical foundation for designing impermeable buffer barriers in geotechnical engineering and geo-environmental engineering safety assessments.

Nonideal Incompressible Lattice-Boltzmann Method for Multicomponent Phase Separating Systems

Understanding and predicting the dynamics of complex fluid systems including liquid–liquid phase separation, relevant to both biological and engineered applications, typically uses a nonideal free energy. Introducing such a thermodynamic constraint into the Lattice-Boltzmann Method can be accomplished by altering either the equilibrium distribution function or the external force. The former requires a lengthy parameterization for a free energy of multiple independent variables which becomes cumbersome for more than three components. The latter has been done for a multicomponent compressible system, but a correction term for the force is required to recover the expected conservation equations. This work builds upon the incompressible single component forcing method from He et al. (Journal of Computational Physics, Vol. 152, No. 2, 1999) by deriving and implementing the required force needed to successfully recover the expected mass conservation from a nonideal free energy with an arbitrary number of components. This allows the simulation of more realistic phase separating fluid systems by including many interacting components, which is demonstrated here for up to five components and phases.

Effect of hollow fiber membrane packing on the performance of modules formed with fiber tows

ABSTRACT Membrane processes are being considered for carbon dioxide capture from concentrated point sources. The processes often use hollow fiber membranes. Modules are formed by gathering fibers together in a bundle. The fibers may be supplied individually or in small groups (i.e. tows) for module fabrication. The use of tows can introduce nonuniform fiber packing due to the size and spacing of the tows. The potential for flow maldistribution and its detrimental effect on module performance is evaluated by solving the governing conservation equations. For the first time, the performance of a module formed from fiber tows is evaluated. Two performance metrics are reported: 1) recovery and 2) required area as a function of carbon dioxide removal. The effects of tow size and tow spacing are evaluated. The results indicate performance is better for smaller tows. Furthermore, lower packings, both intra- and inter-tow, are preferred. The concept of an equivalent planar bundle (EPB) for fiber tow modules is introduced to potentially reduce simulation time and costs. The simple procedure for creating the EPB is validated by comparing the performance metrics calculated using the EPB to the results obtained from full simulations.

Dynamic plastic damage analysis of bending-torsion coupling in cross stiffener systems under mass impact

ABSTRACT This study proposes a simpler theoretical model to predict the bending-torsion coupling dynamic plastic response of cross stiffener systems under mass impact. Traditional methods often rely on torque balance and momentum conservation equations, which involve complex and difficult calculations. To simplify this process, the study uses the law of energy conservation and applies the variational method to solve the dynamic plastic damage of the cross stiffener system. The plastic response of a single stiffener is first analyzed and compared with existing theories, showing consistent results for maximum displacement. The study then extends the model to more complex conditions, performing comparative analyses on stiffeners of different sizes through numerical simulation. The findings show that the maximum displacement at the load and node remains within a reasonable range, with a displacement difference of only 0.9 mm and an error margin of 4.79% in one of the test conditions.

Numerical Investigation of Pyramid Solar Stills with PCM‐Nanoparticles and Absorber Fins: Enhanced Thermal Performance for Sustainable Water Desalination

ABSTRACTIn arid regions facing challenges of limited access to potable water and electricity, solar desalination stands out as a promising solution for producing clean water from brackish sources. This research intends to examine and enhance an innovative and sustainable solar desalination system. A computational study is conducted on a pyramid solar still (PSS) combined with a phase‐change material incorporated with nanoparticles, with a variety of absorber fins. The study investigates three configurations: a traditional pyramid–shaped solar still, one with square absorber fins, and another with circular fins. Paraffin wax mixed with Al2O3 nanoparticles is used beneath the absorber fins. Conservation equations are solved using COMSOL Multiphysics. Simulation outcomes demonstrate that the incorporation of finned absorbers, coupled with elevated water temperatures, significantly enhances the yield of the improved PSS compared with its conventional counterpart. The square‐finned absorber PSS exhibits a substantial increase in total accumulated productivity over the conventional design. Moreover, the PSS with square‐finned absorbers demonstrates an impressive average peak daily thermal efficiency improvement of 49.19% compared with a conventional solar still. This study highlights the effectiveness of the modified PSS as a superior option for household water generation.

