Combined nongray radiative and conductive heat transfer in multiple glazing taking into account specular reflection and absorption

The problem of combined nongray radiative and conductive heat transfer in multiple glazing subjected to solar irradiation is analyzed. A spectral solar model proposed by Bird and Riordan is used to calculate direct and diffuse solar irradiance. The radiation element method by ray emission model, REM², is used to analyze the spectral dependence of radiative heat transfer. Specular reflection at boundary surfaces is taken into account. The spectral dependence of radiation properties of glass such as specular reflectivity, refraction angle, and absorption coefficient is taken into account. The steady‐state temperature and heat flux distributions in the glass layer are obtained and the insulating efficiency of multiple glazing is examined. The overall heat transfer coefficients predicted by the present method are compared with those based on the JIS method. The values obtained by the present method are slightly lower than those obtained by the JIS method. To investigate the spectral variation of radiative heat flux attenuated in the glass layer, the spectral heat flux at the room‐side surface and incident radiation are compared. © 2003 Wiley Periodicals, Inc. Heat Trans Asian Res, 32(8): 712–726, 2003; Published online in Wiley InterScience ( www.interscience.wiley.com ). DOI 10.1002/htj.10125     

An analytical solution of the hyperbolic heat conduction equation for the case of a finite medium symmetrically heated on both sides

This paper presents an analytical solution of the hyperbolic heat conduction equation for the case of a thin slab symmetrically heated on both sides. In the mathematical model adopted, the heating is treated as an internal heat source with capacity dependent on coordinate and time, while walls of the slab are assumed to be insulated. The solution is obtained by Laplace transforms method taking advantage of the method of superposition. The analytical solution is validated by comparison with the results from a numerical model.     

A MONTE CARLO (MC) METHOD APPLIED TO THE MEDIUM WITH NONGRAY ABSORBING-EMITTING-ANISOTROPIC SCATTERING PARTICLES AND GRAY APPROXIMATION

A Monte Carlo (MC) method is applied to calculate radiative transfer in a nongray medium using spectral radiative exchange factor RD ij u . The creditability of the present MC model has been validated by comparing it with the results using other research methods. Meanwhile, the radiative transfer in an isothermal and nonisothermal medium with nongray absorbing-emitting-anisotropic ash particles is calculated by a nongray model and several gray approximation methods. A simplification from a nongray problem to a gray one by Rosseland's mean extinction coefficient, mean albedo y bar 2 , and Planck mean phase function is suggested.     

Radiation heat transfer in a participating medium bounded by specular reflectors

This paper analyzes the problem of one dimensional plane-parallel absorbing emitting and isotropically scattering gray slab with gray, diffusely emitting and specularly reflecting boundaries, under the condition of radiative equilibrium. The problem is exactly formulated. The governing energy equation which is a Fredholm integral equation of the second kind involving the integral functions C_n(t) is solved by an accurate modified quadrature method based on the Gauss-Legendre rule. The temperature distribution and heat flux results are presented. Results of particular cases are compared to that published earlier. A three term exponential curve fit is presented to represent the heat transfer result of diffusely reflecting boundaries case.     

A heat transfer model for the analysis of transient heating of the slab in a direct-fired walking beam type reheating furnace

A mathematical heat transfer model for the prediction of heat flux on the slab surface and temperature distribution in the slab has been developed by considering the thermal radiation in the furnace chamber and transient heat conduction governing equations in the slab, respectively. The furnace is modeled as radiating medium with spatially varying temperature and constant absorption coefficient. The steel slabs are moved on the next fixed beam by the walking beam after being heated up through the non-firing, charging, preheating, heating, and soaking zones in the furnace. Radiative heat flux calculated from the radiative heat exchange within the furnace modeled using the FVM by considering the effect of furnace wall, slab, and combustion gases is introduced as the boundary condition of the transient conduction equation of the slab. Heat transfer characteristics and temperature behavior of the slab is investigated by changing such parameters as absorption coefficient and emissivity of the slab. Comparison with the experimental work show that the present heat transfer model works well for the prediction of thermal behavior of the slab in the reheating furnace.     

