ME 240 Course

Heat Conduction a numerical approach

ME-240 Authors: Zaparoli, E.L.and Andrade, C.R.. Page 1

Heat Conduction A numerical approach

Edson Luiz Zaparoli

Cludia Regina de Andrade



Table of Contents

CHAPTER 1...........................................................................................................................................................4 INTRODUCTION. ................................................................................................................................................4

1.1 BASIC CONCEPTS. ..................................................................................................................................5 1.1.1 First law of thermodynamics............................................................................................................6 1.1.2 Second law of thermodynamics. .......................................................................................................8 1.1.3 Fourier law of microscopic heat transfer.........................................................................................9 1.1.4 Thermal conductivity......................................................................................................................13

1.2 MODES OF HEAT TRANSFER. ................................................................................................................16 1.2.1 Conduction. ....................................................................................................................................16 1.2.2 Convection. ....................................................................................................................................18 1.2.3 Radiation........................................................................................................................................21 1.2.4 Conjugate heat transfer..................................................................................................................23

CHAPTER 2.........................................................................................................................................................26 CONDUCTION HEAT TRANFER: EQUATION, BOUNDARY AND INITIAL CONDITIONS..............26

2.1 HEAT CONDUCTION EQUATION. ...........................................................................................................27 2.2 INITIAL AND BOUNDARY CONDITIONS.................................................................................................31

2.2.1 Initial condition..............................................................................................................................32 2.2.2 Boundary conditions. .....................................................................................................................33

2.2.2.1 Prescribed boundary temperature.........................................................................................................33 2.2.2.2 Prescribed heat flux..............................................................................................................................34 2.2.2.3 Prescribed convection heat transfer coefficient....................................................................................36 2.2.2.4 Heat transfer by radiation.....................................................................................................................41 2.2.2.5 Interface condition. ..............................................................................................................................44

CHAPTER 3.........................................................................................................................................................48



NUMERICAL CONDUCTION HEAT TRANSFER FUNDAMENTALS.....................................................48 3.1 INTRODUCTION ....................................................................................................................................49 3.2 MATHEMATICAL FORMULATION..........................................................................................................51 3.3 NUMERICAL SOLUTION PROCEDURE. ...................................................................................................53

CHAPTER 4.........................................................................................................................................................61 FINITE VOLUME METHOD............................................................................................................................61

4.1 INTRODUCTION ....................................................................................................................................62 4.1.1 Worked-examples: one-dimensional steady-state diffusion ...........................................................67

4.1.1.1 Comparison with the analytical solution. .............................................................................................71 4.1.1.2 Case without internal heat generation. .................................................................................................72

CHAPTER 5.........................................................................................................................................................79 FINITE ELEMENT METHOD..........................................................................................................................79

5.1 INTRODUCTION. ...................................................................................................................................80 5.2 ONE-DIMENSIONAL HEAT CONDUCTION EXAMPLE. ..............................................................................81

5.2.1 Differential formulation (strong formulation)................................................................................81 5.2.2 Weighted residual formulation (weak formulation). ......................................................................82 5.2.3 Worked-example 1: circular rod steady-state conduction. ............................................................91 5.2.4 Worked-example 2: slab steady-state conduction. .........................................................................93

CHAPTER 6.........................................................................................................................................................99 EXAMPLES. ........................................................................................................................................................99

6.1 CIRCULAR FINS ..................................................................................................................................100 6.1.1 Two-Dimensional approach.........................................................................................................100 6.1.2 One-Dimensional approach .........................................................................................................101



Chapter 1 INTRODUCTION.

Objective - this chapter presents introductory concepts related to heat transfer processes:

Basic concepts: first and second laws of thermodynamics; Fourier law;

Heat transfer modes: conduction; convection and radiation.



1.1 BASIC CONCEPTS. Heat transfer is the science that deals with rate of heat transfer as well as the temperature distribution within a system at a specified time. Heat flow is vectorial in the sense that it is in the direction of negative temperature gradients, i.e., from higher toward lower temperatures.

The science of heat transfer is based upon foundations comprising both theory and experiment. As in other engineering disciplines, the theoretical part is constructed from one or more physical (or natural) laws. The physical laws are statements, in terms of various concepts, which have been found to be true through many years of experimental observations. A physical law is called a general law if its application is independent of the medium under consideration. Otherwise, it is called a particular law. There are, in fact, the following four general laws among others upon which all the analyses concerning heat transfer, either directly or indirectly, depend:

1. The law of conservation of mass 2. The first law of thermodynamics 3. The second law of thermodynamics 4. Newton's second law of motion

In addition to the general laws, it is usually necessary to bring certain particular laws into an analysis. Examples are Fourier's law of microscopic heat transfer, the Stefan-Boltzmann law of radiation, Newton's law of viscosity, the ideal-gas law, etc.

We start this chapter with a review of thermodynamic concepts. Thermodynamics can be defined as the science of energy. Although everybody has a feeling of what energy is, it is



difficult to give a precise definition for it. Energy can be viewed as the ability to cause changes.

The name thermodynamics stems from the Greek words therme (heat) and dynamis (power), which is most descriptive of the early efforts to convert heat into power. Today the same name is broadly interpreted to include all aspects of energy and energy transformations, including power production, refrigeration, and relationships among the properties of matter.

1.1.1 FIRST LAW OF THERMODYNAMICS. Although the principles of thermodynamics have been in existence since the creation of universe, thermodynamics did not emerge as a science until the construction of the first successful atmospheric steam engines in England by Thomas Savery in 1697 and Thomas Newcomen in 1712. These engines were very slow and inefficient, but they opened the way for the development of a new science.

The first and second laws of thermodynamics emerged simultaneously in the 1850s primarily out of the works of William Rankine, Rudolph Clausius, and Lord Kelvin (formerly William Thomson). The term thermodynamics was first used in a publication by Lord Kelvin in 1849. The first thermodynamic textbook was written in 1859 by William Rankine, a professor at the University of Glasgow.

The first law of thermodynamics is simply an expression of the conservation of energy principle, and it asserts that energy is a thermodynamic property.

The first law of thermodynamics, also known as the conservation of energy principle, states that energy can neither be created nor destroyed; it can only change forms. Therefore, every bit of energy must be accounted for during a process. The conservation of energy principle (or



the energy balance) for any system undergoing any process may be expressed as follows: The net change (increase or decrease) in the total energy of the system during a process is equal to the difference between the total energy entering and the total energy leaving the system during that process. That is,

Noting that energy can be transferred to or from a system by heat, work, and mass flow, and that the total energy of a simple compressible system consists of internal, kinetic, and potential energies.

In steady operation, the rate of energy transfer to a system is equal to the rate of energy transfer from the system, Fig. 1.1.



Fig. 1.1 Energy balance under steady-state condition.

1.1.2 SECOND LAW OF THERMODYNAMICS. The first law of thermodynamics, which embodies the idea of conservation of energy, gives means for quantitative calculation of changes in the state of a system due to interactions between the system and its surroundings, but tells nothing about the direction that a process might take. On the other hand, observations concerning the unidirectionality of naturally occurring processes have led to the formulation of the second law of thermodynamics, which gives a sense of direction to energy-transfer processes.

The second law of thermodynamics leads to a thermodynamic property entropy. For any reversible process that a system undergoes during a time interval dt, the change in the entropy S of the system is given by:



revTQdS

= Eq. 1.1

For an irreversible process, the change, however, is:

irrevTQdS

Eq. 1.2

where Q is the small amount of heat added to the system during the time interval dt, and T is the temperature of the system at the time of heat transfer. Eq. 1.1 and Eq. 1.2 may be taken as the mathematical statement of the second law, which can also be written in rate form as:

irrevdtQ

TdtdS

1 Eq. 1.3

When one system is isolated (adiabatic, 0=Q ), it may be concluded that 0dtdS , that is,

entropy ever increases if the process is irreversible.

