Time-varying Boundary Conditions Research Articles

A one-dimensional inverse radiative transfer problem with time-varying boundary conditions

In this work are presented the formulation and solution of two inverse radiative transfer problems in one-dimensional homogeneous participating media with time-dependent boundary conditions. In the first one an energy balance for the incoming and exit radiation allows the direct determination of the absorption coefficient. In the second inverse problem we solve the simultaneous estimation of the total extinction and scattering coefficients using the source–detector method. We first consider a more general formulation of the direct radiative transfer problem, and then for the formulation of the inverse problems we use the particular cases of “cold medium” (no internal sources), transparent boundaries (no effects associated with differences on medium and environment refractive indices), and homogeneous media (constant radiative transfer properties). Test case results are presented.

Dolomitization: from conceptual to numerical models

AbstractDolomitization requires not only favourable thermodynamic and kinetic conditions, but also a fluid-flow mechanism to transport reactants to and products from the site of dolomitization. This paper reviews work that seeks to provide a quantitative framework for conceptual models of dolomitization, using analytical and, particularly, numerical simulation models of fluid flow and rock-water interaction. This approach is starting to yield new insights into the major controls on the rate and pattern of fluid flux, and the resultant dolomitization.Three sets of forces can drive the fluid flow required for dolomitization: elevation (topographic) head of meteoric water and/or seawater; gradients in fluid density due to variation in salinity and/or temperature; and pressure due to sedimentological and/or tectonic compaction. However, in many situations individual flow mechanisms may not operate in isolation. Rather fluid flow will commonly be a product of a number of different drives acting simultaneously. The balance between drives will change over time with variations in relative sea-level, climate, platform geometry and palaeogeography (which collectively comprise the critical boundary conditions). The simplistic prediction of dolomite body geometry from a single driving force may be misleading, as fluid flow will critically depend both on the boundary conditions and the distribution of permeability. Indeed, even for single driving forces, model predictions change significantly as simplistic assumptions are relaxed and these key parameters are specified with increasing realism.The coupled modelling of dolomitization reactions within the flow field is less tractable than that of groundwater circulation because the kinetics of dolomitization are less well understood, particularly at lower temperatures. Dolomitization is likely to occur along a reaction front, where a favourable balance is struck between mass transport and reaction kinetics. For instance, in simulations of geothermal convection dolomitization focuses along the 50–60 °C isotherm. Dolomitization reactions are favoured by higher temperatures in deeper zones, but rates are limited by low flow because of lower permeability. Although flow rates are higher in shallow more permeable carbonates, lower temperatures limit reactions. High flow rates during reflux of platform-top brines give rapid dolomitization. This is associated with porosity occlusion in front of and behind the broad zone of replacement dolomitization driven by anhydrite cementation and overdolomitization, respectively. Lithological heterogeneities strongly affect the pattern of dolomitization, which is highly focused within more permeable beds and those with a higher reactive surface area.While we focus here on dolomitization, models can also provide insights into diagenetic processes such as marine calcite cementation and aragonite, calcite and evaporite dissolution by refluxing brines, and by seawater circulation below the aragonite and calcite compensation depths. However, it is important to be aware of the assumptions and limitations of the numerical model(s) used. Particular attention must be paid to specification of boundary conditions, permeability and reactive surface area. The uncritical application of numerical techniques to particular cases of dolomitization is at best uninformative and at worst misleading. Careful application of these techniques offers great promise for well-constrained field problems, with greater inclusion of natural heterogeneity and time-variant boundary conditions. We also need to model feedbacks between diagenesis and porosity-permeability, and to include platform growth in simulations of slower diagenetic processes.

Finite Element Analysis of Tire Curing Process

Curing is one of the most important steps in the tire manufacturing process. During this process, a green tire is formed to the desired shape and the compound is converted to a strong, elastic material to meet tire performance needs. The process of curing is usually accomplished under pressure and an elevated temperature provided by the mold. The curing process is energy consuming and has a strong effect on material properties. To attain an optimal state of cure for different compounds of various dimensions at minimal capital and energy costs requires proper evaluation of the state of cure in a tire. Various numerical models have been proposed to determine the state of cure of rubber compounds in molds. Their applications are limited to simple geometry and boundary conditions. For a tire, which has complex shape and variable boundary conditions and is built from layers of rubber compounds and fiber/rubber composites, the finite element method appears to be an ideal candidate because of its versatility. In this paper, a simulator for the tire curing processes was developed based on the finite element method of an axisymmetric heat transfer problem for composite materials. The anisotropy of heat transfer properties of composite materials, the dependence of properties of rubber compounds on the temperature and/or the extent of cure, the time-varying boundary conditions which include the cooldown of the tire out of the press and the rigorous cure kinetic models are taken into account. The numerical simulation results of a truck tire curing process show that the simulator successfully describes the variation trends in temperature and in state of cure with the tire curing process. It also serves as the core module of an optimization algorithm, which is being developed for the tire curing and rubber formulation designs.