Numerical Analysis and Validation of Horizontal and Vertical Displacements of a Floating Body for Different Wave Periods

This study concentrates on numerically evaluating the behavior of a floating body with a box format. Although research on floating objects has been conducted, the numerical modeling of Wave Energy Converter (WEC) devices, considering the effects of fluctuations, remains underexplored. Therefore, this research intends to facilitate the analysis of floating devices. First, the experimental data served as a benchmark for evaluating the motion paths of the floating box’s centroid. Second, the effects of various wave periods and heights on the floating body’s movement were analyzed. The Volume of Fluid (VOF) multiphase model was applied to simulate the interactions between phases. The computational model involved solving governing equations of mass conservation, volumetric fraction transport, and momentum, employing the Finite Volume Method (FVM). The validation demonstrated that the Normalized Root Mean Square Error (NRMSE) for the x/h ratio was 3.3% for a wave height of 0.04 m and 4.4% for a wave height of 0.1 m. Moreover, the NRMSE for the z-coordinate to the depth of water (z/h) was higher, at 5% for a wave height of 0.04 m and 5.8% for a wave height of 0.1 m. The overall NRMSE remained within acceptable ranges, indicating the reliability of the numerical solutions. Additionally, the analysis of horizontal and vertical velocities at different wave periods and heights showed that for H = 0.04 m, the wave periods had a minimal impact on the amplitude, but the oscillation frequency varied. At H = 0.1 m, both velocities exhibited significantly larger amplitudes, especially for T = 1.2 s and T = 2.0 s, indicating stronger motion with higher wave heights.

The Role of Heater Size and Location in Modulating Natural Convection Behavior in Cu-Water Nanofluid-Loaded Square Enclosures

Enhancing the energy efficiency of thermal systems reduces their consumption, lowers costs, and reduces undesired environmental impact, thus making these systems more sustainable. The current work introduces a passive method for heat transfer enhancement that is carried out using natural convection by nanofluid. This work introduces a computational study of the process of natural convection within a square cavity containing Cu/H2O nanofluid. The cavity wall on the left side undergoes partial isothermal heating, while the opposing side is fully cooled isothermally, with all other boundaries maintained adiabatic. A mathematical model formulated based on a 2-D model was used to provide the solution for the system of governing equations of mass, momentum, and energy conservation, employing the finite element technique. A commercial CFD package is utilized to perform the computational simulation. The present investigation delves into the impact of the Rayleigh number, nanoparticle concentration, heater length, and heater location on the flow field and heat transfer characteristics. The model outcomes were displayed for a wide range of the pertinent parameters as 103 ≤ Ra ≤ 106, 0.25 ≤ lh ≤ 1.0, 0.125 ≤ hc ≤ 0.875, and 0.02 ≤ ϕ ≤ 0.10. Also, correlation equations relating the average Nusselt number to these crucial parameters are derived. These equations are simple and can be applied in practice easily in many fields, such as electric and electronic equipment cooling and thermal management of heat sources. Also, these equations gather all the parameters that affect the heat transfer process. They are shedding light on the intricate interplay between these parameters in the natural convection heat transfer process.

State-of-the-Art Review of Continuum Mechanics-Based Modelling of Soil Surface Erosion

AbstractSoil surface erosion can shape the morphography of rivers and estuaries in the natural environment and induce high potential risks to structures in engineering. Numerical simulations based on continuum mechanics theory can provide reliable assessments of the evolution of surface erosion from the perspective of a large-scale view. However, current studies on continuum mechanics-based modelling are still limited. This paper comprehensively reviews such numerical simulations of soil surface erosion. This review begins by discussing the fundamental physical mechanisms of surface erosion. Subsequently, it explores the basic physics-based conservation equations controlling soils and fluids in surface erosion. Then, the empirical formulae depicting the different stages of surface erosion are presented. Building on these mathematical foundations, this paper reviews various numerical methods for surface erosion modelling from a continuum mechanics perspective. Finally, this paper discusses the advantages and limitations of the numerical methods. This work can provide researchers convenience for using numerical models on surface erosion simulations.

