One-dimensional Finite Difference Research Articles

The effect of nitride formation on the oxidation kinetics of Zry-4 fuel cladding under steam-air atmospheres at 1273–1573 K

The study deals with the oxidation behavior of fuel cladding under mixed steam-air atmospheres. Oxidation tests of Zry-4 were carried out at temperatures of 1373 and 1573 K. Post-test weight gain measurement along with metallographic examination were conducted to study separately the kinetics of the region where nitrides formed and the nitride-free region. The results of oxidation tests at 1273 and 1473 K published previously by the same authors are also addressed in the discussion. The weight gain coming from the nitride-free region was estimated employing one-dimensional finite difference oxygen diffusion model and measured thicknesses of the metallic part of the oxidized specimen, the columnar oxide and the oxygen stabilized α-Zr(O) as well as the fraction of the columnar oxide at the oxide/metal interface. Consequently, the weight gain related to formation of the porous mixed ZrN–ZrO2 regions has been assessed. The results suggested that the nitrides formed beneath the protective columnar oxide layer and that a major part of the weight gain related to formation of the porous mixed ZrN–ZrO2 regions was associated with the formation of ZrO2 from α-Zr(O).

Computational framework for modeling membrane processes without process and solution property simplifications

Accurately modeling membrane processes is critical to evaluating novel process configurations, designing scalable membrane systems, informing process cost estimates, and directing future research. Most membrane process models trade accuracy for computational efficiency by employing simplified approximations of the process (i.e. no salt flux, no pressure drop) and solution properties (i.e. ideal solution, and constant density, viscosity, and diffusivity). This work presents a detailed one-dimensional finite difference model for evaluating membrane processes that avoids these common simplifications. We apply this model to quantify the error introduced by these simplifications for case studies of reverse osmosis, osmotically assisted reverse osmosis, forward osmosis, and pressure retarded osmosis. While the magnitude of error introduced by these simplifications is dependent on the case study parameters and specifications, we find that existing model formulations can underestimate or overestimate average water flux by nearly 50% for some membrane processes operating under standard conditions. Finally, we investigate the error introduced by simplified inlet-outlet models that do not solve the governing system of differential equations, and we assess the accuracy of novel inlet-outlet formulations that use a log and geometric mean, instead of the typical arithmetic mean, to represent non-linear water flux profiles.

Numerical schemes for modelling time-fractional dynamics of non-isothermal diffusion in soils

In this paper we consider the issues for increasing computation schemes performance while modelling non-isothermal convective diffusion using Caputo fractional derivative model. We propose to solve the considered problem using locally one-dimensional finite difference scheme. As the numerical computation of fractional derivative has the biggest influence on computational complexity, an additional approximation procedure of logarithmic complexity is proposed. The procedure represents finite difference form of the fractional derivative as two sets of partial Taylor series which are modified in the process of sequential computations during modelling. As the considered model is non-isothermal, two coupled processes of different speeds are considered. In this case the use of non-uniform time steps can be efficient to improve performance. The proposed approximation procedure is modified for the case of non-uniform time steps and the trial and error method is used to dynamically change time step length during modelling. Time interval in which such schemes are efficient is experimentally determined.

Self-excited vibrations of spring-loaded valves operating at small pressure drops

Flow-induced vibration is a major issue for spring-loaded valves. However, their complex geometry, combined with their diversity of operational conditions, creates a significant challenge for modelling their dynamic response to potential excitation mechanisms. This paper proposes an analytical framework to describe flow–sound–structurecoupling in spring-loaded valves operating at subcritical conditions, therefore without the occurrence of choking at the vena contracta. The model consists of a single-degree of freedom model of the valve structure, a pseudo-force implicit algorithm to calculate impact forces, a one-dimensional unsteady Bernoulli model to describe the flow, and a one-dimensional wave propagation finite difference scheme to account for the acoustic feedback. The model has shown excellent agreement with experimental results as well as promising predictive capabilities for valve life and wear assessment.

