Performance Evaluation of Two Heat Transfer Models of a Walking Beam Type Reheat Furnace

Two different heat transfer models for predicting the transient heat transfer characteristics of the slabs in a walking beam type reheat furnace are compared in this work. The prediction of heat flux on the slab surface and the temperature distribution inside the slab have been determined by considering thermal radiation in the furnace chamber and transient heat conduction in the slab. Both models have been compared for their accuracy and computational time. The furnace is modeled as an enclosure with a radiatively participating medium. In the first model, the three-dimensional (3D) transient heat conduction equation with a radiative heat flux boundary condition is solved using an in-house code. The radiative heat flux incident on the slab surface required in the boundary condition of the conduction code is calculated using the commercial software FLUENT. The second model uses entirely FLUENT along with a user-defined function, which has been developed to account for the movement of slabs. The results obtained from both models have a maximum temperature difference of 2.25%, whereas the computational time for the first model is 3 h and that for the second model is approximately 100 h.     

Efficiency analysis of radiative slab heating in a walking-beam-type reheating furnace

The thermal efficiency of a reheating furnace was predicted by considering radiative heat transfer to the slabs and the furnace wall. The entire furnace was divided into fourteen sub-zones, and each sub-zone was assumed to be homogeneous in temperature distribution with one medium temperature and wall temperature, which were computed on the basis of the overall heat balance for all of the sub-zones. The thermal energy inflow, thermal energy outflow, heat generation by fuel combustion, heat loss by the skid system, and heat loss by radiation through the boundary of each sub-zone were considered to give the two temperatures of each sub-zone. The radiative heat transfer was solved by the FVM radiation method, and a blocked-off procedure was applied to the treatment of the slabs. The temperature field of a slab was calculated by solving the transient heat conduction equation with the boundary condition of impinging radiation heat flux from the hot combustion gas and furnace wall. Additionally, the slab heating characteristics and thermal behavior of the furnace were analyzed for various fuel feed conditions.     

Transient radiative heating characteristics of slabs in a walking beam type reheating furnace

Transient radiative heating characteristics of slabs in a walking beam type reheating furnace is predicted by the finite-volume method (FVM) for radiation. The FVM can calculate the radiative intensity absorbed and emitted by hot gas as well as emitted by the wall with curvilinear geometry. The non-gray weighted sum of gray gas model (WSGGM) which is more realistic than the gray gas model is used for better accurate prediction of gas radiation. The block-off procedure is applied to the treatment of the slabs inside which intensity has no meaning. Entire domain is divided into eight sub-zones to specify temperature distribution, and each sub-zone has different temperatures and the same species composition. Temperature field of a slab is acquired by solving the transient 3D heat conduction equation. Incident radiation flux into a slab is used for the boundary condition of the heat conduction equation governing the slab temperature. The movement of the slabs is taken into account and calculation is performed during the residence time of a slab in the furnace. The slab heating characteristics is also investigated for the various slab residence times. Main interest of this study is the transient variation of the average temperature and temperature non-uniformity of the slabs.     

A Coupled Numerical Study of Slab Temperature and Gas Temperature in the Walking-Beam-Type Slab Reheating Furnace

In the present study, a three-dimensional simulation is performed for the turbulent reactive flow and radiactive heat transfer in the walking-beam-type slab reheating furnace using STAR-CD software. The geometric model takes care of all components of the furnace. To obtain a steady solution, the walking beams are assumed fixed in the furnace and the slab is modeled as a laminar flow having a very high viscosity and thus moving at a nearly constant speed. The temperature distributions of the slab and the gas mixture are obtained through a coupled calculation. The simulation results successfully predict the temperature distribution inside the slab and the heat flux on the slab surfaces, providing an opportunity for a full exploration of the influence of the walking beam system on the skid marks. The simulation results show that the radiative shielding by the static beams is the main cause of the skid marks. The heat loss through the skid button to the cooling system worsens the skid marks.     