Comparison of Contact Angle Models in Two‐Phase Flow Simulations Using a Conservative Phase Field Equation

ABSTRACTIn phase field methods based on a second‐order Allen‐Cahn (AC) equation, contact angles are prescribed mostly via a geometric formulation. However, it is of great interest to utilize the surface‐energy formulation, which is often employed in the Cahn‐Hilliard (CH) phase field method, in the AC phase field method. This article thus put forward a surface‐energy formulation of contact angles. The model was compared with the geometric one in a number of impact problems, including both normal and oblique impacts. The governing equations were discretized using a finite difference method on a half‐staggered grid. The Navier–Stokes equation was tackled using an explicit projection method. The major findings are as follows. First, the geometric model can maintain a fixed contact angle throughout contact line motion, while the surface‐energy one predicts a changeable contact angle, with a fluctuation of about 5°. In the oblique drop impact, contact angle hysteresis was captured even if a static contact angle was applied in the surface‐energy formulation.

A high-order physical-constraints-preserving arbitrary Lagrangian–Eulerian discontinuous Galerkin scheme for one-dimensional compressible multi-material flows

In this paper, a high-order physical-constraints-preserving arbitrary Lagrangian–Eulerian (ALE) discontinuous Galerkin scheme is proposed for one-dimensional compressible multi-material flows. Our scheme couples a conservative equation related to the volume-fraction model with the Euler equations for describing the dynamics of fluid mixture. The mesh velocity in the ALE framework is obtained by using an adaptive mesh method that can automatically concentrate the mesh nodes near the regions with large gradient values and greatly reduce the numerical dissipation near material interfaces. Using this adaptive mesh, the resolution of solution near some special regions such as material interfaces can be improved effectively by our scheme. With the appropriate time step condition and using a bound-preserving and positivity-preserving limiter, our scheme can ensure the positivity of density and pressure and the boundness of volume-fraction, which further ensures the computational robustness and degree of confidence of simulations under large density or pressure ratios and so on. In general, our scheme can be applied to the simulations of compressible multi-material flows efficiently with the essentially non-oscillatory property and physical-constraints-preserving (bound-preserving and positivity-preserving) property, and its steps are more concise compared to some other methods such as the indirect ALE methods. Some examples are tested to demonstrate the accuracy, essentially non-oscillatory property and physical-constraints-preserving property of our scheme.

Conservative wormholes in generalized [formula omitted]- function

We present an exhaustive study of wormhole configurations in κ(R,T) gravity with linear and non-linear functions. The model assumed Morrison-Thorne spacetime where the redshift and shape functions linked with the matter contain and geometry of the spacetime through non-covariant conservation equation of the stress-energy tensor. The first solution was explored assuming a constant redshift function that leads to a wormhole (WH) which is asymptotically non-flat. The remaining solutions were explored in two cases. Firstly, assuming a linear equation of state p(r)=ωρ(r) along with different forms of κ(R,T)-function. This proved enough to derive a shape function of the form b(r)=r0(r0r)1/ω. Secondly, by assuming specific choices of the shape function consistent with the wormhole configuration requirements. All the solutions fulfill flare-out condition, asymptotically flat and supported by phantom energy. Further, the embedding surface and its revolution has been generated using numerical method to see how the length of the throat is affected of the coupling parameters through κ(R,T) function. At the end, we have also calculated the average null energy condition, which is satisfied by all the WH models signifying minimum exotic matter is required to open the WH throats.

