Analytical solutions of transient heat conduction in multilayered slabs and application to thermal analysis of landfills

The study of transient heat conduction in multilayered slabs is widely used in various engineering fields. In this paper, the transient heat conduction in multilayered slabs with general boundary conditions and arbitrary heat generations is analysed. The boundary conditions are general and include various combinations of Dirichlet, Neumann or Robin boundary conditions at either surface. Moreover, arbitrary heat generations in the slabs are taken into account. The solutions are derived by basic methods, including the superposition method, separation variable method and orthogonal expansion method. The simplified double-layered analytical solution is validated by a numerical method and applied to predicting the temporal and spatial distribution of the temperature inside a landfill. It indicates the ability of the proposed analytical solutions for solving the wide range of applied transient heat conduction problems.

THE ASYMPTOTIC BEHAVIOR OF SOLUTION FOR THE NONLINEAR HEAT-CONDUCTION EQUATION AND ITS APPLICATION

In this paper the nonlinear heat-conduction equations rhoc partial derivativew/partial derivativet = partial derivative/partial derivativex (k partial derivativew/partial derivativex) with Dirichlet boundary condition and the nonlinear boundary condition are studied. The asymptotic behavior of the global of solution are analyzed by using Lyapuunov function. As its application, the approximate solutions are constructed.

ON FICTITIOUS DOMAIN METHOD FOR THE NUMERICAL SOLUTION TO HEAT CONDUCTION EQUATION WITH DERIVATIVE BOUNDARY CONDITIONS

This paper investigates the stability and convergence of some knowndifference schemes for the numerical solution to heat conduction equation withderivative boundary conditions by the fictitious domain method.The discrete vari-ables at the false mesh points are firstly eliminated from the difference schemes andthe local truncation errors are then analyzed in detail.The stability and convergenceof the schemes are proved by energy method.An improvement is proposed to obtainbetter schemes over the original ones.Several numerical examples and comparisonswith other schemes are presented.

AN ANALYTICAL SOLUTION OF HEAT CONDUCTION ON A LOCALLY NONUNIFORMLY HEATED SURFACE OF A PLATE WITH FINITE THICKNESS

In engineering and technology, the following problem is often touched upon A certain portion of a plate with finite thickness is heated on its surface, so that the heat flux along the surface is distributed nonuniformly and varying with time. For this kind of heal conduction, there is no available analytical solution given in existing treatises concerning heat conduction. This paper presents an analytical solution of this problem, its calculation results are in good agreement with the experimental data.

Variational Approach to 2D and 3D Heat Conduction Modeling

The paper proposes an approximate solution to the classical (parabolic) multidimensional 2D and 3D heat conduction equation for a 5 × 5 cm aluminium plate and a 5 × 5 × 5 cm aluminum cube. An approximate solution of the generalized (hyperbolic) 2D and 3D equation for the considered plate and cube is also proposed. Approximate solutions were obtained by applying calculus of variations and Euler-Lagrange equations. In order to verify the correctness of the proposed approximate solutions, they were compared with the exact solutions of parabolic and hyperbolic equations. The paper also presents the research on the influence of time parameters τ as well as the relaxation times τ ∗ to the variation of the profile of the temperature field for the considered aluminum plate and cube.

Heat Conduction Analytical Solutions to be Applied in Boundary Conditions Obtained from Discrete Data

Analytical solutions have varied uses. One is to provide solutions that can be used in verification of numerical methods. Another is to provide relatively simple forms of exact solutions that can be used in estimating parameters, thus, it is possible to reduce computation time in comparison with numerical methods. In this paper, an alternative procedure is presented. Here is used a hybrid solution based on Green＇s function and real characteristics （discrete data） of the boundary conditions.

ANALYTICAL SOLUTION OF AXIAL SYMMETRICAL WALL TEMPERATURE DISTRIBUTION FOR HOLLOW CYLINDER

Using the variable transformation method,the formulae of the axial symmetrical wall temperature distribution during steady heat conduction of a hollow cylinder are derived in this paper.The wall temperature distribution and the wall heat flux distribution in both axial and radial direction can be calculated by the temperature distribution of the liquid medium both inside and outside the cylinder with temperature changing in axial direction.The calculation results are almost consistent with the experience results.The applicative condition of the formulae in this paper consists with most of practice.They can be applied to the engineering calculation of the steady heat conduction.The calculation is simple and accurate.

