Unsteady-state Heat Conduction Research Articles

A Mathematical Modeling of Heat Transfer in Continuous Casting Slab

A two-dimensional (2-D) heat transfer and solidification model has been established and applied to calculate the temperature distribution and solid shell thickness profile of a continuous casting slab in a steel plant. A finite difference method was used for the numerical simulation. For thermal analysis, the 2-D slice unsteady-state heat conduction equation with enthalpy convention was used. Meanwhile, non-linear material properties of specific heat and thermal conductivity as well as phase changes during solidification were considered in the model. The temperature distribution and solid shell thickness calculated by mathematical model agree with those predicted by industrial and experimental measurements. The model could also be used to predict the optimum process parameters on casting speed, heat removal rates and the water distribution of secondary cooling zone.

Heating process simulation for a body subjected to an intensive thermal effect on its surface

An approximate method of calculating the nonstationary temperature fields for the case of rapid and intensive heating of bodies is considered. The fast heating of a body is simulated by various boundary conditions at its heated surface. An approximate method is proposed in a one-dimensional formulation to solve the unsteady-state heat conduction equation with these boundary conditions. The applicability of this method in practice is confirmed by comparing the approximate temperature fields with the exact solution. An approximate temperature field is constructed for a particular case with consideration of a temperature dependence of thermal parameters, which allows one to describe the heat localization in a heated zone.

Matrix free meshless method for transient heat conduction problems

In the paper, the element free Galerkin method (EFGM) is applied to calculate two-dimensional unsteady state heat conduction problems. As is well known, most of the meshless methods have higher computational cost than that of finite element method (FEM). In order to overcome this shortcoming especially for transient heat conduction problems, mass lumping procedure is adopted in EFGM, which can decrease the computational cost evidently. Moreover, this technique which can simplify the solution procedure makes the essential boundary conditions enforced directly. The results obtained by EFGM combining mass lumping technique are compared with those obtained by finite element method as well as analytical solutions, which shows that the solutions of the present method are in good agreement with FEM’s and analytical solutions.

A Two-Dimensional Analytical Solution for the Transient Short-Hot-Wire Method

Unlike the conventional transient hot-wire method for measuring thermal conductivity, the transient short-hot-wire method uses only one short thermal-conductivity cell. Until now, this method has depended on numerical solutions of the two-dimensional unsteady heat conduction equation to account for end effects. In order to provide an alternative and to confirm the validity of the numerical solutions, a two-dimensional analytical solution for unsteady-state heat conduction is derived using Laplace and finite Fourier transforms. An isothermal boundary condition is assumed for the end of the cell, where the hot wire connects to the supporting leads. The radial temperature gradient in the wire is neglected. A high-resolution finite-volume numerical solution is found to be in excellent agreement with the present analytical solution.

Analytical solution of heat-conduction problems with a variable initial condition based on determining a temperature perturbation front

With the aid of the integral heat-balance method an analytical solution of the problem of unsteady-state heat conduction has been obtained for an infinite plate with a variable initial condition. To increase the accuracy of solution by the integral method additional boundary conditions are introduced which are determined from the initial differential equation and basic boundary conditions, including those prescribed at the temperature perturbation front.

Determination of thermal conductivity and emissivity of electromagnetically levitated high-temperature droplet based on the periodic laser-heating method: Theory

Recently, a novel method of measuring the thermophysical properties, especially thermal conductivity, of high-temperature molten materials using the electromagnetic levitation technique has been developed by [H. Fukuyama, H. Kobatake, I. Minato, K. Takahashi, T. Tsukada, S. Awaji, Establishment of noncontact AC calorimetry of high-temperature melts using solid platinum spheres as a reference, in: Proceedings of 16th Symposium on Thermophysical Properties, CD-ROM, 2006, p. 937; H. Kobatake, H. Fukuyama, I. Minato, T. Tsukada, S. Awaji, Noncontact AC calorimetry of liquid silicon with suppressing convections in a static magnetic field, in: Proceedings of 16th Symposium on Thermophysical Properties, CD-ROM, 2006, p. 625], where the method was based on periodic laser-heating, and a static magnetic field was superimposed to suppress convection in an electromagnetically levitated droplet. In the present work, the periodic laser-heating method was modeled to estimate the thermal conductivity and emissivity of the electromagnetically levitated droplet using a measured parameter, i.e., the phase lag between the modulated light and the temperature variations detected by a pyrometer, Δ ϕ s, at various frequencies of the modulated light ω. Here, the unsteady-state heat conduction equation for the droplet accompanying radiative heat transfer to the ambient was simplified and transformed to steady-state linear equations. The experimental relation between Δ ϕ s and ω was fitted by the mathematical model proposed here to estimate simultaneously the thermal conductivity and emissivity of molten silicon. Also, the numerical simulations for unsteady thermal field in the electromagnetically levitated droplet which was periodically laser-heated were carried out to demonstrate the validity of the proposed simplified model, and then to investigate the sensitivity of the thermophysical properties to the relation between Δ ϕ s and ω.