Electro-osmotic flow and heat transfer in Jeffery fluid: A multi-membrane microchannel model

Microscale heat transfer is vital for the performance of smart thermal devices like heat sinks, thermosyphons, and microheat pipes. This study introduces a biothermal pumping flow model based on a multi-membrane pumping mechanism that leverages microscale heat transfer. The model describes rhythmic contraction and relaxation of membranes, combined with electro-osmosis in Jeffery fluid flow within a vertical microchannel of finite length. Two membranes on the microchannel walls, with varying amplitudes, diameters, and phase lags, generate pressure that moves fluid in both directions through contraction and expansion cycles. The model is based on the conservation of mass and momentum, using a low Reynolds number approximation to capture microscale transport phenomena at biomedical scales. Dimensionless conservation equations are analytically solved under no-slip boundary conditions, with results computed in MATLAB for clarity. Axial velocity results are simulated and verified using the optimal homotopy analysis method. The model explores the influence of key parameters (UHS, me, λ, Gr, β) on pressure gradient, velocity distribution, volumetric flow rates, skin friction, Nusselt number, and stream function. The findings demonstrate that pressure from membrane motion is significantly affected by thermal effects and buoyancy forces, and flow and pumping characteristics are largely determined by the fluid's rheological qualities and the geometrical features of the membrane. This study provides novel ideas for enhancing the functionality and design of smart thermal devices while also advancing microscale heat transfer technology.

Analytical and numerical models for impedance of a button SOFC anode

We report a physics–based model for the impedance of a button–type anode–supported SOFC. The model includes ion and electron charge conservation equations and a Fick’s diffusion transport equation for hydrogen in the anode support layer. In the limit of small overpotentials, an analytical solution for the anode impedance is derived. For typical SOFC anode parameters, this solution is valid from open–circuit voltage (OCV) up to the cell current density of about 5 mA cm−2. Fast least–squares fitting of the model impedance to an experimental spectrum measured at OCV is demonstrated and the resulting fitting parameters are compared with literature data. A high–current numerical version of the model is also suitable for fast fitting of experimental impedance spectra.

Three-dimensional bioconvection in a nanofluid film over a wedge surface with entropy generation and nonlinear radiation: A revised Buongiorno model

ABSTRACT The advances in the field of nanotechnology tends to improve the rate of heat extraction and transmission from foreseeable causes and has fascinated researchers and scientists. Furthermore, nanofluids are having a useful impact on various fields such as nuclear industries, powered lasers, chilling of microelectronic devices, solar energy capture, transformers oil cooling, and in cancer diagnosis. Hence, magnetohydrodynamic (MHD) three-dimensional bioconvective flow of nanofluid accommodating gyrotactic microorganisms over a wedge surface with zero nanoparticles mass flux boundary conditions is studied, using large Reynolds number approximation. The additional novelty of the study is the implementation of the revised Buongiorno model, which deliberates the effects of thermo-migration and Brownian diffusion of nanoparticles, together with passive control of nanoparticles fraction on the wedge surface. The self-similar form of conservation equations of energy, nanoparticle fraction, gyrotactic microorganisms, and momentum are handled numerically. T The results of the Blasius and the Hiemenz flows are retrieved as special cases of the model. It was found that the temperature of the nanofluid increases considerably when considering the nonlinear thermal radiation compared to the linear radiant heat. Increasing the strength of the Brownian diffusion increases the heat transport but reduces the mass transport.

Multi-Objective Numerical Analysis of Horizontal Rectilinear Earth–Air Heat Exchangers with Elliptical Cross Section Using Constructal Design and TOPSIS

This study presents a numerical evaluation of a Horizontal Rectilinear Earth–air Heat Exchanger (EAHE), considering the climatic and soil conditions of Viamão, Brazil, a subtropical region. The Constructal Design method, combined with the Exhaustive Search, was utilized to define the system constraints, degree of freedom, and performance indicators. The degree of freedom was characterized by the aspect ratio between the vertical and horizontal lengths of the elliptical cross-section duct (H/L). The performance indicators for the EAHE configurations were assessed based on thermal potential (TP) and pressure drop (PD). The Technique for Order Preference by Similarity to Ideal Solution (TOPSIS) was applied for multi-objective evaluation, and a methodology for EAHE is proposed. The problem was solved using FLUENT software (version 2024 R2), which employs the Finite Volume Method to solve the conservation equations for mass, momentum, and energy. The (H/L)T,o = 6.0 configuration showed a 16.4% increase in thermal performance for heating and 15.9% for cooling compared to the conventional circular duct. Conversely, the (H/L)F,o = 1.0 configuration reduced pressure loss by 65.33%. The integration of Constructal Design with TOPSIS facilitated the identification of optimized geometries that achieve a balance between performance indicators and those that specifically prioritize thermal or fluid dynamic aspects, being this approach an original scientific contribution of the present work.

