Engineering topics
Publications and source records attributed to Siegel, R..
In parallel to the industrial application of NDE to flaw detection and material property determination, the medical community has succesfully adapted such methods to the noninvasaive diagnostics and treatment of many conditions and disorders of the human body.
Equilibrium temperatures of an absorbing-emitting layer were obtained for exposure to incident radiation and with the layer boundaries either specular or diffuse. For high refractive indices the surface condition can influence the radiative heat balance if the layer optical thickness is small. Hence for a spectrally varying absorption coefficient the layer temperature is affected if there is significant radiative energy in the spectral range with a small absorption coefficient. Similar behavior was obtained for transient radiative cooling of a layer where the results are affected by the initial temperature and hence the fraction of energy radiated in the short wavelength region where the absorption coefficient is small. The results are a layer without internal scattering. If internal scattering is significant, the radiation reaching the internal surface of a boundary is diffused and the effect of the two different surface conditions would become small.
Some ceramic materials for high temperature applications are partially transparent for radiative transfer. The refractive indices of these materials can be substantially greater than one which influences internal radiative emission and reflections. Heat transfer behavior of single and laminated layers has been obtained in the literature by numerical solutions of the radiative transfer equations coupled with heat conduction and heating at the boundaries by convection and radiation. Two-flux and diffusion methods are investigated here to obtain approximate solutions using a simpler formulation than required for exact numerical solutions. Isotropic scattering is included. The two-flux method for a single layer yields excellent results for gray and two band spectral calculations. The diffusion method yields a good approximation for spectral behavior in laminated multiple layers if the overall optical thickness is larger than about ten. A hybrid spectral model is developed using the two-flux method in the optically thin bands, and radiative diffusion in bands that are optically thick.
A simple set of equations is derived for predicting the temperature distribution and radiative energy flow in a semitransparent layer consisting of an arbitrary number of laminated sublayers that absorb, emit, and scatter radiation. Each sublayer can have a different refractive index and optical thickness. The plane composite region is heated on each exterior side by a different amount of incident radiation. The results are for the limiting case where heat conduction within the layers is very small relative to radiative transfer, and is neglected. The interfaces are assumed diffuse, and all interface reflections are included in the analysis. The thermal behavior is readily calculated from the analytical expressions that are obtained. By using many sublayers, expressions provide the temperature distribution and heat flow for a diffusing medium with a continually varying refractive index, including internal reflection effects caused by refractive index gradients. Temperature and heat flux results are given to show the effect of variations in refractive index and optical thickness through the multilayer laminate.
A simple set of equations is derived for predicting the temperature distribution and radiative energy flow in a semitransparent layer consisting of an arbitrary number of laminated sublayers that absorb, emit, and scatter radiation. Each sublayer can have a different refractive index and optical thickness. The plane composite region is heated on each exterior side by a different amount of incident radiation. The results are for the limiting case where heat conduction within the layers is very small relative to radiative transfer, and is neglected. The interfaces are assumed diffuse, and all interface reflections are included in the analysis. The thermal behavior is readily calculated from the analytical expressions that are obtained. By using many sublayers, the analytical expressions provide the temperature distribution and heat flow for a diffusing medium with a continuously varying refractive index, including internal reflection effects caused by refractive index gradients. Temperature and heat flux results are given to show the effect of variations in refractive index and optical thickness through the multilayer laminate.
Heat transfer characteristics are analyzed for a plane layer of semitransparent material with refractive index not less than 1. Energy transfer in the material is by conduction, emission, absorption, and isotropic scattering. Each side of the layer is heated by radiation and convection. For a refractive index larger than unity, there is internal reflection of some of the energy within the layer. This, coupled with scattering, has a substantial effect on distributing energy across the layer and altering the temperature distribution from when the refractive index is unity. The effect of scattering is examined by comparisons with results from an earlier paper for an absorbing layer. Results are given for a gray medium with a scattering albedo up to 0.999, and for a two-band spectral variation of the albedo with one band having low absorption. Radiant energy leaving the surface as a result of emission and scattering was examined to determine if it could be used to accurately indicate the surface temperature.