PaRallel Algorithms for Modelling Two-Dimensional Non-Equilibrium Salt Transfer Processes on the Base of Fractional Derivative Model

Abstract Modelling of salt transfer processes in fractal structured media has been considered on the base of fractional derivative equations with Caputo-Gerasimov derivatives with respect to space variables. Initial-boundary problem has been solved using locally one-dimensional finite difference scheme. Procedure of fractional derivative approximation has been proposed to lower computational complexity of solution process. Parallel algorithms for distributed memory systems and GPU have been considered. Analysis of using one-dimensional and red-black data partitioning schemes is presented and new parametric scheme which have better characteristics in the determined conditions has been proposed.

A one-dimensional model to describe flow localization in viscoplastic slender bars subjected to super critical impact velocities

In this paper we investigate flow localization in viscoplastic slender bars subjected to dynamic tension. We explore loading rates above the critical impact velocity: the wave initiated in the impacted end by the applied velocity is the trigger for the localization of plastic deformation. The problem has been addressed using two kinds of numerical simulations: (1) one-dimensional finite difference calculations and (2) axisymmetric finite element computations. The latter calculations have been used to validate the capacity of the finite difference model to describe plastic flow localization at high impact velocities. The finite difference model, which highlights due to its simplicity, allows to obtain insights into the role played by the strain rate and temperature sensitivities of the material in the process of dynamic flow localization. Specifically, we have shown that viscosity can stabilize the material behavior to the point of preventing the appearance of the critical impact velocity. This is a key outcome of our investigation, which, to the best of the authors’ knowledge, has not been previously reported in the literature.

Investigation of pork fat as potential novel phase change material for passive cooling applications in photovoltaics

This paper considers pork fat as a novel potential phase change material (PCM) for photovoltaic applications. The PV-PCM configuration was numerically analysed in order to investigate the main performance parameters and its general thermal behaviour regarding passive cooled photovoltaic panels (a simplified one-dimensional finite difference code was applied for the numerical investigation). Two specific PV-PCM systems were compared (convectional organic PCM and pork fat), i.e. their performance response was examined by applying the simplified numerical model. Performance analysis was obtained for specific geographical location with Mediterranean climate conditions (City of Split, Croatia). The obtained results showed that there is no significant difference in performance benefit when the two considered PV-PCM systems are compared. The difference in annual simulated electricity delivery is negligible when comparing the PV-PCM systems, i.e. it was slightly less for the PV-pork fat configuration. The previous finding was important as it proved that pork fat has potential from a performance point of view. A general economic aspect was also addressed in order to check the pork fat's economic viability as a PCM material. It was found that in order to get reasonable LCOE (Levelized Cost of Electricity), the pork fat unit price should be below 0.50 €/kg for specific considered case. Regarding the current market price of pork fat, the considered passive cooling option is currently not economically viable for the general given circumstances. However, economic viability could be established in the case of active cooling configurations where waste heat could be utilized and which could lead to a more favourable LCOE. The main issue with pork fat is one related to the long-term degradation of its physical properties which certainly requires intense further experimental investigation.

The discontinuous Galerkin method with higher degree finite difference compatibility conditions and arbitrary local and global basis functions

This paper focuses on the discontinuous Galerkin (DG) method in which the compatibility condition on the mesh skeleton and Dirichlet boundary condition on the outer boundary are enforced with the help of one-dimensional finite difference (FD) rules, while in the standard approach those conditions are satisfied by the penalty constraints. The FD rules can be of arbitrary degree and in this paper the rules are applied up to fourth degree. It is shown that the method presented in this paper gives better results in comparison to the standard version of the DG method. The method is based on discontinuous approximation, which means that it can be constructed using arbitrary local basis functions in each finite element. It is quite easy to incorporate some global basis functions in the approximation field and this is also shown in the paper. The paper is illustrated with a couple of two-dimensional examples.