On the onset of stationary convection on Jeffrey nanofluid layer saturated with a porous medium: Brinkman model

In this paper, we investigate the onset of convection in a Jeffrey nanofluid layer saturated with the porous medium using Darcy-Brinkmann model. Normal mode analysis and Galerkin type weighted residual method (GWRM) are used to analyse conservation equations. Effects of Brownian motion and thermophoresis are taken into account in the Jeffrey nanofluid model. The Buongiorno model deployed for nanoparticles incorporates the influences of thermophoresis and Brownian motion. Three cases of free-free, rigid-rigid and rigid-free boundaries are considered. For stationary convection, the effects of Darcy number, Jeffrey parameter, Lewis number, nanoparticle Rayleigh number, porosity and modified diffusivity ratio for all the above mentioned boundary conditions are investigated analytically and graphically. The numerical computed values of stationary thermal Rayleigh number are presented graphically for three distinct combinations of boundary conditions. The study is of great significance in many different areas such as automotive, pharmaceutical, geophysics, soil sciences, food processing, oceanography, limnology, etc., and excellent coincidence is found regarding the present paper and earlier published work.

Optimal control of Hydrocarbon Reducer (HC) injection based on Trust-Region-Reflective Algorithm (TRRA) and physical models for Diesel Oxidation Catalyst (DOC)

The aim of this paper is to control the DOC outlet gas temperature between 600 ± 15 °C by optimizing the hydrocarbon (HC) injection into the Diesel Oxidation Catalyst (DOC) for active regeneration of the downstream Diesel Particulate Filter (DPF). First, based on the physical model of the DOC thermal dynamics, the energy conservation equation for the gas phase temperature is simplified by using variable substitution, and the energy conservation equation for the solid phase temperature is simplified by considering only the heat generated by the HC injection. By solving the simplified model using the Trapezoidal Rule-Backward Differentiation Formula 2 (TR-BDF2) method and optimizing the model parameters in combination with the Gauss-Newton method, the computational efficiency and accuracy were significantly improved. Next, the DOC downstream temperature observer is designed by combining the solution characteristics of the DOC model with the unscented Kalman filter (UKF) algorithm to ensure that the catalyst operates within the optimal temperature range. Then, this paper verifies the accuracy of the model and observer through bench tests covering four steady-state operation conditions and World Harmonized Transient Cycle (WHTC) test cycle, and the results show that the model and observer exhibit high accuracy in both steady-state and transient operation conditions. Finally, the DOC temperature was successfully maintained between 600 ± 15 °C by optimizing the control of the HC injection rate, thus achieving the expected temperature control target. These research results provide theoretical and practical support for diesel engine emission control and DPF active regeneration.

Solidification process of hollow metal droplets impacting a substrate

Recently, there has been a renewed interest in hollow droplet impact, as the internal bubble makes a difference in the spreading morphology during thermal spraying and three-dimensional printing. This paper is dedicated to the solidification process of a hollow metal droplet impacting a cold substrate. The influence of initial impact velocity and substrate temperature on the spreading process is numerically investigated by employing a phase-field method coupled with a solidification model. Solidification was shown to strongly alter droplet spreading and the flow state of the center liquid column. To describe the flow state of the center liquid column, three patterns, namely ‘no-jet’, ‘transition region’, and ‘counter-jet’ are defined. The simulation results show that the formation of the center counter-jet is inhibited by the solidification layer when Péclet number Pe<357, where Pe is the ratio of convection rate to diffusion rate. In addition, the critical Pe at which the phenomenon of counter-jet occurs gradually rises with the substrate temperature falling. Finally, a quantitative analysis of the maximum spreading diameter is proposed by establishing an energy conservation equation. The predicted diameters are in good agreement with the simulated results. It allows us to provide a general expression for the hollow droplets' maximum spreading diameter, using the impact and temperature parameters.