The temperature field of an infinite solid containing a cylindrical channel with a thermally thin surface coating

The Laplace method of integral transformation is used to find an analytical solution of the problem of unsteady-state heat conduction for an infinite isotropic solid containing a cylindrical channel with a thermally thin surface coating filled with a high-temperature gas.

The Temperature Uniformity During Air Drying of a Colloidal Liquid Droplet

ABSTRACT The mathematical analysis for drying of individual droplets containing dissolved or suspended solids has been given significant importance due to the increasing popularity of the spray drying operation in the production of various chemicals, ceramics, drugs, food, and dairy powders as well as nanoparticles. The physical and biological qualities of the final products primarily depend on the history experienced by the droplet within the dryer. It is, therefore, desirable to ‘estimate’ the droplet's behavior and various characteristics such as moisture content and temperature profiles accurately. In the literature, several models have been presented to estimate moisture and temperature profiles inside a particle considering various assumptions. One common assumption is the uniform temperature distribution within the droplets being dried. The present article has presented an estimation procedure to evaluate the temperature distribution within a porous skim milk droplet to determine whether the uniform temperature distribution assumption is reasonable. Here, the surface-center temperature differences were estimated by considering the one-dimensional, unsteady-state heat conduction equation for a spherical droplet. Shrinkage of droplets was taken into consideration during modeling. A new concept of the Biot number has also been applied in the current article to assist in the determination of the rate-limiting process.

Temperature dependent electrical conductivities of fruit purees during ohmic heating

Ohmic heating takes its name from Ohm’s law; the food material switched between electrodes has a role of resistance in the circuit. In this study, the apricot and peach purees were heated on a laboratory scale static ohmic heater by applying voltage gradients in the range of 20–70 V/cm. The voltage gradient was statistically significant on the ohmic heating rates for both purees ( P < 0.05). The linear temperature dependent electrical conductivity relations were obtained. Bubbling was observed above 60 °C especially at high voltage gradients. The ohmic heating system performance coefficients were in the range of 0.49–1.00. The unsteady-state heat conduction equation for negligible internal resistance was solved with an ohmic heating generation term by the finite difference technique. The predictions of the mathematical model using obtained electrical conductivity equations were found to be very accurate.

Two-dimensional thermomechanical analysis of continuous casting process

In this work the two-dimensional analysis of continuous casting of low carbon steel was presented. The interaction between moved ingot, copper mould and transport rolls was modeling. The influence of liquid steel ferrostatic pressure and coupled between temperature and deformation fields were taken into consideration.For thermal analysis (with phase change), the two-dimensional unsteady-state heat conduction equation with enthalpy convention was used. For non-linear mechanical analysis the elasto-plastic material model with bilinear isotropic hardening was used. The Von Mises yield criterion was determined by the yield stress and the hardening modulus.For numerical solution the commercial system ANSYS®, which based on finite elements method was applied. The temperature distributions and thickness of solid shell were computed. Additionally, inside the solid shell, the distribution of stresses caused by thermal stresses and influence of mould and transport rolls were calculated.