Numerical Modelling of Frost Formation of Flat Surface

Frost formation begins when humid air comes in contact with a cold surface kept below freezing temperature of water. Objective of the present study is to develop a numerical model which can predict frost formation parameters such as rate of frost growth, frost densification and to study effect of ambient conditions on these parameters. The one-dimensional pure implicit finite difference method is adopted for solving differential equations. Numerical code is written in MATLAB 2013. The proposed numerical model is validated against two independent published experimental data with 6.6 and 16.8% deviation. Effect of ambient parameters like wall temperature, ambient temperature, humidity and Reynolds number on frost growth and densification are investigated. Also effect of variable wall temperature and variable ambient temperature on frost growth rate is discussed.

Mangroves buffer marine protected area from impacts of Bovoni Landfill, St. Thomas, United States Virgin Islands

One of the many ecosystem services that mangrove systems provide is their ability to act as buffers between the land and sea, protecting human development from storm surges while also trapping terrestrial pollutants. In St. Thomas, United States Virgin Islands, an ecologically-important mangrove system sits between Bovoni Landfill and a marine protected area, the St. Thomas East End Reserves. To characterize the physical processes driving this mangrove system, groundwater hydraulic head, sediment cores, sediment surface temperatures, and water and sediment chemistry were analyzed. Hydraulic head data from January to November 2014 were used to determine vertical and horizontal groundwater flow directions. Water and sediment samples were tested for heavy metals potentially originating from Bovoni Landfill. Stratigraphic context was provided by the sediment cores and used to infer past environmental conditions. Subsamples were taken from these cores and analyzed for dry bulk density, organic matter content (through loss on ignition), and heavy metals using electron microscopy. Vertical groundwater velocity and sediment porosity were determined by calibrating a one-dimensional finite difference heat transport model to near surface temperature data from depths of 0, 7, 14, and 21 cm. Groundwater was found to flow from the terrestrial upland, through the mangroves, and toward the ocean for the majority of the study. Flow reversal was seen after long periods of little precipitation. In the surface and shallow groundwater samples, trace metal concentrations were measured from 23 to 105 μg/L for Cr, Ni, Sn, and Zn. Sediment samples collected near the landfill contained Bi, Cr, Sn, Ti, and Zn. Very slow flushing of sediment pore water was indicated by the vertical groundwater velocities produced from the heat transport model, which ranged from ±10–7 to ±10–9 m/s. This study revealed that the mangrove system is an important buffer system protecting the outer lagoon of the marine protected area from terrestrial contaminants via sediment trapping and slowing of water fluxes from the upland area into the lagoon. The results presented here can be used as a baseline for future studies and are relevant to local managers and to landfill closure plans.

Design of Thermal Protection System for Reusable Hypersonic Vehicle Using Inverse Approach

In this study, an inverse heat conduction analysis based on the Levenberg-Marquardt method is applied to reconstruct the aerothermal heating and thermal protection material response of a reusable hypersonic vehicle (RHV) from the knowledge of temperature measurements taken within the thermal protection system (TPS). In addition to reconstruction of TPS material response, a lightweight passive TPS for the RHV based on these predictions is designed. In this analysis, the experimental temperature data are generated by a direct approach; this is due to the lack of real experimental setup that is used to measure the temperature. The direct analysis has performed by an explicit one-dimensional finite difference code to analyze the in-depth temperature response of the TPS. The temperature data determined from the direct approach are used to simulate the temperature measurements. The influence of measurement errors upon the accuracy of the inverse results is also investigated. Finally, results show that the Levenberg-Marquardt method is an effective technique to design a lightweight passive TPS for an RHV by using the inverse approach.