Assessment of gas radiative property models in the presence of nongray particles

In this study, a radiation code based on the method of lines solution of the discrete ordinates method for the prediction of radiative heat transfer in nongray gaseous media is developed by incorporation of two different spectral gas radiative property models, banded spectral line-based weighted sum of gray gases (banded SLW) and gray wide band (GWB) approximation in the presence of nongray absorbing–emitting–scattering particles. The aim is to introduce an accurate and CPU efficient spectral gas radiation model, which is compatible with spectral fuel/ash particle property models. Input data required for the radiation code and its validation are provided from two combustion tests previously performed in a 300 kWt atmospheric bubbling fluidized bed combustor test rig burning low calorific value Turkish lignite with high volatile matter/fixed carbon (VM/FC) ratio in its own ash. The agreement between wall heat fluxes and source term predictions obtained by global and banded SLW models reveal that global SLW model can be converted to an accurate wide band gas model (banded SLW) which can directly be coupled with spectral particle radiation. Furthermore, assessment of GWB approximation by benchmarking its predictions against banded SLW model shows that GWB gives reasonable agreement with a higher CPU efficiency when the particle absorption coefficient is at least in the same order of magnitude with the gas absorption coefficient.     

Nongray inverse boundary design problems in enclosures at radiative equilibrium

In this paper, an inverse analysis is used to find an appropriate heat flux distribution over the heater surface of radiant enclosures, filled with nongray media at radiative equilibrium from the knowledge of desired (prespecified) temperature and heat flux distributions over the given design surface. Regular and irregular 2D enclosures filled with nongray combustive gas products are considered. Radiation is considered the dominant mode of heat transfer and the medium temperature is obtained from the energy equation. To evaluate the nongray behavior of the participating gases properly, the spectral‐line weighted‐sum‐of‐gray‐gases (SLW) model with updated correlations is used. The dependence of absorption coefficients and the weights of the SLW model on the temperature of the medium makes the inverse problem nonlinear and difficult to handle. Here, the inverse problem is formulated as an optimization problem and the Levenberg‐Marquardt method has been used to solve it. The finite volume method is exploited for the discretization of the energy equation and the spatial discretization of the radiative transfer equation (RTE). The discrete ordinates method (TN quadrature) is used for the angular discretization of RTE. Five test cases, including homogeneous and inhomogeneous media, are investigated to prove the ability of the present methodology for achieving the desired conditions.     

Integral equation solutions based on exact ray paths for radiative transfer in a participating medium with formulated refractive index

The exact analytical path length of radiation traveling in a slab with formulated variable refractive index is derived. Based on the analytical path lengths, the integral equations in terms of intensity moments for radiative transfer in a participating slab with one of the family of spatially varying refractive indices are developed. We solve the integral equations for radiative transfer in a slab at radiative equilibrium or for radiative transfer in an isothermal slab. The boundaries are assumed to be black for the slab at radiative equilibrium and the index jumps at both boundaries for the isothermal slab are considered. For comparison purpose, we also solve the radiative equilibrium problems by the discrete ordinates method (DOM). The nondimensional emissive power and nondimensional radiative heat flux obtained by solving integral equations show an excellent agreement with those obtained by the DOM. For the slab at radiative equilibrium and with positive gradient of refractive index, the jump of the emissive power at bottom boundary decreases with the increase of optical thickness for the cases with slightly varying refractive index, but the trend may not hold for the cases with significantly varying refractive index. For the non-scattering slab with positive gradient of refractive index and fixed refractive indices at the boundaries, the directional emittances at both boundaries for the case with linear refractive index are smaller than those for the case with a refractive index of slope-increasing profile. Effects of the scattering albedo and the scattering phase function coefficient are investigated too.     

The Re-Plating Algorithm for Radiation Total Exchange Area Calculation

Great efforts have been made to date toward modeling nongray radiative heat transfer accurately. In this article, a new version of the plating algorithm, designated the re-plating algorithm, for total exchange areas (TEAs) calculation from direct exchange areas (DEAs) for nongray radiative problems is presented. The re-plating algorithm calculates TEAs for a given band number b from those of band number b ? 1 by performing successive re-plating procedures. The effectiveness of the new algorithm is demonstrated for thermal modeling of an aluminum brazing furnace and a glass treatment furnace. CPU requirements for TEA calculation were reduced significantly.     

Modeling of Flow and Heat Transfer in a Molten Glass Mini-Film for High Temperature Heat Collection in a Falling-Film Solar Central Receiver