1.1.3 FOURIER LAW OF MICROSCOPIC HEAT TRANSFER. A physical law is called a particular law if its application is dependent of the medium or process under consideration.



Since the mechanism of diffusive (microscopic) heat transfer on the molecular level is thought to be the exchange of kinetic energy between the molecules, the most fundamental approach in analyzing heat conduction in a substance would be to apply the laws of motion to each individual molecule or a statistical group of molecules, subsequent to some initial state. In most engineering problems, however, primary interest does not lie in the molecular behavior of a substance, but rather in how the substance behaves as a continuous medium. In heat conduction (and, therefore, heat convection) studies, the molecular structure is neglected and the matter is considered to be a continuous medium (continuum), which fortunately is a valid approximation to many practical problems where only macroscopic information is of interest. Such a model may be used provided that the size and the mean free path of molecules are small compared with other dimensions existing in the medium. This approach, also known as the phenomenological approach to heat transfer problems, is simpler than microscopic approaches and usually provides the answers required in engineering.

Fourier's law of microscopic heat transfer, which is the basic law governing heat conduction based on the continuum concept, states that the heat flux (i.e., the rate of heat transfer per unit area) due to microscopic heat transfer in a given direction at a point within a medium (solid, liquid or gaseous) is proportional to the temperature gradient in the same direction at the same point. For heat conduction in any direction n, this law is given by:

Eq. 1.4

where is the magnitude of the heat flux in the n direction, and is the temperature gradient in the same direction. Here k is a proportionality constant known as the thermal conductivity of the material of the medium under consideration, and is a positive quantity.



The minus sign is included so that heat flow is positive in the direction of a negative temperature gradient.

Thermal conductivity is a thermophysical property and has the units W/(m K) in the SI system. A medium is said to be homogeneous if its thermal conductivity does not vary from point to point within the medium, and heterogeneous if there is such a variation. Further, a medium is said to be isotropic if its thermal conductivity is the same in all directions, and anisotropic if there exists directional variation in thermal conductivity.

In anisotropic media the heat flux due to conduction in a given direction may also be proportional to the temperature gradients in other directions, and therefore Eq. 1.4 may not be valid.

Heat transfer from one hot place to a cold place in an opaque solid medium occurs due to molecular activity. Molecules at the hot place exchange their kinetic and vibrational energies with neighboring layers through random motion and collisions.

A temperature gradient, or slope, is established with energy continuously being transported in the direction of decreasing temperature. This mode of energy transfer is called microscopic or diffusive thermal energy transport. Since molecular activity is characteristic of all materials, this microscopic transport also takes place in liquids and gases.

The microscopic transport in a homogeneous and isotropic medium is modeled by the Fouriers law that states: The microscopic heat flux is proportional to the gradient of the temperature field.



The constant of proportionality, which is called the thermal conductivity k, is an inherent property of the material itself and usually depends on temperature. Fouriers Law is stated as:

Eq. 1.5

Fig. 1.2 Isotherms and heat flux vectors.

In rectangular coordinates Eq. 1.5, can be expanded to:

xTkqx = ;

yTkqy = ;

zTkqz = Eq. 1.6

Temperature is a scalar quantity that can be defined at every point in a material. The gradient of a scalar field is a vector. Hence, the heat flux has a direction, which is the negative of the temperature gradient, Fig. 1.2. If the temperature field is thought of as contour lines of equal temperature (isotherms), then the heat flux is everywhere normal (perpendicular) to these contours and in the direction of flowing from high temperature to low temperature.

Three observations are worth noting here:



(i) The negative sign means that when the gradient is negative, heat flow is in the positive direction, i.e., towards the direction of decreasing temperature, as dictated by the second law of thermodynamics,

(ii) The conductivity k need not be uniform since Eq. 1.5 applies at a point in the material and not to a finite region. In reality the thermal conductivity can vary with temperature. However, Eq. 1.5 is limited to isotropic material, i.e., k is invariant with direction,

(iii) Remembering that heat transfer deals with the determination of the temperature distribution in a region, once T(x,y,z) is known, the heat flux can be easily determined by simply differentiating the function T and using Eq. 1.5.

(iv) If there is a temperature gradient in any medium, then a microscopic thermal energy transport will be established.

1.1.4 THERMAL CONDUCTIVITY. Thermal conductivity is the physical property denoting the ease with which a particular substance can accomplish the microscopic thermal energy transport (heat flux). The thermal conductivity of a material is found to depend on the chemical composition of the substance, or substances, of which it is composed, the phase (i.e., gas, liquid, or solid) in which it exists, its crystalline structure if a solid, the temperature and pressure to which it is subjected, and whether or not it is a homogeneous material.

The microscopic thermal energy transport can occur by different mechanisms as:

Gases.



Gas molecules are in a state of random motion. When collisions occur, there is an exchange of kinetic energy that results in thermal energy being transmitted down a temperature gradient from a hot (higher molecular kinetic energy) to a cold (lower molecular kinetic energy) region.

Dielectric Liquids. In liquids (excluding liquid metals such as mercury at normal temperatures and sodium at high temperatures), the molecules are relatively closely packed, and thermal energy transport occurs primarily by longitudinal vibrations, similar to the propagation of sound. The structure of liquids is not well understood at present, and there are no good theoretical formulas for conductivity.

Pure Metals and Alloys. The primary mechanism of heat conduction is the movement of free electrons. A smaller contribution is due to the transfer of atomic motions by lattice vibrations, or waves; however, this contribution is unimportant except at cryogenic temperatures. In alloys, the movement of free electrons is restricted, and thermal conductivity decreases markedly as alloying elements are added. The lattice wave contribution then becomes more important but is difficult to predict owing to the variable effects of heat treatment and cold working.

Dielectric Solids. Thermal energy transport is almost entirely due to atomic motions being transferred by lattice waves; hence, thermal conductivity is very dependent on the crystalline structure of the material.



Nonhomogeneous materials. Many building materials and insulators have anomalously low values of conductivity because of their porous nature. The conductivity is an effective value for the porous medium and is low due to air or a gas filling the interstices and pores, through which heat is transferred rather poorly by conduction and radiation. Expanded plastic insulations, such as polystyrene, have a large molecule refrigerant gas filling the pores to reduce the conductivity. These materials due to its porous structure are considered to have an apparent thermal conductivity. Nonisotropic materials. Some materials may exhibit nonisotropic thermal conductivity that result from a directional preference by a fibrous structure (as in the case of wood, asbestos, composite materials, etc).



1.2 MODES OF HEAT TRANSFER. Heat is the form of energy that can be transferred from one system to another as a result of temperature difference. A thermodynamic analysis is concerned with the amount of heat transfer as a system undergoes a process from one equilibrium state to another. The science that deals with the determination of the rates of such energy transfers is the heat transfer. The transfer of energy as heat is always from the higher-temperature medium to the lower-temperature one, and heat transfer stops when the two mediums reach the same temperature.

It is customary to categorize the various heat transfer processes into three basic types or modes, although, as will become apparent as one studies the subject, it is certainly a rare instance when one encounters a problem of practical importance which does not involve at least two, and sometimes all three, of these modes occurring simultaneously. The three modes are conduction, convection, and radiation. All modes of heat transfer require the existence of a temperature difference, and all modes are from the high-temperature medium to a lower-temperature one.

1.2.1 CONDUCTION. Heat conduction is the term applied to the mechanism of internal energy exchange from one body to another, or from one part of a body to another part, by the exchange of the kinetic energy of motion of the molecules by direct communication or by the drift of free electrons in the case of heat conduction in metals.

This flow of energy or heat passes from the higher energy molecules to the lower energy ones (i.e., from a high temperature region to a low temperature region, Fig. 1.3.). The distinguishing feature of conduction is that it takes place within the boundaries of a body, or



across the boundary of a body into another body placed in contact with the first, without an appreciable displacement of the matter comprising the body.

