METHODOLOGY TO GENERATE ACCURATE SOLUTIONS FOR VERIFICATION IN TRANSIENT THREE-DIMENSIONAL HEAT CONDUCTION

This article describes the development of accurate solutions for transient three-dimensional conductive heat transfer in Cartesian coordinates for a parallelepiped which is homogeneous and has constant thermal properties. The intended use of these solutions is for verification of numerical computer programs which are used for solving transient heat conduction problems. Verification is a process to ensure that a computer code is free of errors and accurately solves the mathematical equations. The exact solutions presented in this article can have any combination of boundary conditions of specified temperature, prescribed heat flux, or imposed convection coefficient and ambient temperature on the surfaces of the parallelepiped. Additionally, spatially uniform nonzero initial condition and internal energy generation are treated. The methodology to obtain the analytical solutions and sample calculations are presented.     

Temperature Distribution in Flux Channels with Discrete Contact Boundary Conditions

An analytical approach for the thermal behavior of two-dimensional rectangular flux channels with arbitrary boundary conditions on the source plane is presented. The boundary condition along the source plane can be a combination of the first kind boundary condition (Dirichlet or prescribed temperature) and the second kind boundary condition (Neumann or prescribed heat flux). To model the boundary conditions along the source plane, the method of least squares is used. The proposed solution is in the form of Fourier series expansion and can be applied to both symmetrical and non-symmetrical channels. This method is more general than other approaches and there is no need to use equivalent heat flux distributions to model isothermal heat sources. The general approach for obtaining the multidimensional temperature profile in flux channels and the advantages of the least-square method is discussed. The proposed solution can be used to calculate the temperature at any specified point in the flux channel. Two case studies are presented. The first case study is a flux channel with five discretely specified contact temperatures along the source plane. The second case study has both of the first kind and second kind boundary conditions on the source plane. The analytical results for both systems are compared with finite element method using a commercial software package. It is shown that the proposed approach can precisely model the temperature profile over the flux channel.     

Solution of an initial-boundary value problem for heat conduction in a parallelepiped by time partitioning

An initial-boundary value problem for transient heat conduction in a rectangular parallelepiped is studied. Solutions for the temperature and heat flux are represented as integrals involving the Green's function (GF), the initial and boundary data, and volumetric energy generation. Use of the usual GF obtained by separation of variables leads to slowly convergent series. To circumvent this difficulty, the dummy time interval of integration is partitioned into a short time and a long time subintervals where the GFs are approximated by their small and large time representations. This paper deals with the analysis and implementation of this time partitioning method.     

Numerical solution of the three-dimensional diffusion equation in leaching processes

A mathematical model is presented to account for the leaching of a solute during the heating of a parallelepiped solid with the diffusion coefficient being a function of temperature. The alternating direction implicit method developed by Douglas and Brian was chosen to solve the resulting equations. Details of the finite difference scheme are given with special reference to the variable diffusion coefficient and the third kind boundary condition. The model was applied to predict the retention of water soluble vitamins during the blanching of potato parallelepipeds.     

A generalized analysis for liquid-fuel vaporization and burning

A generalized method is presented for vaporization and combustion of multiple-droplet arrays, liquid films, pools, and streams. Conditions are explained for the existence of a mass flux potential function that is independent of fuel type and scalar boundary conditions and satisfies the three-dimensional Laplace’s equation. Gas-phase properties, composition, and flame location are functions only of the potential function, specified scalar boundary conditions, and fuel type. Variable properties are considered. Flame stand-off distances for liquid interfaces near wet-bulb temperatures are predicted more accurately with variable properties. Flame location and transport properties are found for decane, heptane, and methanol fuels, with different ambient conditions. The analysis also applies to vaporization without combustion and to combustion with transient liquid-phase heating.     

Determination of an unknown radiation term in an inverse problem of heat equation

A linear conduction equation with radiative boundary condition is considered, in which the function representing radiative flux is unknown, and is to be determined from overspecified data. Exact and approximate explicit solutions are presented for the temperature and radiation term. Some uniqueness and stability results are presented. Finally some numerical results will be given.     

Green's function solution of temperature field for flow in porous passages

Recently, the heat transfer in porous passages has received attention from many investigators. The Green's function solution method can serve as a powerful tool to accomplish this task of providing solutions to this type of problems with or without the effect of axial conduction. The study of heat transfer with emphasis on frictional heating, in the absence of axial conduction, is the subject of this presentation. As a simple example, consideration is given to the numerical study of the heat transfer in flow between two impermeable parallel plates. The individual effects of temperature change at the walls, frictional heating, and the combined effects are examined. The data shows that the combined effects can produce removable singularities under certain boundary conditions. To avoid the occurrence of singularities in these types of applications, certain heat transfer parameters are presented in different but basic forms.     