ABSTRACT

In concentrating solar power plants, there is a strong incentive to increase the collection temperature and the overall exergy efficiency of the system. Some molten glass mixtures are attractive working fluids for high temperature solar thermal heat collection because optimized glass mixtures can be more stable, less-toxic, and less-corrosive than, for example, molten salts at high temperatures (≥1000°C). A specific phosphorous pentoxide glass mixture is considered in this study to explore its performance in a molten glass falling film central receiver design for collection of heat at conditions resulting in a mini-film with a thickness less than 3mm. In our falling molten glass thin film, the phosphate glass flow is treated as a laminar, Newtonian and gravity-driven flow over a slightly inclined flat plate using an explicit finite difference scheme to evaluate its heat transfer performance for a direct absorption receiver concept. One of the main challenges of modeling transport in the molten glass is the strong dependence of its viscosity on temperature. To incorporate this effect in our numerical analysis, a temperature-dependent viscosity model is used in the momentum equation to model the fluid behavior as it flows down the surface and is progressively heated. An exponential function is used to model the viscosity as it changes with temperature to properly fit the measured the viscosity data provided by Halotechnics. Also, a variable film thickness model analysis is implemented to treat the variation in film thickness that results from the viscosity variation with temperature. In order to avoid stability issues, the finite difference scheme is organized in terms of nondimensional parameters that include all important properties that govern the system. The results of our model indicate that thinning of the film as it flows over the heated surface enhances the heat transfer performance on the lower portion of the receiver system. The heat transfer coefficient increases instead of remaining constant (as normally expected for fully developed laminar flows) on the lower portion of the heated surface. The results further indicate that using a thin mini-film of molten glass for solar thermal heat collection provides high heat transfer performance and enhances the exergy collection.     

Estimation of the transient surface temperature and heat flux of a steel slab using an inverse method

In the steel industry it is of great importance to be able to control the surface temperature and heating- or cooling rates during heat treatment processes. An experiment was performed in which a steel slab was heated up to 1250 °C in a fuel fired test furnace. The transient surface temperature and heat flux of a steel slab is calculated using a model for inverse heat conduction. That is, the time dependent local surface temperature and heat flux of a slab is calculated on the basis of temperature measurements in selected points of its interior by using a model of inverse heat conduction. Time- and temperature histories were measured at three points inside a steel slab. Measured temperature histories at the two lower locations of the slab were used as input to calculate the temperature at the position of the third location. A comparison of the experimentally measured and the calculated temperature histories was made to verify the model. The results showed very good agreement and suggest that this model can be applied to similar applications in the Steel industry or in other areas where the target of investigation for some reason is inaccessible to direct measurements.     

Nongray radiation from gas and soot mixtures in planar plates based on statistical narrow-band spectral model

The nongray behavior of combustion products plays an important role in various areas of engineering. Based on the statistical narrow-band (SNB) spectral model with an exponential-tailed inverse intensity distribution and the ray-tracing method, a comprehensive investigation of the influence of soot on nongray radiation from mixtures containing H₂O/N₂+soot, CO₂/N₂+soot, or H₂O/CO₂/N₂+soot was conducted in this paper. In combustion applications, radiation transfer is significantly enhanced by soot due to its spectrally continuous emission. The effect of soot volume fraction up to 1×10^-6 on the source term, the narrow-band radiation intensities along a line-of-sight, and the net wall heat fluxes were investigated for a wide range of temperature. The effect of soot was significant and became increasingly drastic with the increase of soot loading.     

Solar pond ground heat loss to a moving water table

An analytical solution is presented that calculates the heat loss from the bottom of a solar pond (or any heated object) to a soil that contains a moving water table. The water table is treated as a fluid slab moving as a slug flow in one dimension. Edge effects and horizontal heat conduction are ignored. Both steady-state and time-dependent solutions are presented. Results are presented in terms of an effectiveness ratio—the actual heat flux divided by the steady-state heat flux resulting from a constant temperature heat sink at the depth of the water table. The only water-table parameter that strongly affects the effectiveness is the fluid capacity rate. Thus, for any potential solar pond site, a measurement of the mass flow rate of the water table combined with knowledge of the soil thermal properties will allow a good estimation of the ground heat loss expected over the lifetime of the pond.     

Experimental study of the dynamic behaviour of a porous medium submitted to a wall heat flux in view of thermal energy storage by sensible heat

This study concerns the storage of thermal energy in a porous bed mainly formed by a vertical channel, filled with glass beads, heated on one of the vertical walls by a constant heat flux. The use of glass beads is motivated by the possible use of such a system for storage of solar energy by sensitive heat and its optimization. The medium is characterized by a large thermal inertia which favors a slow return of the heat stored during the heating phase. After model reduction, the system can be dynamically modeled as a simple first order which allows us to evaluate the heat transfer coefficient and to predict its response to real values of the solar flux. The system efficiency defined as the ratio of the stored energy to the energy provided increases with the storage volume and decreases with the return time for fixed width and storage time.     

Gas heat conduction in an evacuated tube solar collector

We investigated experimentally the pressure dependency of the gas heat conduction in an evacuated plate-in-tube solar collector. A stationary heat loss experiment was built up with an electrically heated real-size collector model. The gas pressure was varied from 10⁻³ to 10⁴ Pa, the temperatures of the absorber and the casing were held at 150°C (electrical heaters) and 30°C (water cooling), respectively. Losses by radiation and solid conduction were determined experimentally at pressures below 0.1 Pa. At higher pressures these background losses were subtracted from the total heat losses, to receive the heat losses by gas heat conduction. The experimental results were compared with approximative theoretical models. The onset of convection is in agreement with the usual theories for parallel plates, taking the largest distance between the absorber and the glass tube as the plate distance. As a first approximation the pressure dependency of the gas heat conduction is described by the usual theory for parallel plates, taking the smallest distance between the absorber and the glass tube as the plate distance.     