A novel modeling and simulation technique of photo–thermal interactions between lasers and living biological tissues undergoing multiple changes in phase

Knowledge of heat transfer in biological bodies has many therapeutic applications involving either raising or lowering of temperature, and often requires precise monitoring of the spatial distribution of thermal histories that are produced during a treatment protocol. Extremes of temperature into the freezing and burning ranges are useful in surgical procedures for selective killing and/or removal of target tissues. For example, the primary objective of hyperthermia is to raise the temperature of the diseased tissue to a therapeutic value, typically 41–44°C, and then thermally destroy it. The present paper therefore aims to develop a mathematical model for effective simulation of photo–thermal interactions between laser rays and biological tissues. In particular, damage of biological tissues when subjected to single point laser diathermy is numerically investigated using a unique enthalpy-based approach for modeling multiple phase change, (namely, melting of fat and vaporization of water content of the tissues) and the associated release/absorption of latent heat in conjunction with unsteady state heat conduction mechanisms. The governing equations of bio-heat transfer coupled with initial and boundary conditions are solved using a finite volume approach in conjunction with line by a line tri-diagonal matrix algorithm (TDMA) solver. Temperature responses of tissues subject to laser heating are quantitatively investigated in detail using the present model, and the resultant solutions are expected to be immensely useful in a variety of Bio-thermal practices in medicine and surgery.

A novel modeling and simulation technique of photo?thermal interactions between lasers and living biological tissues undergoing multiple changes in phase

Knowledge of heat transfer in biological bodies has many therapeutic applications involving either raising or lowering of temperature, and often requires precise monitoring of the spatial distribution of thermal histories that are produced during a treatment protocol. Extremes of temperature into the freezing and burning ranges are useful in surgical procedures for selective killing and/or removal of target tissues. For example, the primary objective of hyperthermia is to raise the temperature of the diseased tissue to a therapeutic value, typically 41–44°C, and then thermally destroy it. The present paper therefore aims to develop a mathematical model for effective simulation of photo–thermal interactions between laser rays and biological tissues. In particular, damage of biological tissues when subjected to single point laser diathermy is numerically investigated using a unique enthalpy-based approach for modeling multiple phase change, (namely, melting of fat and vaporization of water content of the tissues) and the associated release/absorption of latent heat in conjunction with unsteady state heat conduction mechanisms. The governing equations of bio-heat transfer coupled with initial and boundary conditions are solved using a finite volume approach in conjunction with line by a line tri-diagonal matrix algorithm (TDMA) solver. Temperature responses of tissues subject to laser heating are quantitatively investigated in detail using the present model, and the resultant solutions are expected to be immensely useful in a variety of Bio-thermal practices in medicine and surgery.

Measurement of thermal environment in Kyoto city and its prediction by CFD simulation

In the summer of 2002, measurements were simultaneously performed to investigate the characteristics of heat flow in urban areas at three locations in Kyoto city: (1) a commercial urban area mixed with low-rise traditional residential buildings that represents the urban area of Kyoto; (2) a university campus area with lots of green zones; and (3) a plaza covered with a concrete slab which was used as a reference point of measurement. Heat flux of boundary layer over the three locations and the surface temperatures of building walls and streets were measured to investigate the urban thermal environment. For the analysis, a new simulation code was developed by combining unsteady state heat conduction of building walls and grounds, radiation heat exchange between them, and airflow by computational fluid dynamics (CFD). By using this code, the thermal environment of the urban areas such as air temperature, humidity, wind velocity, and boundary layer heat flux was predicted and compared with the measured results. It was found that this model could predict the real thermal environment of the urban area. Using this code, the effect of additional green on roofs and grounds can be investigated in order to mitigate urban heat island and to improve urban thermal environment at the street level.

Excelの表計算機能を利用した数値計算プロセスの可視化 : 第5報 2次元円筒座標系で記述された非定常熱伝導問題へのアイコン化セルの拡張(一般セッション 伝熱解析及び理論 I)

Excelの表計算機能を利用した数値計算プロセスの可視化 : 第5報 2次元円筒座標系で記述された非定常熱伝導問題へのアイコン化セルの拡張(一般セッション 伝熱解析及び理論 I)

B211 Excelの表計算機能を利用した数値計算プロセスの可視化

A newly developed visual-oriented numerical method for solving the two-dimensional unsteady-state heat conduction problem which applies the spreadsheet of Excel is explained through a simple physical model. The present unsteady-state numerical calculation has been realized by introducing a dummy sheet with empty cells and a sheet with icon cells, in which difference equations concerning the inner, boundary and singular points and so on are embedded. By organizing and expanding such sorts of icon cells, the proposed method is expected to get applied against much wider range of heat transfer problems in the future.