Investigation of the slab heating characteristics in a reheating furnace with the formation and growth of scale on the slab surface

In this work, the development of a mathematical heat transfer model for a walking-beam type reheating furnace is described and preliminary model predictions are presented. The model can predict the heat flux distribution within the furnace and the temperature distribution in the slab throughout the reheating furnace process by considering the heat exchange between the slab and its surroundings, including the radiant heat transfer among the slabs, the skids, the hot combustion gases and the furnace wall as well as the gas convection heat transfer in the furnace. In addition, present model is designed to be able to predict the formation and growth of the scale layer on the slab in order to investigate its effect on the slab heating. A comparison is made between the predictions of the present model and the data from an in situ measurement in the furnace, and a reasonable agreement is found. The results of the present simulation show that the effect of the scale layer on the slab heating is considerable.     

基于瞬态非线性分析的中厚板坯加热过程应力场仿真研究

为提高中厚板坯加热效率和防止出现热应力裂纹,需要精确研究中厚板坯加热升温热应力规律。本文在热弹塑性分析理论基础上,采用有限元方法,研究了中厚板坯加热过程中全时域内热应力场变化规律。本文针对实际生产加热工艺,建立了3500板坯加热过程分析模型,考虑了导热系数、比热变化对加热的影响,仿真了中厚板坯的加热过程,得出加热炉温度与加热速度对板坯加热过程热应力的影响规律,具有重要工程应用意义。     

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.     

Comparisons of different heat transfer models of a walking beam type reheat furnace

Four different heat transfer models (Model-1 to -4) for the prediction of temperature of the slabs of a walking beam type reheat furnace have been compared. The models are classified based on the solution methodology and simplifications. In the first three models (Model-1 to -3), the furnace is modelled as radiating medium with spatially varying known temperature. Model-1 solves the 3D transient conduction in the slab and radiation in the furnace separately and is coupled via the boundary condition. In the second model, both radiation in the furnace and conduction in the slab are solved simultaneously. A user defined function (UDF) programme has been developed to process the movement of the slabs. Model-3 is similar to Model-2 but it includes additionally the skid support systems for the slabs. In the Model-4, convection in the furnace has been included in addition to all the features considered in Model-3. The convection has been modelled with the consideration of flow of hot gas through the inlet of the burners. All the models have been compared for their performance and computational time. Model-1 has been found to be quite economical and accurate. The inclusion of skid supporting system has little effect in the temperature distribution in the slab.     

Dual-phase-lag heat conduction in a furnace wall made of functionally graded materials

In this article, the transient heat transfer in a furnace wall, which is made of functionally graded materials (FGMs), is investigated based on the hyperbolic-type dual-phase-lag (DPL) heat conduction model to consider the microstructural interactions in the fast transient process of heat conduction. All material properties of the furnace wall are assumed to vary following a power-law form along the radial direction with arbitrary non-homogeneity indices. For simplicity, the values of the phase lags are taken constant. A semi-analytical solution for the temperature field is obtained in the Laplace domain. The transformed temperature solution is inverted to the physical quantity by using numerical inversion of the Laplace transform. A comparison between the hyperbolic-type DPL model and thermal wave model in the temperature responses of the furnace wall is made. Effects of different phase-lag values on the behavior of heat transfer are also investigated.     

Semi-analytical solutions for the dual phase lag heat conduction in multilayered media

A semi-analytical solution procedure for transient heat transfer in composite mediums consisting of multi-layers within the framework of the dual phase lag model is presented. The procedure is then used to derive solutions for the temperature-, temperature gradient-, and heat flux distributions in a two-layer composite planar slab, a bi-layered solid-cylinder and sphere. The solutions obtained are applicable to the classical Fourier heat diffusion, hyperbolic heat conduction, phonon–electron interaction, and phonon scattering models with perfect or imperfect contact and with layers of different materials. The interfacial contact resistance, the heat flux and temperature gradient phase lags, thermal diffusivities and conductivities, initial temperatures of the composite medium and a general time-dependent boundary heat flux enter the solutions as parameters, allowing the solutions obtained to be applicable to a wide range of arrangements including perfect and imperfect contact. Analysis of thermal wave propagation, transmission and reflection in planar, cylindrical and spherical geometries with imperfect interfaces are presented, and geometrical—as well as the temperature gradient phase lag—effects on the thermal lagging behavior in different layered media are discussed.     

