Thermal radiation boundary conditions

From time to time we have mentioned that thermal conductivities of materials vary with temperature however, over a temperature range of 100 to 200°C the variation is not great (on the order of 5 to 10 percent) and we are justified in assuming constant values to simplify problem solutions. Convection and radiation boundary conditions are particularly notorious for their nonconstant behavior. Even worse is the fact that for many practical problems the basic uncertainty in our knowledge of convection heat-transfer coefficients may not be better than 20 percent. Uncertainties of surface-radiation properties of 10 percent are not unusual at all. For example, a highly polished aluminum plate, if allowed to oxidize heavily, will absorb as much as 300 percent more radiation than when it was polished. 

Consider a spherical shell of inner radius r outer radius Tj, thermal conductivity k, and emisslvity e. The outer surface of llie shell is subjected to radiation to surrouuding surfaces at but the direction of heat transfer is not known. Express the radiation boundary condition on the outer surface of the shell. 

Thermal/structural response models are related to field models in that they numerically solve the conservation of energy equation, though only in solid elements. Finite difference and finite element schemes are most often employed. A solid region is divided into elements in much the same way that the field models divide a compartment into regions. Several types of surface boundary conditions are available adiabatic, convection/radiation, constant flux, or constant temperature. Many ofthese models allow for temperature and spatially dependent material properties. 

On the surface of the tubes, furnace, etc., the boundary conditions require that the flow of any substance across a material surface be equal to zero so that the normal surface of the component of the corresponding vector of the flow is zero. These conditions are identical for all the substances. The boundary conditions for temperature will be the same as the boundary conditions for a,..., h, if we do not remove heat from the plume, i.e., if only thermally insulated, not thermally radiating surfaces are introduced or if all walls with temperature T0 are located where the gas temperature is equal to T0. 

It should be noted that when the enclosure contains a gas, the convective heat transfer rates can be low and radiant heat transfer can be significant. Some gases, such as carbon dioxide and water vapor, absorb and emit radiation and in such cases the energy equation has to be modified to account for this. However, even when the gas in the enclosure is transparent to radiation, there can be an interaction between the radiant and convective heat transfer. For example, for the case where the end walls can be assumed to be adiabatic, if grab and qKm are the rates at which radiant energy is being absorbed and emitted per unit wall area at any point on these end walls then the actual thermal boundary conditions on these walls are ... 

Consider a 20-cm-ihick large concrete plane wall k 0.77 V/in °C) subjected to convection on both sides with r, = 27"C and A, = 5 W/m °C on the inside, and = 8°C and A2 = 12 W/m °C on the outside. Assuming constant thermal conductivity with no heat generation and negligible radiation, [a) express the differential equations and the boundary conditions for steady one-dimensional heal conduction through the wall, (A) obtain a relation for the variation of temperature in the wall by solving the differential equation, and (c) evaluate the temperatures at the inner and outer surfaces of the wall. 

A solar heat flux q, is incident on a sidewalk whose thermal conductivity is k, solar absorptivity is a and convective heat transfer coefficient is h. TaWng the positive x direction to be towards the sky and disregarding radiation exchange with the surroundings surfaces, the correct boundary condition for this sidewalk surface is... 

Here A is the thermal conductivity of the solid (not the fluid ) at the wall. Eq. (2.23) stipulates a linear relationship between the temperature t)w and the slope of the temperature profile at the surface, this is also known as the 3rd type of boundary condition. As in (2.18), d d/dn is the derivative in the normal direction outward from the surface. The fluid temperature t F, which can change with time, and the heat transfer coefficient a must be given for the solution of the heat conduction problem. If a is very large, the temperature difference (t w — i9F) will be very small and the boundary condition (2.23) can be replaced by the simpler boundary condition of a prescribed temperature (i9w = i9F). The boundary condition in (2.23) is only linear as long as a is independent of w or ( w — f F), this factor is very important for the mathematical solution of heat conduction problems. In a series of heat transfer problems, for example in free convection, a changes with (i w — F), thereby destroying the linearity of the boundary condition. The same occurs when heat transfer by radiation is considered, as in this... 