Analysis of the solvent diffusion in glassy polymer films using a set inversion method

Within the framework of solvent diffusion in glassy polymers, this paper concerns an experimental study of toluene sorption and desorption in P(MMA/ nBMA) copolymer films. Gravimetric experiments (quartz microbalance) are performed in a pressure and temperature controlled chamber. Coupling between solvent diffusion and viscoelastic relaxation is taken into account through the time-dependent solubility model, based on the Fick diffusion equation inside the film and a time variable boundary condition at the film/vapor interface. Viscoelastic relaxation is described by a first order model or by a stretched exponential. In the present paper, a special focus is given on the set inversion method used to analyze the data and to derive well-defined uncertainty intervals upon each determined quantity, taking all the uncertainties on the weight measurements into account. We find that the mutual diffusion coefficient strongly decreases in the glassy state, of about two orders of magnitude for a 0.05 decrease in the solvent weight fraction.

Determination of thermal diffusivity of mortadella using actual cooking process data

Thermal properties are very important to simulate heat transfer during heat treatment of foods. The aim of this work was to study heat transfer aspects of the mortadella cooking process and the use of actual temperature transient profiles to estimate the effective thermal diffusivity ( α). The Ball and Olson equation was used to determine α behavior during the cooking process, wherein it was verified that, α increases twofold after protein denaturation, at close to 70 °C. Another objective was to verify the ability of a conduction heat transfer model to predict the heat penetration curve. The results showed that it is possible to estimate the effective thermal diffusivity using experimental temperature history and the heat conduction model, even for time-varying boundary conditions.

One-dimensional simulation of solute transfer in saturated–unsaturated porous media using the discontinuous finite elements method

A one-dimensional transport model for simulating water flow and solute transport in homogeneous–heterogeneous, saturated–unsaturated porous media is presented. The model is composed of a combination of accurate numerical algorithms for solving the nonlinear Richard's and advection–dispersion equations (ADE). The mixed form of Richard's equation is solved using a standard finite element method (FEM) with primary variable switching. The transport equation is solved using operator splitting, with the discontinuous finite element method (DFE) for discretization of the advective term. A slope limiting procedure for DFE avoids numerical instabilities but creates very limited numerical dispersion for high Peclet numbers. An implicit finite differences scheme (FD) is used for the dispersive term. The unsaturated flow and transport model (Wamos-T) is applied to a variety of rigorous problems including transient flow, heterogeneous medium and abrupt variations of velocity in magnitude and direction due to time-varying boundary conditions. It produces accurate and mass-conservative solutions for a very large range of grid Peclet numbers. The Wamos-T model is a good and robust alternative for the simulation of mass transport in unsaturated domain.

Method of Measuring Thermal Diffusivity of Composites with Thick Fillers and Reinforced Rubbers

The thermal diffusivity is a property of materials necessary for the mathematical modelling of unsteady heat transfer proceeding during their processing. A new method of measuring thermal diffusivity of solid materials was developed. In estimating the thermal diffusivity of solid materials by this method the experimental temperature field of the sample must be analysed. The values of the thermal diffusivity at different temperatures are obtained by solving the equation of unsteady heat conduction for time variable boundary condition. This method can be applied for composites with thick fillers and reinforced rubbers and enables us to determine the thermal diffusivity of elastomers and reinforced rubbers at temperatures ranging from 20 to 200°C.

Design and manufacture of a gradient-index axicon

A gradient-index axicon with its initial focus offset from the back surface was designed with the thin-lens approximation. Two samples were fabricated by means of the time-varying boundary condition diffusion method, which is based on the modified quasi-chemical diffusion model. Intensity profile measurements were taken along the focal region of the axicons. The samples produced extended line foci. From the intensity measurements, the central spot widths and back focal lengths were determined. The peak widths matched theoretical predictions made with the diffraction theory for the samples and showed good agreement with the predicted widths for a pseudo-Bessel beam, showing that the axicon produced a pseudo-diffractionless beam.

