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At least 415 records · Page 23
All sourcesNASA NTRSDOE OSTI
NASA NTRS2014

CVB: the Constrained Vapor Bubble Capillary Experiment on the International Space Station MARANGONI FLOW REGION

The Constrained Vapor Bubble (CVB) is a wickless, grooved heat pipe and we report on a full- scale fluids experiment flown on the International Space Station (ISS). The CVB system consists of a relatively simple setup a quartz cuvette with sharp corners partially filled with either pentane or an ideal mixture of pentane and isohexane as the working fluids. Along with temperature and pressure measurements, the two-dimensional thickness profile of the menisci formed at the corners of the quartz cuvette was determined using the Light Microscopy Module (LMM). Even with the large, millimeter dimensions of the CVB, interfacial forces dominate in these exceedingly small Bond Number systems. The experiments were carried out at various power inputs. Although conceptually simple, the transport processes were found to be very complex with many different regions. At the heated end of the CVB, due to a high temperature gradient, we observed Marangoni flow at some power inputs. This region from the heated end to the central drop region is defined as a Marangoni dominated region. We present a simple analysis based on interfacial phenomena using only measurements from the ISS experiments that lead to a predictive equation for the thickness of the film near the heated end of the CVB. The average pressure gradient for flow in the film is assumed due to the measured capillary pressure at the two ends of the liquid film and that the pressure stress gradient due to cohesion self adjusts to a constant value over a distance L. The boundary conditions are the no slip condition at the wall interface and an interfacial shear stress at the liquid- vapor interface due to the Marangoni stress, which is due to the high temperature gradient. Although the heated end is extremely complex, since it includes three- dimensional variations in radiation, conduction, evaporation, condensation, fluid flow and interfacial forces, we find that using the above simplifying assumptions, a simple successful model can be developed.

heat pipes↗
NASA NTRS1997

The Effect of Gravity on the Combustion Synthesis of Porous Ceramics and Metal Matrix Composites

Combustion synthesis (self propagating, high temperature synthesis-SHS) is a novel technique that is capable of producing many advanced materials. The ignition temperature (Tig) of such combustion synthesis reactions is often coincident with that of the lowest melting point reactant. The resultant liquid metal wets and spreads around the other solid reactant particles of higher melting points, thereby improving the reactant contact and kinetics, followed by formation of the required compounds. This ignition initiates a combustion propagating wave whose narrow reaction front rapidly travels through the reactants. Since this process is highly exothermic, the heat released by combustion often melts the reactant particles ahead of the combustion front and ignites the adjacent reactant layer, resulting in a self-sustaining reaction. Whenever a fluid phase (liquid or gas) is generated by the reaction system, gravity-driven phenomena can occur. Such phenomena include convective flows of fluid by conventional or unstable convection and settling of the higher density phases. A combustion process is often associated with various kinds of fluid flow. For instance, if the SHS reaction is carried out under inert or reactive gas atmospheres, or a volatile, e.g., B2O3, is deliberately introduced as a reactant, convective flows of the gas will occur due to a temperature gradient existing in the atmosphere when a combustion wave is initiated. The increased gas flow will produce a porous (or expanded) SHS product. Owing to the highly exothermic nature of many SHS reactions, liquid phase(s) can also form before, at, or after the combustion front. The huge temperature gradient at the combustion front can induce convective flows (conventional or unstable) of the liquid phase. Each of these types of convective fluid flow can change the combustion behavior of the synthesizing reaction, and, therefore, the resultant product microstructure. In addition, when two or more phases of different density are produced at or ahead of the propagating combustion front settling of the higher density phase will occur resulting in a non-uniform product microstructure and properties.

Moore, J. J.↗
NASA NTRS1982

Manufacturing in space: Fluid dynamics numerical analysis

Numerical computations were performed for natural convection in circular enclosures under various conditions of acceleration. It was found that subcritical acceleration vectors applied in the direction of the temperature gradient will lead to an eventual state of rest regardless of the initial state of motion. Supercritical acceleration vectors will lead to the same steady state condition of motion regardless of the initial state of motion. Convection velocities were computed for acceleration vectors at various angles of the initial temperature gradient. The results for Rayleigh numbers of 1000 or less were found to closely follow Weinbaum's first order theory. Higher Rayleigh number results were shown to depart significantly from the first order theory. Supercritical behavior was confirmed for Rayleigh numbers greater than the known supercritical value of 9216. Response times were determined to provide an indication of the time required to change states of motion for the various cases considered.