Additional parameters specified in the numerical model include the electrode exchange current densities and several gap electrical contact resistances. These quantities were determined empirically by comparing FLUENT predictions with stack performance data. The FLUENT model uses the electrode exchange current densities to quantify the magnitude of the activation overpotentials via a Butler-Volmer equation [1], A radiation heat transfer boundary condition was applied around the periphery of the model to simulate the thermal conditions of our experimental stack, situated in a high-temperature electrically heated radiant furnace. The edges ofthe numerical model are treated as a small surface in a large enclosure with an effective emissivity of 1.0, subjected to a radiant temperature of 1 103 K, equal to the gas-inlet temperatures. 

Heat Transfer on Walls With Radiation. The local Nusselt numbers normalized with respect to Nu H have been obtained by Kadaner et al. [8] for thermally developing flow with the radioactive duct wall boundary condition . This is expressed as ... 

Abstract Results from the four-year long heating phase of the Drift-Scale Heater Test at the Exploratory Studies Facility at Yucca Mountain, Nevada, provide a basis to evaluate conceptual and numerical models used to simulate thermal-hydrological coupled processes expected to occur at the proposed repository. A three-dimensional numerical model was built to perform the analyses. All model simulations were predicated on a dual (fracture and matrix) continuum conceptualization. A 20-percent reduction in the canister heat load to account for conduction and radiation heat loss through the bulkhead, a constant pressure boundary condition at the drift wall, and inclusion of the active fracture model to account for a reduction in the number of fractures that were hydraulically active provided the best agreement between model results and observed temperatures. The views expressed herein are preliminary and do not constitute a final judgment of the matter addressed or of the acceptability of its use in a license application... 

The statistical collection and representation of the weather conditions for a specified area during a specified time interval, usually decades, together with a description of the state of the external system or boundary conditions. The properties that characterize the climate are thermal (temperatures of the surface air, water, land, and ice), kinetic (wind and ocean currents, together with associated vertical motions and the motions of air masses, aqueous humidity, cloudiness and cloud water content, groundwater, lake lands, and water content of snow on land and sea ice), nd static (pressure and density of the atmosphere and ocean, composition of the dry ir, salinity of the oceans, and the geometric boundaries and physical constants of the system). These properties are interconnected by the various physical processes such as precipitation, evaporation, infrared radiation, convection, advection, and turbulence, climate change... 

For the temperature component of the modeling framework, the initial temperature is set to be the ambient temperature (300 K). Thermal boundary conditions on the tooling surfaces are considered in the forms of a specified boundary temperature or heat flux. The temperatures of the two extreme upper and lower surfaces of the external spacers are assumed to be constant of 300 K. Heat loss due to radiation is considered at the lateral surfaces to be / = — Wq), where/is the heat flux... 

A one-dimensional thermal response model was developed to predict the temperature of FRP structural members subjected to fire. Complex boundary conditions can be considered in this model, including prescribed temperature or heat flow, as well as heat convection and/or radiation. The progressive changes of thermophysical properties including decomposition degree, density, thermal conductivity, and specific heat capacity can be obtained in space and time domains using this model. Complex processes such as endothermic decomposition, mass loss, and delatnina-tion effects can be described on the basis of an effective material properties over the whole fire duration. 

In this case, the wall thermal resistance is considered negligible. This kind of thermal boundary condition can be used, for example, for micro-radiators working in space and manufactured with materials having a large value of thermal conductivity. 

This effect, which had been predicted by quantum electrodynamics, can intuitively be understood as follows in the resonant case, that part of the thermal radiation field that is in the resonant cavity mode can contribute to stimulated emission in the transition ) n — ), resulting in a shortening of the lifetime (Sect. 11.3). For the detuned cavity the fluorescence photon does not fit into the resonator. The boundary conditions impede the emission of the photon. If the rate dN/dt of atoms in the level n) behind the resonator is measured as a function of the cavity resonance frequency cu, a minimum rate is measured for the resonance case co = coo (Fig. 14.49). 

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