A Finite Element Approach to Transient Thermal Analysis of Work Rolls in Rolling Process

The three-dimensional transient thermal problem of work rolls in the entire rolling process has been formulated. It includes the time-varying boundary conditions specified at the roll surface taking the schedule of both rolling and idling cycles into consideration. The corresponding finite element equations are derived and solved by the Runge-Kutta-Verner method. The finite element solutions indicate that the temperature variations in the circumferential direction are overwhelming. Case studies unveil the thermal characteristics of the work rolls in various kinds of mill operations. The effects of the specific heat and the angular velocity of the work rolls are presented. Numerical results are compared with Guo’s analytical solutions. [S1087-1357(00)01604-X]

Meshfree simulation of deforming domains

Spatial discretization of the domain and/or boundary conditions prevents application of many numerical techniques to physical problems with time-varying geometry and boundary conditions. By contrast, the R-functions method (RFM) for solving boundary and initial value problems discretizes not the domain but the underlying functional space, while the prescribed boundary conditions are satisfied exactly. The clean and modular separation of geometric information from the numerical procedures results in a solution technique that is essentially meshfree and allows an almost effortless modification of geometrical shapes, boundary conditions, and the governing equations. We show that these properties of the RFM make it highly suitable for automated modeling and simulation of non-stationary physical problems with time-varying geometries and boundary conditions.

Analysis of unsaturated/saturated water flow near a fluctuating water table

Periodically changing boundary conditions influence the water flow at the interface of the unsaturated/saturated zone. In this study we investigate periodically time-variant lower boundary conditions in a one-dimensional soil column with a water table, using numerical solutions of Richards equation. The signals used for the simulations are triangular and rectangular wave functions as well as a sine function with periods ranging from minutes to years and amplitudes of 25, 50, and 100 cm. The upper boundary was described by a zero or a constant downward flux of 0.01 and 1.0 cm d −1. The hydraulic properties were described by the van Genuchten–Mualem model with parameters representing soils from the textural classes sand, loamy sand, sandy loam, and loam. Water content and pressure head profiles exhibited sharper fronts for upward movement of the water table than for downward movement. This also resulted in asymmetric variations of matric head and saturation as a function of time. By analyzing the simulation results in terms of the maximum, minimum, and average height of the water table, we have shown that the water table was strongly damped for the upward movement but weakly damped for the downward movement. We present a criterion for determining the period and amplitude of the boundary condition in which the damping of the water table movement is negligible. The unsaturated zone above the water table was characterized by the height at which the water flow changed from a transient to a steady state. By applying scaling [Youngs, E.G., 1990. Application of scaling to soil-water movement considering hysteresis. In: Elrick, D.E., Hillel, D. (Eds.), Scaling in Soil Physics, Principles and Applications. SSSA Special Publication No. 25, pp. 23–37] and using a macroscopic length scale derived from steady unsaturated water flow, we found that the variation of the simulation results caused by textural differences was greatly reduced.

The impact of flooding on modelling salt transport processes to streams

The development of many of the world's arid and semi-arid regions has resulted in the salinisation of land and water resources. In these areas, soils and groundwaters are often naturally saline and any disturbance of the delicate hydrological balance results in mobilisation of the stored salt. The salt transport mechanisms are often highly complex, the understanding of which necessitates the use of computer modelling in combination with field studies. In this paper the transport of salt between groundwater and streams on the Chowilla floodplain in south-eastern Australia was modelled and compared with available field data. The large salinity contrast between the fresh stream and floodwater and the saline groundwater results in density-dependent flow behaviour, and hence necessitated the use of a variable density flow and solute transport model (SUTRA). The model was applied in cross-section over a 6.1-km-long transect across the floodplain. Time varying boundary conditions were employed at the locations of three streams on the transect to simulate the interaction between the rising and falling streams and the adjacent aquifer during and after floods. The model was used to assess the importance of overbank floods in the transport of salt to floodplain streams by carrying out simulations under various recharge scenarios. The simulations showed that the mixing of floodwater and groundwater within the bank storage adjacent to the streams could predict the observed short-term (<12 months) salt load recessions. In order to predict the observed long-term (12–24 months) salt load recessions, the inclusion of localised recharge during overbank floods is required, as hypothesised by previous field-based studies.