Robertson, S. J.↗
DOE OSTI2024

Transport from electron-scale turbulence in toroidal magnetic confinement devices

Plasma transport driven by turbulence ultimately determines the energy confinement performance of controlled fusion devices regardless of their confinement schemes and configurations. A large variety of plasma instabilities have been proposed for driving turbulence responsible for anomalous plasma transport beyond classical/neoclassical transport due to collisions. Although ion-scale turbulence usually dominates due to its large eddy size and saturation level, electron-scale turbulence has been recognized to be important in regions where ion-scale turbulence is suppressed (e.g., in internal transport barrier and in spherical tokamak H-mode plasmas) or is close to marginality. Electron-scale turbulence has been shown to nonlinearly interact with ion-scale turbulence, which modifies the dynamics of both and affects the resulting plasma transport, particularly when ion-scale instability is weakly driven. In this review paper, we focus on electron-scale turbulence that is believed to operate in magnetically confinement fusion devices and aim to provide a review of theoretical, numerical, and experimental developments in understanding electron-scale turbulence and its role in driving anomalous plasma turbulence. In particular, we focus on the electrostatic electron temperature gradient (ETG) mode which is the most widely recognized plasma instability underlying electron-scale turbulence observed in magnetically confined plasmas. Finally, we note that there are other less studied instabilities that might be responsible for observed electron-scale turbulence, most notably ubiquitous mode, and short-wavelength ion temperature gradient (SWITG) mode, which will be briefly touched on in this review.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
NASA NTRS1999

The Role of Velocity Redistribution in Enhancing the Intensity of the He II 304 A Line in the Quiet Sun Spectrum

We present observational evidence of the effect of small scale ("microturbulent") velocities in enhancing the intensity of the He II lambda304 line with respect to other transition region emission lines, a process we call "velocity redistribution". We first show results from the 1991 and 1993 flights of SERTS (Solar EUV Rocket Telescope and Spectrograph). The spectral resolution of the SERTS instrument was sufficient to infer that, at the spatial resolution of 5", the line profile is nearly gaussian both in the quiet Sun and in active regions. We were then able to determine, for the quiet Sun, a lower limit for the amplitude of non-thermal motions in the region of formation of the 304 A line of the order of 10 km/s. We estimated that, in the presence of the steep temperature gradients of the solar Transition Region (TR), velocities of this magnitude can significantly enhance the intensity of that line, thus at least helping to bridge the gap between calculated and observed values. We also estimated the functional dependence of such an enhancement on the relevant parameters (non-thermal velocities, temperature gradient, and pressure). We then present results from a coordinated campaign, using SOHO/CDS and H-alpha spectroheliograms from Coimbra Observatory, aimed at determining the relationship between regions of enhanced helium emission and chromospheric velocity fields and transition region emission in the quiescent atmosphere. Using these data, we examined the behavior of the He II lambda304 line in the quiet Sun supergranular network and compared it with other TR lines, in particular with O III lambda600. We also examined the association of 304 A emission with the so-called "coarse dark mottle", chromospheric structures seen in H-alpha red wing images and associated with spicules. We found that all these observations are consistent with the velocity redistribution picture.

Andretta, Vincenzo↗
NASA NTRS2003

Numerical Calculation of the Morphology of a Solid/Liquid Interface Near an Insoluble Particle

A numerical mathematical model capable of accurately describing the evolution of the shape of the solid/liquid interface in the proximity of a foreign particle is presented in this paper. The model accounts for the influence of the temperature gradient and the Gibbs-Thomson and disjoining pressure effects. It shows that for the systems characterized by k(sub P) < k(sub L) the disjoining pressure causes the interface curvature to change its sign in the close-contact particle/interface region. It also shows that the increase of the temperature gradient diminishes the effect of the disjoining pressure. Calculated critical solidification velocities for the pushing/engulfment transition are compared with experimental measurements performed in microgravity conditions.