Heat transfer in a reaction–diffusion system with a moving heat source

A general formulation is presented for a moving boundary problem in which heat is generated at the boundary due to an exothermic reaction involving a species which diffuses into a dispersed phase from an external medium of finite volume. The speed of the moving boundary is prescribed based on the solution of the mass diffusion problem and an analysis is presented of the thermal dynamics of the system. The set of equations describing heat transport leads to a Green’s function type problem with time dependent boundary conditions and the Galerkin finite element method is employed to develop a numerical solution. Transformations are introduced to freeze the moving boundary and partition the domain for ease of computation, and an iterative scheme is defined to satisfy the heat flux jump boundary condition and match the temperature field across the moving boundary. The numerical results are used to set the limits of applicability of an analytical perturbation solution. Essential aspects of thermal dynamics in the system are described and parametric regions resulting in a local temperature hot spot are delineated. Computed contour plots describing thermal evolution are presented for different combinations of parameter values. These may be of utility in the prediction of thermal development, for control and avoidance of hot spot formation, and in physical parameter estimation.     

Optimal Heating Control of Thermosensitive Rectangular Domain under Restrictions on Stresses in a Plastic Zone

A numerical algorithm to solve the problem on optimal heating control for a long isotropic homogeneous rectangular parallelepiped under plane strain has been proposed. The control (the surrounding temperature at one of the boundary surfaces of a parallelepiped) which in a minimal time, carries the body from the initial thermal state to the final one characterized by the given mean-integral temperature has been determined. In addition, restrictions both to the control function and to the maximal tangential stress intensity have been considered. The case of elastoplastic deformation of the material has been studied in the framework of the theory of the body element deformation along small curvature paths.     

Average effects of forced convection over a flat plate with an unheated starting length

Parallel flow over a flat plate with an unheated starting length is a good approximation of actual conditions in many engineering applications. A single general closed-form analytical solution for the average heat transfer coefficient for either an isothermal or isoflux boundary condition and for either laminar or turbulent flow is presented. For a smooth surface experiencing a uniform heat flux, the average surface temperature is also of interest. Individual analytical expressions are presented for the average temperature of a flat plate with an isoflux boundary condition for both laminar and turbulent flow. These expressions provide an expeditious means of determining average conditions for convective flow over a partially heated flat plate.     

Conjugated heat transfer in unsteady channel flows

The exact analytical solution of the unsteady impulsive Thermo-Fluid Dynamic field arising in a two-dimensional channel with thick solid walls is presented when the thermal field in the fluid is coupled with the thermal field in the solid (conjugated heat transfer). The cases studied in this paper depend on the boundary conditions imposed on the unwetted sides of the channel walls: assigned temperature and adiabatic condition. Moreover the case of a given heat loss at the unwetted wall is also considered in an appendix. The temperature and heat flux at the solid–fluid interface are analyzed as function of time and of the nondimensional parameters governing the problem.     

Improving the accuracy of heat balance integral methods applied to thermal problems with time dependent boundary conditions

In this paper the two main drawbacks of the heat balance integral methods are examined. Firstly we investigate the choice of approximating function. For a standard polynomial form it is shown that combining the heat balance and refined integral methods to determine the power of the highest order term will either lead to the same, or more often, greatly improved accuracy on standard methods. Secondly we examine thermal problems with a time-dependent boundary condition. In doing so we develop a logarithmic approximating function. This new function allows us to model moving peaks in the temperature profile, a feature that previous heat balance methods cannot capture. If the boundary temperature varies so that at some time t > 0 it equals the far-field temperature, then standard methods predict that the temperature is everywhere at this constant value. The new method predicts the correct behaviour. It is also shown that this function provides even more accurate results, when coupled with the new CIM, than the polynomial profile. Analysis primarily focuses on a specified constant boundary temperature and is then extended to constant flux, Newton cooling and time dependent boundary conditions.     

A numerical study of single-phase convective heat transfer in microtubes for slip flow

The steady-state convective heat transfer for laminar, two-dimensional, incompressible rarefied gas flow in the thermal entrance region of a tube under constant wall temperature, constant wall heat flux, and linear variation of wall temperature boundary conditions are investigated by the finite-volume finite difference scheme with slip flow and temperature jump conditions. Viscous heating is also included, and the solutions are compared with theoretical results where viscous heating has been neglected. For these three boundary conditions for a given Brinkman number, viscous effects are presented in the thermal entrance region along the channel. The effects of Knudsen and Brinkman numbers on Nusselt number are presented in graphical and tabular forms in the thermal entrance region and under fully developed conditions.     

Developing Laminar Flow and Heat Transfer in Annular-Sector Ducts

For annular-sector ducts, steady, laminar, and constant-property forced-convection flow and heat transfer in the entrance region have been analyzed numerically using a general, marching procedure. Two types of thermal boundary conditions have been considered: (1) uniform temperature both axially and peripherally (T boundary condition); (2) uniform axial heat flux with uniform peripherally temperature at any cross section (H1 boundary condition). Numerical analysis has been conducted in the following range of parameters: D_i/D_o = 0.00, 0.25, 0.50, apex angle of the sector 2 alpha = 18 degrees, 20 degrees, 24 degrees, 30 degrees, 40 degrees, and Pr = 0.707. The solutions of the developing Nusselt number and friction factor are presented as functions of nondimensional axial distance. Comparisons are made between the computed results and the analytical or numerical results available in the literature. For all cases compared, satisfactory agreement is obtained.     