Experimental and numerical investigation of the steady periodic solid-liquid phase-change heat transfer

A numerical and experimental investigation is presented of a periodic phase-change process dominated by heat conduction. In the experimental arrangement a plane slab of PCM is periodically heated from above. A one-dimensional control volume computer code has been developed for the solution of the corresponding mathematical model. The comparison between numerical predictions and experimental data shows good agreement, even though appreciable effects are produced by free convection and heat transfer to the environment, neglected in the model but unavoidable in the experiment. Finally, in order to study the energy stored in the process, parameters like amplitude and mean value of the oscillations are discussed as functions of the significant dimensionless numbers of the problem.     

A mathematical model of a solar chimney

A simple mathematical model of a solar chimney is proposed. The physical model is similar to the Trombe wall. One side of the chimney is provided with a glass cover which with the other three solid walls of the chimney form a channel through which the heated air could rise and flow by natural convection. Openings provided at the bottom and top of the chimney allow room air to enter and leave the channel. Steady state heat transfer equations were set up to determine the boundary temperatures at the surface of the glass cover, the rear solar heat absorbing wall and the air flow in the channel using a thermal resistance network. The equations were solved using a matrix-inversion solution procedure. The thermal performance of the solar chimney as determined from the glass, wall and air temperatures, air mass flow rate and instantaneous heat collection efficiency of the chimney are presented. Satisfactory correlation was obtained with experimental data from other investigators. Further experimental investigation is currently under way.     

Numerical Simulation of Buoyancy-Induced Turbulent Flow Between Two Concentric Isothermal Spheres

Buoyancy-induced turbulent flow and natural convection heat transfer between two differentially heated concentric isothermal spheres is studied numerically. The low-Reynolds-number k–ω model is used for turbulence modeling. The two-dimensional governing equations are discretized using control volume method and solved by employing the alternating direction implicit scheme. Results are presented in the form of streamline and temperature patterns, and local and average Nusselt numbers, over the heated and cooled boundaries for a wide range of Rayleigh numbers (10²–10¹⁰), extending the previous studies to the turbulent flow regime and for the radius ratio of 2. The results of the flow pattern and average Nusselt numbers were compared with the previously published experimental and numerical investigations and very good agreements were observed. For low values of Rayleigh numbers, regions with conduction-dominated flow pattern accompanied with low values of Nusselt numbers were observed, while for higher Rayleigh numbers, the flow pattern was changed to the convection dominated boundary layer type flow, resulting in an increase in the rate of heat transfer and flow velocities adjacent to both inner and outer boundaries. The average Nusselt numbers were correlated against Rayleigh number and a 1/4 power dependence of Ra in both laminar and turbulent regimes is obtained.     

Numerical investigation of parabolic trough receiver performance with outer vacuum shell

The useful heat gain of a parabolic collector system is directly dependent on the heat loss from the absorber at its operating temperature. Selective coatings with evacuated/non evacuated glass tubes are employed to control radiative and convective heat losses. A concentric glass shell under vacuum is investigated for its thermal performance as this method circumvents the need for direct sealing between the glass envelope and the metal receiver to maintain vacuum and its related technical challenges. The performance is compared against a non evacuated receiver and its influence under different wind velocities; emissivities are calculated by a one dimensional theoretical model and solved by an iterative method.     

Three-dimensional Mathematical Model for Flow and Heat Transfer in Electric Glass Furnaces

A three-dimensional mathematical model was developed for calculating Joule heat release, glass flow, and heat transfer in electric glass furnaces. The model developed here allows for multiple electrodes, multiphase firing, and the feeding and withdrawal of molten glass. The model is fairly general with respect to the arrangement of the electrodes, the firing pattern, and the choice of boundary conditions, and it allows for the temperature dependence of the glass properties. The model was used to calculate electric potential, rate of heat release, flow pattern, and temperature distribution in the melting of flint and amber glasses in an all-electric melter with side-entering electrodes. Calculations were performed for the industrial conditions of pull, power, and electric firing scheme. The bulk glass temperature was found to be very uniform with large temperature gradients near the boundaries. The calculated flow pattern was in general quite complex, with several circulation loops. The temperature and maximum velocity for the amber glass were found to be higher than the corresponding values for the flint glass.