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.     

Effect of boundary phonon scattering on Dual-Phase-Lag model to simulate micro- and nano-scale heat conduction

In this paper, one-dimensional heat conduction within a thin slab for Knudsen numbers more than 0.1 is implemented using the Dual-Phase-Lag (DPL) model including phonon scattering boundary condition. The Dual-Phase-Lag equation is solved with a stable and convergent finite difference scheme. Also the Laplace transformation technique is employed to solve DPL equation analytically. The results show that in the smaller values of Knudsen number, the results of the DPL model lay very close to the solution of the Boltzmann equation. Also, it is shown that moving towards the steady state, the DPL model reduces to the Cattaneo and Vernotte (CV) model and has results more accurate than the Ballistic-Diffusive Equations (BDE). It is also shown that the temperature distribution is closer to the results of Boltzmann equation relative to the heat flux distribution. Due to the simplicity of derivation of the DPL model formulation and its possibility for developing to higher dimensions, using the DPL model with new boundary condition is recommended to simulate nano- and micro-scale heat conduction. To investigate the accuracy of the DPL model, its results are compared with the results obtained from BDE model, and Boltzmann equation.     

Rewetting of Hot Vertical Surfaces by a Falling Liquid Film

In this work, a transient heat conduction model is developed for rewetting a hot wall surface by a falling liquid film. In the model, the heat conduction in the rewetted wall is assumed to be two‐dimensional. Convection heat transfer from the hot surface to rewetting fluid is considered negligible in the dry surface region ahead of the wet front. The numerical solution indicates that the rewetting process is mainly controlled by two‐dimensional heat conduction in the rewetted wall, even for the walls of low Biot number, especially at low initial temperatures. The effects of Biot number and initial wall temperature on the rewetting velocity are investigated. Comparison of the results with previous studies is presented.     

Thermodynamic stability analysis of non-Fourier model-based bio-heat transfer in laser-irradiated cylindrical-shaped multilayered biological tissue

The present research focuses on examining the thermic response of living tissue in the form of a triple-layered cylindrical structure when subjected to laser light and the compatibility analysis of non-Fourier heat transfer with thermodynamics second law. The temperature field in the triple-layered cylindrical living tissue subjected to laser light is determined by numerically solving the transient radiative transfer equation in conjunction with the dual phase lag (DPL) based bioheat equation. Once the temperature field is known, the equilibrium and nonequilibrium entropy production rate (EPR) is calculated based on the hypothesis of classical irreversible thermodynamics and extended irreversible thermodynamics, respectively. The present results are verified against the data from the literature and found a good match between them. A comparative analysis of the Fourier and non-Fourier models is accomplished. The equilibrium and nonequilibrium EPR values for the Fourier model are found to be positive. While the equilibrium EPR is negative for non-Fourier heat conduction and does not satisfy the thermodynamics second law, nonequilibrium EPR is always a positive value for Fourier, DPL, and hyperbolic models and satisfies thermodynamics second law. It has been investigated how thermal relaxation times affect the temperature field and EPRs in tissue are subjected to laser light.     

Notable physical anomalies manifested in non-Fourier heat conduction under the dual-phase-lag model

In this study, a number of notable physical anomalies concerning non-Fourier heat conduction under the dual-phase-lag (DPL) model are observed and investigated. It is found that, during the transient heat transfer process, the over-diffusion mode predicts a “hyper-active” to “under-active” transition in thermal behavior. The main cause behind it lies in the time-varying effect of τ_T (the phase lag of the temperature gradient) on the thermal response. Also, change of polarity in reflected thermal waves can be observed when a constant-temperature boundary is involved, which hints that a heating process may be followed by a spontaneous cooling effect. A fairly strong connection is present between the τ_T-induced dispersive effect and an unusual thermal accumulation phenomenon in an on–off periodic heating process. Furthermore, a paradox involving a moving medium is detected in the DPL model, which can be solved by replacing the temporal partial derivatives in the DPL equation with the material derivatives. During the process of analysis, a high-order characteristics-based TVD scheme is relied on to provide accurate and reliable numerical simulations to the DPL heat conduction equation under various initial-boundary conditions.     

Numerical Simulation of Thermal Damage to Living Biological Tissues Induced by Laser Irradiation Based on a Generalized Dual Phase Lag Model

A generalized dual phase lag (DPL) bioheat model based on the nonequilibrium heat transfer in living biological tissues is applied to investigate thermal damage induced by laser irradiation. Comparisons of the temperature responses and thermal damages between the generalized and classical DPL bioheat model, derived from the constitutive DPL model and Pennes bioheat equation, are carried out in this study. It is shown that the generalized DPL model could predict significantly different temperature and thermal damage from the classical DPL model and Pennes bioheat conduction model. The generalized DPL equation can reduce to the classical Pennes heat conduction equation only when the phase lag times of temperature gradient (τ _T ) and heat flux vector (τ _q ) are both zero. The effects of laser parameters such as laser exposure time, laser irradiance, and coupling factor on the thermal damage are also studied.     

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.     

