Implicit Finite Difference Model Research Articles

An expansion of the Fisher model for concentration dependent grain boundary diffusion

In accounting for concentration dependent grain boundary diffusion, an extra term is derived and added to the boundary condition in Fisher’s Model for atomic transport in fast diffusing boundaries under the condition of low diffusant concentration resulting in an expanded model. The validity of this is evaluated using an implicit finite difference model which shows that a minimum of around 0.5–0.6 order of magnitude difference between the minimum and maximum grain boundary diffusion values is necessary for a significant effect to be observed. This was investigated for a grain boundary diffusion coefficient that exponentially increased and decreased with concentration. Whereas effects on measured grain boundary diffusion coefficients are evaluated to be negligible, the concentration dependency of the grain boundary diffusion substantially affects the elemental distribution in the surrounding of the grain boundary. The latter directly influences grain boundary related metallurgical phenomena such as grain growth or phase transformations in nanocrystalline materials.

Experimental Investigation of Fractional Order Behavior in an Oscillating Disk

Abstract An experimental investigation is presented of an oscillating disk submerged in water. The system is adapted from one modeled by the Bagley-Torvik equation, being modified to more closely approximate idealizations in the derivation. The modified system eliminates alignment problems that would violate the assumption of fluid forces being due only to shear in the fluid. An implicit finite difference model based on the Riemann-Liouville fractional derivative was used to model the observed oscillatory free response resulting from a non-homogeneous initial condition. The numerical model accounted for motion well until the amplitude was small.

Thermo-chemical-mechanical simulation of low temperature nitriding of austenitic stainless steel; inverse modelling of surface reaction rates

A multi-physics thermo-chemical-mechanical 1-dimensional implicit finite difference model is presented to simulate the evolution of composition and residual stress profiles over the expanded austenite case developing during low temperature nitriding of austenitic stainless steels. The model combines 1-dimensional diffusion of nitrogen in the depth direction with a concentration-dependent diffusivity, elasto-plastic accommodation of the lattice expansion, stress gradient-induced diffusion of nitrogen, solid solution-strengthening by nitrogen and trapping of nitrogen by chromium atoms. The rate of the surface reaction governing the transfer of nitrogen from the gas to the solid is unknown and was evaluated by inverse modelling. The modelling was applied adopting the surface reaction rate as the only fitting parameter and taking mass-uptake curves (thermogravimetry) as the constraint, while all other data were taken from established literature values. Very good agreement is achieved between the predicted and experimental composition-depth profiles. Further, the applicability of the present model to plasma nitriding was verified by simulating (not fitting) the evolution of composition-depth profiles obtained after plasma nitriding of stainless steel. The good to very good agreement of the present model's predictions with experimental data for gaseous and plasma nitriding, indicates that the essential multi-physics influences and parameters are taken into account, with a minimum of adjustable parameters.

An Implicit Finite Difference Model for Electrokinetic Remediation of Cd-Spiked Kaolinite Under Acid-Enhanced and Unenhanced Conditions

A numerical model based on an implicit finite difference method is presented to describe multi-species transport in cadmium-contaminated kaolin under electrokinetic remediation process. A set of algebra and partial differential equations were used to mathematically describe the reactive transport of the target species. We considered different coupled transport phenomena such as diffusion, electroosmotic flow, and ionic migration. Geochemical and electrochemical reactions including precipitation/dissolution, adsorption/desorption, water auto ionization, and electrolysis in anode and cathode cells were included in the model. Simulation results were compared with the available experimental data for electrokinetic remediation of cadmium-contaminated kaolinite. The coefficient of determination (R2) and index of agreement (IA) were calculated to indicate the model accuracy and to compare the present model with the existing explicit finite difference model in the literature. The R2 and IA calculation described that the present model had a good capability of simulating the experimental measurement. In addition, it illustrated that the presented implicit finite difference model had a better simulation result than the explicit model. The sensitivity analysis was conducted to understand the impact of tortuosity and retardation factors on the model output. Sensitivity analysis indicated that instead of the acid-enhanced test, the retardation factor had more impact on the unenhanced test. In addition, the sensitivity analysis of tortuosity factor revealed that this parameter had more significant effect on the cadmium concentration rather than pH profile.

Blockage location in pipelines using an implicit nonlinear finite-difference model optimization

In this paper, a new approach for punctual blockage location in pipelines is proposed. This approach is based on an implicit finite-difference modeling of pipeline dynamics including the effect of punctual blockage under study, and on a related optimization: using measurements at both ends of the pipeline, together with an appropriate excitation, the parameters of the blockage are directly identified from a bank of simulators. This methodology is shown to be pretty efficient via simulation results, even in the presence of additive measurement noise.

