Two-dimensional Heat Conduction Equation Research Articles

Effect of absorption of solar radiation by water of different optical types on convection and heat transfer just under the air–water interface: the case of zero wind speed

A horizontally infinite water layer cooled from above and absorbing solar radiation is considered. Optical water types are classified according to Jerlov (Marine Optics, Scientific, 1976). The convection and the heat transfer are simulated numerically based on the linear equation of state and the time-dependent two-dimensional heat-conduction and Navier-Stokes equations. To take into account the effect of solar radiation, the conventional heat-conduction law for water with a free surface is corrected by introducing a particular function of heat and solar-radiation fluxes. Analytical expressions for this function are fitted.

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.

Heat transfer process simulation by finite differences for online control of ladle furnaces

A numerical method for solving the two-dimensional transient heat conduction equation is presented to determine temperature profiles of ladle furnaces during the casting of special steels. The method is based on an explicit formulation of the heat transfer process in finite differences. In order to improve the efficiency of the method, two auxiliary algorithms are added, one for correcting finite difference results from coarse mesh calculations and the other to perform time extrapolation of the temperature behaviour between furnace batches. The calculated steel temperatures are found to be in good agreement with experimental data, obtained at a Brazilian thin steel casting facility. Currently, about 80% of the ladle batches at this facility come out within their commercial specifications, against 45% previously observed without the use of such a model. With average processing times of the order of 100 s in an AMD K6II 300 MHz computer, the method is suitable for real time thermal tracking of ladles and steel.

Lattice-free Finite Difference Method ForBackward Heat Conduction Problems

We construct a high order finite difference method in which quadrature points do not need to have a lattice structure. In order to develop our method we show two tools using Fourier transform and Taylor expansion, respectively. On the other hand, the backward heat conduction problem is a typical example of ill-posed problems in the sense that the solution is unstable for errors in data. Our aim is to create a measles method which can be applied to the ill-posed problem. From numerical experiments we confirmed that our method is effective in order to solve the two-dimensional backward heat conduction equation subject to mixed boundary conditions.

An Analysis of Temperature Distributions in Microwave Heating of Foods with Non-Uniform Dielectric Properties

Using flat cylindrical samples having non-uniform dielectric properties, we studied the heat transfer in food products when only the side surfaces of the samples were irradiated with microwaves. The temperature distributions in the food were measured by means of an infrared thermometer in two-dimensional form. Additionally, a transient two-dimensional heat conduction equation was induced and the temperature distributions in the food were determined using a numerical calculation. When the experimental results were compared with the results of the numerical calculations, their tendency of temperature distribution closely agreed, and it was confirmed that the mathematical model is useful in predicting the temperature distributions of food with non-uniform dielectric properties during microwave heating.

Numerical solutions of conjugate heat transfer and thermal stresses in a circular pipe externally heated with non-uniform heat flux

Conjugate heat transfer by forced convection through an externally heated pipe has many important engineering applications. In the present work, the radial and axial heat conductions and thermal stresses in a pipe with uniform or non-uniform wall heat flux of fully developed laminar forced convective conjugate heat transfer have been considered for analysis. The analysis is based on the two-dimensional steady-state heat conduction equation and laminar boundary layer equation for the flowing fluid by using a finite difference scheme. Water has been used as a fluid. Numerical calculations have been performed by using the FLUENT 4.5 and HEATING7 computer codes. The temperature and stress ratio distributions inside the pipe wall, heated from the outer surface by applying uniform and non-uniform heat fluxes, have been presented for two different mean flow velocities. The temperature distributions of the flowing fluid inside the pipe have also been presented for all investigated cases.

A model to predict fire resistance of non‐load bearing wood‐stud walls

AbstractThe author has developed a series of computer models to predict the fire resistance of wood‐framed walls and floors. The models utilize two‐dimensional heat‐conduction equations and thermo‐physical property data to describe heat transfer through the assemblies. The model for wall assemblies WALL2D, the basic version of the wall model, has already been published in Fire and Materials. Recently, WALL2D has been extended to WALL2DN to analyse heat transfer through insulated walls and walls that experience openings at the joints between adjacent sheets of gypsum board. Since gypsum board shrinks at high temperatures, the joints between adjacent sheets of gypsum board open. Hot fire gases, thereby, enter the openings and heat the edge of the gypsum board and wood studs. The new model WALL2DN simulates the joint opening and describes the resultant effect of openings. The model also calculates heat transfer through insulation in the stud cavity and depicts the effect of insulation on the fire resistance of non‐load bearing wall assemblies. Insulation selected in WALL2DN is glass‐fibre insulation, rock‐fibre insulation, polystyrene foam and polyurethane foam. When walls are exposed to fire, the insulation in the cavity shrinks (and/or melts) and an empty space appears at the interface between insulation and gypsum board. The model simulates this shrinking behaviour of insulation in the cavity. Finally, the model was validated by comparing the predicted results to those from full‐scale standard fire‐endurance tests. Copyright © 2003 John Wiley & Sons, Ltd.