A simple set of equations is derived for predicting temperature radiative energy flow in a two-region semitransparent laminated layer in the limit of zero heat conduction. The composite is heated on its two sides by unequal amounts of incident radiation. The two layers of the composite have different refractive indices, and each material absorbs, emits, and isotropically scatters radiation. The interfaces are diffuse, and all interface reflections are included. To illustrate the thermal behavior that is readily calculated from the equations, typical results an given for various optical thicknesses and refractive indices of the layers. Internal reflections have a substantial effect on the temperature distribution and radiative heat flow.
An analytical solution was obtained by Siegel (1991, 1992) for local boundary heat fluxes by a radiating medium at uniform temperature in a 2D rectangular region. It is shown here that, after local fluxes from the medium to the walls have been evaluated, it is very easy to compute local fluxes arriving from the adjacent and opposite walls. This extends the previous analysis and provides convenient relations to include radiation from a black boundary, each side of the rectangle being at a different uniform temperature. The final expressions are helpful in performing spectral calculations that must be made for many spectral bands.
Transient solutions were obtained for a square region of heat conducting semitransparent material cooling by thermal radiation. The region is in a vacuum environment, so energy is dissipated only by radiation from within the medium leaving through its boundaries. The effect of heat conduction during the transient is to partially equalize the internal temperature distribution. As the optical thickness of the region is increased, the temperature gradients increase near the boundaries and corners, unless heat conduction is large. The solution procedure must provide accurate temperature distributions in these regions to prevent error in the calculated radiation losses. Two-dimensional numerical Gaussian integration is used to obtain the local radiative source term. A finite difference procedure with variable space and time increments is used to solve the transient energy equation. Variable spacing was used to concentrate grid points in regions with large temperature gradients.
The effect of the index of refraction on the temperature distribution and radiative heat flux in semitransparent materials, such as some ceramics, is investigated analytically. In the case considered here, a plane layer of a ceramic material is subjected to external radiative heating incident on each of its surfaces; the material emits, absorbs, and isotropically scatters radiation. It is shown that, for radiative equilibrium in a gray layer with diffuse interfaces, the temperature distribution and radiative heat flux for any index of refraction can be obtained in a simple manner from the results for an index of refraction of unity.
The index of refraction can considerably influence the temperature distribution and radiative heat flow in semitransparent materials such as some ceramics. For external radiant heating, the refractive index influences the amount of energy transmitted into the interior of the material. Emission within a material depends on the square of its refractive index, and hence this emission can be many times that for a biackbody radiating into a vacuum. Since radiation exiting through an interface into a vacuum cannot exceed that of a blackbody, there is extensive reflection at the internal surface of an interface, mostly by total internal reflection. This redistributes energy within the layer and tends to make its temperature distribution more uniform. The purpose of the present analysis is to show that, for radiative equilibrium in a gray layer with diffuse interfaces, the temperature distribution and radiative heat flux for any index of refraction can be obtained very simply from the results for an index of refraction of unity. For the situation studied here, the layer is subjected to external radiative heating incident on each of its surfaces. The material emits, absorbs, and isotropically scatters radiation. For simplicity the index of refraction is unity in the medium surrounding the layer. The surfaces of the layer are assumed diffuse. This is probably a reasonable approximation for a ceramic layer that has not been polished. When transmitted radiation or radiation emitted from the interior reaches the inner surface of an interface, the radiation is diffused and some of it thereby placed into angular directions for which there is total internal reflection. This provides a trapping effect for retaining energy within the layer and tends to equalize its temperature distribution. An analysis of temperature distributions in absorbing-emitting layers, including index of refraction effects, was developed by Gardon (1958) to predict cooling and heat treating of glass plates. The interfaces were optically smooth; the resulting specular reflections were computed from the Fresnel reflection laws. This provides a somewhat different behavior than for diffuse interfaces. A similar application was for heating that occurs in a window of a re-entry vehicle (Fowle et al., 1969). A number of recent papers (Rokhsaz and Dougherty, 1989; Ping and Lallemand, 1989; Crosbie and Shieh, 1990) further examined the effects of Fresnel boundary reflections and nonunity refractive index. Other examples of analyses of both steady and transient heat transfer to single or multiple plane layers (Amlin and Korpela, 1979; Tarshis et al., 1969) have used diffuse assumptions at the interfaces as in the present study
Reference is made to the work of Shah (1979) which demonstrated the possibility of partially integrating the radiative equations analytically to obtain an 'exact' solution. Shah's solution was given as a double integration of the modified Bessel function of order zero. Here, it is shown that the 'exact' solution for a rectangular region radiating to cold black walls can be conveniently derived, and expressed in simple form, by using an integral function, Sn, analogous to the exponential integral function appearing in plane-layer solutions.
