有限元方法详解：热力学与流体流动入门

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"《有限元基础及传热应用》是一本深入浅出介绍有限元方法在热力学和流体流动领域应用的书籍，适合初学者入门。作者包括来自University of Wales Swansea的Roland W. Lewis、Perumal Nithiarasu以及马来西亚Universiti Sains Malaysia的Kankanhalli N. Seetharamu。本书由John Wiley & Sons Ltd于2004年出版，并强调了对版权的尊重和保护。" 《有限元基础及传热应用》详细阐述了有限元方法的基础知识，这是计算工程和科学中广泛采用的一种数值分析技术。该方法的核心在于将复杂的连续区域分解成许多简单的单元，通过这些单元的组合来近似整个问题的解决方案。对于热传导和流体流动这类偏微分方程控制的物理现象，有限元方法能够有效地求解其数学模型。书中的内容可能涵盖了以下几个方面： 1. **有限元理论**：介绍有限元的基本概念，包括变分原理、Galerkin方法和弱形式。书中可能详细解释了如何从连续问题构建离散问题，以及如何通过线性代数求解离散方程组。 2. **网格生成**：讲解如何将复杂几何形状划分成有限个简单元素，如三角形或四边形，并讨论网格质量对解的影响。 3. **热传导**：讨论一维、二维和三维热传导问题，涵盖稳态和瞬态情况。可能包括导热方程的有限元离散化，以及边界条件的处理。 4. **流体流动**：涉及Navier-Stokes方程和连续性方程的有限元解法，包括流体动力学的基础概念和流动类型的分类。 5. **数值实验与实例**：提供实际问题的案例研究，帮助读者理解和应用所学理论，可能包括传热问题（如热交换器设计）和流体流动问题（如管道内的流动或翼型周围的流场）。 6. **软件应用**：可能介绍了一些常用的有限元软件，如ANSYS、COMSOL等，以及如何利用它们进行求解和后处理。 7. **误差分析**：讨论收敛性和精度问题，解释如何评估有限元解的准确度，并提供误差估算的方法。 8. **高级主题**：可能深入探讨了如非线性问题、时间依赖问题、多物理场耦合等进阶话题。通过阅读《有限元基础及传热应用》，读者不仅可以掌握有限元方法的基本原理，还能了解到如何将其应用于实际的热力学和流体力学问题，从而为解决工程中的实际挑战奠定坚实的基础。同时，书中的例子和练习题有助于巩固理解并提升应用技能。

Introduction

1.1 Importance of Heat Transfer

The subject of heat transfer is of fundamental importance in many branches of engineering.

A mechanical engineer may be interested in knowing the mechanisms of heat transfer

involved in the operation of equipment, for example boilers, condensers, air pre-heaters,

economizers, and so on, in a thermal power plant in order to improve their performance.

Nuclear power plants require precise information on heat transfer, as safe operation is an

important factor in their design. Refrigeration and air-conditioning systems also involve

heat-exchanging devices, which need careful design. Electrical engineers are keen to avoid

material damage due to hot spots, developed by improper heat transfer design, in electric

motors, generators and transformers. An electronic engineer is interested in knowing the

efﬁcient methods of heat dissipation from chips and semiconductor devices so that they can

operate within safe operating temperatures. A computer hardware engineer is interested in

knowing the cooling requirements of circuit boards, as the miniaturization of computing

devices is advancing at a rapid rate. Chemical engineers are interested in heat transfer

processes in various chemical reactions. A metallurgical engineer would be interested

in knowing the rate of heat transfer required for a particular heat treatment process, for

example, the rate of cooling in a casting process has a profound inﬂuence on the quality

of the ﬁnal product. Aeronautical engineers are interested in knowing the heat transfer rate

in rocket nozzles and in heat shields used in re-entry vehicles. An agricultural engineer

would be interested in the drying of food grains, food processing and preservation. A

civil engineer would need to be aware of the thermal stresses developed in quick-setting

concrete, the effect of heat and mass transfer on building and building materials and also the

effect of heat on nuclear containment, and so on. An environmental engineer is concerned

with the effect of heat on the dispersion of pollutants in air, diffusion of pollutants in soils,

thermal pollution in lakes and seas and their impact on life. The global, thermal changes

and associated problems caused by El Nino are very well known phenomena, in which

energy transfer in the form of heat exists.

Fundamentals of the Finite Element Method for Heat and Fluid Flow R. W. Lewis, P. Nithiarasu and K. N. Seetharamu

 2004 John Wiley & Sons, Ltd ISBNs: 0-470-84788-3 (HB); 0-470-84789-1 (PB)

2 INTRODUCTION

The previously-mentioned examples are only a sample of heat transfer applications to

name but a few. The solar system and the associated energy transfer are the principal

factors for existence of life on earth. It is not untrue to say that it is extremely difﬁcult,

often impossible, to avoid some form of heat transfer in any process on earth.