Analysis of the heat load of thin metal foil during electron beam scanning

The thermal loading of thin metal foils used as extraction windows in high current electron accelerators is one of the most important processes which should be analysed during the design of such devices. Heat loads at metal surface are also of special interest in many other plasma applications. The distribution of heat loads at the surface of thin metal foil is evaluated from the energy absorption in the foil and forced cooling with air flow. Time dependence of temperature distribution at the foil surface is calculated from the two-dimensional heat conduction equation. The simplified heat transient analysis is also given to determine the influence of scanning frequencies, electron beam size and shape ,shape of current impulses in scanning coils, energy loss in foil, the beam current and air cooling coefficient on time evolution of temperature for an arbitrary fixed point at the foil in first period of electron beam scanning. In most d.c. high current electron accelerators the concentrated electron beam generated at the cathode is accelerated along multi-gap structure and after that is dispersed over the surface of thin metal window (usually titanium (1)) by two sets of perpendicular scanning coils to obtain as uniform distribution of thermal load as possible (2). Thin metal foil (usually 50 µm thick) separates vacuum region of the accelerator from the atmosphere and is supported by a rigid frame usually cooled by water flow . The energy of the electron beam is absorbed in the foil and causes serious heating of the material, so thin foil is cooled from atmosphere side by transverse forced air stream delivered from manifolds located alongside the foil (3). The amount of the energy absorbed in window material from electron beam with given energy and current is fundamental information for design of reliable extraction system. The modification of the energy spectrum and angular distribution of primary electron beam is the result of electron interaction with the matter. In case when scattering foil is 'thin' i.e. small compared to the incident electron range, the multiple Coulomb scattering is observed together with other processes of less importance. The Monte Carlo SHOWME code was used to calculate the absorption, transmission and reflection of 800 keV electrons falling on 50 µm flat titanium foil perpendicular to its surface. The following results were obtained from the analysis (4): TE =0 .95 ± 0.01 (the energy transmission coefficient) RE =0 .005 ± 0.001 (the energy reflection coefficient) φA =0 .042 ± 0.004 (the energy absorption coefficient)

TEHD Behavior of Non-Newtonian Dynamically Loaded Journal Bearings in Mixed Lubrication for Direct Problem

A transient non-Newtonian TEHD analysis for dynamically loaded journal bearings in mixed lubrication is developed for direct problem. A mass conserving cavitation is included and viscoelasticity subsuming shear thinning is characterized by the upper converted Maxwell fluid and the power law fluid models. The Reynolds equation, the transient two-dimensional energy equation in the film, the quasi-steady-state two-dimensional heat conduction equation in the bushing and heat balance in the journal are solved using finite difference formulations. The thermal and elastic deformation equations are solved using the finite element method. The relation between the average contact pressure and the average gap is numerically determined in micro scale. Effects of roughness texture, degree of asperity contact, shear index, relaxation time, thermal and elastic deformations on bearing behavior are investigated.

Two-dimensional quench propagation model for a three-dimensional high-temperature superconducting coil

The results of an experimental and analytical study of quench propagation in a three-dimensional, layer-wound, high-temperature superconducting magnet are presented. The test magnet is wound "dry" with silver-sheathed Bi-2223/Ag tape. It operates in zero background field under quasi-adiabatic conditions at 20 K and above, with a transport current of 50-100 A. Quenching is initiated by a stainless steel heater attached to one turn of the outermost layer. The resulting thermal response was recorded through voltage measurements across regions of the conductor near the heater. Quench events are simulated by a numerical code using the finite-difference method to solve the two-dimensional transient heat conduction equation. The model indicates that thermal contact resistance has a dominant effect on propagation in the transverse direction (across layers).

