Fixed Temperature Boundary Conditions Research Articles

Torques on the Inner and Outer Spheres Induced by the Boussinesq Thermal Convection in a Rotating Spherical Shell

We investigate the directions and amplitudes of torques on the inner and outer spheres induced by the stable finite-amplitude traveling wave solutions which bifurcate supercritically at the critical points and have four-fold symmetry in the azimuthal direction under the impermeable, no-slip and fixed-temperature boundary conditions. The ratio ratio of inner and outer radii η=0.4 and the Prandtl number P =1, while the Taylor number is varied from 52 2 to 500 2 and the Rayleigh number is from about R c to 1.2-2 R c , where R c is the critical Rayleigh number. It is shown that the direction of the torque on the inner sphere is prograde at small Taylor numbers, while it becomes retrograde at large Taylor numbers. Weakly nonlinear analyses show that the nonlinear term in the energy equation is most effective to generate the global distribution of mean zonal flows, however, the azimuthal component of the nonlinear term in the Navier–Stokes equation becomes most important for generation of the torque on the inne...

Simulations of long‐column flow experiments related to geologic carbon sequestration: effects of outer wall boundary condition on upward flow and formation of liquid CO2

AbstractImproving understanding of CO2 migration, phase change, and trapping processes motivates the development of large‐scale laboratory experiments to bridge the gap between bench‐scale experiments and field‐scale studies. Critical to the design of such experiments are defensible configurations that mimic relevant subsurface flow scenarios. We use numerical simulation with TOUGH2/ECO2M and ECO2N to design flow and transport experiments aimed at understanding upward flows including the transition of CO2 from supercritical to liquid and gaseous forms. These experiments are designed for a large‐scale facility such as the proposed laboratory for underground CO2 investigations (LUCI). LUCI would consist of one or more long‐column pressure vessels (LCPVs) several hundred meters in length filled with porous materials. An LCPV with an insulated outer wall corresponds to the column being at the center of a large upwelling plume. If the outer wall of the LCPV is assigned fixed temperature boundary conditions corresponding to the geothermal gradient, the LCPV represents a narrow upwelling through a fault or well. Numerical simulations of upward flow in the columns reveal complex temporal variations of temperature and saturation, including the appearance of liquid CO2 due to expansion cooling. The results are sensitive to outer thermal boundary conditions. Understanding of the simulations is aided by time‐series animations of saturation‐depth profiles and trajectories through P‐T (pressure‐temperature) space with superimposed phase saturations. The strong dependence of flow on hydrologic properties and the lack of knowledge of three‐phase relative permeability and hysteresis underlines the need for large‐scale flow experiments to understand multiphase leakage behavior. © 2012 Society of Chemical Industry and John Wiley & Sons, Ltd

Convection in rotating spherical fluid shells with inhomogeneous heat flux at the outer boundary

The Earth's core is subject to a laterally varying heat flux at the outer boundary, which may account for correlations between the geomagnetic field and lower mantle structure. Studies of nonmagnetic, rotating convection in a spherical shell with fixed temperature boundary conditions have revealed flows resonating with, or locked to, the boundary anomalies when the length scales of the convection are close to those of the boundary condition. Here we study a similar system but for fixed heat flux upper boundary conditions, as in the Earth's core. We first map out the onset of thermal instability in a rotating shell of aspect ratio 0.4 for uniform outer boundary cooling with both rigid and stress-free boundaries. A preference for large scale (azimuthal wavenumber m = 1) flows, not observed for the uniform temperature case, persists to Ekman numbers down to almost 10− 4. The preference for large scales is greatest for rigid boundaries and high (≥ 1) Prandtl numbers. Hemispheric asymmetry appears in the weakly nonlinear regime, with small scale columnar convection coexisting with larger scale flows. We next study the effect of heterogeneous cooling of strength ε proportional to the Y 2² spherical harmonic, which resembles the assumed heat flow at the core-mantle boundary. We illustrate the results with two sets of parameters, (1) when the most unstable mode in the uniform case is m = 2, the same as the heterogeneous boundary condition, and (2) when it is m = 1. We follow the numerical solutions from steady flows dominated by boundary heating, through periodic flows drifting at non-uniform rates, to chaotic flows. In case (1) both thermal convection and boundary-driven flow are dominated by the same azimuthal wavenumber m = 2 as the boundary condition; they give way to periodic flows of the same symmetry (even m) at low boundary heterogeneity. At higher ε the symmetry is broken as modes with odd m are excited. In case (2) at low ε the m = 1 and m = 2 modes compete, while higher ε imposes an m = 2 symmetry. Boundary effects depend strongly on the most unstable wavenumber at onset of convection with uniform boundary cooling: these simple linear results are a good guide to the probable behaviour of more complex, nonlinear regimes and have already been used to find suitable parameter ranges in a geodynamo calculation.