Horizontal concrete slabs as passive solar collectors

Natural convection, radiation and conduction heat transfer on horizontal concrete slab systems is experimentally studied. The system was a 0.78 m long, 0.40 m wide and 0.10 m thick concrete slab. A heat source was used to impose a constant heat flux which could be varied from about 200 to 700 W/m². Temperatures at various points and heat flux by natural convection at the horizontal surface were measured. Using various assumptions, the system was also analyzed theoretically. It is found that the mathematical model to study the transient heat transfer in the slab system was satisfactory to predict its thermal behavior in various conditions. An empirical correlation for natural convection on the horizontal concrete slab was derived and used in the analysis. The results showed that the incident energy on the concrete slab was not a parameter affecting strongly the absorbed heat by the slab, the radiation heat losses made about 60% while those by natural convection 40%, and the major energy storage–restitution was taking place during the first 3 to 4 h. The optimization of energy storage density and the thermal performance was also discussed and various parameters affecting them were defined.     

Coupled thermal and hydrodynamic models of flat micro heat pipes for the cooling of multiple electronic components

An analytical solution for both the liquid and vapour flows inside a flat micro heat pipe (MHP) coupled to an analytical solution for the temperature inside the MHP wall is presented. The maximum heat transfer capability of a flat MHP, on which several heat sources and heat sinks are located, is calculated. The capillary structure inside the MHP is modeled by considering a porous medium, which allows to take into account capillary structures such as meshes or sintered powder wicks. The thermal model is able to calculate the part of heat flux transferred only by heat conduction in the MHP wall from the heat transferred by change of phase.     

An experimental and theoretical study of the nonlinear heat conduction in dry porous media

Heat transfer in porous media is important in various engineering fields, including contaminated soil incineration. Most heat transfer models are theoretical in nature. Consequently, this study was undertaken to perform both theoretical and experimental studies of heat transfer in two different sand matrices. A mathematical model based on Fourier's law of heat conduction for a one‐dimensional system with the variable thermal conductivity was developed. The experimental part included heating sand samples placed in a small reactor within an infrared furnace. The transient temperature profiles of the sand layers were monitored by thermocouples. The bulk thermal conductivity was estimated to be linearly proportional to the temperature. The temperature profiles predicted by the model of heat conduction with a variable bulk thermal conductivity was compared by the observed temperatures in Quartz and Sea sands matrices up to 1300 K. Copyright © 1999 John Wiley & Sons, Ltd.     

A simple solution for transient heat transfer during heating of a slab product

A theoretical and experimental study of transient heat transfer in the heating of an individual slab product, subjected to an air flow at a temperature of 50°C and a velocity of 1 m/s, is presented. Experimental temperature measurements at the centre of the slab product were made, and the experimental heat-transfer rates were derived from the temperature data. A simplified analytical technique, using the boundary condition of the third kind in transient heat transfer, was used to predict the theoretical heat transfer rates for two cases, the first considering that the heat transfer coefficient is a convective heat transfer coefficient, and the second considering that heat transfer coefficient is the sum of the convective and radiative heat transfer coefficients. The experimental heat-transfer rates were compared with the predictions for two cases, and a very good agreement was obtained.     

Inverse Radiation Problem of Boundary Incident Radiation Heat Flux in Semitransparent Planar Slab with Semitransparent Boundaries

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High heat flux in glass

Transient conduction and nongray radiation in an absorbing, emitting and isotropically scattering glass slab with reflective boundaries is solved by the application of an implicit finite difference/discrete ordinates method. Solutions are presented for predicting temperatures and heat fluxes in a one dimensional slab geometry being externally heated by a constant high heat flux at one boundary. The boundaries are assumed to be optically smooth with Fresnel reflection and have a uniform relative real refractive index. The window material is intensely heated to levels that exceed the softening/melting point temperature in order to categorized the transient onset of these conditions without phase or volume change. The numerical results reveal the effects of the system parameters on the heat transfer characteristics. Comparisons are made between a gray model and a 16 band, evenly spaced, nongray model in the 1 to 4 μm wavelength region. Precise knowledge of the thermal behavior of the windows is essential to the development of glass and ceramic materials in diathermal and optical applications.     