Thermal modeling of continental subduction and exhumation constrained by heat flow and seismicity in Taiwan

The thermal evolution of crustal exhumation subsequent to the subduction of the continental margin is modeled through numerical solutions to the two-dimensional heat conduction equation. The boundary conditions used in the modeling are basically constrained by available geophysical and geological observations in the Taiwan area. Temperature distributions are calculated at one-million-year intervals over a 5 Ma duration. Although this study presents a tentative estimation based on some reasonable assumptions, the preliminary results not only explain the sharp change in seismicity within the crust but also agree with the general heat flow pattern on the surface of Taiwan. More specifically, both the high heat flow and the aseismic belt beneath the Eastern Central Range, which are probably attributed to higher temperatures within the crust, are modeled by crustal exhumation after the subduction of the continental margin. However, the occurrence of deeper earthquakes east of the Central Range is a result of the lower thermal gradient, which is due to the subduction of the cold continental crust into the uppermost mantle. In summary, preliminary thermal modeling supports the possibility of a tectonic model of active continental subduction and crustal exhumation for the Taiwan Orogen.

A Model for Simulating the Performance of a Pavement Heating System as a Supplemental Heat Rejecter With Closed-Loop Ground-Source Heat Pump Systems

The thermal loads of commercial and institutional buildings are generally cooling-dominated. When such buildings use ground source heat pump systems (GSHP) they reject more heat to the ground-loop heat exchanger than they extract over the annual cycle. In these situations, supplemental heat rejecters can be used to reduce the required size of the ground-loop heat exchanger, thereby reducing the first cost of the system. This paper describes the development, validation, and use of a design and simulation tool for modeling the performance of a hydronic pavement heating system as a supplemental heat rejecter in ground-source heat pump systems. The model uses a finite difference method to solve the transient two-dimensional heat conduction equation and has been formulated for use in component based system simulation. Full-scale experiments were conducted concurrently with the development of the model, the results of which have been used for validation purposes. An example application simulation is presented to demonstrate the use of the model as well as the viability of the use of pavement heating systems as supplemental heat rejecters in GSHP systems. [S0199-6231(00)00404-4]

Two-dimensional energy transfer and plasma formation under laser beam irradiation of a subcritical-density material

Based on two-dimensional numerical simulations, a study was made of the radiation absorption, the energy transfer, and the plasma formation upon the interaction of a laser beam with a homogeneous medium consisting of light elements with a density not exceeding the critical plasma density, which corresponds to the plasma resonance for the wavelength of the driving laser radiation. The calculations were performed by the HEAT-3D code, which involved the solution of a two-dimensional heat conduction equation with an energy source describing the absorption of laser radiation due to inverse bremsstrahlung in the material. The simulations were performed of the interaction of laser beams of radii 10-2—6×10-2 cm, 1014—5×1015 W cm-2 in intensity, and with wavelengths of 1.053 and 0.527 μm with materials composed of light elements with densities of 1–10 mg cm-3. An analysis of the simulations showed that the spatial temperature distribution of the resultant plasma is determined by the anisotropy of energy transfer. In its turn, the degree of anisotropy depends on the relation between the beam radius and the laser radiation absorption length, which is a function of the density and the temperature of the resultant plasma. The results of simulations are compared with the findings of experiments on laser irradiation of targets composed of low-density materials.

Q-conditional symmetry of a nonlinear two-dimensional heat-conduction equation

We investigate theQ-conditional symmetry of a nonlinear two-dimensional heat-conduction equation. By using ansatzes, we obtain reduced equations.

Thermal reliability testing of functionally gradient materials using thermal shock method

We found a solution of an unsteady two-dimensional heat conduction equation in a functionally gradient material (FGM) which is subjected to a double thermal shock, namely, a local heating of a specimen by a power laser beam and cooling of a heated surface by a water-air spray. We developed an analytical method whereby a coating is described as a laminated plate composed of n layers with the constant material properties within a layer. Temperature distribution in a nonhomogeneous laminated plate is obtained in a form of series using the Laplace–Hankel integral transforms. In order to extend the model of a laminated plate to describe FGM where thermal physical characteristics are continuous functions of spatial coordinate, we considered the limiting case of the obtained temperature distribution when the thickness of the layer iΔ i → 0, and the number of layers n→∞. This allowed us to obtain the temperature distribution in an easy-to-use analytical form which can be used for determining thermal stresses in FGM. The dependence of the temperature distribution in FGM on the operating parameters of a double thermal shock method, e.g., a duration of heating, laser beam radius, the rate of a spray cooling, is discussed.