Linear Stability of Thermal Convection in Rotating Systems with Fixed Heat Flux Boundaries

Linear stability of rotating thermal convection in a horizontal layer of Boussinesq fluid under the fixed heat flux boundary condition is examined by the use of a vertically truncated system up to wavenumber one. When the rotation axis is in the vertical direction, the asymptotic behavior of the critical convection for large rotation rates is almost the same as that under the fixed temperature boundary condition. However, when the rotation axis is horizontal and the lateral boundaries are inclined, the mode with zero horizontal wavenumber remains as the critical mode regardless of the rotation rate. The neutral curve has another local minimum at a nonzero horizontal wavenumber, whose asymptotic behavior coincides with the critical mode under the fixed temperature condition. The difference of the critical horizontal wavenumber between those two geometries is qualitatively understood by the difference of wave characteristics; inertial waves and Rossby waves, respectively.

Estimating the thermal conductivity of chars and porous residues using thermal resistor networks

A conceptually simple approach is used to estimate the thermal conductivity of a composite solid. The approach uses a three-dimensional (3D) network of thermal resistors to model a unit cube of material. Fixed temperature boundary conditions are applied on two opposing faces of the cube and the other four faces are well insulated. The net heat flux through one of the fixed temperature faces is then used to compute the effective thermal conductivity. The results are compared with analytical estimates and experimental results for Al 2O 3 with spherical inclusions and the differences between 2D and 3D results are investigated. Correlations are also given for the numerical results. The cube itself may be composed of any number of different phases (provided of course that this number is much less than the total number of thermal resistors in the network) but the discussion is restricted to two phase systems with some reference to a three phase system.

Transient thermal simulations of a three-dimensional unit cell in power control systems and high-power microwave devices

Detailed numerical assessment is given supporting the need to go beyond two-dimensions for numerically analyzing the electrothermal characteristics of power control devices employed in high-performance paralleling layout topology, and high-power microwave devices in monolithic microwave integrated circuits (MMIC). An accurate understanding of transient thermal characteristics, particularly the transient power-overload failure mechanism, is needed and requires the use of three-dimensional (3-D) transient electrothermal simulations. This will enable the design optimization of these power control devices for hardening against failure due to external fault conditions. We have compared the characteristic thermal speed response of Si, SiC, and GaN power control devices under short-circuit conditions, using 3-D time-dependent simulations. Such improved thermal simulation is also very important for high-power microwave devices and MMIC. Calibration of our numerical techniques with experiment is implemented using atomic force microscopy temperature measurements. The consistent agreement between measured values and simulated results indicates the need to couple detailed numerical simulation with high-resolution measurement technique, especially in identifying heat-flow bottlenecks. This will allow taking into account the actual thermal coupling of semiconductor substrate with the base plate and surrounding environment, where the fixed-temperature boundary condition usually applied at the substrate is not valid.

FINITE-ELEMENT ANALYSIS OF HEAT CONDUCTION PROBLEMS WITH IRREGULAR REGION BOUNDARY USING MULTILEVEL TECHNIQUES

An effective finite-element algorithm with fast computation and low memory storage is developed to solve heat transfer problems with an irregular domain and a fixed temperature boundary condition. This algorithm combines the use of the Gauss-Seidel method and a multigrid algorithm and introduces an adjustable factor to correct the irregular boundary to obtain a more accurate result. To illustrate the algorithm, the computational procedures are conducted for heat transfer problems in a quarter circle region. The results show that the numerical solutions are in very good agreement with the exact solutions. The phase-change problem is also investigated in this study. Compared with Tao?s results, a good result is presented. This algorithm also can be easily extended to three-dimensional heat transfer problems with fixed temperature boundary conditions.

On the heating of a two-dimensional slab in a microwave cavity: aperture effects

AbstractThe steady-state heating of a two-dimensional slab by the TE10 mode in a microwave cavity is considered. The cavity contains an iris with a variable aperture and is closed by a short. Resonance can occur in the cavity, which is dependent on the short position, the aperture width and the temperature of the heated slab.The governing equations for the slab are steady-state versions of the forced heat equation and Maxwell's equations while fixed-temperature boundary conditions are used. An Arrhenius temperature dependency is assumed for both the electrical conductivity and the thermal absorptivity. Semi-analytical solutions, valid for small thermal absorptivity, are found for the steady-state temperature and the electric-field amplitude in the slab using the Galerkin method.With no-iris (a semi-infinite waveguide) the usual S-shaped power versus temperature curve occurs. As the aperture width is varied however, the critical power level at which thermal runaway occurs and the temperature response on the upper branch of the S-shaped curve are both changed. This is due to the interaction between the radiation, the cavity and the heated slab. An example is presented to illustrate these aperture effects. Also, it is shown that an optimal aperture setting and short position exists which minimises the input power needed to obtain a given temperature.