(a) (b)

Fig. 1.3 Heat flux and isotherms in a finned tube wall.

Conduction is the transfer of energy from the more energetic particles of a substance to the adjacent less energetic ones as a result of interactions between the particles. Conduction can take place in solids, liquids, or gases. In gases and liquids, conduction is due to the collisions and diffusion of the molecules during their random motion. In solids, it is due to the combination of vibrations of the molecules in a lattice and the energy transport by free



electrons. A cold canned drink in a warm room, for example, eventually warms up to the room temperature as a result of heat transfer from the room to the drink through the aluminum can by conduction.

The rate of heat conduction through a medium depends on the geometry of the medium, its thickness, and the material of the medium, as well as the temperature difference across the medium. We know that wrapping a hot water tank with glass wool (an insulating material) reduces the rate of heat loss from the tank. The thicker the insulation, the smaller the heat loss. We also know that a hot water tank will lose heat at a higher rate when the temperature of the room housing the tank is lowered. Further, the larger the tank, the larger the surface area and thus the rate of heat loss.

1.2.2 CONVECTION. Convection is the term applied to the heat transfer mechanism which occurs in a fluid by the mixing of one portion of the fluid with another portion due to gross movements of the mass of fluid. The actual process of energy transfer from one fluid particle or molecule to another is still one of conduction, but the energy may be transported from one point in space to another by the displacement of the fluid itself.

The fluid motion may be caused by external mechanical means (e.g., by a fan, pump, etc.), in which case the process is called forced convection (Fig. 1.4). If the fluid motion is caused by density differences which are created by the temperature differences existing in the fluid mass, the process is termed free convection or natural convection, Fig. 1.5.



Fig. 1.4 Cooling of a rack of electronic components by forced convection.

Fig. 1.5 - The warming up of a cold drink in a warmer environment by natural convection.

The circulation of the water in a pan heated on a stove is an example of free convection. It is virtually impossible to observe pure heat conduction in a fluid because as soon as a temperature difference is imposed on a fluid, natural convection currents will occur as a result of density differences.

The presence of a temperature gradient in a fluid in a gravity field always gives rise to natural convection mechanism. Therefore, forced convection is always accompanied by natural convection inducing a combined or mixed convection heat transfer process.



Natural convection may help or hurt forced convection heat transfer, depending on the relative directions of buoyancy induced and the forced convection motions:

i) In assisting flow, the buoyant motion is in the same direction as the forced motion. Therefore, natural convection assists forced convection, and enhances heat transfer. An example is upward forced flow over a hot surface (Fig. 1.6a).

ii) In opposing flow, the buoyant motion is in the opposite direction to the forced motion. Therefore, natural convection resists forced convection, and decreases heat transfer. An example is upward forced flow over a cold surface (Fig. 1.6b).

iii) In transverse flow, the buoyant motion is perpendicular to the forced motion. Transverse flow enhances the fluid mixing, and thus increases heat transfer. An example is horizontal forced flow over a hot or cold cylinder (Fig. 1.6c).

The basic laws of heat conduction must be coupled with those of fluid motion in order to describe, mathematically, the process of heat convection. The mathematical analysis of the resulting system of differential equations is perhaps one of the most complex fields of applied mathematics.



(a) (b) (c)

Fig. 1.6 Mixed convection: (a) assisted flow; (b) opposing flow; (c) transverse flow.

1.2.3 RADIATION. Thermal radiation is the term used to describe the electromagnetic radiation which has been observed to be emitted at the surface of a body which has been thermally excited (Fig. 1.7a). Everything around us constantly emits thermal radiation. This electromagnetic radiation is emitted in all directions; and when it strikes another body, part may be reflected, part may be transmitted, and part may be absorbed.



(a) (b) (c)

Fig. 1.7 Examples of thermal radiation heat transfer.

Thus, in a manner completely different from the two modes previously discussed, heat may pass from one body to another without the need of a medium of transport between them (Fig. 1.7b). In some instances there may be a separating medium, such as air, which is unaffected by this passage of energy (Fig. 1.7c). The heat of the sun is the most obvious example of thermal radiation.

There will be a continuous interchange of energy between two radiating bodies, with a net exchange of energy from the hotter to the colder. Even in the case of thermal equilibrium, an energy exchange occurs, although the net exchange will be zero.



1.2.4 CONJUGATE HEAT TRANSFER. Conjugate heat transfer occurs when there is a coupling among different heat transfer modes in the interfaces separating the materials involved in the energy transfer process.

Fig. 1.8 shows an electronic components cooling example.

Fig. 1.8 Examples of conjugate heat transfer problems: interface coupled conduction and convection.



The conduction and convection occur in different materials, solid domains (printed circuit board, electronic chips) and fluid domain (airflow heat sink). Energy equations must be solved simultaneously in both solid and fluid. Such problems are called conjugate or coupled. To design heat transfer equipment, it is important to know the heat transfer rate usually encountered along a solid-fluid interface, Fig. 1.9. Thus, conduction in a solid and convection in a fluid simultaneously occur in nature, rather than occurring separately.

Fig. 1.9 - Separation of an actual thermal problem into a conduction problem and a convection problem.



However, because of our inability to solve this coupled actual problem, the problem is cut (Fig. 1.9) along the interface and artificially separated into a conduction problem and a convection problem, which is named classical or decoupled approach. Then we try to solve each problem separately after replacing the real interface boundary conditions with simpler but somewhat artificial boundary conditions, expressed in terms of a heat transfer coefficient h, defined as:

)( = TThq wconv Eq. 1.7



Chapter 2 CONDUCTION HEAT TRANFER: EQUATION, BOUNDARY AND INITIAL CONDITIONS.

Objective - this chapter presents the mathematical model to the heat conduction problem:

Energy balance: heat conduction equation. Boundary conditions. Initial conditions.



2.1 HEAT CONDUCTION EQUATION. The mechanism of heat conduction at the molecular level is visualized as the exchange of kinetic energy between the micro-particles in the high and low temperature regions. In phenomenological heat conduction theory, however, the molecular structure of a medium is disregarded and the medium is considered to be a continuum. Analytical investigations into heat conduction based on the concept of continuum start with the derivation of the heat conduction equation. The heat conduction equation is a mathematical relation, expressed by a differential equation, between temperature and time and space coordinates.

To derive the general heat conduction equation, it is considered a stationary and opaque solid as shown in Fig. 2.1a. The temperature distribution in this solid is represented by T(r, t), and thermal conductivity by k which may be a function of space coordinates and/or temperature T. For the solid, density is denoted by and constant pressure specific heat by cp. Consider a point P at any location r in the solid:

Fig. 2.1 An opaque solid and the control volume for the derivation of the general heat conduction equation.



Assuming that the point P is enclosed by any surface S lying entirely within the solid. The volume of the space enclosed by S is represented by v, as illustrated in Fig. 2.1b. Also assuming that internal energy is generated inside the solid at a rate of qe = qe(r, t) per unit volume. Since, there is no work through the surface S, the first law of thermodynamics (energy balance) is represented by:

+= S v em dvqdSnqdmet r" Eq. 2.1Where m is the solid mass and e is the energy per unit mass of the solid. Since the solid is stationary, Eq. 2.1 can also be written as:

+= S v em dvqdSnqdmtu r" Eq. 2.2where u is the internal energy per unit mass of the solid. In general, for a substance that is homogeneous and invariable in composition:

dTcdppudT

Tudp

pudu p

TpT

+

=

+

= Eq. 2.3

Since for a solid, 0

Tpu , results that dTcdu p and:



+= S v em p dvqdSnqdmtTc r" Eq. 2.4The surface integral in the above equation can be converted into a volume integral by using the divergence theorem as:

( ) += v v em p dvqdvqdmtTc " Eq. 2.5Using this relation, dvdm = , for the mass element:

( ) += v v ev p dvqdvqdvtTc " Eq. 2.6Or:

( ) 0" = +v ep dvqqtTc Eq. 2.7Since the integral in the above relation vanishes for every volume element v and the integrand is a continuous function, its integrand must vanish everywhere, thus yielding:

( ) 0" =+ ep qqtTc Eq. 2.8



Assuming that, the solid is isotropic, Fourier's law then gives:

Tkq =" Eq. 2.9Substituting the Eq. 2.9 into Eq. 2.8, then gives:

( ) ep qTktTc += Eq. 2.10

This is the general heat conduction equation for isotropic solids. If the thermal conductivity k is a function of space coordinates only, then Eq. 2.10 is a linear partial differential equation. On the other hand, when k depends on temperature, Eq. 2.10 becomes a nonlinear partial differential equation.



