Transient Equation Research Articles

Verification оf Non-Stationary Mathematical Model оf Concrete Hardening in Thermal Technological Installations

Thermo-technical installations consuming significant amounts of thermal energy are used in order to intensify precast and reinforced concrete production processes under industrial conditions. Despite significant progress in the study of concrete hardening in accelerated hydration devices, a prominent lack of reliable and cost-effective research and optimization methods of their operation is observed. The methods used in real production processes are mainly based on empirical dependences obtained for specific technological conditions. These methods can not always be applied for other modes and technologies. The present paper develops calculation methods based on fundamental laws that make it possible to obtain functions for evolution of concrete product hydration process. Methods of mathematical modeling permit to develop new ways directed on improvement of modes for heat treatment of concrete products and accelerated hydration technologies. The paper describes a mathematical model for calculating a hardening process of a concrete product that includes a transient three-dimensional heat conductivity equation, a function of internal heat release due to behavior of exothermic reactions of cement hydration and also a system of initial and boundary conditions. A numerical simulation for temperature and hydration coefficient of a concrete product having shape of a 0.1´0.1´0.1 m cube has been performed in the paper. Verification of the non-stationary mathematical model for calculating temperature fields and hydration degree while using experimental data on concrete product strength obtained under industrial conditions. Investigations on hydration degree function of time have shown that experimentally obtained values of compressive strength correlate with hydration coefficient and hydration rate functions of heat treatment time which are calculated on the basis of the proposed non-stationary mathematical model of concrete product hardening. Satisfactory agreement of experimental and calculated data confirms adequacy of the proposed non-stationary mathematical model for calculating temperature fields and hydration degree with accelerated heat treatment of concrete products.

The role of particle dopant to the thermal conductivities of PCM coconut oil by means of the T-history method

We described in this paper the role of chemical dopant to the thermal conductivity of organic phase change material (PCM) of coconut oil (co_oil) based on T-history method. We used the nanoparticle dopant, namely graphite, CuO, and ZnO. Each kind of dopant was added to co_oil in a certain amount of 1 wt% and 2 wt%. For comparison, the data of pure co_oil was also taken and compared to the data from direct measurement. Because of the smaller diameter to height ratio, the lumped capacitance method is applicable here. Hence, the heat transfer between PCM and water during solidification or melting process is one dimensional, so that the one-dimensional transient heat diffusion equation for cylindrical geometry is applicable. Analysis of the data for solidification and melting processes led to the values of solid and liquid thermal conductivities. We note that in general 1 wt% dopants have effectively increased the thermal conductivities of co_oil, which is important for the effective heat transport of the material in response to the heat from the environment. More increase of dopant concentration has resulted in the reduction of thermal conductivities, which might be due to the agglomeration of particle dopant due to van der Waals interaction between particles.

CFD analysis of drug uptake and elimination through vascularized cancerous tissue

Today, one of the biggest challenges for the physicians and scientists is optimizing the drug transport to the solid tumors. Predicting the behavior of the formation of the vasculature in tumor and normal tissues is the hallmark for the drug delivery to the cancerous cells. In this study, we assume a geometry with microvessels in tumor and its surrounding tissue. We employed Darcy’s law and starling’s law to obtain the steady-state interstitial fluid pressure and velocity. Then, these values are used for the transient drug transport equation. Our aim is to investigate the drug transport in a vascularized medium by considering the plasma concentration as a variable. Our results show that the high interstitial fluid pressure and heterogeneity of tumor’s vasculature are the main barriers to drug penetration and cell killing. In this study, we used a denser network of microvasculature in both tumor and normal tissues. Different doses and time periods for the drug administration are also investigated under the tri-exponential plasma concentration. Interstitial drug concentration is highly depended on the pressure gradient. The convection term is negligible in comparing with diffusion owing to the low interstitial fluid velocity. Two methods of drug injection are compared that both suffered from the low convection rate through the cancerous tissue. The elimination rate of the drug is shown based on the different amount of dosages.