Transient hygrothermoelastic response in a porous cylinder subjected to ramp-type heat-moisture loading

Abstract

The classical theory of heat conduction (Fourier theory) predicts an infinite speed for thermal disturbance propagation, which is physically unrealistic. By extending the classical Fourier heat conduction and Fick’s diffusion, this article develops hyperbolic diffusion/heat conduction laws with phase lags of heat/moisture flux to simulate coupled heat-moisture diffusion-propagation behavior with the Defour and Soret effects. A porous cylinder subjected to a ramp-type heating and humidifying at the surface is studied. The Laplace transform is used to obtain a closed-form solution of the temperature, moisture, displacements and stresses in the cylinder. Numerical results are calculated via the inversion of the Laplace transform. Obtained results show that the thermal/moisture relaxation time or phase lag plays a significant role in affecting transient hygrothermoelastic field. For a non-vanishing phase lag, non-Fourier and non-Fickian effects exist and hygrothermal waves have finite propagation speeds. The influences of the phase lag of heat/moisture flux and ramp-type time parameter on the transient response of hygrothermoelastic field are presented graphically. A comparison of the numerical results based on the classical model and the present one is made. The non-Fourier heat conduction and non-Fickian diffusion can effectively avoid the shortcomings induced by the classical Fourier and Fick laws.     

Numerical Technique for Resolving the Dual Phase Lag Heat Conduction in Thin Film Metal

Abstract

In this article, a three-time level finite difference scheme is used to resolve the dual phase lag’s (DPL) heat conduction in a micro scale gold film subjected to spontaneous temperature boundary conditions without knowing the heat flux. Finite difference analog of DPL equation on applying to the intermediate grid points of the computational domain results into a system of linear, algebraic equations which can be solved using Thomas’ algorithm to finally obtain the transient temperature solution distributions in the film. The solution predicted by the DPL model is compared with that obtained by the single-phase Cattaneo–Vernotte’s model. Further, the way in which non-Fourier’s temperature distributions affected by the diffusion due to the increase in Heat Conduction Model numbers agree with the predecessor’s published results. The results by both the models revealed a finite thermal wave speed in the film contrasting the infinite speed of heat propagation as stated by the classical Fourier’s thermal model. Low spatial step and higher order finite difference schemes are recommended for better accurate numerical results of the non-Fourier’s temperature distributions occurring in the very short transient period between the instants of the suddenly applied spatial temperature gradient and the reaching of the steady state conditions.     

Hybrid formulation and solution for transient conjugated conduction–external convection

This work presents a hybrid numerical–analytical solution for transient laminar forced convection over flat plates of non-negligible thickness, subjected to arbitrary time variations of applied wall heat flux at the fluid–solid interface. This conjugated conduction–convection problem is first reformulated through the employment of the coupled integral equations approach (CIEA) to simplify the heat conduction problem on the plate by averaging the related energy equation in the transversal direction. As a result, an improved lumped partial differential formulation for the transversally averaged wall temperature is obtained, while a third kind boundary condition is achieved for the fluid from the heat balance at the solid–fluid interface. From the available steady velocity distributions, a hybrid numerical–analytical solution based on the generalized integral transform technique (GITT), under its partial transformation mode, is then proposed, combined with the method of lines implemented in the Mathematica 5.2 routine NDSolve. The interface heat flux partitions and heat transfer coefficients are readily determined from the wall temperature distributions, as well as the temperature values at any desired point within the fluid. A few test cases for different materials and wall thicknesses are defined to allow for a physical interpretation of the wall participation effect in contrast with the simplified model without conjugation.     

Real-time solution of heat conduction in a finite slab for inverse analysis

Laplace transform is used to solve the problem of heat conduction over a finite slab. The transfer functions relating the temperature and heat flux on the front and back surfaces of the finite slab are developed. Although there are many competing methods for constructing the inverse Laplace transform, we use polynomial approximation of the transfer function. Therefore, transient solutions for given boundary conditions are easily obtained using SIMULINK. This process is much simpler than other numerical solution methods for the heat equation. Most importantly, our method of solution allows us to obtain, in real-time, the front surface temperature and heat flux based on the thermodynamic measurements on the back surface. We also demonstrate the feasibility of reconstructing the front surface temperature when sensor noise is incorporated to the back surface measurements.     

Monitoring of transient 3D temperature distribution and thermal stress in pressure elements based on the wall temperature measurement

Abstract

Two methods for monitoring the thermal stresses in pressure components of thermal power plants are presented. In the first method, the transient temperature distribution in the pressure component is determined by measuring the transient wall temperature at several points located on the outer insulated surface of the component. The transient temperature distribution in the pressure component, including the temperature of the inner surface is determined from the solution of the inverse heat conduction problem (IHCP). In the first method, there is no need to know the temperature of the fluid and the heat transfer coefficient. In the second method, thermal stresses in a pressure component with a complicated shape are computed using the finite element method (FEM) based on experimentally estimated fluid temperature and known heat transfer coefficient. A new thermometer with good dynamic properties has been developed and applied in practice, providing a much more accurate measurement of the temperature of the flowing fluid in comparison with standard thermometers. The heat transfer coefficient on the inner surface of a pressure element can be determined from the empirical relationships available in the literature. A numerical-experimental method of determination of the transient heat transfer coefficient based on the solution of the 3D-inverse heat conduction problem has also been proposed. The heat transfer coefficient on the internal surface of a pressure element is determined based on an experimentally determined local transient temperature distribution on the external surface of the element or the basis of wall temperature measurement at six points located near the internal surface if fluid temperature changes are fast. Examples of determining thermal and pressure stresses in the thick-walled horizontal superheater header and the horizontal header of the steam cooler in a power boiler with the use of real measurement data are presented.     