Time fractional dual-phase-lag heat conduction equation

We build a fractional dual-phase-lag model and the corresponding bioheat transfer equation, which we use to interpret the experiment results for processed meat that have been explained by applying the hyperbolic conduction. Analytical solutions expressed by H-functions are obtained by using the Laplace and Fourier transforms method. The inverse fractional dual-phase-lag heat conduction problem for the simultaneous estimation of two relaxation times and orders of fractionality is solved by applying the nonlinear least-square method. The estimated model parameters are given. Finally, the measured and the calculated temperatures versus time are compared and discussed. Some numerical examples are also given and discussed.

Momentum and heat transfer of a special case of the unsteady stagnation-point flow

This paper investigates the unsteady stagnation-point flow and heat transfer over a moving plate with mass transfer,which is also an exact solution to the unsteady Navier-Stokes(NS)equations.The boundary layer energy equation is solved with the closed form solutions for prescribed wall temperature and prescribed wall heat flux conditions.The wall temperature and heat flux have power dependence on both time and spatial distance.The solution domain,the velocity distribution,the flow field,and the temperature distribution in the fluids are studied for different controlling parameters.These parameters include the Prandtl number,the mass transfer parameter at the wall,the wall moving parameter,the time power index,and the spatial power index.It is found that two solution branches exist for certain combinations of the controlling parameters for the flow and heat transfer problems.The heat transfer solutions are given by the confluent hypergeometric function of the first kind,which can be simplified into the incomplete gamma functions for special conditions.The wall heat flux and temperature profiles show very complicated variation behaviors.The wall heat flux can have multiple poles under certain given controlling parameters,and the temperature can have significant oscillations with overshoot and negative values in the boundary layers.The relationship between the number of poles in the wall heat flux and the number of zero-crossing points is identified.The difference in the results of the prescribed wall temperature case and the prescribed wall heat flux case is analyzed.Results given in this paper provide a rare closed form analytical solution to the entire unsteady NS equations,which can be used as a benchmark problem for numerical code validation.

Heat transfer characteristics of thin power-law liquid films over horizontal stretching sheet with internal heating and variable thermal coefficient

The effect of internal heating source on the film momentum and thermal transport characteristic of thin finite power-law liquids over an accelerating unsteady horizontal stretched interface is studied. Unlike most classical works in this field, a general surface temperature distribution of the liquid film and the generalized Fourier＇s law for varying thermal conductivity are taken into consideration. Appropriate similarity transformations are used to convert the strongly nonlinear governing partial differential equations （PDEs） into a boundary value problem with a group of two-point ordinary differential equations （ODEs）. The correspondence between the liquid film thickness and the unsteadiness parameter is derived with the BVP4C program in MATLAB. Numerical solutions to the self-similarity ODEs are obtained using the shooting technique combined with a Runge-Kutta iteration program and Newton＇s scheme. The effects of the involved physical parameters on the fluid＇s horizontal velocity and temperature distribution are presented and discussed.

Solutions of the modification heating conduction equations of a kind of laser thermal effect

This paper has solved the Chester modified heat conduction equation of the different relaxation time r value under different temperature conditions, different boundary conditions and the different initial conditions by different means of methods. These solutions can help to obtain temperature field of laser thermal effects.

Explicit analytical solutions of non-Fourier heat conduction equation for IC chip

Four basic explicit analytical solutions of non_Fourier heat conduction equation simulating IC chip thermal condition are derived. Some other analytical series solutions can also be derived with the above_mentioned basic solutions. Analytical solution has its own irreplaceable important meaning in theory. In addition, as the standard solution to check the accuracy, convergence and effectiveness of various numerical computational methods and their differencing schemes, grid generation ways and so on, the analytical solution is very useful also for the newly rapidly developing computational heat transfer.

一维热传导方程推导及定解条件小结

以均匀且各向同性的细棒为例,用两种方法推导细棒在传热过程中温度所满足的偏微分方程,且对一些常见的定解条件进行总结说明.

A Comprehensive Review on Multi-Dimensional Heat Conduction of Multi-Layer and Composite Structures:Analytical Solutions

Heat conduction in multi-layer and composite materials is one of the fundamental heat transfer problems in many industrial applications.Due to different materials types,interface conditions,and various geometries of these laminates,the heat conduction mechanism is more complicated than that of one-layer isotropic media.Analytical solutions are the best ways to study and understand such problems in depth.In this study,different existing analytical solutions for heat conduction in multi-layer and composite materials are reviewed and classified in rectangular,cylindrical,spherical,and conical coordinates.Applied boundary conditions,internal heat source,and thermal contact resistance as the most critical parameters in the solution complexity investigated in the literature,are discussed and summarized in different tables.Various types of multi-layer structures such as isotropic,anisotropic,orthotropic,and reinforced laminates are included in this study.It is found that although more than half a century has passed since the beginning of the research on heat transfer in multi-layer composites,new researches that can help with a better understanding in this area are still being offered.The challenges and shortcomings in this area are also discussed to guide future researches.