The transient cooling of an absorbing, emitting, and scattering cylinder is studied by using the forward integration of energy transfer equations. The analysis presented here provides cooling curves for a cylindrical region, initially at uniform temperature and then suddenly exposed to a much cooler environment. Results are presented for several optical radii and different values of the scattering albedo.
An analysis is carried out for radiative cooling of a plane layer initially filled with liquid drops that solidify and then continue to cool by loss of sensible heat. This is in connection with a proposed lightweight radiator system for heat dissipation in space. Hot liquid drops would be ejected and then cooled by direct exposure in space; they would then pass into a collector for reuse. The cooling analysis contains three transient zones. In the first, the drops cool by losing latent heat until the outermost drops become solid. Then the cooling continues by loss of both latent and sensible heat. A similarity behavior is eventually achieved in which the transient emittance of the layer depends only on the optical thickness.
A proposed lightweight radiator system for waste heat dissipation in space would eject streams of coolant in the form of small, hot liquid droplets. The droplets would lose radiative energy by direct exposure to the very low-temperature environment of space, and would then be collected for reuse. The cooling behavior of a layer composed of many small droplets was studied by numerical solution of the radiative integral equations. Since there is mutual interference for radiative energy dissipation, an array droplet will cool more slowly than if each drop is exposed individually. Since liquid metal droplets may be used, the study includes results for conditions with high scattering. For optically thin regions, especially with high scattering, the temperature distribution is sufficiently uniform that the cooling can be computed using the approximation of a constant layer emittance. For optically thick layers starting at uniform temperature, the temperature distributions become nonuniform with time. It was found that the cooling process goes through a starting transient; a constant emittance condition is then achieved where the emittance is lower than that for a layer at uniform temperature.
The effects of flow oscillations on axial energy diffusion in a porous medium, in which the flow is continuously disrupted by the irregularities of the porous structure, are analyzed. The formulation employs an internal heat transfer coefficient that couples the fluid and solid temperatures. The final relationship shows that the axial energy transport per unit cross-sectional area and time is directly proportional to the axial temperature gradient and the square of the maximum fluid displacement.
An analysis of the effect of flow oscillations on laminar flow heat transfer in a channel with uniform heat addition is presented. It is shown that the effect of flow oscillations will be to reduce the channel heat transfer coefficient. This effect is due to the fact that the heat addition along the channel wall produces an increasing fluid temperature along the channel length. The flow oscillations interacting with this positive temperature gradient will induce a heat flow back toward the channel inlet. This will tend to inhibit the heat transfer process and will raise the wall temperature required to transfer away a given amount of heat at the channel wall.
Novel mathematical techniques are here used to treat general boundary conditions in convectively cooled continuous ingot casting problems, applying a constant convective heat transfer coefficient at the ingot sides (so that the temperature distribution along the sides is unknown). The problem is first inverted to treat the physical coordinates as dependent variables of temperature and heat flow, mapping the ingot into a region that is nearly rectangular. A boundary perturbation method is then used to obtain an analytical solution in this near-rectangular region. Solidification interface shapes depend on two dimensionless parameters: one is a function of the casting velocity, while the other is dependent on the convective heat transfer coefficient at the cooled wall.