Dual-Phase-Lag Heat Conduction in Finite Slabs With Arbitrary Time-Dependent Boundary Heat Flux

Applying a constant or transient heat flux on a plane slab is a common technique in microelectronics technology and material processing, including laser patterning, micromachining, and laser surface treatment processes. Although Fourier's law is typically very precise for evaluating temperatures in solids, a number of experimental observations suggest the existence of non-Fourier transient conduction in these applications. Since the dual-phase-lag (DPL) model of heat conduction can be compatible with the hypothesis of local equilibrium thermodynamics (as shown here), the effects of temperature gradient relaxation time on the non-Fourier hyperbolic conduction in a finite slab subjected to an arbitrary time-dependent surface heat flux is examined by this model. The combination of diffusion- and wave-like features in heat conduction process is properly monitored by the DPL model for two types of heat flow regimes, namely, gradient precedence and flux precedence. The results indicate considerable deviations between the predictions of these regimes.     

Extending the validity of lumped capacitance method for large Biot number in thermal storage application

In a typical thermal energy storage system, a heat transfer fluid is usually used to deposit/extract heat when it flows through a packed bed of solid thermal storage material. A one-dimensional model of the heat transfer and energy storage/extraction for a packed-bed thermal storage system has been developed previously by the authors. The model treats the transient heat conduction in the thermal storage material by using the lumped capacitance method, which is not valid when the Biot number is large. The current work presents an effective heat transfer coefficient between the solid and fluid for large Biot numbers. With the corrected heat transfer coefficient, the lumped capacitance method can be applied to model the thermal storage in a wide range of Biot numbers. Four typical structures for the solid thermal storage material are considered. Formulas for the effective heat transfer coefficient (and effective Biot number) are presented. To verify the prediction by the lumped capacitance method using the effective heat transfer coefficient, we compare the results to the corresponding analytical solutions. The results are in very good agreement. The effective heat transfer coefficient extended the validity of the lumped capacitance method to large Biot numbers, which is of significance to the analysis of thermal energy storage systems.     

Analytical solution of the parabolic and hyperbolic heat transfer equations with constant and transient heat flux conditions on skin tissue

In this article, the parabolic (Pennes bioheat equation) and hyperbolic (thermal wave) bioheat transfer models for constant, periodic and pulse train heat flux boundary conditions are solved analytically by applying the Laplace transform method for skin as a semi-infinite and finite domain. The bioheat transfer analysis with transient heat flux on skin tissue has only been studied by Pennes equation for a semi-infinite domain. For modeling heat transfer in short duration of an initial transient, or when the propagation speed of the thermal wave is finite, there are major differences between the results of parabolic and hyperbolic heat transfer equations. The non-Fourier bioheat transfer equation describes the thermal behavior in the biological tissues better than Fourier equation. The outcome of transient heat flux condition shows that by penetrating into the depths beneath the skin subjected to heat, the amplitude of temperature response decreases significantly. The blood perfusion rate can be predicted using the phase shift between the surface temperature and transient surface heat flux. The thermal damage of the skin is studied by applying both the parabolic and hyperbolic bioheat transfer equations.     

Frost behavior on a fin considering the heat conduction of heat exchanger fins

A mathematical model is proposed for predicting frost behavior on a heat exchanger fin under frosting conditions, taking into account fin heat conduction. The change in the three-dimensional airside airflow caused by frost growth is reflected in this model. The numerical estimates of frost thickness are consistent with experimental data, with an error of less than 10%. Due to fin heat conduction, frost thickness decreases exponentially toward the fin tip, while considerable frost growth occurs near the fin base. When a constant fin surface temperature is assumed, the predicted frost thickness was larger by more than 200% at maximum, and the heat flux by more than 10% on average, compared to results obtained with fin heat conduction taken into account. Therefore, fin heat conduction could be an essential factor in accurately predicting frost behavior. To improve prediction accuracy under the assumption of constant fin surface temperature, the equivalent temperature (for predicting frost behavior) is defined to be the temperature at which the heat transfer rate neglecting fin heat conduction is the same as the heat transfer rate with fin heat conduction taken into consideration. Finally, a correlation for predicting the equivalent temperature is suggested.