A critical evaluation of the variability induced by different mathematical equations on hydraulic conductivity determination using disc infiltrometer

Infiltration measurements are mandatory input for hydrological modelling. Disc infiltrometer is used for determining infiltration in the field by allowing three-dimensional flow of water under the negative head at the surface. There are steady-state and transient mathematical equations for obtaining hydraulic characteristics based on disc infiltrometer measurements. Different assumptions and formulations adopted by these equations may induce analysis-dependent variability in hydraulic parameter determination from the disc infiltrometer measurements. In this study, a critical evaluation of nine mathematical equations used for determining near-surface saturated hydraulic conductivity based on mini-disc infiltrometer (MDI) measurements in the field for two different seasons is carried out. The saturated hydraulic conductivity determined by Guelph permeameter was used as the reference for evaluating the appropriateness of equations considered in this study. Considering different statistical procedures, Wooding–Gardner, Weir’s Refinement, van Genuchten Zhang, Ankeny, and Haverkamp equations identified by Bland–Altman plot are recommended as the most reliable mathematical equations that can be used for analysing MDI measurements. The appropriateness of the mathematical equation for MDI analysis with respect to soil type needs to be investigated further.

Rotorcraft blade-vortex interaction noise prediction using the Lattice-Boltzmann method

The aim of this paper is to assess the accuracy, capabilities and computational performances of the Lattice-Boltzmann/Very Large Eddy Simulation Method to predict the unsteady aerodynamic loads, the rotor wake development and the noise radiation of helicopter rotors in strong Blade-Vortex Interaction conditions. The numerical flow solution is obtained by solving the explicit, transient and compressible Lattice-Boltzmann equation implemented in the high-fidelity CFD/CAA solver Simulia PowerFLOW®. The acoustic far-field is computed by using the Ffwocs-Williams & Hawkings integral solution applied to a permeable surface encompassing the whole helicopter geometry. The employed benchmark configuration is the 40% geometrically and aeroelastically scaled model of a BO-105 4-bladed main rotor tested in the open-jet anechoic test section of the German-Dutch wind tunnel in the framework of the HART-II project. In the present study, only the baseline operating condition of the experimental campaign, without Higher-Harmonic Control enabled, is considered. All simulations are performed by assuming a rigid blade motion, but a computational strategy based on a combination of a rigid blade pitching motion and a transpiration velocity boundary condition applied on the blade surface is employed to take into account the blade elastic deformation motion measured during the experiments. As expected, modeling the blade elastic deformation leads to more accurate predictions of control settings, unsteady air-loads and noise footprint. The effects of the computational grid on the aerodynamic and aeroacoustic prediction is documented as well.

Numerical study on the effect of the location of the phase change material in a concentric double pipe latent heat thermal energy storage unit

Latent heat thermal energy storage (LHTS) represents an intriguing solution to the problem of variability of supply that exists in solar thermal power systems. One such storage system consists of a double pipe, where a phase change material (PCM) is enclosed in either the central pipe, or the annulus surrounding it. The heat transfer fluid fills the other void in the system. Whether the PCM is used in the central pipe or the annulus could, potentially, significantly alter the thermal performance of the system. Thus, a comparison between the PCM mounted in the annulus (case A) and the inner tube (case B) was conducted numerically, to investigate the advantages and disadvantages of each case with regard to the melting and solidification performance. A horizontal double pipe latent heat thermal energy storage device was considered. The numerical simulation solved the transient balance equations in a two-dimensional system. The enthalpy-porosity method was used to simulate the phase change and the Boussinesq approximation, which accounts for the small changes in density that drive natural convection, was applied. The effect of the initial temperature of the heat transfer surface (HTS) on the sensible and latent heat changes of the model PCM, RT-50, was tested for both the melting and solidification processes. Aiming to assess the differences in the storage performance, the average PCM temperature, the liquid fraction, and the average velocity of both the melting and solidification processes were examined. The results of the simulation showed that for both cases, convection was dominant after only a short period of the melting process. The melting time was significantly different in the two cases, i.e. it was shorter in case B than case A by almost 50%. Furthermore, an increase in the temperature of the HTS by 5 °C notably affected the melting time of both cases by as much as 20%. This effect was more pronounced in case B, which had a melting time which was 41% shorter than case A. Finally, the results revealed that the solidification process in case A was more rapid than case B with the total solidification time of case A being lower by 43.4%.

Modeling of electrokinetic remediation combining local chemical equilibrium and chemical reaction kinetics.