The study of heat transfer provides economical and efﬁcient solutions for critical prob-

lems encountered in many engineering items of equipment. For example, we can consider

the development of heat pipes that can transport heat at a much greater rate than copper or

silver rods of the same dimensions, even at almost isothermal conditions. The development

of present day gas turbine blades, in which the gas temperature exceeds the melting point of

the material of the blade, is possible by providing efﬁcient cooling systems and is another

example of the success of heat transfer design methods. The design of computer chips,

which encounter heat ﬂux of the order occurring in re-entry vehicles, especially when the

surface temperature of the chips is limited to less than 100

◦

C, is again a success story for

heat transfer analysis.

Although there are many successful heat transfer designs, further developments are still

necessary in order to increase the life span and efﬁciency of the many devices discussed

previously, which can lead to many more new inventions. Also, if we are to protect our

environment, it is essential to understand the many heat transfer processes involved and, if

necessary, to take appropriate action.

1.2 Heat Transfer Modes

Heat transfer is that section of engineering science that studies the energy transport between

material bodies due to a temperature difference (Bejan 1993; Holman 1989; Incropera and

Dewitt 1990; Sukhatme 1992). The three modes of heat transfer are

1. Conduction

2. Convection

3. Radiation.

The conduction mode of heat transport occurs either because of an exchange of energy

from one molecule to another, without the actual motion of the molecules, or because of

the motion of the free electrons if they are present. Therefore, this form of heat transport

depends heavily on the properties of the medium and takes place in solids, liquids and

gases if a difference in temperature exists.

Molecules present in liquids and gases have freedom of motion, and by moving from

a hot to a cold region, they carry energy with them. The transfer of heat from one region

to another, due to such macroscopic motion in a liquid or gas, added to the energy transfer

by conduction within the ﬂuid, is called heat transfer by convection. Convection may be

free, forced or mixed. When ﬂuid motion occurs because of a density variation caused by

temperature differences, the situation is said to be a free, or natural, convection. When

the ﬂuid motion is caused by an external force, such as pumping or blowing, the state is

INTRODUCTION 3

deﬁned as being one of forced convection. A mixed convection state is one in which both

natural and forced convections are present. Convection heat transfer also occurs in boiling

and condensation processes.

All bodies emit thermal radiation at all temperatures. This is the only mode that does

not require a material medium for heat transfer to occur. The nature of thermal radiation

is such that a propagation of energy, carried by electromagnetic waves, is emitted from the

surface of the body. When these electromagnetic waves strike other body surfaces, a part

is reﬂected, a part is transmitted and the remaining part is absorbed.

All modes of heat transfer are generally present in varying degrees in a real physical

problem. The important aspects in solving heat transfer problems are identifying the sig-

niﬁcant modes and deciding whether the heat transferred by other modes can be neglected.

1.3 The Laws of Heat Transfer

It is important to quantify the amount of energy being transferred per unit time and for that

we require the use of rate equations.

For heat conduction, the rate equation is known as Fourier’s law, which is expressed

for one dimension as

=−k

(1.1)

where q

is the heat ﬂux in the x direction (W/m

); k is the thermal conductivity (W/mK,

a property of material, see Table 1.1)and dT/dx is the temperature gradient (K/m).

For convective heat transfer, the rate equation is given by Newton’s law of cooling as

q = h(T

− T

) (1.2)

where q is the convective heat ﬂux; (W/m

); (T

− T

) is the temperature difference

between the wall and the ﬂuid and h is the convection heat transfer coefﬁcient, (W/m

(ﬁlm coefﬁcient, see Table 1.2).

The convection heat transfer coefﬁcient frequently appears as a boundary condition in

the solution of heat conduction through solids. We assume h to be known in many such

problems. In the analysis of thermal systems, one can again assume an appropriate h if not

available (e.g., heat exchangers, combustion chambers, etc.). However, if required, h can

be determined via suitable experiments, although this is a difﬁcult option.

The maximum ﬂux that can be emitted by radiation from a black surface is given by

the Stefan–Boltzmann Law,thatis,

q = σT

(1.3)

where q is the radiative heat ﬂux, (W/m

); σ is the Stefan–Boltzmann constant (5.669 ×

−8

), in W/m

and T

is the surface temperature, (K).

The heat ﬂux emitted by a real surface is less than that of a black surface and is given by

q = σT

(1.4)

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