Catalina, Adrian V.↗
NASA NTRS2004

Analytical Solution for the Critical Velocity of Pushing/Engulfment Transition

The distribution of ceramic particles in a metal matrix composite material depends primarily on the interaction of the particles with the solid/liquid interface during the solidification process. A numerical model that describes the evolution of the shape of the solid/liquid interface in the proximity of a foreign particle will presented in this paper. The model accounts for the influence of the temperature gradient and the Gibbs-Thomson and disjoining pressure effects. It shows that for the systems characterized by k(sub p) < k(sub L) the disjoining pressure causes the interface curvature to change its sign in the close-contact particle/interface region. It also shows that the increase of the temperature gradient diminishes the effect of the disjoining pressure. The analysis of the numerical results obtained for a large range of processing conditions and materials parameters has led to the development of an analytical solution for the critical velocity of pushing/engulfinent transition. The theoretical results will be discussed and compared with the experimental measurements performed under microgravity conditions.

Catalina, Adrian V.↗
NASA NTRS1984

Maximization of Growth Rates During Czochralski Pulling

It was suggested from theory(1-4) that silicon can be grown from the melt at rates far exceeding the current state of the art. Previous theoretical and experimental investigations which predict maximum rates of pulling during Czochralski growth are reviewed. Several experimental methods are proposed to modify the temperature distribution in a growing crystal to achieve higher rates of pulling. A physical model of a Czochralski crystal of germanium in contact with its melt was used to quantitatively determine, by direct measurement of the axial temperature distribution in the solid, the increase in axial temperature gradients effected by an inverted conical heat reflector located above the melt and coaxially about the physical model. Preliminary results indicate that this is an effective method of increasing the thermal resistance between the hot melt and crucible wall and a growing crystal. Under these conditions the enhancement of the interfacial temperature gradients permit a commensurate increase in the rate of crystal pulling.

Wargo, M. J.↗
DOE OSTI2023

Linear instabilities in the hot-ion regime in a high-field spherical tokamak

Abstract The present operating high-field compact spherical tokamak ST40 is an important step toward an ST-based fusion reactor (Gryaznevich and Asunta 2017 Fusion Eng. Des. 123 177; McNamara 2023 Nucl. Fusion 63 054002). Temperature and density profiles and their uncertainties in recent hot ion ST40 plasmas with central ion temperatures exceeding 8.6 keV, have been inferred using an integrated analysis of several diagnostics including line-of-sight integration and volume average measurements, as well as limited profile information from a charge-exchange-recombination spectrometer. A linear gyrokinetic stability analysis has been carried out to identify the most unstable micro-instabilities in these hot ion plasmas. In one of these plasmas, it is found that linear growth rates of both ion- and electron-scale ( k θ ρ s ⩾ 0.2 where k θ is the poloidal wavenumber and ρ s is the ion gyro-radius with sound speed) modes decrease from the edge toward the core of the plasma (i.e. from ρ = 0.8 to 0.3 where ρ is the square root of the normalized magnetic flux), while the change of linear growth rates at k θ ρ s < 0.2 is non-monotonic. In particular, at ρ = 0.3 no unstable mode was found at k θ ρ s > 5 , and about an order of magnitude reduction in the maximum linear growth rate in the ion-scale wavenumber range of 0.2 ⩽ k θ ρ s < 1 is seen at ρ = 0.3 compared with the ρ = 0.8 location. It is found through parametric scans that while the ion temperature gradient (ITG) mode/trapped electron mode (TEM) and kinetic ballooning mode (KBM)/kinetic shear Alfvén (KSA) mode are more important in the plasma core, ubiquitous mode (UM) is found to be the important ion-scale instability at ρ ⩾ 0.6 . At ρ = 0.8, two instabilities are found at the electron scale, with one being the typical electrostatic electron temperature gradient (ETG) mode at ρ e scale and beyond and the other being an unidentified low-real-frequency instability (at the intermediate scale between electron and ion gyroradii) which can propagate in the ion direction and is driven by ETG. The existence of UM at the ion scale and the low-real-frequency instability and ETG mode at the electron scale at ρ = 0.8 is supported by an extensive linear gyrokinetic stability analysis of another ST40 hot ion plasma in a wider parametric range, taking into account the uncertainties in the experimental profiles.

ubiquitous mode↗
NASA NTRS1986

Analytical and numerical studies of the thermocapillary flow in a uniformly floating zone