A general analytical solution to the equation of transient forced convection with fully developed flow

A general solution to the energy equation under zero wall temperature or zero heat flux boundary condition for the decay of an inlet and initial temperature distribution of an incompressible transient turbulent flow heat transfer between two parallel plates is given. It is shown that these solutions may then be used to obtain solutions due to unit steps in wall temperature or wall heat flux which is sufficient to sort out prescribed wall temperature and prescribed wall heat flux boundary condition. The results are confirmed experimentally by the frequency method. An experimental apparatus has been designed, built and used for this purpose.     

Model for boiling explosion during rapid liquid heating

The process of rapid liquid heating with a linearly increasing boundary temperature condition has been simulated by applying the analytical solution of 1D semi-infinite heat conduction in association with the molecular theory of homogeneous nucleation boiling. A control volume having the size of a characteristic critical cluster at the liquid boundary is considered, and the corresponding energy balance equation is obtained by considering two parallel competing processes that take place inside the control volume, namely, transient external energy deposition and internal energy consumption due to bubble nucleation and subsequent growth. Depending on the instantaneous rate of external energy deposition and boiling heat consumption within the control volume, a particular state is defined as the boiling explosion condition in which bubble generation and growth cause the liquid sensible energy to decrease. The obtained results are presented in terms of the average liquid temperature rise within the control volume, maximum attainable liquid temperature before boiling explosion and the time required to achieve the condition of boiling explosion. The model is applied for the case of water heating at atmospheric pressure with initial and boundary conditions identical to those reported in the literature. Model predictions concerning boiling explosion are found to be in good agreement with the experimental observations. The boiling explosion condition as predicted by the present model is verified by comparing the heat flux across the liquid–vapor interface with the corresponding limit of maximum possible heat flux, q_max,max, at the time of boiling explosion. A comparative study between the actual heat flux and the limit of maximum heat flux, q_max,max, at the time of boiling explosion for different rates of boundary heating indicates that, with much higher boundary heating rates, it is possible to heat the liquid to a much higher temperature before theoretical instantaneous boiling explosion occurs.     

一种处理无限空间自然对流边界条件的新方法

数值求解了常热流边界条件下竖直平板在无限空间里的层流自然对流问题.在10^3＜Ra＜10^7,Pr=0.71范围内得到了竖平板周围的流场和温度场分布以及平板的平均对流换热系数.提出了一种处理外部虚拟边界的新方法,将边界上的已知压力转换为速度,与已有数值解及相似解的比较表明该方法可以用较小的计算区域得到准确的结果.     

Heat Transfer Characteristics of Laminar Flow in Internally Finned Tubes under Various Boundary Conditions

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Heat transfer for Herschel–Bulkley fluids in the entrance region of a rectangular duct

In this study, we present a numerical solution for combined laminar fluid flow and heat transfer of Herschel–Bulkley non-Newtonian fluids in the entrance region of a rectangular duct. The governing equations are solved iteratively by using finite difference method to obtain temperature, bulk temperature, and Nusselt number. Two cases of the thermal boundary conditions are considered; (i) T thermal boundary condition “constant temperature at the wall” and (ii) H2 thermal boundary condition “constant heat flux at the wall”. The results are presented in tables and figures for different parameters for the fluid and the duct geometry.     

Determination of skin temperature distribution and heat flux during simulated fires using Green's functions over finite-length scales

One of the current practices for measuring heat flux during flash fire testing, forest fires, and other industrial cases focuses on the use of semi-infinite models to predict the heat flux during exposure through surface temperature measurements on simulated skin sensors. For short time frames, these models can be shown to have acceptable accuracy. However, when considering longer time exposures at reduced heat fluxes, such as with firefighters in a forest fire, the accuracy of these models could be brought into question. A one-dimensional, finite length scale, transient heat conduction model was developed using a Green's function approach on a rectangular sensor. The model was developed using transient temperature boundary conditions to avoid the use of complicated radiation and convection conditions at each boundary. For comparison, a semi-infinite model utilizing the same boundary condition on the exposed face was solved using both the Laplace transform method and Green's function method. Experimental data was obtained during exposure to a cone calorimeter. All measurements were taken for a minimum duration of 2 min. This temperature data was used to develop appropriate curves for the boundary conditions and validate the analytical models. It was found that the temperature obtained from the one-dimensional transient heat conduction model based on Green's functions agreed well with the experimental results over longer exposure times, and with reduced error when compared with the semi-infinite model. This suggests that modeling the problem on a finite-length scale will produce more accurate or more conservative temperature and heat flux results over extended periods of exposure in high heat load applications.