Numerical modeling of two species radionuclide transport in a single fracturematrix system with variable fracture aperture

A variable aperture model, instead of a conventional parallel plate model, is utilized to study the transport of radionuclides in a single coupled fracture-matrix system. A fully implicit finite difference model has been developed, which incorporates fracture aperture width variation in the numerical study of two species radionuclide transport. Two distinct geometric profiles namely, sinusoidal and logarithmic have been used to capture the variation of aperture width. The dependence of advection, hydrodynamic dispersion, linear sorption, and matrix diffusion on aperture width is considered in the analysis of radionuclides transport. Two species (parent and daughter) radioactive decay chain is also incorporated. There is a greater retardation of radionuclides in fracture for the variable aperture model than the parallel plate model. Sensitivity analysis on fracture surface sorption coefficient, longitudinal dispersivity, matrix porosity, and matrix diffusion coefficient shows that the conventional parallel plate model overestimate the radionuclide concentration in the fracture when compared to the variable aperture model.

Twenty-four hour simulation of solar chimneys

A model for time dependent analysis of solar chimneys is presented. The energy balance equations for three components of solar chimneys, absorbing plate, cover glass and air-gap are discretized with respect to time using an implicit finite difference model. The discretized nonlinear energy balance equations are solved for numerous time steps over a 24-h period using the Newton–Raphson method. The time dependent solar irradiation is determined using the clear sky model. The model's performances for various parameters of solar chimney that affect the thermal mass of the absorbing plate are tested. It was determined that a solar chimney with a relatively large thermal-mass produces airflow well into night and early morning when no solar irradiance is present. Also, a high thermal mass of the absorbing plate results in smaller variations in the airflow rate. The results of the present model for a solar chimney with no thermal storage compare well with the previously published data.

Fast-Scan Controlled-Adsorption Voltammetry for the Quantification of Absolute Concentrations and Adsorption Dynamics

Fast-scan cyclic voltammetry has depended on background subtraction to quantify small changes in neurotransmitter concentration. Because of this requirement, measurements of absolute concentrations using fast-scan cyclic voltammetry have been limited. Here we develop and characterize fast-scan controlled-adsorption voltammetry (FSCAV), which enables direct measurements of absolute concentrations in vitro without the use of flow injection to change the concentration. This enables probing the diffusion-controlled adsorption dynamics of biogenic amines and other adsorbing species. An implicit finite-difference model of mass-transport-limited adsorption was developed and is in agreement with experimental results. Optimization of FSCAV yielded a sensitivity of 81 ± 11 nA/μM for dopamine, corresponding to a limit of detection of 3.7 ± 0.5 nM. Through the combination of novel instrumentation and validated computer simulations, we show that FSCAV is an important measurement tool that can be used to determine absolute concentrations and study mass-transport-limited adsorption.

Simulation of wellbore stability with thermo-hydro-chemo-mechanical coupling in troublesome formations: an example from Ahwaz oil field, SW Iran

Wellbore stability is a main concern in drilling op- eration. Troublesome drilling issues are chemically active for- mations and/or high-pressure-high-temperature environments. These are mainly responsible for most of wellbore instabilities. Wellbore failure is mostly controlled by the interaction between active shales and drilling fluid in shale formations. The factors influencing this interaction consist of fluid pressure, tempera- ture, composition of drilling fluid, and exposure time. In this paper, a non-linear fully coupled chemo-thermo-poroelasticity model is developed. At first, a fully implicit finite difference model is presented to analyze the problem, and then, it is verified through comparison with analytical models. For this purpose, a computer code has been written in MATLAB envi- ronment. Results indicate that chemical osmosis and thermal osmosis significantly influence the wellbore stability in shale formations. Cooling and high salinity in mud decrease pore pressure and total tangential and radial stresses in the formation and vice versa. Moreover, CaCl2 solutions result in higher osmotic pressure than NaCl solutions. The type of constitutive model is investigated on wellbore failure. In addition, the proposed model is applied for wellbore instability in Ahwaz oil field, SW Iran. Results show that the proposed model considerably predicts the wellbore instability in this field.