One Dimensional Numerical Simulation of Aggradation-Degradation in a Channel Using Finite Difference Method Case Study Chashma Right Bank Canal (CRBC)

A one-dimensional MacCormack explicit finite difference model is developed for simulating hydraulics and bed changes in irrigation channels. The Saint-Venant equations describing unsteady flow in open channels and the continuity equation for the conservation of sediment mass are numerically solved. These equations are highly nonlinear and therefore do not have analytical solutions. For this purpose the MacCormack scheme is used. The scheme is second order accurate; it is a coupled solution as it is a two-step predictor corrector method. Model gives results in terms of bed level changes, flow depth and discharge provided physical boundaries of the system are valid for simulation time. Model execution and accuracy is very sensitive to time step and stability. The simulated results show a good agreement with previous studies in the downstream section, and can predict an average value of measured profile in the upstream section. The application of this model to Chashma Right Bank Canal in Pakistan and results of the model are compared with the published results gave very convincing result.

Co-simulation of a HVAC system-integrated phase change material thermal storage unit

A co-simulation environment, consisting of a detailed mathematical model of a thermal energy storage unit which is incorporated with an EnergyPlus simulation model of a full building HVAC system, is described. The two models are integrated using the user-defined plant component feature in EnergyPlus and the Building Controls Virtual Test Bed (BCVTB) environment. The thermal energy storage unit, which consists of encapsulated phase change material in a series of flat plates and a heat transfer working fluid (water), is modelled using a transient one-dimensional forward finite difference method. The thermal storage model is executed within MATLAB and is verified against experimental data, showing a discharging heat transfer accuracy to within 2.5%. The building model, which incorporates a retrofitted ground source heat pump system within a thermally massive building, is simulated in the EnergyPlus environment. The co-simulation arrangement allows for in-depth analysis of the integrated system under dynamic operating conditions, which is currently not possible within the EnergyPlus environment. Moreover, the overall adopted approach, based on generic integration of a detailed mathematical model, using a third party generalised programming environment, into an established building simulation environment, serves as a successful exemplar for other researchers and practitioners working in the field.

Numerical Simulation of the Freeze-Thaw Behavior of Mortar Containing Deicing Salt Solution.

This paper presents a one-dimensional finite difference model that is developed to describe the freeze-thaw behavior of an air-entrained mortar containing deicing salt solution. A phenomenological model is used to predict the temperature and the heat flow for mortar specimens during cooling and heating. Phase transformations associated with the freezing/melting of water/ice or transition of the eutectic solution from liquid to solid are included in this phenomenological model. The lever rule is used to calculate the quantity of solution that undergoes the phase transformation, thereby simulating the energy released/absorbed during phase transformation. Undercooling and pore size effects are considered in the numerical model. To investigate the effect of pore size distribution, this distribution is considered using the Gibbs-Thomson equation in a saturated mortar specimen. For an air-entrained mortar, the impact of considering pore size (and curvature) on freezing was relatively insignificant; however the impact of pore size is much more significant during melting. The fluid inside pores smaller than 5 nm (i.e., gel pores) has a relatively small contribution in the macroscopic freeze-thaw behavior of mortar specimens within the temperature range used in this study (i.e., +24 °C to -35 °C), and can therefore be neglected for the macroscopic freeze-thaw simulations. A heat sink term is utilized to simulate the heat dissipation during phase transformations. Data from experiments performed using a low-temperature longitudinal guarded comparative calorimeter (LGCC) on mortar specimens fully saturated with various concentration NaCl solutions or partially saturated with water is compared to the numerical results and a promising agreement is generally obtained.

Theoretical investigation of the effects of the cushion on reducing underwater noise from offshore pile driving

Offshore pile driving noise has gained increasing attention due to its adverse influence on marine animals. Two models are established to investigate the influence of the cushion on the underwater noise from pile driving. A one-dimensional nonlinear finite difference model is developed to compute the impact force and the soil penetration. The radiated sound pressure is predicted by the second model based on a semi-analytical variational formulation. For the examined pile, a proper cushion can significantly dampen the high-frequency impact force and reduce the peak of the sound pressure. It is found that there exists a stiffness range within which the decline of the cushion stiffness effectively reduces the single sound exposure level but hardly increases the required hammer blows to achieve a certain penetration depth.