A mathematical model for reactive-transport processes in porous media is presented. The modeled system includes diffusion, electromigration and electroosmosis as the most relevant transport mechanisms and water electrolysis at the electrodes, aqueous species complexation, precipitation and dissolution as the chemical reactions taken place during the treatment time. The model is based on the local chemical equilibrium for most of the reversible chemical reactions occurring in the process. As a novel enhancement of previous models, the local chemical equilibrium reactive-transport model is combined with the solution of the transient equations for the kinetics of those chemical reactions that have representative rates in the same order than the transport mechanisms. The model is validated by comparison of simulation and experimental results for an acid-enhanced electrokinetic treatment of a real Pb-contaminated calcareous soil. The kinetics of the main pH buffering process, the calcite dissolution, was defined by a simplified empirical kinetic law. Results show that the evaluation of kinetic rate entails a significant improvement of the model prediction capability.

A global random walk on spheres algorithm for transient heat equation and some extensions

Abstract We suggest in this paper a global Random Walk on Spheres (gRWS) method for solving transient boundary value problems, which, in contrast to the classical RWS method, calculates the solution in any desired family of m prescribed points. The method uses only N trajectories in contrast to mN trajectories in the conventional RWS algorithm. The idea is based on the symmetry property of the Green function and a double randomization approach. We present the gRWS method for the heat equation with arbitrary initial and boundary conditions, and the Laplace equation. Detailed description is given for 3D problems; the 2D problems can be treated analogously. Further extensions to advection-diffusion-reaction equations will be presented in a forthcoming paper.

Speed Simulation of Pig Restarting from Stoppage in Gas Pipeline

Pigging is a common operation in the oil and gas industry. Because of the compressibility of the gas, starting up a pipeline inspection gauge (pig) from a stoppage can generate a very high speed of the pig, which is dangerous to the pipe and the pig itself. Understanding the maximum speed a pig achieves in the restarting process would contribute to pig design and safe pigging. This paper presents the modeling of a pig restarting from a stoppage in gas pipeline. In the model, the transient equations of gas flow are solved by method of characteristics (MOC). Runge-Kutta method is used for solving the pig speed equation. The process of a pig restarting from a stoppage in a horizontal gas pipe is simulated. The results indicate that the maximum speed a pig achieves from a stoppage is primarily determined by the pressure of the pipe and the pressure change caused by the obstructions. Furthermore, response surface methodology (RSM) is used to study the maximum speed of pig. An empirical formula is present to predict the maximum speed of a pig restarting from a stoppage in gas pipeline.

On snowpack heating by solar radiation: A computational model

A new approach to calculate the solar radiation transfer in absorbing and scattering snow and the resulting heating and possible partial melting of snow is suggested. The propagation of both the directional solar radiation and diffuse radiation from the sky is calculated using the transport approximation for the scattering phase function and two-flux method for radiative transfer of a diffuse component of the scattered radiation. A comparison with the direct Monte Carlo simulation of solar radiation transfer in a snowpack confirmed sufficiently good accuracy of the differential method. The calculated distribution of the absorbed radiation power is considered as a heat source in a transient energy equation to obtain the evolution of temperature profile in a snowpack during several solar days. The computational results obtained for the model problem as applied to heat transfer conditions typical of polar regions make clear a relative contribution of the visible and near-infrared spectral ranges and also the role of both radiative and convective cooling as well as continuous conductive heating of a deep snow layer. The effect of a sloping surface on solar heating of snow is also analyzed in the paper.

Investigation of thermal energy storage systems in concentrated solar power

This study presents computational fluid dynamics (CFD) modelling results obtained for the first cycle on a sensible thermal energy storage system based on rocks and pebbles freely poured into a mild steel tank. This study is primarily focused on proposing a plan to find an optimized particle size for the packed bed storage using various heat transfer fluids namely water, therminol-vp1, and air. The computational domain consists of an axisymmetric tank of cylindrical shape filled with a porous bed coupled with the wall. The governing equations have been solved for incompressible turbulent flow and fully developed forced convection, based on the two-phase transient model equation (SSTKW) to calculate the temperature of the fluid and solid phases. The porosity has been considered as 0.28 for the packed bed while the thermodynamic properties of both phases are temperature-dependent. The influence of the thermal dispersion within the porous bed, as well as the effective conductivity between the beads, were considered. The heat transfer coefficient was calculated according to the correlation for forced convection within porous media. Experimental temperature profile at the inlet of the bed is applied as a boundary condition in the simulation. Simulation results showed a good agreement with experimental data.