2.2 INITIAL AND BOUNDARY CONDITIONS. The heat conduction equations were developed using an energy balance on a differential element inside the medium, and they remain the same regardless of the thermal conditions on the surfaces of the medium. That is, the differential equations do not incorporate any information related to the conditions on the surfaces such as the surface temperature or a specified heat flux. Yet we know that the heat flux and the temperature distribution in a medium depend on the conditions at the surfaces, and the description of a heat transfer problem in a medium is not complete without a full description of the thermal conditions at the bounding surfaces of the medium. The mathematical expressions of the thermal conditions at the boundaries are called the boundary conditions. To obtain a unique solution to a problem, we need to specify more than just the governing differential equation. We need to specify some conditions (such as the value of the function or its derivatives at some value of the independent variable) so that forcing the solution to satisfy these conditions at specified points will result in unique values for the arbitrary constants and thus a unique solution.

But since the differential equation has no place for the additional information or conditions, we need to supply them separately in the form of boundary or initial conditions.

To complete the formulation of a problem, we need to have certain conditions to determine the constants of integration. These will be the initial and boundary conditions of the problem. That is to say, there is an infinite number of solutions to the differential equation Eq. 2.10, but there is usually only one solution for the prescribed initial and boundary conditions.



The number of conditions in the direction of each independent variable is equal to the order of the highest derivative of the governing differential equation in the same direction. That is, we need to specify one initial condition (for time-dependent problems) and two boundary conditions in each coordinate direction.

Physically, the heat conduction equation represents the heat transfer process inside the domain. So, the interaction between the domain and the environment must be represented by the boundary conditions. Furthermore, initial condition must represent all the phenomenon features that have been occurred before the time t = 0.

2.2.1 INITIAL CONDITION. The initial condition for a time-dependent problem is the prescribed temperature distribution in the medium under consideration at some instant of time, usually at the beginning of the heating or cooling process; that is, at t = 0.

Mathematically speaking, if the initial condition is given by then the solution T(r, t) of the problem must be such that, at all points of the medium,

Eq. 2.11

where r is the position vector.



2.2.2 BOUNDARY CONDITIONS. The body, inside which occurs conduction heat transfer, interacts with the environment exchanging heat by either heat transfer mode: conduction, convection, radiation or simultaneous heat transfer. This heat transfer interaction can be approximately represented by the boundary conditions described bellow.

Boundary conditions specify the temperature or the heat flow at the boundary surfaces of a region. For convenience in the analysis we separate the boundary conditions into the following categories: prescribed boundary temperature, prescribed heat flux, heat transfer by convection, heat transfer by radiation, and interface condition.

2.2.2.1 Prescribed boundary temperature. The distribution or the value of temperature may be prescribed at a boundary surface, as shown in Fig. 2.4. This prescribed surface temperature may, in general, be a function of space and time variables; that is:

Eq. 2.12

Fig. 2.2 Prescribed surface temperature at boundary surface S.



A boundary condition of this form is also called the boundary condition of the first kind. The prescribed temperature distribution at the boundary surface S, in special cases, can be a function only of position or time or it can be a constant. If the temperature on the boundary surface vanishes, that is, if:

Eq. 2.13

then this boundary condition is called the homogeneous boundary condition of the first kind. This boundary condition represents approximately the interaction of part of the body surface with fluid in movement undergoing boiling or condensation phase change. The convective heat transfer for this flow is very high, resulting in a nearly uniform temperature in the interface.

2.2.2.2 Prescribed heat flux. The distribution or the value of the heat flux across a boundary may be specified to be constant or a function of space and/or time. Consider a boundary surface at r = rs and let n be the outward-drawn unit vector normal to this surface, as shown in Fig. 2.3a.

If the heat flux leaving the surface is specified, then the boundary condition can be written as:

Eq. 2.14a

which is a statement of energy balance at any location rs on the surface.



When the heat flux crossing the surface into the region is prescribed, as shown in Fig. 2.3b, the boundary condition becomes:

Eq. 2.14b

When the heat flux is specified either from the surface or to the surface, it mathematically means that we are given the normal derivative of temperature at the boundary. A boundary condition of this form is also called the boundary condition of the second kind. If the derivative of temperature normal to the boundary surface is zero, that is,

Eq. 2.15

then the boundary condition is called the homogeneous boundary condition of the second kind, and such a boundary condition indicates either a thermally insulated boundary (i.e., no heat transfer) or a symmetry condition at the boundary.

Eq. 2.14b represents approximately the domain surface heating by an electric resistor device suitable insulated to impose a nearly uniform input heat flux at the boundary surface S, Fig. 2.3.

When the surface S, Fig. 2.3, is completely wrapped by an insulating blanket, the heat flux is nearly zero and can be represented by Eq. 2.15.



Fig. 2.3 - Prescribed heat flux at a boundary surface S.

2.2.2.3 Prescribed convection heat transfer coefficient. Convection has already been defined as the process of heat transport in a fluid in macroscopic motion. As a mechanism of heat transfer it is important not only between the molecules of a fluid but also between a fluid and a solid surface when they are in contact. This rate of heat transfer between a fluid in motion and a solid surface is major information to thermal equipment design.

When a fluid flows over a solid surface as illustrated in Fig. 2.4, it is an experimental observation that the fluid particles adjacent to the surface stick to it and therefore have zero velocity relative to the surface. Other fluid particles attempting to slide over the stationary ones at the surface are retarded as a result of viscous forces between the fluid particles. The



velocity of the fluid particles thus asymptotically approaches that of the undisturbed free stream over a distance (velocity boundary-layer thickness) from the surface.

Fig. 2.4 - Velocity and thermal boundary layers along a solid surface.

As illustrated in Fig. 2.4, if > TTw , then heat will flow from the solid to the fluid particles at the surface. The energy thus transmitted increases the internal energy of the fluid particles (thermal energy storage) and is carried away by the motion of the fluid. The temperature distribution in the fluid adjacent to the surface will asymptotically approaches the free-stream value, T , in a short distance T (thermal boundary-layer thickness) from the surface. Since the fluid particles at the surface are stationary, the heat flux from the surface to the fluid will be:



==

nT

kqq ffnconv Eq. 2.16

where kf is the thermal conductivity of the fluid, Tf is the temperature distribution in the fluid, the subscript w means the derivative is evaluated at the surface, and n denotes the normal direction from the surface.

From Eq. 1.7 and Eq. 2.16, it is obtained that:

== TTn

Tk

TTqh

w

w

ff

w

conv Eq. 2.17

The heat transfer coefficient has the units W/(m2K) in the SI system.

To obtain h, the convective heat transfer coefficient, the value of temperature gradient at solid/fluid interface is necessary. This value may be calculated solving the convective heat transfer mathematical model (a system of partial differential equations that represents the fluid flow and heat transfer) as will be seeing in convections chapters. Traditionally, the convective heat transfer coefficient is calculated doing experimental measurements.

