Transient Conduction Equation Research Articles

Inverse time-dependent perfusion coefficient identification

The identification of the time-dependent perfusion coefficient in the transient bio-heat conduction equation is investigated. In this inverse coefficient identification problem, the additional measurement necessary to render a unique solution may be a heat flux, interior temperature or mass measurement which is taken permanently along the time interval of interest. A numerical approach based on the boundary element and mollification methods is developed. Numerical results are presented and discussed.

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.

가열로 내 슬랩의 온도 예측을 위한 2차원 열전달 모델

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 and transient conduction governing equations in the slab, respectively. The furnace is modeled as radiating medium with spatially varying temperature and constant absorption coefficient. The slab is moved with constant speed through non-firing, charging, preheating, heating, and soaking zones in the furnace. Radiative heat flux which is 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 applied 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 shows that the present heat transfer model works well for the prediction of thermal behavior of the slab in the reheating furnace.

A simple method for estimating transient heat transfer in slab-on-ground floors

The problem of calculating transient heat transfer in concrete floor slabs is complicated due to ground coupling, which can require the numerical solution of two or three-dimensional transient conduction equations. This paper presents a simplified method for calculating transient slab-on-ground heat transfer that can be incorporated within hourly simulation programs. The method assumes that there are two primary one-dimensional paths for heat transfer from a ground-coupled floor slab: (1) one-dimensional heat transfer from the perimeter of the slab to the ambient and (2) one-dimensional heat transfer between the slab interior surface and a portion of the soil beneath the slab. The perimeter heat transfer is assumed to occur at quasi-steady state and is characterized in terms of a perimeter heat loss factor (F p). Transient heat transfer within the slab and ground are modeled using a simple thermal circuit employing three nodes with an adiabatic boundary condition at a specified depth within the soil underneath the slab. Although some simulation models consider this type of two-path model, there appears to be no validation of this approach and there is no guidance for specifying perimeter heat loss factors and underfloor soil depths and node locations for the thermal circuit. In the current paper, results from detailed two-dimensional finite-element models for typical floor constructions and soil properties were used to identify (1) locations for nodes within the slab and soil, (2) correlations for soil depth as a function of soil properties associated with the underfloor adiabatic boundary condition, and (3) correlations for perimeter heat loss factor as a function of soil properties and edge insulation levels for different constructions. Transient heat transfer results from the simple model compared well with results from the finite-element program for different floor constructions, edge insulation, soil properties, locations, and times of year.

Sensitivity study of time delay coefficient of heat transfer formulations for insulated steel members exposed to fire

Heat transfer analysis for an insulated steel member exposed to fire conditions, in general, involves solving a 2D transient conduction equation with well-posed conditions. For design purpose, the current Eurocode 3 provisions adopt a closed-form solution from the SP approach which employs a simplified 1D characteristic heat transfer model. A virtual “time delay” has been incorporated in the analytical solution based on the assumption that the heat capacitance of insulation material is small compared to that of steel section. A sensitivity study is presented in this paper to assess the validity of the time delay coefficient. Results of the study indicate that for insulation with large heat capacitance, the time delay estimation in the SP approach yields significant discrepancy compared against exact solution. An exact formulation of time delay is recommended, which gives accurate results for insulation material with either small or large heat capacitance.

Thermal Performance of Next Generation of Modulated Pump Lasers in Telco Equipment

The present investigation numerically modeled the thermal behavior of the laser diode using the parabolic transient conduction equation. In addition, the current study compared the thermal performance of the continuous wave pump lasers versus the modulated wave pump lasers. This comparison revealed that the temperature of the modulated wave pump laser can approach the temperature of the continuous wave pump laser with the same average power dissipation when the frequency approaches infinity. Finally, the resulting thermal behavior was correlated and expressed in an empirical form, which physically described the thermal performance of the modulating pump laser.

Numerical heat transfer in a cavity with a solar control coating deposited to a vertical semitransparent wall

A transient two-dimensional computational model of combined natural convection, conduction, and radiation in a cavity with an aspect ratio of one, containing air as a laminar and non-participating fluid, is presented. The cavity has two opaque adiabatic horizontal walls, one opaque isothermal vertical wall, and an opposite semitransparent wall, which consists of a 6-mm glass sheet with a solar control coating of SnS–CuxS facing the cavity. The semitransparent wall also exchanges heat by convection and radiation from its external surface to the surroundings and allows solar radiation pass through into the interior of the cavity. The momentum and energy equations in the transient state were solved by finite differences using the alternating direction implicit (ADI) technique. The transient conduction equation and the radiative energy flux boundary conditions are coupled to these equations. The results in this paper are limited to the following conditions: 104≤Gr≤106, an isothermal vertical cold wall of 21°C, outside air temperatures in the range 30°C≤T0≤40°C and incident solar radiation of AM2 (750 W m−2) normal to the semitransparent wall. The model allows calculation of the redistribution of the absorbed component of solar radiation to the inside and outside of the cavity. The influences of the time step and mesh size were considered. Using arguments of energy balance in the cavity, it was found that the percentage difference was less than 4 per cent, showing a possible total numerical error less than this number. For Gr=106 a wave appeared in the upper side of the cavity, suggesting the influence of the boundary walls over the air flow inside the cavity. A Nusselt number correlation as a function of the Rayleigh number is presented. Copyright © 2000 John Wiley & Sons, Ltd.