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.     

Exact analytical solution of unsteady axi-symmetric conductive heat transfer in cylindrical orthotropic composite laminates

This study presents an exact analytical solution of transient heat conduction in cylindrical multilayer composite laminates. This solution is valid for the most generalized linear boundary conditions consisting of the conduction, convection and radiation heat transfer. Here, it is supposed that the fibers are winded around the cylinder and their direction can be changed in each lamina. Laplace transformation is applied to change the domain of the solutions from time into the frequency. An appropriate Fourier transformation has been derived using the Sturm–Liouville theorem. Here, a set of equations for Fourier coefficients are obtained based on the boundary conditions both inside and outside the cylinder, and the continuity of temperature and heat flux at boundaries between adjacent layers. The exact solution of this set of equations is obtained using Thomas algorithm and Fourier coefficients are expressed by recessive relations. Due to the difficulty of applying the inverse Laplace transformation, the Meromorphic function method is utilized to find the transient temperature distribution in laminate. Some industrial examples are presented to investigate the ability of current solution for solving the wide range of applied steady and unsteady problems.     

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.     

Temporal analysis of dual phase‐lag double‐diffusive MHD flow within a porous microchannel with chemical reaction

A novel investigation is carried out to capture the transient effects of a dual phase‐lag (DPL) model for combined heat and mass transfer magnetohydrodynamic (MHD) flow within a porous microchannel in the presence of Dufour effects and homogenous first‐order chemical reaction. The governing equations for the fluid flow problem are solved using the Laplace transform method, which is a powerful technique for solving partial differential equations. Its inversion is done by using the INVLAP subroutine of MATLAB. The numerical values of fluid velocity, fluid temperature, and species concentration are demonstrated graphically and those of skin friction, heat transfer rate, and mass transfer rate are presented through tables. It is for the first time that the actual time gap between the DPL model, the Cattaneo‐Vernotte model, and the classical Fourier?s model has been deciphered and the results unique to the DPL model are presented. We observe a clear difference between the DPL and the other two models at a dimensionless time , which gradually diminishes as time progresses, and all models coincide together at , that is, where a steady state temperature is reached. An important contribution of this study lies in discovering the time‐bound effects of the phase‐lag parameters of the DPL model on fluid temperature, species concentration, and fluid velocity and support them by physical justification. A similar discussion is provided for all other flow parameters. The results conveyed through this study would undoubtedly help researchers to advance the design of mechanical systems in microdevices involving MHD flow in porous media.     

Investigation for the dual phase lag behavior of bio-heat transfer

The success of hyperthermia treatment depends on the precise prediction and control of temperature distribution in the tissue. It was absolutely a necessity for hyperthermia treatment planning to understand the heat transport occurring in biological tissue. The tissue is highly non-homogenous, and non-Fourier thermal behavior in biological tissue has been experimentally observed. The dual phase lag model of heat conduction has been used to interpret the non-Fourier thermal behavior. This work attempts to be an extension study of Antaki [12] and explore whether the DPL thermal behavior exists in tissue. The inverse non-Fourier bio-heat transfer problem in the bi-layer spherical geometry is analyzed. In order to further address whether the dual phase lag model of bio-heat transfer merits additional study, the comparisons of the history of temperature increase among the present calculated results, the calculated values from the classical bio-heat transfer equation, and the experimental data are made for various measurement locations.     

用渐近分析法研究蜂窝蓄热体温度分布

1前言在冶金、机械和石化工业锻造炉、均热炉、连续加热炉、热处理炉、钢包烘烤炉、辐射管和熔铝炉上应用的高温空气燃烧(High Temperature Air Combustion,Hi-TAC)[1],具有热效率高、低NOx排放和燃烧放热均匀等特点。大多数的HiTAC应用了蜂窝蓄热系统[2]。温度变化和温度效率(     

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.     

Unsteady conjugate laminar heat transfer of a rotating non-uniformly heated disk: Application to the transient experimental technique

Presented in the paper are the results of an investigation of 2D heat conduction effects on the transient heat transfer of a rotating disk heated up to a non-uniform initial temperature and suddenly subjected to unsteady cooling by still air. A self-similar solution of the transient laminar convective heat transfer confirmed that the heat transfer coefficient rapidly becomes time-independent and equal to its value at steady-state conditions. An analytical solution of the unsteady two-dimensional heat conduction inside a disk made of Plexiglas^® confirmed that the known infinite-slab approach can still be used as a transient technique for determining heat transfer coefficients. Use of the regular heat transfer regime theory for the same purpose can be recommended only for the cases with the moderate initial temperature non-uniformity.