Microwave heating of materials with impurities

The microwave heating of materials is important in many industrial processes. For example, it is used for the smelting of metals and the sintering of ceramics. Hot-spots (localised areas of high temperature) can develop in the material being heated or in the microwave oven itself, with disastrous consequences. Impurities in the material or in a component of the microwave oven can have different electromagnetic and thermal properties to the surrounding material. Different rates of heating occur at these sites, which gives rise to differential heating, which can lead to the generation of hot-spots. The generation of hot-spots by this mechanism is considered for a finite one-dimensional slab with a single impurity at its centre. A fixed-temperature boundary condition is applied at both ends of the slab and one end of the slab is irradiated by microwaves of constant amplitude. The heat absorption at the impurity is assumed to have a power-law dependence on temperature (hence hot-spot generation can occur via thermal runaway). Depending on the electrical and thermal properties of the material there are two possibilities; either a hot-spot occurs or a steady-state solution occurs due to a balance between heat absorption in the material and heat loss through the boundaries. These steady-state solutions are found for both linear and non-linear thermal absorptivity and constant and decaying electric-field amplitude. If possible the region of parameter space in which they occur (in the rest of the parameter space hot-spots occur) is also found. In addition, numerical solutions are developed to verify the steady-state solutions and to investigate cases where analytical solutions are difficult to derive, such as for materials with multiple impurities.

Pore water pressure at an ice lens: Its measurement and interpretation

One dimensional freezing tests with fixed temperature boundary conditions and an applied back-pressure to the pore water were carried out on Devon silt in order to determine the pore pressure at an ice lens. It was established that tests with an applied back-pressure result in the same freezing characteristics as conventional tests. During closed-system freezing, the pore pressure dropped continuously and reached an ultimate value which is related to the temperature at the base of the warmest ice lens at thermal steady state. For a given soil, the temperature is dependent upon the amount of cooling produced by the removal of heat released at the ice lens by the migratory water, and upon the amount of warming resulting from the vertical settlement of the base of the ice lens caused by the consolidation of the unfrozen soil. For Devon silt under zero applied load, the maximum drop in pore pressure varied between 210 and 258 kPa. According to the Clapeyron equation, the temperatures at the ice lens inferred from the pore pressure measurements varied then between -0.17 and -0.21°C. Comparison with temperature data obtained from conventional tests associated with the X-ray photography technique suggests that the temperature inferred from the change in pore pressure is somewhere between the temperature of ice lens formation and the lowest temperature at which the same ice lens can grow for a rate of cooling of the frozen fringe smaller than 0.1°C/day.

The short-term influence of various concentrations of atmospheric carbon dioxide on the temperature profile in the boundary layer

A radiative-conductive model is constructed to study short-term effects of various carbon dioxide concentrations on the atmospheric boundary layer for different seasons. The distribution of the exchange coefficient is modeled with the aid of the KEYPS formula. Infrared radiation calculations are carried out by means of the emissivity method and by assuming that water vapor and carbon dioxide are the only radiatively active gases. Global radiation is computed by specification of Linke's turbidity factor. It is found that doubling the carbon dioxide concentration increases the temperature near the ground by approximately one-half of one degree if clouds are absent. A sevenfold increase of the present normal carbon dioxide concentration increases the temperature near the ground by approximately one degree. Temperature profiles resulting from presently observed carbon dioxide concentration and convective cloudiness of 50% or less are compared with those resulting from doubled carbon dioxide concentrations and the same amounts of cloud cover. Again, it is found that a doubling of carbon dioxide increases the temperature in the lower boundary layer by about one-half of one degree. The present results are obtained on the basis of fixed temperature boundary conditions as contrasted to the study ofManabe andWetherald (1967). Howeve, the conclusions are not addressed to global climate change, but to the distribution of the temperature of the air layer near the ground.

Linear Simulations of Boussinesq Convection in a Deep Rotating Spherical Shell

Abstract We present extensive linear numerical simulations of Boussinesq convection in a rotating spherical shell of finite depth. The motivation for the study is the problem of general circulation of the solar convection zone. We solve the marching equations on a staggered grid in the meridian plane for the amplitudes of the most unstable Fourier mode of longitudinal wavenumber m between 0 and 24, for Taylor number T between 0 and 106, at a Prandtl number P=1, for a shell of depth 20% of the outer radius. Stress-free, fixed-temperature boundary conditions are used at the inner and outer bounding surfaces. Modes of two symmetries, symmetric and antisymmetric about the equator, are studied. The principal results are as follows: Increasing Taylor number T splits the most unstable solutions for each m into two classes: a broad band of high m solutions which peak at or near the equator, and a small number of low m solutions which peak at or near the poles. The equatorial modes are unstable at lower Rayleigh n...