A Model of Bubble Nucleation on a Micro Line Heater

The bubble nucleation mechanism on a cavity-free micro line heater surface was studied by using the molecular cluster model. A finite difference numerical scheme for the three-dimensional transient conduction equation for the liquid was employed to estimate the superheated volume where homogeneous bubble nucleation could occur due to heat diffusion from the heater to the liquid. Calculation results revealed that bubble formation on the heater is possible when the temperature at the hottest point in the heater is greater than the superheat limit of the liquid by 6°C–12°C, which is in agreement with the experimental results. Also it was found that the classical bubble nucleation theory breaks down near the critical point where the radius of the critical bubble is below 100 nm.

Solution of nonlinear transient conduction equation by a modified boundary integral procedure

We apply a modified boundary integral procedure known as the Green element method (GEM) to solve problems arising from nonlinear conduction in a homogeneous isotropic medium. The method is based on the solution of an integral version of the governing partial differential equation over a priori defined subdomains. In addition, a piecewise interpolation of the dependent variable is associated with every nodal positions of the problem domain. By adopting this finite element concept in addition to its boundary integral formulation, GEM arrives at a slender coefficient matrix which is easy to handle numerically, and at the same time offers a more straightforward methodology for handling nonlinearity with the boundary integral theory. This procedure resumes with the conversion of the governing partial differential equation into its integral representative followed by typical finite element features such as scalar interpolation, and domain discretization. The integral equations are solved for each element, and the results assembled to yield the scalar profile over the problem domain. To establish the numerical accuracy of the numerical method thus developed, examples dealing with 1-D and 2-D scalar conduction in a homogeneous domain are solved and compared with those existing in literature.

Local heat transfer coefficients on the circumference of a tube in a gas fluidized bed

A heated horizontal heat transfer tube was installed 14.8 cm above the distributor plate in a square fluid bed measuring 30.5 × 30.5 cm. Four different Geldart B sized particle beds were used (sand of two different distributions, an abrasive and glass beads) and the bed was fluidized with cold air. The tube was instrumented with surface thermocouples around half of the tube circumference and with differential pressure ports that can be used to infer bubble presence. Numerical execution of the transient conduction equation for the tube allowed the local time-varying heat transfer coefficient to be extracted. Data confirm the presence of the stagnant zone on top of the tube associated with low superficial velocities. Auto-correlation of thermocouple data revealed bubble frequencies and the cross-correlation of thermal and pressure events confirmed the relationship between the bubbles and the heat transfer events. In keeping with the notion of a “Packet renewal” heat transfer model, the average heat transfer coefficient was found to vary in sympathy with the root-mean square amplitude of the transient heat transfer coefficient.

A Single Expression for the Solution of the One-Dimensional Transient Conduction Equation for the Simple Regular-Shaped Solids

For transient conduction in certain simple regular-shaped solids, a dimensionless geometric parameter {omega} can be defined and used for writing the governing equations for each case as a single equation. Analytical solutions for the infinite slab, for a very long cylinder, and for a sphere are well known and can be found in several references including Carslaw and Jaeger (1947), Arpaci (1966), and Ozisik (1993). The solutions given are specific to each geometric shape. This article gives a single solution which can be applied to each of the three basic geometries by a mere substitution of the appropriate value of the geometric parameter {omega}. The Heisler charts (Heisler, 1947) provide solutions for each of the three basic geometries in a graphical form and can be found in several undergraduate heat transfer textbooks. A definite advantage of the single solution is that an appropriate algorithm based on the generalized solution can be written and incorporated into a computer program for transient conduction analysis involving any of the three basic geometries as an alternative to the use of separate equations.

NUMERICAL MODELING OF A NUCLEATE BOILING SURFACE

ABSTRACT A computer program developed to analyze nucleate boiling over a heated surface is described. The model solves the three-dimensional transient conduction equation within the heater. The conduction solution is coupled with closure relationships to mimic the bubble dynamics and the associated heat transfer coefficients. Sample problems are run using a copper surface subject to partial nucleate boiling in saturated water at atmospheric pressure. The results are shown to be in good qualitative agreement with the pertinent experimental observations.

Evaporative cooling due to a gently deposited droplet

The transient thermal behavior of a single water droplet gently deposited on the surface of a semi-infinite solid is investigated. A coupled model that solves simultaneously the transient conduction equation for the solid and the liquid to yield the surface temperature and heat flux distributions as well as the description of the droplet evaporation transient is proposed. The predictions of the evaporation time are compared with experimental data. An additional model is presented which assumes constant heat flux at the liquid-solid interface. This model provides a closed form solution for the solid surface transient temperature distribution.