The microgravity environment of an orbiting vehicle permits crystal growth experiments in the presence of greatly reduced buoyant convection in the liquid melt. Crystals grown in ground-based laboratories do not achieve their potential properties because of dopant variations caused by flow in the melt. The floating zone crystal growing system is widely used to produce crystals of silicon and other materials. However, in this system the temperature gradient on the free sidewall surface of the melt is the source of a thermocapillary flow which does not disappear in the low-gravity environment. The idea of using a uniform rotation of the floating zone system to confine the thermocapillary flow to the melt sidewall leaving the interior of the melt passive is examined. A cylinder of fluid with an axial temperature gradient imposed on the cylindrical sidewall is considered. A half zone and the linearized, axisymmetric flow in the absence of crystal growth is examined. Rotation is found to confine the linear thermocapillary flow. A simplified model is extended to a full zone and both linear and nonlinear thermocapillary flows are studied theoretically. Analytical and numerical methods are used for the linear flows and numerical methods for the nonlinear flows. It was found that the linear flows in the full zone have more complicated and thicker boundary layer structures than in the half zone, and that these flows are also confined by the rotation. However, for the simplified model considered and for realistic values for silicon, the thermocapillary flow is not linear. The fully nonlinear flow is strong and unsteady (a weak oscillation is present) and it penetrates the interior. Some non-rotating flow results are also presented. Since silicon as a large value of thermal conductivity, one would expect the temperature fields to be determined by conduction alone. This is true for the linear and weakly nonlinear flows, but for the stronger nonlinear flow the results show that temperature advection is also important. Uniform rotation may still be a means of confining the flow and the results obtained define the procedure to be used to examine this hypothesis.

Fowlis, W. W.↗
DOE OSTI2024

Accelerating CO2 Storage Site Characterization through a New Understanding of Favorable Formation Properties and the Impact of Core-Scale Heterogeneities

CO2 sequestration in deep geologic formations can permanently reduce atmospheric CO2 emissions and help to abate climate change. Target formations must undergo a time- and resource-intensive site evaluation process, assessing storage capacity, environmental safety, and suitability for CO2 trapping via reactive transport models based on data from a limited number of core samples. As such, simulations are often simplified and omit heterogeneities in formation properties that may be significant but are not well understood. To facilitate more rapid site assessment, this work first defines the aquifer properties of favorable storage formations through the analysis of promising and active storage sites. Data show quartz is the most prevalent formation mineral with carbonate minerals, highly reactive with injected CO2, present in over 75% of formations. Porosity and permeability data are highly clustered at 10–30% and 10–1000 mD. Field-scale reactive transport simulations are then constructed and used to analyze CO2 trapping efficiency. The models consider porosity and carbonate mineral heterogeneity as well as the impacts of typical temperature gradients. Simulated sequestration efficiencies are compared to results from a comparable homogenous model to understand the implications of aquifer non-uniformities. The results show a lower sequestration efficiency in the homogeneous model during the injection phase. During the post-injection phase, the homogenization of porosity and carbonate mineralogy results in a higher sequestration efficiency. Incorporating the temperature gradient also increases the sequestration efficiency. Importantly, the maximum deviation between the homogeneous and heterogeneous simulations at the end of the 50-year study period is only ~10%. Larger impacts may be incurred for properties outside the defined, promising ranges suggested here.

Environmental Sciences & Ecology↗
NASA NTRS1987

Direct numerical simulation of buoyantly driven turbulence

Numerical simulations of homogeneous turbulence subject to buoyant forcing were performed. The presence of a mean temperature gradient combined with a gravitational field results in a forcing term in the momentum equations. The development of the turbulence was studied and compared to the decay of similar fields in the absence of gravity. In the buoyantly driven field, the vorticity is preferentially aligned with the intermediate eigenvector of the strain-rate tensor and the local temperature gradient is more likely to be aligned with the most compressive eigenvector. These relationships are qualitatively similar to those observed in previous shear flow results studied by Ashurst (1987). A tensor diffusivity model for passive scalar transport developed from shear flow results in Rogers, Moin, and Reynolds (1986) also predicts this buoyant scalar transport, indicating that the relationship between the scalar flux and the Reynolds stress is similar in both flows.