ON BOUNDARY ELEMENTS FOR SIMULATION OF CATHODIC PROTECTION SYSTEMS WITH DYNAMIC POLARIZATION CURVES

The Boundary Element Method (BEM) is applied to predict the evolution of current density and potential distributions with respect to time that appear when cathodic protection systems are employed to prevent corrosion of metallic structures in contact with an electrolyte. This phenomenon is governed by the Laplace or Poisson equation, subjected to non-linear time-varying boundary conditions, described by dynamic polarization curves. Numerical techniques that consider the dependency of these curves upon time and formation potential have been developed based on potentiostatic data obtained from in-situ experiments. © 1997 John Wiley & Sons, Ltd.

The effects of forced boundary conditions on flow within a cubic cavity using digital particle image thermometry and velocimetry (DPITV)

Buoyancy-driven convection within a cavity, whose sidewalls are heated and cooled, is a problem of great interest, because it has applications in heat transfer and mixing. Most studies to date have studied one of two cases: the steady-state case or the development of the transient flow as it approaches steady state. Our main concern was to study the response of the cavity to time-varying thermal boundary conditions. We therefore decided to observe the flow phenomena within a convection cavity under sinusoidal thermal forcing of the sidewalls. To map the flow properly, it is necessary to have simultaneous kinematic and thermal information. Therefore, the digital particle image thermometry and velocimetry (DPITV) is used to acquire data. Implementing this technique requires seeding the flow with encapsulated liquid crystal particles and illuminating a cross section of the flow with a sheet of white light. Extraction of the thermal and kinematic content is in two parts. For the first, the liquid crystals will reflect different colors of the visible spectrum, depending on the temperatures to which they are subjected. Therefore, calibrating their color reflection with temperature allows for the extraction of the thermal content. For the second part, the kinematic information is obtained through the use of a digital cross-correlation particle image velocimetry technique. With the use of DPITV, the flow within a convection cavity is mapped and studied under steady forcing and sinusoidally forced boundary conditions at the Brunt-Väisälä frequency. For the sinusoidally forced case, three cases are studied. In the first, the heating between the two walls is in phase. In the second, the heating between the two walls is 180° out of phase. In the third, the heating between the two walls is 90° out of phase. For steady forcing, the thermal plots show that the flow develops a linearly stratified profile within the center of the cell. At the sidewalls, however, owing to forcing, hot/cold thermal boundary layers develop at the left/right walls. These hot/cold thermal boundary layers then turn around the upper-left/lower-right corners and develop into intrusion layers that extend across the top and bottom walls. The vorticity and streamlines show that the bulk of the fluid motion is concentrated around the walls, whereas the fluid within the center of the cell remains stationary. For the sinusoidally forced cases, the thermal plots show the existence of many thermal “islands,” or pockets of fluid where the temperature is different with respect to its surroundings. The vorticity plots show that the center of the cell is mostly devoid of vorticity and that the vorticity is mainly confined to the sidewalls, with some vorticity at the top and bottom walls. For the 0° forcing, the streamlines show the development of two counterrotating rollers. For the 180° forcing, the streamlines show the development of only one roller. Finally, for the 90° forcing, the streamlines show the development of both a two-roller and a one-roller system, depending on the position within the forcing cycle.

A SEMI-EMPIRICAL APPROACH TO HANDLE BROKEN-LINE HEATING: DETERMINATION OF EMPIRICAL PARAMETERS AND EVALUATION OF PROCESS DEVIATIONS

An existing semi-empirical model for simulating product temperature profiles during thermal processing of conduction or convection heating foods under time varying boundary conditions (variable retort temperatures) was extended for the case of broken-line heating products. The use of the method for determination of the empirical heat penetration parameters for broken-line heating curves as defined by Ball (j h , f h1 , f h2 , x bh ) was evaluated. Starch solutions, showing broken-line heating behavior, were used as a food simulant. To investigate the consistency of the determined broken-line heating parameters, and to test the applicability ofthe method when boundary conditions are time dependent, process deviations consisting of drops on the heating medium temperature during the holding phase of a process were evaluated. The model is a promising approach, if the correct empirical parameters are used.

Modelling of temperature and weight loss kinetics during meat chilling for time variable conditions using an analytical based method — III. Calculations versus measurements on pork carcass hindquarters

A heat transfer model of an infinite cylinder, adapted to account for time-variable boundary conditions (part I — Kuitche etal, 1996a) and validated by comparison with measurements carried out on wet plaster samples (part II — Kuitche etal, 1996b), has been tested against data from pork carcass hindquarters. Experiments under constant chilling conditions were used to fit a shape factor by comparing measured and simulated core temperature, and an equivalent exchange surface area, by comparing weight loss kinetics, which were both related to carcass weight: 52 kg < W < 100 kg; 0 ° C < T a < 8 ° C; 0.5 m/ s < V a < 2 m/ s and 70% < RH < close to saturation. The temperature profiles were then well predicted at other locations within the samples. The values of the fitted parameters are compared with those obtained in part II and with theoretical determinations. The agreement between calculations and measurements recorded during time-variable chilling condition experiments, typical of those observed in industrial chillers, was good enough to enable cost comparison of different processes in practice.