Virus Transport through Unsaturated Zone: Analysis and Parameter Identification

AbstractThis paper presents numerical and parameter estimation models for the analysis of virus transport and the identification of transport parameters in the unsaturated zone. The numerical model couples a mass conservative fully implicit finite difference model simulating moisture flow in the unsaturated zone with the hybrid finite volume model for virus transport. The accuracy of the numerical scheme is tested for both advection- and dispersion-dominated transport. The comparison of the numerical model with the analytical solution indicates that the numerical model’s predictions are in excellent agreement with the analytical predictions. The parameter estimation is formulated as a nonlinear least-squares minimization problem in which the parameters are estimated by minimizing the deviations between the model-predicted and experimentally observed virus concentrations. A parameter estimation procedure is developed by coupling the numerical model simulating one-dimensional virus transport in the unsatura...

Laboratory Validation of Ultra-Soft Soil Deformation Model

Ultra-soft soil with high moisture content will experience large strain deformation under one-dimensional compression with little or no gain in effective stress. Such deformation behaviour does not comply with Terzaghi’s effective stress gain theory. The e-log \( \sigma_{v}^{\prime } \) relationship of ultra-soft soil is non-linear with large compression index in the first order of log cycle. This paper proposes three compression indices (\( C_{c1}^{*} \), \( C_{c2}^{*} \) and \( C_{c3}^{*} \)) for stresses covering three log cycles. Good prediction of settlement magnitude is possible with these newly proposed compression parameters for ultra-soft soil. In addition, implicit finite difference model applying the large strain theory is also proposed and validated with results from laboratory measurements. The time factor curves for ultra-soft soil with large strain compression are also proposed and validated.

Simulation of solute transport under oscillating groundwater flow in homogeneous aquifers

In this paper, we focus on the influence of temporal variations in the regional hydraulic gradient, particularly, on the impact of temporal variations in the boundary conditions on the spreading of solute plumes in homogenous aquifers. We examined the problem by numerical simulations. Twodimensional fully implicit finite difference model TRANS_GW_2D for the unsteady groundwater flow and a random walk particle tracking model TRANS_RW_2D for solute transport have been developed to solve the governing equations without restriction on the values of aquifer storativity or on the magnitude of the temporal fluctuations. It has been shown that transient flow conditions (in terms of gradient magnitude variability) have a significant impact on contaminant transport only if the amplitude and period of the oscillations are relatively large. For relatively small oscillations, a steady state flow field can be justified. Transient conditions may be relevant in coastal aquifers with high tidal amplitudes. This tidal variation can have an effect on the spreading of solutes and on salt-water intrusion. Our numerical experiments demonstrate that in case of relatively high storativity values, the dispersion coefficient is amplified in time as the plume moves towards the fluctuating boundary.

A two‐dimensional vertical non‐hydrostatic σ model with an implicit method for free‐surface flows

AbstractAn implicit finite difference model in the σ co‐ordinate system is developed for non‐hydrostatic, two‐dimensional vertical plane free‐surface flows. To accurately simulate interaction of free‐surface flows with uneven bottoms, the unsteady Navier–Stokes equations and the free‐surface boundary condition are solved simultaneously in a regular transformed σ domain using a fully implicit method in two steps. First, the vertical velocity and pressure are expressed as functions of horizontal velocity. Second, substituting these relationship into the horizontal momentum equation provides a block tri‐diagonal matrix system with the unknown of horizontal velocity, which can be solved by a direct matrix solver without iteration. A new treatment of non‐hydrostatic pressure condition at the top‐layer cell is developed and found to be important for resolving the phase of wave propagation. Additional terms introduced by the σ co‐ordinate transformation are discretized appropriately in order to obtain accurate and stable numerical results. The developed model has been validated by several tests involving free‐surface flows with strong vertical accelerations and non‐linear waves interacting with uneven bottoms. Comparisons among numerical results, analytical solutions and experimental data show the capability of the model to simulate free‐surface flow problems. Copyright © 2004 John Wiley & Sons, Ltd.

Modelled response of Gulf St Vincent (South Australia) to evaporation, heating and winds

In Gulf St Vincent, Australia, the salinity of the head waters can exceed 42 in summer when evaporation is maximum and the rainfall is minimal. A depth-integrated implicit finite difference model is extended to simulate the summer–autumn evolution of salinity, temperature, and density distributions, with climatological evaporation, rainfall, air temperature, and wind stress as inputs. Advection of salt and heat by the density- and wind-driven circulation is modelled by the Quick scheme, whereas horizontal mixing by tidal circulation is parameterised by a dispersion coefficient related to the oscillatory vertical shear. Simulated distributions and seasonal variations compare well with available observations, which feature the flow of highly saline water along the eastern side of the gulf, while the western side is bathed by less saline shelf water. Model results show that, despite the increasing salinity gradients in summer, opposing temperature gradients can stifle the shelfward density currents in the southern parts of the region. Autumn cooling intensifies these density currents so that at the end of the season the flushing of highly saline water, accumulated in the gulf throughout the summer, is enhanced. It was found that the variability in the general circulation brought about by the directional variability in the prevailing winds is an important factor in maintaining the observed salinity distributions in the region.