When a solid boundary surface is subjected to convective heat transfer with an ambient fluid at a prescribed temperature as shown in Fig. 2.5a, the boundary condition at this boundary surface can be expressed, using Newton's law of cooling, as:



Eq. 2.18a

Or

Fig. 2.5 - Boundary surface subjected to convective heat transfer into an ambient fluid at temperature T with a heat transfer coefficient h.

Eq. 2.18b



where h is the heat transfer coefficient between this surface and the surrounding fluid. If n represents the inward-drawn normal of the boundary surface as shown in Fig. 2.5b, the boundary condition can be written as:

Eq. 2.19a

or

Eq. 2.19b

A boundary condition of the form of either Eq. 2.18b or Eq. 2.19b is also called the boundary condition of the third kind. Here we note that when h the boundary condition of the third kind reduces to the boundary condition of the first kind (prescribed temperature). The ambient fluid temperature T may be a constant or a function of space and/or time. If the ambient fluid temperature T = 0, that is, if:

Eq. 2.19b



where the plus and minus signs correspond to the differentiations along outward and inward normals, respectively, then the boundary condition is called the homogeneous boundary condition of the third kind.

2.2.2.4 Heat transfer by radiation. If two isothermal surfaces A1 and A2, Fig. 2.6, having emissivities 1 and 2 and absolute temperatures T1 and T2, respectively, exchange heat by radiation only, then the net flux of heat exchange between these two surfaces is given by:

( )42412121, TTFqr = Eq. 2.20If A1 and A2 are two large parallel surfaces with negligible losses from the edges as shown in Fig. 2.6, then the factor 21F in Eq. 2.20 is given by:

1111

2121

+= F Eq. 2.21

If A1 is completely enclosed by the surface A2 as shown in Fig. 1.4b, then:

+

+=

111111

22

1

12121 AA

fF Eq. 2.22



(a)

(b)

Fig. 2.6 - Two isothermal surfaces A1 and A2 exchanging radiation energy.

where f1-2 is a purely geometric factor called radiation shape factor or configuration factor between the surfaces A1 and A2, and is equal to the fraction of the radiation leaving surface A1 that directly reaches surface A2.

In certain applications it may be convenient to define a radiation heat transfer coefficient hr as:

( )2121, TThq rr = Eq. 2.23When this concept is applied to Eq. 2.20, hr becomes:

( )( )21

42

4121

TTTTFhr

= Eq. 2.24



When the solid boundary surface is subjected to radiative heat transfer with an environment at an "effective" blackbody temperature Te, as shown in Fig. 2.7, the boundary condition at this boundary can be written as:

( )[ ]4421 , ess

TtrTFnTk =

Eq. 2.25

where 21F is the radiation shape factor of the surface and its surrounding surfaces and is the Stefan-Boltzmann constant.

Fig. 2.7 - Boundary surface losing heat only by radiation into an environment maintained at an "effective" blackbody temperature Te.

Since it involves the fourth power of the surface temperature (dependent variable), Eq. 2.25 is a nonlinear boundary condition.



2.2.2.5 Interface condition. At an interface between two materials, such as shown in Fig. 2.8, the rate of heat flow must be continuous since energy cannot be destroyed or generated there; that is,

Eq. 2.26

Fig. 2.8 - Two media in contact.

Temperature distribution and heat flow lines along two solid plates pressed against each other for the case of perfect and imperfect contact are illustrated in Fig. 2.9.



If the two materials are in perfect thermal contact (Fig. 2.9a), the temperature of the surfaces at the interface will be equal to each other, that is,

Eq. 2.27

If the contact is not thermally perfect (Fig. 2.9b), some form of thermal contact resistance or heat transfer coefficient (hc) must be introduced at the interface as:

ss

cc trTtrT

qh),(),( 21

= Eq. 2.28

Where hc is experimentally determined.



(a) (b)

Fig. 2.9 - (a) Ideal (perfect) thermal contact; (b) Actual (imperfect) thermal contact.

At the interface, the heat flux is determined by:

ssc n

TknTkq

== 2211 Eq. 2.29

and the temperature difference is given by:



c

css h

qtrTtrT = ),(),( 21 Eq. 2.30

The definitions of this two heat transfer coefficients (h and hr) make possible approximately decouple the conjugate heat transfer phenomenon and study only the method to solve the conduction heat transfer problems.

These coefficients are determined by the solution of their respective mathematical model:

convective system of nonlinear partial differential equations. radiative - system of nonlinear integro-differential equations.



Chapter 3 NUMERICAL CONDUCTION HEAT TRANSFER FUNDAMENTALS.

Objective - this chapter presents basic fundamentals of numerical heat conduction analysis.



3.1 INTRODUCTION The process to solve a theoretical conduction heat transfer problem involves:

physical problem features to be considered. obtainment of differential equations (energy conservation and Fourier Law). initial conditions (t = 0) and boundary conditions (coupling of the domain with the

neighboring environment);

To obtain an approximate solution it is necessary to use a discretization method which approximates the differential equations by a system of algebraic equations, which can then be numerically solved.

The approximations are applied to small domains in space and/or time so the numerical solution provides results at discrete locations in space and time. These most usual techniques are: finite difference, finite volume and finite element.

At this context, the numerical heat transfer technique can be used and three steps are required:

1. pre-processor: a. drawing of geometric domain (space occupied by the body); b. grid generation (domain subdivision in small finite elements or finite

volumes); c. physics, mathematical and numerical setups (mathematical model, solid

properties, numerical approach, boundaries and initial conditions);



2. processor: computational solution of the algebraic equations system obtained after the governing equation discretization;

3. post-processor: quantitative and qualitative (visualization) analysis of the obtained results.



3.2 MATHEMATICAL FORMULATION It comprises: domain delimitation, differential equation, initial and boundary conditions.

Heat conduction occurs inside the body shown in Fig. 3.1 with the external environment interaction represented by appropriated boundary conditions.

Fig. 3.1 Domain (V) and boundary conditions on a heat-conducting body ( = T = temperature field). In the body interior, the first law of thermodynamic and Fourier law are represented by the heat conduction differential equation given by:

V



inside V volume: 0q)Tk(tTc ep = Eq. 3.1

with the following boundary conditions (prescribed by the body-environment interaction):

at S1 surface: T = Ts Eq. 3.2a

at S2 surface: ns

qnTk =

2

(assuming entering heat flux as positive) Eq. 3.2b

at S3 surface: 3

3sfs

)TT(hnTk = Eq. 3.2c

The exact solution of the above problem is represented by a temperature field T(x,y,z,t). When this function is substituted in Eq. 3.1 and Eq. 3.2a to Eq. 3.2c, the left hand side is equal to the right one of these equations. This formulation is known in the literature as strong form.

In the numerical method, an approximated solution to the above problem, )t,z,y,x(T~ , is obtained by the procedure described in the next item.



3.3 NUMERICAL SOLUTION PROCEDURE. It comprises: discretization method for the differential equation, solid modelling, mesh generation and solution of the algebraic equations system.

The first step to obtaind the weak form of the strong form (Eq. 3.1 and Eq. 3.2a to Eq. 3.2c) is to substitute the approximated function )t,z,y,x(T~ in Eq. 3.1, resulting:

=

ep q)T~k(

tT~c

Eq. 3.3

where the residual () represents the error incurred by approximating T(x,y,z,t) with )t,z,y,x(T~ in the governing differential equation.

The second step is to enforce nulls values to several weighted-residual average residual over the entire domain, expressed by:

0)( = dVWV

j , j = 1,2,, N Eq. 3.4

0)~(~

=

dVqTktTcWV epj , j = 1,2,, N Eq. 3.5

Where Wj = linearly independent weighting functions.



The weighted-residuals method is a general procedure for obtaining approximate solutions of linear and nonlinear differential equations.