基于慢烤实验的混合炸药热传导系数计算方法

热传导系数作为炸药的重要热力学参数,显著影响炸药装药的点火响应特性。在未开展炸药热传导系数测定实验的情况下,为了快速有效确定一种新型炸药热传导系数,针对典型圆柱段装药结构,建立了轴对称热传导理论模型,获得混合炸药热传导方程的稳态解析解,提出了基于慢烤实验数据的混合炸药热传导系数计算方法。确定了新型不敏感炸药GOL-1(HMX/Al/AP/Binder)的热传导系数,典型烤燃条件下小尺寸装药结构点火响应数值模拟计算结果显示:不同加热速率下装药中心温升曲线计算结果与实验结果均吻合较好,点火时刻装药中心位置温度和点火时间的计算结果与实验偏差最大为2.27%和1.12%,表明GOL-1炸药热传导系数的有效性及数值模拟方法的可行性。建立的热传导计算方法揭示了慢烤装药温度-时间曲线中包含的炸药导热特性及规律,相比体积加权方法和串/并联导热模型方法更适用于计算混合炸药热传导系数,在缺少新型炸药热传导系数测定实验数据的情况下,不失为一种有效的确定方法,为弹药热安全性设计与评估提供了炸药基础参数,推动安全弹药数字化设计与量化评估发展。

二维慢烤模型点火位置及其温度的理论推演

为理论分析凝聚炸药慢速烤燃过程,并为慢速烤燃点火点位置研究奠定理论基础。根据炸药的热传导理论方程,利用叠加原理和分离变量法,将炸药非反应性热传导项与自热反应热传导项拆分,推导得到凝聚炸药二维慢速烤燃模型的温度分布解析解。计算并分析了自热反应温度场随时间和炸药尺寸的变化规律。根据慢烤试验结果,对理论计算得出的点火点位置、点火温度及时间进行了验证。研究结果表明:理论确定的点火时间、点火点位置与试验测量的点火时间、点火点位置相符。炸药自热反应最高温度所在位置随其尺寸的增加,由中心移动至边缘角落。随着试验时间的增加,自热反应温度分布趋于稳定。就尺寸为∅76×190 mm的RDX炸药而言,发生点火时自热反应温度分布与点火前80000 s时的自热反应温度分布一致。

非稳态导热问题分析解特征函数正交性的讨论

非稳态导热问题分析解是传热学中十分重要的内容之一,多数传热学教材对特征函数正交性的论述较为简略,影响了对求解过程的理解.本文以大平板非稳态导热问题分析解求解过程为例,重点讨论了特征函数正交性的含义、特征函数正交性的证明以及特征函数正交性的应用,给出了保持求解过程连贯、帮助学生理解的教学建议,为非稳态导热问题分析解及其数学解析过程的教学提供了参考.

带有移动边界的热传导方程反问题基本解方法

对带有移动边界的非齐次热传导方程反问题进行了研究,采用基本解方法求解此问题。利用积分变换将非齐次热传导方程转化成齐次方程,并应用基于奇异值分解的正则化方法对所得的不适定方程组进行求解,将方程中的未知热源项和部分初值进行了同时反演。通过数值算例显示,文中的方法对于求解此类带有移动边界的热传导方程反问题是稳定有效的。

基于局部观测数据的一维热传导方程系数与初值同时反演问题的数值计算

利用局部观测数据研究了一维热传导方程系数与初值同时反演问题。首先利用函数代换及热传导方程解的解析延拓性,证明了基于局部观测数据下系数与初值同时反演问题的唯一性;然后基于深度神经网络,构造了同时反演问题的反演算法;最后通过构造具有解析解和不具有解析解的数值算例,验证了反演算法的有效性。

基于有限元分析方法对高温作业专用服装设计的研究

本文针对的主要问题是在时间和厚度等可变因素的共同影响下,运用有限元分析方法,设计出合理的一维热传导模型,并应用到实际的高温防护服设计中。考虑到三维模型较为复杂,本文采用降低模型维度的方法进行分析,由简入繁。根据题目中给定的基本数据,利用Fourier定律、能量守恒定律、合理的数学建模知识以及使用Matlab计算软件,解决了题目中的两个问题。