Influence of bed proximity on the three-dimensional characteristics of the wake of a sharp-edged bluff body

The gap flow effect in a wake is investigated to develop an improved picture of the formation of fluid structures via a numerical simulation of flow past a bluff body with two different clearances from the bed. These two cases are compared with the no-gap case which is considered as a reference case. The transient three-dimensional Navier-Stokes equations are numerically solved using a finite volume approach with the detached eddy simulation as the turbulence model. The effect of the free surface is included in the model by using the volume of fluid method. The fluid structures that are generated in the wake are identified using the λ2-criterion. It is found that the gap flow influences the formation of various types of fluid structures in the wake region. The horseshoe vortex appears to be attenuated as the gap size increases. The turbulent structures at the core of the wake appear to be disorganized and have the ability to extend further in the transverse direction in the absence of the horseshoe vortex. A new structure is identified in the wake flow when the gap size exceeds a threshold value. This structure acts to enhance the positive wall-normal velocity and accelerate the restoration of the free surface to its original position at shorter distances downstream of the bluff body. The lateral entrainment to the wake region is also enhanced as the gap is introduced in the wake flow.

Cell Renormalized Fokker–Planck equation method (CR-FPK) for fractional order nonlinear system

In this paper, the random response of nonlinear dynamic systems with fractional order term are examined. In this context, an efficient approach called CR-FPK method is proposed to numerically obtain transient stochastic response for those nonlinear systems under white-noise excitation in which damping is of fractional order. This approach is based on the extension of stochastic averaging method and the process of cell renormalization. By first revisit the process of equivalent linearization and stochastic averaging, the Markov property of envelope response is firstly settled. Then the idea of cell renormalization is introduced to reconstruct the derivative moments and the corresponding Fokker–Planck–Kolmogorov (FPK) equation of the envelope process. The transient and stationary probability density function (PDF) of the envelope process is finally obtained by solving the transient FPK equation. The reliability of this method is verified by using the benchmark results of analytical method and Monte Carlo Simulation. The conclusion and the issues to be further studied are presented in the last part.

Ultrasonic liquid metal processing: The essential role of cavitation bubbles in controlling acoustic streaming

The acoustic streaming behaviour below an ultrasonic sonotrode in water was predicted by numerical simulation and validated by experimental studies. The flow was calculated by solving the transient Reynolds-Averaged Navier-Stokes equations with a source term representing ultrasonic excitation implemented from the predictions of a nonlinear acoustic model. Comparisons with the measured flow field from Particle Image Velocimetry (PIV) water experiments revealed good agreement in both velocity magnitude and direction at two power settings, supporting the validity of the model for acoustic streaming in the presence of cavitating bubbles. Turbulent features measured by PIV were also recovered by the model. The model was then applied to the technologically important area of ultrasonic treatment of liquid aluminium, to achieve the prediction of acoustic streaming for the very first time that accounts for nonlinear pressure propagation in the presence of acoustic cavitation in the melt. Simulations show a strong dependence of the acoustic streaming flow direction on the cavitating bubble volume fraction, reflecting PIV observations. This has implications for the technological use of ultrasound in liquid metal processing.

Prediction of microstructure composition in steel plate heated using high power Yb:YAG laser radiation

This work concerns numerical modelling and computer simulations of temperature field and phase transformations during Yb:YAG laser heating of sheets made of S355 steel. The distribution of laser power emitted by Trumpf laser head D70 is used in the analysis. The heat source is modelled on the basis of interpolation algorithms using geostatistical kriging method. Coupled heat transfer and fluid flow in the fusion zone are described respectively by transient heat transfer equation with convective term and Navier-Stokes equation. The kinetics of phase transformations and volumetric fractions of arising phases are obtained on the basis of Johnson-Mehl-Avrami (JMA) and Koistinen-Marburger (KM) models. Continuous Heating Transformation (CHT) diagram is used for heating process and Continuous Cooling Transformation (CCT) diagram is used for heated steel with the decomposition of final volume fractions of phases transformed form austenite dependant on cooling rates.