The term )~( TkWj can be expanded as:

)~()()~()~( TkWTkWTkW jjj += Eq. 3.6

Employing Eq. 3.6, the weak form of the heat conduction equation is:

0)~()()~(~

=

+

dVTkWTkWqtTcWV jjepj , j = 1,2,,N Eq. 3.7

( )[ ]

+

S

jV

jepj dSsTkWdVTkWqtTcW r~)~()(~

j = 1,2,,N Eq. 3.8

where the Greens theorem was employed to obtain Eq. 3.8 from Eq. 3.7.

The next step is the mesh generation that comprises the domain (V) representation by a large number of small subvolumes (finite volumes or finite elements) as shown in Fig. 3.2.

In the finite volume method, the weighting function, Wj (j = P node) is equal to 1 inside the control volume (finite volume), Fig. 3.3, and it is null otherwise. Then, Eq. 3.8 becomes:



Fig. 3.2 Example of a 3-D body discretized by tetrahedrals.

( ) 0~~ =

jj SV

ep dSsTkdVqtTc r , j = 1,2,,M Eq. 3.9

where: Vj = domain of control volume around the j node. (j = P node, in Fig. 3.3), Sj = surface of volume Vj and M = mesh node number.



Fig. 3.3 Control volume around P node of a two-dimensional triangular mesh.

In the finite element method, the temperature field is evaluated by the function )t,z,y,x(T~ , Eq. 3.10, inside the domain:

),,()(~),,,(~1

zyxntTtzyxTM

iii

== , i = 1,2,,M Eq. 3.10

Where Ti = temperature at i= P node of the mesh; ni = interpolation function associated with i node, Fig. 3.4. The interpolation function, ni, must have the following properties:

unitary value at i node; null outside of the elements that doesnt contain i node; null at all nodes except i.



Fig. 3.4 - Tree types of ni (interpolatating function): 1 boundary corner node; 2 boundary side node and 3 interior node.

Also, in finite element, the weighting function is equal to nj ( = 0 for all nodes located at boundaries where the value of T is specified and = 1 elsewhere). Substituting Eq. 3.10 into Eq. 3.8 and replacing Wj with nj results:

( ) ( ) 0~)~(~,,

=

+

jeje S

jV

jepj dSsTkndVTknqtTcn r j = 1,2,,M Eq. 3.11

P



Where: ni = nj when i = j; jeV , = volume of all elements that contains node j and jeS , = outside surface (boundary) of all elements that contains node j.

Eq. 3.9 (finite volume) and Eq. 3.11 (finite element) convert the partial differential equation (Eq. 3.1) into an algebraic equation linking the j = P node temperature with its neighboring nodes temperature.

When this procedure is applied to all mesh nodes, an algebraic equation system, Fig. 3.5, is obtained that is solved employing direct (Gausss elimination method) or iterative schemes (conjugate gradient method, splitting methods, multigrid methods, Lanczos method, GMRES).

Fig. 3.5 Example of an algebraic equation system for unknown nodes temperatures ( ti ).



In the last step, the results are processed to obtain derived quantities as heat flux, heat rate and different visualizations (domain geometry and mesh display, vector plots, line and shaded plots, 2D and 3D surface plots), as shown in Fig. 3.6.

(a)

(b)

(c)

(d)

Fig. 3.6 Examples of results post-processing: (a) mesh; (b) isothermals; (c) heat flux and (d) three points temperature profiles.



In two subsequent chapters, the finite volume and finite element methods will be applied to solve examples of steady state one-dimensional heat conduction problems.



Chapter 4 FINITE VOLUME METHOD.

Objective - this chapter presents an example of finite volume method application.



4.1 INTRODUCTION The first step in the finite volume method is to divide the domain into discrete control volumes (grid generation), as shown in Fig. 4.1. Let us place a number of nodal points in the space between A and B. The boundaries (or faces) of control volumes are positioned mid-way between adjacent nodes. Thus each node is surrounded by a control volume or cell. It is common practice to set up control volumes near the edge of the domain in such a way that the physical boundaries coincide with the control volume boundaries.

Fig. 4.1 Finite volume method notation.

At this point it is appropriate to establish a system of notation that can be used in future developments. The usual convention of CFD methods is shown in Fig. 4.2. A general nodal point is identified by P and its neighbours in a one-dimensional geometry, the nodes to the west and east, are identified by W and E, respectively. The west side face of the control



volume is referred to by w and the east side control volume face by e. The distances between the nodes W and P, and between nodes P and E, are identified by xWP and xPE respectively. Similarly the distances between face w and point P and between P and face e are denoted by xwP and xPe respectively. Fig. 4.2 shows that the control volume width is x = xwe.

Fig. 4.2 Example of a control volume for a 1-D geometry.

The key step of the finite volume method is the integration of the governing equation (or equations) over a control volume to yield a discretised equation at its nodal point P. The governing equation (Eq. 3.9) can be rewritten as:

( ) 0dSnTkdVSii SV

=+ r , i = 1,2,,M Eq. 4.1



Employing the notation indicated in Fig. 4.2, Eq. 4.1 can be expressed by:

( ) +

+=

+=+ dSiidxdTkdVSdSnidxdTkdVSdSnTkdVS eVSVSV iiiiirrrrr

+ dS)i(idxdTk

w

rr

= we

AdxdTkA

dxdTkVS + = 0

Eq. 4.2

where A is the cross-sectional area of the control volume face, V is the volume and S is the average value of source S over the control volume.

It is a very attractive feature of the finite volume method that the discretised equation has a clear physical interpretation. Eq. 4.2 states that the diffusive flux of the temperature field T leaving the east face minus the diffusive flux of T entering the west face is equal to the generation of T, i.e. it constitutes a balance equation for T over the control volume.

It is important to note that Eq. 4.2 can represent the steady-state diffusion of an arbitrary property with a diffusion coefficient . In order to derive useful forms of the discretised equations, the interface diffusion coefficient and the gradient d/dx at east (e) and west (w) are required.

Following well-established practice, the values of the property and the diffusion coefficient are defined and evaluated at nodal points. To calculate gradients (and hence fluxes) at the control volume faces an approximate distribution of properties between nodal points is used. Linear approximations seem to be the obvious and simplest way of calculating interface values



and the gradients. This practice is called central differencing. In a uniform grid linearly interpolated values for e and w are given by:

Eq. 4.3

And the diffusive flux terms are evaluated as

Eq. 4.4

In practical situations, as illustrated later, the source term S may be a function of the dependent variable. In such cases the finite volume method approximates the source term by means of a linear form:

Eq. 4.5

Substitution of equations Eq. 4.3 and Eq. 4.4 into Eq. 4.2 gives

Eq. 4.6



This can be re-arranged as

Eq. 4.7

Identifying the coefficients of W and E in Eq. 4.7 as aW and aE, and the coefficient of P as aP, the above equation can be written as:

Eq. 4.8

Where:

The values of Su and Sp can be obtained from the source model (Eq. 4.5). Note that Eq. 4.8 represents the discretised form of Eq. 3.1.



4.1.1 WORKED-EXAMPLES: ONE-DIMENSIONAL STEADY-STATE DIFFUSION

Consider the one-dimensional steady-state heat conduction in a slab of thickness L = 2 cm with uniform heat generation q = 1000 kW/m3, as schematized in Fig. 4.3. The thermal conductivity is constant, k = 0.5 W/m/K.

Faces A and B are at temperatures of 100 C and 200 C, respectively.

Assume that the dimensions in the y- and z-directions are so large and that temperature gradients are significant only in the x-direction.

Fig. 4.3 Diagram of a 1-D heat conduction problem in a slab with internal heat generation.

The domain is divided into five control volumes (see Fig. 4.4) giving x = 0.004 m; a unit area is considered in the y-z plane.

Fig. 4.4 Grid used to solve the 1-D stady-state conduction in a slab.