Saturated pool nucleate boiling mechanisms at high heat fluxes

A new model has been developed where the coupled transient two-dimensional conduction equation is solved for the heater and the liquid macrolayer, while allowing for the time-wise thinning of the macrolayer. The major conclusions are: (1) dominant evaporation occurs at the liquid-vapor-solid contact point (triple point) and is required to match boiling curve behavior quantitatively; (2) evaporation at the stem interface and bubble-macrolayer interface is negligible (except near critical heat flux); and (3) the results are sensitive to closure relationships, especially to the active site-density correlation.

A Second-Law Study on Packed Bed Energy Storage Systems Utilizing Phase-Change Materials

Thermal modeling of packed bed, thermal energy storage systems has traditionally been limited to first-law considerations. The exceptions include a few second-law studies, noted in the Introduction, of sensible heat storage systems and the latent heat storage systems. The cited second-law studies treat the storage and removal processes essentially as “batch” heating and cooling. The approximation effectively ignores the significant temperature gradient, especially in the axial direction, in the storage medium over a substantial portion of both the storage and removal processes. The results presented in this paper are for a more comprehensive model of the packed bed storage system utilizing encapsulated phase-change materials. The fundamental equations for the system are similar to those of Schumann, except that a transient conduction equation is included for intraparticle conduction in each pellet. The equations are solved numerically, and the media temperatures obtained are used for the determination of the exergy (or availability) disposition in complete storage-removal cycles. One major conclusion of the study from both the first-law and second-law perspectives is that the principal advantage in the use of phase-change storage material is the enhanced storage capacity, compared with the same size of packed bed utilizing a sensible heat storage material. Thermodynamically, however, it does not appear that the system employing phase-change storage material will always, or necessarily, be superior to that using a sensible heat-storage material. The latter conclusion is reached only on the basis of the second-law evaluation.

Transient heat transfer in a boiled potato: a study related to food process engineering

This paper examines the transient heat transfer within a potato while it is being cooked in boiling water. For spherically shaped potatoes it has been shown that for most of the heating process the potato behaves as a homogenous conducting material whose temperature rise is satisfactorily described by Fourier's equation for transient conduction. Microscopic examination of the potato structure shows how, when its temperature rises above about 65°, the gelatinization and swelling of the starch granules cause mechanical disintegration of the tuber tissue. This process corresponds to the potato beginning to change from raw to cooked, and it also leads to a deviation from the theory of the ideal solid. It has been shown, for the variety of potato used, that for the center to cook before the outside overcooks the potato should have a maximum diameter of 40 mm.

Finite element modeling of laser-induced hole formation in optical storage media

A finite element model has been used to solve the transient thermal conduction equation for multilayer dye-polymer optical storage media. An accurate computer model is required to determine the laser heat input throughout the absorbing layers due to optical interference effects through the thickness of the dye. Only by this technique can accurate temperature distributions be calculated. From the resulting temperature distributions finite element analysis is used to determine the resulting thermal stresses. These stresses may be sufficiently high to cause intcrlayer failure by fracture.

Comparison of transient thermal conduction in tellurium and organic dye based digital optical storage media

Considerable research and development has been done, in recent years, on combinations of optically absorbing thin films and substrates for digital optical storage media. Recent theoretical models have considered the formation of pits, by laser heating, in such media in two stages: (i) a preinitiation stage before pit formation; (ii) a pit growth stage. To compare the validity of suggested mechanisms for initiation accurate temperature calculations are required. In this paper the full transient thermal conduction equation is solved by finite element methods for both a tellurium and an organic dye layer system and the results compared.

Thermal Aalysis of Quench Propagation In Superconducting Coils

The transient three-dimensional conduction equation with a source term that accounts for the propagating resistive wave front has been solved for a superconducting magnet coil by the control volume finite-difference method with floating nodes. Coupled with this, the code solves the electrical and magnetic circuit equations. Because of the large number of nodes dictated by the geometric structure of the coil, an alternate direction implicit (ADI) technique was used to overcome the computer storage problem. The behavior of the propagating front and the highly nonlinear material properties limit the size of the solution time steps. A variable time step and variable number of nodes have been used to reduce the overall computational time. The results of the analysis are in excellent agreement with experimental data.

Experiment on transient heat transfer in closed narrow channel.

Heat transfer coefficients and transient pressures in closed narrow channels were obtained experimentally, in order to assess the gap heat transfer models in the computer code WTRLGD which were devised to analyze the internal pressure behavior of waterlogged fuel rods. Gap widths of channels are 0.1∼0.5 mm to simulate the gap region of waterlogged fuel rods, and test fluids are water (7∼89.2°C) and Freon-113 (9.2°C). The results show that the heater temperature and the pressure measured in the experiments without the DNB occurrence are simulated fairly well by the calculational model of WTRLGD where the heat transfer in a closed narrow channel is evaluated with one-dimensional transient thermal conduction equation and Jens & Lottes' correlation for nucleate boiling. Consequently, it is also suggested that the above equations are available for evaluation of heat flux from fuel to internal water of waterlogged fuel rods. The film boiling heat transfer coefficient was in the same order of that evaluated by B...