Ashurst, William T.↗
NASA NTRS1974

On the distribution of He+ in the plasmasphere from observations of resonantly scattered He II 304-A radiation

The intensity of the He II 304-A radiation in the night sky was measured from a sounding rocket launched from White Sands Missile Range on June 9, 1972. The data obtained on this flight are compared with predictions of theoretical models of the He(+) altitude distribution within the plasmasphere. The observations are not consistent with an isothermal equilibrium model of the helium ion in the plasmasphere, confirming earlier results. The data are best fit by a diffussive equilibrium model of exospheric He(+) having a plasma temperature gradient of 0.5 K per km and a number density at 1600 km of 200,000 He(+) ions per cu cm. In this model the ion densities are calculated without allowance for thermal diffusion. A similar model that allows for thermal diffusion and that has a He(+) number density at 1600 km of 1500 per cu cm and a temperature gradient of 1.2 K per km yields a less satisfactory fit to the observations but is more consistent with in situ satellite observations.

Paresce, F.↗
NASA NTRS1992

Eclipse maps of the accretion disk in the classical nova V Persei

The surface brightness distribution in the accretion disk of V Per is reconstructed using a maximum entropy technique. The resulting brightness temperature gradient is much less steep than that from a standard steady state, optically thick disk for all reasonable mass accretion rates. A model including a rim which obscures part of the disk does not resolve this discrepancy. However, if the inner disk is disrupted out to about 0.15-0.25R sub L1 (for example by a magnetic field), the brightness temperature gradient can be very close to T sub b varies as R exp -3/4 (1-b(Rin/R)exp 1/2) law. The value of Rin needed to match this steady state distribution depends on the distance to the system, its size and brightness, and the rotation rate of the white dwarf.

Wood, Janet H.↗
NASA NTRS1974

Physical properties of solar chromospheric plages. II - Chromospheric plage models

Chromospheric models of plages are proposed to explain profiles of the Ca II H, K, 8498-A, 8542-A, and 8662-A lines. These models are consistent with boundary conditions imposed by the photosphere and the Lyman continuum. It is found that increasing emission in these lines is consistent with a picture of increasing temperature gradient in the low chromosphere and the resulting increase in pressure and electron density at similar line optical depths. With this picture it is shown how the distribution of chromospheric parameters across the solar disk can be empirically determined directly from Ca II filtergrams. It is also proposed that the high-density aspects of solar activity are produced by steep temperature gradients in the low chromosphere and thus by the enhanced heating mechanisms that steepen these gradients.

Shine, R. A.↗
NASA NTRS1985

The importance of Soret transport in the production of high purity silicon for solar cells

Temperature-gradient-driven diffusion, or Soret transport, of silicon vapor and liquid droplets is analyzed under conditions typical of current production reactors for obtaining high purity silicon for solar cells. Contrary to the common belief that Soret transport is negligible, it is concluded that some 15-20 percent of the silicon vapor mass flux to the reactor walls is caused by the high temperature gradients that prevail inside such reactors. Moreover, since collection of silicon is also achieved via deposition of silicon droplets onto the walls, the Soret transport mechanism becomes even more crucial due to size differences between diffusing species. It is shown that for droplets in the 0.01 to 1 micron diameter range, collection by Soret transport dominates both Brownian and turbulent mechanisms.

Srivastava, R.↗
NASA NTRS1999

Experimental Investigation of Pool Boiling Heat Transfer Enhancement in Microgravity in the Presence of Electric Fields