The governing equation and boundary conditions are given by Eq. 3.1 and Eq. 3.2a, respectively, with its corresponding discretized form, Eq. 4.2, where the source term is evaluated by calculating the average generation (i.e. S V = qV) within each control volume, resulting that:

Eq. 4.9

For each one of nodes 2, 3 and 4 temperature values to the east and west are available as nodal values. So, Eq. 4.9 can be written as:

Eq. 4.10

or

Eq. 4.11

Eq. 4.11 can be compared with Eq. 4.7 and since ke = kw = k, we have the following coefficients:



With: Nodes 1 and 5 are boundary nodes, and therefore require special treatment. To incorporate the boundary conditions at nodes 1 and 5 we apply the linear approximation for temperatures between a boundary point and the adjacent nodal point. At node 1 the temperature at the west boundary is known. Integration of Eq. 4.1 at the control volume surrounding node 1 and the introduction of the linear approximation for temperatures between A and P yields

Eq. 4.12

The above equation can be re-arranged, using ke = kA = k, to yield the discretised equation for boundary node 1:

With:

At nodal point 5, the temperature on the east face of the control volume is known. The node is treated in a similar way to boundary node 1. At boundary point 5 we have



Eq. 4.13

The above equation can be re-arranged, noting that kB = kw = k, to give the discretised equation for boundary node 5:

With:

Substitution of numerical values for A = 1, k = 0.5 W/m/K, q = 1000 kW/m3 and x = 0.004 m everywhere gives the coefficients of the discretised equations summarised in Table 4.2. Tab. 4.1 Grid nodes and the coefficients values of the discretised equation.

Given directly in matrix form the equations are:



The solution to the above set of equations is

4.1.1.1 Comparison with the analytical solution. The analytical solution to this problem may be obtained by integrating Eq. 4.1 twice with respect to x and by subsequent application of the boundary conditions. This gives

Eq. 4.14

The comparison between the finite volume solution and the exact solution is shown in Tab. 4.2 and Fig. 4.5 and it can be seen that, even with a coarse grid of five nodes, the agreement is very satisfactory.



Tab. 4.2 Numerical and exact solutions for the heat conduction in a slab with source-term.

Fig. 4.5 - Comparison of the numerical results with the analytical solution (heat diffusion with internal heat generation).

4.1.1.2 Case without internal heat generation. If the source-term in Eq. 3.1 is null, this equation reduces to:



Eq. 4.15

where the analytical solution is given by Eq. 4.14 with q = 0:

AAB Tx

LTTT +

= Eq. 4.16

On the other hand, the numerical solution is similar to the case with internal heat generation (Eq. 4.2) but with S = 0.

Consider the problem of source-free heat conduction in an insulated rod whose ends are maintained at constant temperatures of 100 C and 500 C respectively, as schematized Fig. 4.6. Let us divide the length of the rod into five equal control volumes as shown in Fig. 4.7. This gives x = 0.1 m.

Fig. 4.6 - Diagram of a 1-D heat conduction problem in a rod without internal heat generation.

The grid consists of five nodes. For each one of nodes 2, 3 and 4 temperature values to the east and west are available as nodal values.



Fig. 4.7 Grid used to solve the 1-D stady-state conduction in a rod.

Consequently, discretised equations can be readily written for control volumes surrounding these nodes as:

Eq. 4.17

The thermal conductivity (ke = kw = k), cross-sectional area (Ae = Aw = A) and node spacing (x) are constants. Therefore the discretised equation for nodal points 2, 3 and 4 is:

With:

Su and Sp are zero in this case since there is no source term in the governing Eq. 4.15. Nodes 1 and 5 are boundary nodes. Therefore, the integration of Eq. 4.15 over the control volume surrounding point 1 gives:



Eq. 4.18

This expression shows that the flux through control volume boundary A has been approximated by assuming a linear relationship between temperatures at boundary point A and node P. We can re-arrange Eq. 4.18 as follows:

Eq. 4.19

Comparing Eq. 4.18 with Eq. 4.7 it can be easily identified that the fixed temperature boundary condition enters the calculation as a source term (Su + SpTp) with Su = (2kA/x)TA and Sp = -2kA/x and that the link to the (west) boundary side has been suppressed by setting coefficient aw to zero.

Eq. 4.18 may be cast in the same form as Eq. 4.8 to yield the discretised equation for boundary node 1:

With:

The control volume surrounding node 5 can be treated in a similar manner. Its discretised equation is given by



Eq. 4.20

As before, it is assumed a linear temperature distribution between node P and boundary point B to approximate the heat flux through the control volume boundary. Eq. 4.20 can be re-arranged as

Eq. 4.21

The discretised equation for boundary node 5 is

With:

The discretisation process has yielded one equation for each of the nodal points 1 to 5. Substitution of numerical values gives kA/x = 100 and the coefficients of each discretised equation can easily be worked out. Their values are given in Tab. 4.3.



Tab. 4.3 Grid nodes and the coefficients values of the discretised equation.

The resulting set of algebraic equations for this example is

This set of equations can be re-arranged as



The above set of equations yields the steady state temperature distribution of the given situation. The resulting matrix equation can be solved by using, for example, Gaussian elimination. For TA = 100 and TB = 500 the solution is

The exact solution is a linear distribution between the specified boundary temperatures (see Eq. 4.16): T = 800 x + 100. Fig. 4.8 shows that the exact solution and the numerical results coincide.

Fig. 4.8 - Comparison of the numerical result with the analytical solution (heat diffusion without internal heat generation).



Chapter 5 FINITE ELEMENT METHOD.

Objective - this chapter presents the application of finite element method to solve onedimensional heat conduction problem.



5.1 INTRODUCTION.

The finite element method (FEM), sometimes referred to as finite element analysis (FEA), is a computational technique used to obtain approximate solutions of boundary value problems in science. A boundary value problem is a mathematical problem in which one or more dependent variables must satisfy differential(s) equation(s) everywhere within a known domain of independent variables and satisfy specific conditions on the boundary of the domain. Boundary value problems are also called field problems. The field is the domain of interest and most often represents a physical body. The field variables are the dependent variables of interest governed by differential(s) equation(s). The boundary conditions are the specified values of the field variables (or related variables such as derivatives) on the boundaries of the field. Depending on the type of physical problem bein analyzed, the field variables may include temperature, heat flux, fluid velocity, pressure



5.2 ONE-DIMENSIONAL HEAT CONDUCTION EXAMPLE. Application of the Galerkin finite method to the problem of one-dimensional, steady-state heat conduction is developed with reference to Fig. 5.1, which depicts a solid body undergoing heat conduction in the direction of the x axis only. Surfaces of the body normal to the x axis are assumed to be perfectly insulated, so that no heat loss occurs through these surfaces.

Fig. 5.1 One-dimensional heat conduction in a circular rod.

5.2.1 DIFFERENTIAL FORMULATION (STRONG FORMULATION). Assuming that the heat conduction process is in steady state (T/t = 0) and that the thermal conductivity is constant, Eq. 3.1, for steady-state, one-dimensional conduction becomes:



0)( 22

=+=+ ee qx

TkqTk Eq. 5.1

In this example, the extreme right end of the cylinder is maintained at a temperature of Tb, while the left end is subjected to a heat input rate qa. These two boundary interations can be mathematically represented by the following boundary conditions (using Eq. 3.2a and Eq. 3.2a):

aa

bb

qxTk

nTkxxat

TTxxat

==

===

Eq. 5.2

5.2.2 WEIGHTED RESIDUAL FORMULATION (WEAK FORMULATION). Eq. 3.11, can now be simplified for this one-dimensional problem. For this steady-state one-dimensional conduction:

( ) ( )[ ] 0~)~(~ =

+

S

jV

jepj dSsTkndVTknqtTcn r j = 1,2,,M+1 Eq. 3.11

0~=

tT ; x

jj ex

nn r

= )( ; xexTT r=

~)~( ; dxAdV = Eq. 5.3



Substituting Eq. 5.3 in Eq. 3.11, the weak form formulation for this one-dimensional conduction is:

0)(~

)~

( =

+ AeexTkndxAexTkexnqna

b

ax

xxj

x

x xxj

ejrrrr j = 1,2,,M+1 Eq. 5.4

where A = rod cross-section area and xesrr = at axx = .