Prediction of Weld Interface Depth and Width at Optimum Laser Welding Temperature for Polypropylene

In this preliminary research, Through Transmission Laser Welding (TTLW) of polypropylene is systematically studied through finite element method (FEM) and response surface methodology (RSM) combined approach. The thermal field simulated by solving a three-dimensional transient heat diffusion equation using COMSOL Multiphysics. Statistical software Minitab v17 has been used to develop an experimental design as per central composite design. The objective of the work is in twofold. The first objective of the study aims to develop mathematical models to predict the weld interface depth (MD), weld interface width (MW) and maximum temperature (Tmax), primarily in terms of the process line-energy. The developed mathematical models are examined by analysis-of-variance (ANOVA) technique to check their adequacy. The second objective of the study is to establish an optimised combination of laser welding parameters (laser power, scanning speed and spot diameter) to prevent polypropylene thermal degradation. The desirable MD and MW conditions have been predicted for the established optimised set of laser welding parameters. The results from this preliminary investigation using a FEM approach, can provide the basis for future design of experiments to help predict and obtain enhanced weld quality for polypropylene specifically and other polymer material system in general.

A new model of self-excited induction generator to feed a single phase load with an application in lighting animal farm

This paper presents a new model of self-excited single-phase induction generator, used by a three-phase machine for lighting animal farm through biogas energy. The designs and applications of the previous generator models have been found to be very complex, for this reason simple and effective model of SEIG is proposed. The transient equations of the proposed model were represented by SIMULINK blocks, while the steady state equations by electrical equivalent circuit. The model is derived from Steinmetz connection circuit, stationary reference frame qd-axes theory and symmetrical component theorem. Experimental and simulation using MATLAB are the two methods used to prove the validity of the present model. Results obtained from the two methods were closed to each other. Also, the perfect balance point was achieved at unity power factor. This application proved that the output voltage and current have sinusoidal waves which implying that the generator was operating efficiently.

A New Model of Self-Excited Induction Generator (SEIG) to Feed a Single Phase Load with an Application in Lighting Animal Farm

This paper presents a new model of self-excited single-phase induction generator, used by a three-phase machine for lighting animal farm through biogas energy. The designs and applications of the previous generator models have been found to be very complex, for this reason simple and effective model of SEIG is proposed. The transient equations of the proposed model were represented by SIMULINK blocks, while the steady state equations by electrical equivalent circuit. The model is derived from Steinmetz connection circuit, stationary reference frame qd-axes theory and symmetrical component theorem. Experimental and simulation using MATLAB are the two methods used to prove the validity of the present model. Results obtained from the two methods were closed to each other. Also, the perfect balance point was achieved at unity power factor. This application proved that the output voltage and current have sinusoidal waves which implying that the generator was operating efficiently.

Inlet pressure simulation of pigging in uphill gas pipeline

Pipe cleaning is a common operation in the oil and gas industry. In this paper, the governing equation of the pipeline inspection gauge (PIG, lowercase pig is commonly used) speed is combined with the gas flow equations. The method of characteristics (MOC) is used to solve the transient equations of gas flow. And the process of a pig passing over an uphill section of a gas pipeline is simulated. The results indicate that a pig may get stuck in uphill gas pipeline, due to the coupling of the gas and the pig. Under these circumstances, a higher pressure of the upstream could be helpful for driving the pig in motion. Additionally, the ratio of inlet pressure rise during the pigging process is primarily determined by the inclination of the uphill section. In addition, a formula to predict the inlet pressure during pigging in an uphill pipe is presented. Furthermore, the proposed method and solution can be utilized to predict the speed and position of the pig, as well as the gas pressure and the stoppage of the pig in hilly gas pipelines.

Explicit Solution To Extract Self-Diffusion and Surface Exchange Coefficients from Isotope Back-Exchange Experiments

Multistep 18O isotope exchange procedures and subsequent analytical techniques can be used to elucidate the effect of ambient gas atmospheres on the transport properties of oxide ion-conducting materials utilized in high-temperature solid oxide devices for electrochemical energy conversion. In this contribution, we provide an explicit solution to the one-dimensional transient diffusion equation to estimate oxygen self-diffusion and surface exchange coefficients of oxide ion conducting materials exposed to multistep 18O exchange procedures. Although an analytical solution exists for representing the diffusion profiles of labeled species obtained from a single-step isotope exchange procedure, it is not applicable to the diffusion profiles resulted from consecutive procedures with dynamically altered initial and surface boundary conditions. Hence, a new analytical solution is found for the diffusion problem representing the isotope back-exchange procedure in a semi-infinite spatial domain. The explicit solut...