Comparison of Numerical Procedures for Gate Stroking

A study was conducted to compare three existing solution methods for the inverse problem of unsteady open-channel flow (i.e., gate stroking). Both the accuracy of the results and the robustness of the methods were examined. Similar solutions were computed with a method-of-characteristics and an implicit finite-difference model, except under extreme transients, in which case the method-of-characteristics model performed better. However, the characteristic model was unable to find a solution for some proposed examples. An explicit finite-difference model was also examined and, although it produced satisfactory results for some examples, it is inherently unstable. A nonlinear, implicit finite-difference gate-stroking method was developed, which was sufficiently accurate and more robust than the existing methods.

Unsteady Flow Analysis in Large-scale Canal Networks Using the Implicit Finite Difference Model with Computational Ordering Procedures

Unsteady Flow Analysis in Large-scale Canal Networks Using the Implicit Finite Difference Model with Computational Ordering Procedures

A 3-dimension implicit finite-difference model for wind-driven water current in Moshui Lake

A 3-dimension implicit finite-difference model for wind-driven water current in Moshui Lake

Heat transfer in the hot rolling of metals

The heat-transfer coefficient (HTC) in the roll gap during the hot rolling of AA5XXX-series (aluminum-magnesium) alloys has been measured in a laboratory mill with the aid of thermocouples attached to the surface and embedded in the interior of test samples. The heat-transfer coefficient was calculated from the sample temperature response using an implicit finite-difference model over a range of temperatures, strain rates, and pressures. Values of 200 to 450 kW/m2 °C were obtained by backcalculation. A comparison of the results from this study with those measured in a previous investigation on two steel alloys has led to the development of an equation which characterizes the HTC as a function of the ratio of the rolling pressure to the flow stress at the surface of the workpiece. This relationship has been employed to explain the apparent differences in the heat-transfer behavior of different metals at similar rolling pressures.

CROSOWAT: A decision tool for irrigation schedule

A numerical model has been developed to predict the soil moisture content over the root zone depth as a function of time and to estimate the irrigation water requirements in humid basins under various management options. An implicit finite difference model for transient one dimensional flow, backed up by routines for computing various components of the field hydrological cycle such as evapotranspiration, effective rainfall, surface runoff, capillary rise, etc. is proposed. The paper describes the assumptions made, theoretical considerations of various submodules, system flowchart, hardware requirements and details of input-output. Comparison of the model predicted soil moisture profiles with the field observations is also given along with the results of numerical experiments carried out with the model in the form of sensitivity analysis for various model parameters. Statistical error analysis is also given to prove reliability of the model for estimation of irrigation water requirements.

Modeling the effect of mulch optical properties and mulch-soil contact resistance on soil heating under plastic mulch culture

A numerical model was developed to simulate the effect of plastic mulches on the field energy balance and soil temperature regime. Newton-Raphson methods were used to solve the energy balances of the soil and mulch simultaneously. The resultant temperature of the soil surface was used as a boundary condition for an implicit finite difference model of soil heat flow. The model contained a detailed radiation section to account for mulch optical properties in shortwave and longwave bands. Heat transfer between the mulch and soil surface was modeled using a thermal contact resistance, and transport in the boundary layer was quantified using flux profile theory. The effect of evaporation and condensation on the underside of the mulch was not considered. The accuracy of the model was verified by comparing simulated soil temperatures with field data collected under five commercially available mulches. Simulated average daily temperatures were within ±1.3°C of the observed values and root mean square errors were less than 2.5°C in all cases. The model showed that mulch optical properties strongly influenced the way energy was partitioned at the surface. Complete characterization of the mulches in the longwave spectrum was required for realistic simulation. Changing longwave reflectance by only 0.3, for example, caused maximum soil temperature to change by 5°C under some mulches. Thermal contact resistance between the mulch and soil affected temperatures by up to 20°C beneath mulches that strongly transmitted or absorbed radiation. However, contact resistance was not an important factor for mulches that had nearly equal shortwave transmittance and absorptance. Results demonstrate that a complex relationship exists between mulch optical properties and thermal contact resistance. Modification of mulch longwave properties during manufacturing and control of thermal contact resistance during mulch installation appear to be two promising areas for research. An alternative numerical scheme was evaluated that utilized linear forms of the energy balance equations. The linearized model produced results almost identical to the original model, but required less computing time and could be adapted easily to simulate multiple mulch layers.