Effects of Bond Roll Pattern and Temperature on the Microstructure and Properties of Polyethylene Nonwovens

This paper investigates the effects of bond roll pattern and bonding temperature on the microstructure, tensile properties, flexural rigidity, and abrasion of polyethylene (PE) nonwovens. Bond roll variables include percent bond area, bond point concentration, and side-wall angle. Temperature greatly affects nonwoven mechanical properties. Of the various bond roll variables, bond point concentration significantly affects the elongation and flexural rigidity of PE nonwovens, whereas the side-wall angle affects tenacity. Microstructure analysis indicates that PE nonwovens fail by disintegration of the bond area rather than by fibers breaking at the bond perimeter. A one-dimensional, unsteady-state heat conduction model of the area under a bond point confirms that partial melting occurs in the center of thermally bonded areas.

The onset of transient convection in bottom heated porous media

The theory of transient convection in bottom heated porous media under constant heat flux (CHF) condition or fixed surface temperature (FST) condition is advanced and verified by computational fluid dynamics (CFD) simulations. The use of κ ∗ , instead of κ m tends to artificially inflate the value of Rayleigh number by about 30%. A new transient Rayleigh number for unsteady-state heat conduction was defined to predict the onset of transient convection in porous media, which were successfully simulated. The critical transient Rayleigh number from the simulation for CHF was about 29.60, which is close to the theoretical value of 27.1 calculated by Ribando and Torrance in 1976. In the case of FST, the critical transient Ra c was found to be 30.9, which is close to the theoretical value of 32.3. The critical times of onset for simulations were predicted with good accuracy. The prediction of the critical wavelengths of the emerging plumes were fair for the 2D simulations. Any experiment to verify the linear stability analysis for thermal instability must simultaneously concur in the three eigenvalue parameters, namely the Biot number, the critical wavenumber and the corresponding critical Rayleigh number, apart from the physical boundaries. The average maximum transient Nusselt number was found to be 3.41 for CHF and 3.5 for FST respectively.

Hydrolytic Degradation of Nylon 66 Pile Carpet Fibers

Hydrolytic degradation of nylon 66 fibers for sculpturing pile carpets is investigated in this study. Applying an aqueous solution of para-toluenesulfonic acid catalyzes the hydrolysis of nylon 66 and degrades the fibers. The reduced strength of the fibers enables sculpturing of pile carpets. The morphology of the degraded fiber tips is reported as a function of treatment conditions. Insufficient steamer residence time leads to an agglom eration of fiber tips within a tuft rather than fiber degradation, and results in a rougher tactile feel. The first-order kinetic parameters reported in the literature are used in conjunction with a simple, one-dimensional, unsteady-state heat conduction model to represent the sculpturing process. The extents of degradation predicted for the longer treatment times agree with experimentally measured values.

The moving boundary problems from melting and freezing to drying and frying of food

It was possible to describe large number of processes such as melting, solidification, microwave thawing, spray-drying, and freeze-drying by a single theoretical model. All these applications and some others have one thing in common, that is the formation of a moving interface during the process of heat and mass transfer. This interface may be defined by a sharp boundary or by a region, depending on the application considered. The unsteady-state heat conduction equation was found to describe well the transport of heat in these processes. Two different numerical solutions have been developed, one for the sharp interface and the other for materials that undergo phase transformation within a range of temperatures. In spite of the large variation in the applications considered in this paper, the agreement was good between the theoretical predictions and our experiments, and some of the measurements available in the literatures.

Solar pond conception — experimental and theoretical studies

A one dimensional transient mathematical model for predicting the thermal performance of the salt gradient solar pond is developed and presented. In this paper, the natural solar ponds and different artificial solar pond systems found in the literature are introduced. Necessary modifications are made on the experimental stand located in Istanbul Technical University, the experimental stand is introduced and natural phenomena produced in the pond by the different solar pond variations under natural conditions are observed. In the theoretical work based on a one dimensional unsteady state heat conduction model with internal heat generation, the energy and mass balance equations for the upper convective zone, the non-convective zone and the lower convective zone, all of which form the solar pond, are written in terms of differential equations. These equations are solved analytically and numerically. The results obtained from the analysis are compared with the experimental results. The temperature and the concentration profiles are separately presented in the figures.