Optimization of building envelope design for nZEBs in Mediterranean climate: Performance analysis of residential case study

Design criteria for a residential nZEB in Mediterranean climate are discussed. The integrated design procedure focuses on the problem of a large number of available building variants concerning the building envelope. The aim is to search the ones that minimize winter and summer energy demand without compromising thermal comfort.The adopted methodological approach combines the use of dynamic energy simulation tool (EnergyPlus), based on a one-dimensional conduction finite difference solution method, and a constrained multi-objective optimization algorithm. For four cities (Madrid, Nice, Naples, Athens), several passive strategies are compared: thermal properties of the building envelope, adoption of phase change materials with different melting temperatures, cool roof solutions, several window/wall ratio values, some external and internal shading systems. The results allow to evidence that it is difficult to understand the best trade-off between summer and winter performance, by assuring high standard of thermal comfort when the aim is to reach NZEB objectives in Mediterranean climate. However, some guidelines are indicated, starting from the discussed results.

Equivalence of one-dimensional second-order linear finite difference operators

The direct, gauge and projective equivalence problems for one-dimensional second-order linear finite difference operators are solved using the method of equivariant moving frames.

Energies and wave functions of an off-centre donor in hemispherical quantum dot: Two-dimensional finite difference approach and ritz variational principle

Eigenvalues equation solutions of a hydrogen-like donor impurity, confined in a hemispherical quantum dot deposited on a wetting layer and capped by an insulating matrix, are determined in the framework of the effective mass approximation. Conduction band alignments at interfaces between quantum dot and surrounding materials are described by infinite height barriers. Ground and excited states energies and wave functions are determined analytically and via one-dimensional finite difference approach in case of an on-center donor. Donor impurity is then moved from center to pole of hemispherical quantum dot and eigenvalues equation is solved via Ritz variational principle, using a trial wave function where Coulomb attraction between electron and ionized donor is taken into account, and by two-dimensional finite difference approach. Numerical codes developed enable access to variations of donor total energy, binding energy, Coulomb correlation parameter, spatial extension and radial probability density with respect to hemisphere radius and impurity position inside the quantum dot.

Reconstruction of aero-thermal heating and thermal protection material response of a Reusable Launch Vehicle using inverse method

A transient inverse heat conduction analysis based on Lavenberg–Marquardt method is presented to reconstruct the aero-thermal heating and thermal protection material response of a Reusable Launch Vehicle (RLV), from the knowledge of temperature measurements taken within the thermal protection system. In this analysis, due to the lack of real experimental set-up to measure the temperature however it is generated by direct approach using real heating profile on the surface of RLV developed at NASA Langley Research Center. The direct analysis is performed by an explicit one-dimensional finite difference code to determine the in-depth temperature response of Integrated Thermal Protection System (ITPS) of RLV. The temperature data determined from direct approach is used to simulate the temperature measurements. The effects of thermocouple arrangement and measurement error upon the stability of inverse results are investigated. Results show that the Lavenberg–Marquardt is an effective technique to predict aero-thermal heating of RLV by using inverse approach.

A study of high-order grid-characteristic methods on unstructured grids

In this paper, grid-characteristic methods for solving hyperbolic systems using approximation with high-order interpolation on unstructured tetrahedral and triangular grids are studied. Interpolation methods of order from 1 through 5 inclusive are considered. One-dimensional finite difference schemes for these methods are presented. The stability of these schemes is investigated. The grid-characteristic methods on unstructured triangular and tetrahedral grids are successfully used in seismic prospecting, specifically under the Arctic shelf and permafrost conditions. They are also used to solve problems of seismics, dynamic deformation and destruction, and to study anisotropic composite materials.