Modelling of temperature and weight loss kinetics during meat chilling for time-variable conditions using an analytical-based method — II. Calculations versus measurements on wet plaster cylinders and cast

Calculations carried out using the heat-transfer model in an infinite cylinder, adapted to account for time-variable boundary conditions (see part I of this series of papers), are compared to experimental temperature and weight loss kinetics. The model is first validated by comparing the calculations with measurements carried out on wet plaster cylinders (diameters 12.7, 10 and 7.5 cm) chilled under the following conditions: (1) under constant conditions (34 trials) covering a wide range of variations in air properties: 1 < V a < 2.2 m/ s, 0 < T a < 10 ° C and 44 < RH < close to saturation, and (2) under time-variable conditions (12 trials) for an air velocity varying between 0.6 and 2.2 m/s and an air temperature varying between −15 and 10 °C. Without adjustments, the temperature kinetics at different points within the sample are predicted under constant and time-variable conditions with an error margin of 1 °C. The relative error in chilling time prediction is 5%. Weight loss at the end of chilling, which is around 2% of the original weight, is calculated with an absolute error of 0.1%. The ability of the model to simulate the chilling of a plaster mould of a pork carcass hindquarter is then tested. Five trials carried out with air velocities ranging from 0.5 to 2.2 m/s yielded a shape factor adjustment. The mean temperature variation of the thickest section of the sample was then successfully simulated. A discussion on the difficulties encountered in simulating the weight loss kinetics with the same precision as for cylinders, reveals the limits of the analytical approach.

The effects of temperature-dependent viscosity on mantle convection with the two major phase transitions

We have studied the consequences of including temperature-dependent viscosity in mantle convection with both of the major phase transitions included. We have employed a two-dimensional cartesian model in a box of aspect ratio four. The time-dependent equations have been cast within an extended Boussinesq framework in which each of the phase changes has been simulated by an enhanced thermal expansivity. Both latent heat release and viscous dissipation have been considered. Two different forms of temperature-dependent viscosity, the Arrhenius and the Frank-Kamenetzky forms, have been employed. We have considered both models with a constant temperature (equilibrium) boundary condition at the core-mantle boundary and models with a time-varying (non-equilibrium) temperature boundary condition from cooling of the core. The effects of temperature-dependent viscosity are to enhance the tendency of the convective system to be layered because of the influences of the latent heat release and the mechanical heating on creating a sharp viscosity drop across the transition zone. We have found that there is a non-linear coupling between latent heat release from the interaction of the hot plume with the endothermic phase transition and the temperature-dependent viscosity. This positive feedback process can give rise to the production of very high temperatures in the transition zone, which may cause melting in the region between the two major phase transitions. Flushing events produced by the non-equilibrium boundary condition are much more dramatic than those generated by the equilibrium boundary condition. The non-linear coupling between latent heat release and temperature-dependent viscosity is also stronger in the non-equilibrium models. Comparison with models without the presence of latent heat release and viscous dissipation shows that the models without the heating terms produce a greater degree of mass flux across the endothermic phase boundary because of the lack of the viscosity stratification in the interior of the convective flow. Results of this moded can be applied to other types of endothermic phase transition found in the lithosphere or to the spinel-perovskite transition in the deep portion of the Martian mantle.

Interaction between a moving mirror and radiation pressure: A Hamiltonian formulation.

We present a nonrelativistic Hamiltonian of the interaction between a moving mirror and radiation pressure. This Hamiltonian is derived directly from the equation of motion of a moving mirror, and the wave equation with time-varying boundary conditions. We discuss the canonical quantization of both the field and the motion of the mirror.

Analytical solutions of transient heat conduction in semi-infinite solid with time varying boundary conditions by means of similarity transformation

An anlytical solution to the transient one dimensional linear heat conduction with time varying boundary conditions is obtained by means of similarity transformation via one-parameter groups. Two types of time varying boundary conditions are considered. These are; time varying surface temperature and time varying surface heat flux boundary conditions. In the analysis, thermal conductivity and thermal diffusivity, are assumed to be constant. The solution is generalized for a general polynomial type of time varying boundary conditions.