In boiling high heat fluxes are possible driven by relatively small temperature differences, which make its use increasingly attractive in aerospace applications. The objective of the research is to develop ways to overcome specific problems associated with boiling in the low gravity environment by substituting the buoyancy force with the electric force to enhance bubble removal from the heated surface. Previous studies indicate that in terrestrial applications nucleate boiling heat transfer can be increased by a factor of 50, as compared to values obtained for the same system without electric fields. The goal of our research is to experimentally explore the mechanisms responsible for EHD heat transfer enhancement in boiling in low gravity conditions, by visualizing the temperature distributions in the vicinity of the heated surface and around the bubble during boiling using real-time holographic interferometry (HI) combined with high-speed cinematography. In the first phase of the project the influence of the electric field on a single bubble is investigated. Pool boiling is simulated by injecting a single bubble through a nozzle into the subcooled liquid or into the thermal boundary layer developed along the flat heater surface. Since the exact location of bubble formation is known, the optical equipment can be aligned and focused accurately, which is an essential requirement for precision measurements of bubble shape, size and deformation, as well as the visualization of temperature fields by HI. The size of the bubble and the frequency of bubble departure can be controlled by suitable selection of nozzle diameter and mass flow rate of vapor. In this approach effects due to the presence of the electric field can be separated from effects caused by the temperature gradients in the thermal boundary layer. The influence of the thermal boundary layer can be investigated after activating the heater at a later stage of the research. For the visualization experiments a test cell was developed. All four vertical walls of the test cell are transparent, and they allow transillumination with laser light for visualization experiments by HI. The bottom electrode is a copper cylinder, which is electrically grounded. The copper block is heated with a resistive heater and it is equipped with 6 thermocouples that provide reference temperatures for the measurements with HI. The top electrode is a mesh electrode. Bubbles are injected with a syringe into the test cell through the bottom electrode. The working fluids presently used in the interferometric visualization experiments, water and PF 5052, satisfy requirements regarding thermophysical, optical and electrical properties. A 30kV power supply equipped with a voltmeter allows to apply the electric field to the electrodes during the experiments. The magnitude of the applied voltage can be adjusted either manually or through the LabVIEW data acquisition and control system connected to a PC. Temperatures of the heated block are recorded using type-T thermocouples, whose output is read by a data acquisition system. Images of the bubbles are recorded with 35mm photographic and 16mm high-speed cameras, scanned and analyzed using various software packages. Visualized temperature fields HI allows the visualization of temperature fields in the vicinity of bubbles during boiling in the form of fringes. Typical visualized temperature distributions around the air bubbles injected into the thermal boundary layer in PF5052 are shown. The temperature of the heated surface is 35 C. The temperature difference for a pair of fringes is approximately 0.05 C. The heat flux applied to the bottom surface is moderate, and the fringe patterns are regular. In the image a bubble penetrating the thermal boundary layer is visible. Because of the axial symmetry of the problem, simplified reconstruction techniques can be applied to recover the temperature field. The thermal plume developing above the heated surface for more intensive heating is shown. The temperature distribution in the liquid is clearly 3D, and tomographic techniques have to be applied to recover the temperature distribution in such a physical situation. A sequence of interferometric images showing the temperature distribution around the rising bubble, recorded with a high-speed camera is shown. Again, the temperature distribution is 3D, and a more complex approach to the evaluation, the tomographic reconstruction has to be taken. Measurement of the temperature distribution from the fringe pattern temperature distributions that yield important information regarding heat transfer are determined. Two algorithms that allow the quantitative evaluation of interferometric fringe patterns and the reconstruction of temperature fields during boiling have been developed at the Heat Transfer Laboratory of the Johns Hopkins University. In the first algorithm the bubble is assumed to be axially symmetrical, which significantly reduces the computational effort for quantifying the temperature distribution around the bubble. For this purpose the thermal boundary layer around the bubble is divided into equidistant concentric shells, and the refractive index is assumed to be constant in each of the shells. Since large temperature gradients are expected in the vicinity of the bubble during boiling, the deflection of the light beam cannot be neglected in boiling experiments. Since the exit angle of the light beam is known, this allows to account for the deflections and phase shifts outside the boundary layer (in the bulk fluid and in the windows of the test cell). Three dimensional temperature distributions in the vicinity of the bubble are reconstructed using tomographic techniques. In tomography, the measurement volume is sliced into 2D planes. In the present study these planes are parallel to the heated surface. The objective is to determine the values of the field parameter of interest in form of the field function in these 2D planes. The field parameter is the change of the refractive index of the liquid in the measurement volume caused by temperature changes. By superimposing data for many 2D planes recorded at the same time instant, the 3D temperature distribution in the measurement volume is recovered.

Herman, Cila↗
NASA NTRS1991

Thermal behavior of a titanium honeycomb-core sandwich panel

Finite element thermal stress analysis was performed on a rectangular titanium honecomb-core sandwich panel which is subjected to thermal load with a temperature gradient across its depth. The distributions of normal stresses in the face sheets and the face-sheet/sandwich-core interfacial shear stresses are presented. The thermal buckling of the heated face sheet was analyzed by assuming the face sheet to be resting on an elastic foundation representing the sandwich core. Thermal buckling curves and thermal buckling load surface are presented for setting the limit for temperature gradient across the panel depth.

Ko, William L.↗