An experimental and theoretical investigation into the heat transfer of a finned water wall tube in a circulating fluidized bed boiler

Heat transfer improvement in a water wall tube with fins was investigated in a circulating fluidized bed (CFB) boiler. Experiments were first conducted in a 6 MWth CFB boiler then a model was developed to analyse and interpolate the results. Temperatures at some discrete points within the wall cross-section of the tube were measured by burying 0.8 mm thermocouples within a tube. Experimental data showed an increase in heat absorption up to 45 per cent. A good agreement between measured and predicted values was noted. The distribution of temperature in the metal wall and of heat flux around the outer wall of a tube with longitudinal and lateral fins was analysed by numerical solution of a two-dimensional heat conduction equation. Effects of bed-to-wall heat transfer coefficient, water-to-tube inside heat transfer coefficient, bed temperature, water temperature and thermal conductivity of the tube material on the heat flux around the water tube are discussed. The present work also examines the influence of the length of the longitudinal fin and the water tube thickness. Heat flux was highest at the tip of the longitudinal fin. It dropped, but increased again near the root of the lateral fin. Copyright © 2000 John Wiley & Sons, Ltd.

Sputtering and thermal effect during ion microbeam patterning of polymeric films

The purpose of this article is to investigate the machinability of polymers and the main parameters involved during ion microbeam processing of polymers. Investigations were made for the case of ion microbeam patterning of polyethyleneterephthalate and nylon-6 using a focused ion beam system with a needle type ion source. The extracted beam diameter was 0.2 μm, the current density on the target was 1.0 A/cm2, and the beam energy was 30 keV. The removal rate, which shows the number of atoms removed when an ion impinges on the surface, was obtained experimentally. By comparing the removal rate and the theoretical sputtering yield, it became clear that effects other than sputtering have a great influence on the machinability of polymers. Two-dimensional unsteady heat conduction equations were computed using the finite difference method in order to estimate the temperature increment within and around the beam irradiated area. The results show that localized heating and a change of chemical bonding state during ion microbeam processing strongly influence the patterning of polymers.

Finite time thermal analysis of ground integrated‐collector‐storage solar water heater with transparent insulation cover

A transparent honeycomb insulated ground integrated-collector-storage system has been investigated for the engineering design and solar thermal performance. The system consists of a network of pipes embedded in a concrete slab whose surface is blackened and covered with transparent insulation materials (TIM) and the bottom is insulated by the ground. Heat may be retrieved by the flow of fluid through the pipe. A simulation model has been developed; it involves the solution of the two-dimensional transient heat conduction equation using an explicit finite-difference scheme. Computational results have been used to determine the effect of such governing parameters as depth as well as pitch of the pipe network and collector material on the thermal performance of the system. The pipe network depth of 10 cm and the TIM cover made of 5 cm compounded honeycomb seem suitable for the proposed system. Solar gain (solar collection efficiency of 30–50% corresponding to collection temperature of 40–60°C) and the diurnal heat storage characteristics of the system are found to be of the right order of magnitude for solar water heating applications. Copyright © 1999 John Wiley & Sons, Ltd.

Finite time thermal analysis of ground integrated-collector-storage solar water heater with transparent insulation cover

A transparent honeycomb insulated ground integrated-collector-storage system has been investigated for the engineering design and solar thermal performance. The system consists of a network of pipes embedded in a concrete slab whose surface is blackened and covered with transparent insulation materials (TIM) and the bottom is insulated by the ground. Heat may be retrieved by the flow of fluid through the pipe. A simulation model has been developed; it involves the solution of the two-dimensional transient heat conduction equation using an explicit finite-difference scheme. Computational results have been used to determine the effect of such governing parameters as depth as well as pitch of the pipe network and collector material on the thermal performance of the system. The pipe network depth of 10 cm and the TIM cover made of 5 cm compounded honeycomb seem suitable for the proposed system. Solar gain (solar collection efficiency of 30–50% corresponding to collection temperature of 40–60°C) and the diurnal heat storage characteristics of the system are found to be of the right order of magnitude for solar water heating applications. Copyright © 1999 John Wiley & Sons, Ltd.

Computation of boiling water on circular finned tubes

This study investigates the maximum heat transfer rate and the critical temperature of circular finned tubes in a boiling liquid. The analysis is conducted numerically by solving a two-dimensional heat conduction equation in a steady state. The dependence between heat flux and the temperature at the inner wall of the tube is quantified. Varying the width, height, and pitch of fins, an optimal finned tube with efficient heat transfer rate can be obtained. In the theoretical approach the successive over relaxation (S.O.R.) accompanied with Multi-grid scheme is used. The local heat transfer rates are assumed to follow power-law-type temperature dependence. The initial guess at very high temperatures or so-called a cooling process is also executed in a same way. The results reveal that increasing either the width or the height of a fin increases the total heat transfer rate.