Or, after the dot product:

0~

)~

( =

+

+

a

b

ax

j

x

x

jej x

TkndxxTk

xn

qn j = 1,2,,M+1 Eq. 5.5

Now a mesh is generated, that is, the problem domain is divided into M elements ( e = 1,2,M ), Fig. 5.2, bounded by (M+1) values xi of the independent variable, so that x1 = xa and xM+1 = xb to ensure inclusion of the global boundaries.

Fig. 5.2 Domain ba xxx discritized into M elements.



Each element e contains several nodes, Fig. 3.2. In this ease onedimensional problem, Fig. 5.2, each element has only two nodes. Note that, element e extend from xe to xe+1 in this simple onedimensional problem, each element e contains two nodes placed at xe and xe+1. In two and threedimensional problems, there isnt a simple relationship between the element and its nodes. This information is stored in the connectivity matrix linking each element and its nodes.

An approximate solution is assumed in the form:

)(~)(~1

1xnTxT

M

iii+

== , i = 1,2,,M+1 Eq. 5.6

where iT~ is the approximate value of T at x = xi and ni(x) is a corresponding interpolation

(trial) function. These interpolation functions are nonzero over only the elements that contains the node i, as shown in Fig. 5.3 (note the overlap of only two interpolation functions in each element).



Fig. 5.3 First four interpolation functions.

Specifically, for an interior node, a trial function ni(x) is nonzero only in the interval 1i1-i x x x +



As the trial functions are linear interpolation functions in this example, the value of the solution )(~ xT in xi < x < xi+1 is a linear combination of adjacent nodal values iT

~ and 1~+iT .In

the interval x3 < x



Using nj (ni) properties (Fig. 3.4, Fig. 5.3 and Eq. 5.7) there are three particular cases for Eq. 5.10: j = 1, Mj 2 and j = M+1: Case j = 1 (n1 is non-null for 21 xxx - element: e = 1): For node j = 1:

1=

12

21 )( xx

xxxn = , 21 xxx - element: e = 1

21 0)( xxwhenxn >=

boundary condition at x = x1 = xa (Eq. 5.2 ax

qxTk

a

= )

12

12

12

21

1

11

~~)(~)(~xxxxT

xxxxTxnTxelT

M

iii

+== +

= when 21 xxx

[ ] 0))(1(1~1~121 12

212

11212

2 =+

+

+

axx e qdxxxTkxxTkxxqxx xx j

= 1

Eq. 5.11

Integrating this equation, a finite element discretized form for Eq. 5.1 is obtained:



( ) ( )( )

+

=

ae qxxqT

xxkT

xxk

2~~ 12

212

112

j = 1 Eq. 5.12

Case Mj 2 (nj is non-null for 11 + jj xxx - elements: e = j-1 and e = j): For this node range:

1= jj

jj

jj xxxwhenxx

xxxn

=

1

1

1)( - element e = j-1

11

1)( ++

+



01~1~1

1~1~1

1

1

11

111

1

111

11

1

=

+

+

+

+

+

+

++

+++

+

dxxx

Tkxx

Tkxx

qxxxx

dxxx

Tkxx

Tkxx

qxxxx

j

j

j

j

x

xjj

jjj

jjj

ejj

j

x

xjj

jjj

jjj

ejj

j

j = 2,3,,M Eq. 5.13

Integrating this equation, a finite element discretized form for Eq. 5.1 is obtained:

( ) ( ) ( ) ( )( ) ( )

+=

=

++

+

+++

22

~~~

11

1111

11

jjjje

jjj

jjjjj

jjj

xxxxq

Txx

kTxx

kxx

kTxx

k

j = 2,3,,M Eq. 5.14

Case j = M+1(nM+1 is non-null for 1+ MM xxx - element: e = M) Due to specified temperature boundary condition at x = xb the value of is null ( 0= ). Here j = M+1 node is retained to shown the boundary condition imposed if nothing is specified at a domain surface in the finite element method. This boundary condition is called natural boundary condition. For node j = M+1:

0=



MM

MM xx

xxxn =+

+1

1 )( when 1+ MM xxx - element: e = M

MM xxwhenxn



5.2.3 WORKED-EXAMPLE 1: CIRCULAR ROD STEADY-STATE CONDUCTION. Now a numeric example will be solved using the above formulation. The circular rod depicted in Fig. 5.1 has an outside diameter of 60 mm, length of 1 m, and is perfectly insulated on its circumference. The left half of the cylinder is aluminum, for which k = 200 W/m-C and the right half is copper having k = 389 W/m-C. The extreme right end of the cylinder is maintained at a temperature of 80C, while the left end is subjected to a heat input rate 4000 W/m2. Due to Joule effect (electrical to internal energy conversion), this circular rod is subject to a uniform heat generation q = 1000 W/m3. Using four equal-length elements, the steady-state temperature distribution in the cylinder will be determined. For this problem: M = 4, x1 = 0.00 m, x2 = 0.25 m, x3 = 0.50 m, x4 = 0.75 m, x5 = 1.00 m, k = 200 W/moC for e = 1 - 2, k = 389 W/moC for e = 3 4, 2/4000 mWqa = and qe = 0. For j = 1, Eq. 5.12:

4000225.01000~

25.0200~

25.0200

21 +=

TT j = 1 Eq. 5.18

For j = 2, Eq. 5.14:

225.01000

225.01000~

25.0200~

25.0200

25.0200~

25.0200

321 +=

++

TTT j = 2 Eq. 5.19



For j = 3, Eq. 5.14:

225.01000

225.01000~

25.0389~

25.0389

25.0200~

25.0200

432 +=

++

TTT j = 3 Eq. 5.20

For j = 4, Eq. 5.14:

225.01000

225.01000~

25.0389~

25.0389

25.0389~

25.0389

543 +=

++

TTT j = 4 Eq. 5.21

For j = 5, Eq. 5.16:

80~1 5 =T j = 5 Eq. 5.22

Now, the following algebraic equation system can be assembled:

=

++

++

802502502504125

~~~~~

10000155631121556000155623568000008001600800000800800

5

4

3

2

1

TTTTT

Eq. 5.23



Solving this algebraic equation system, approximate values for temperature distribution are obtained:

=1~T 96.730 oC; =2~T 91.574 oC; =3~T 86.105 oC; =4~T 83.133 oC; =5~T 80.000 oC Eq. 5.24

5.2.4 WORKED-EXAMPLE 2: SLAB STEADY-STATE CONDUCTION.

Consider the one-dimensional steady-state heat conduction in a slab of thickness L = 2 cm with uniform heat generation qe = 1000 kW/m3, as schematized in Fig. 5.4. The thermal conductivity is constant, ke = 0.5 W/m/K. Faces A and B are at temperatures of 100 C and 200 C, respectively.

Assume that the dimensions in the y- and z-directions are so large and that temperature gradients are significant only in the x-direction.

Fig. 5.4 Diagram of a 1-D heat conduction problem in a slab with internal heat generation.

The domain is divided into five control volumes (see Fig. 5.5) giving x = 0.004 m; a unit area is considered in the y-z plane.



Fig. 5.5 Grid used to solve the 1-D stady-state conduction in a slab.

This problem was solved in Chapter 4 using finite volume technique. To compare results, now it is solved by finite element method using nodes at same x coordinates: x0 = 0.000 m; x1 = 0.002 m; x2 = 0.006 m; x3 = 0.010 m; x4 = 0.014 m; x5 = 0.016 m and x6 = 0.020 m. Due to specified temperature boundary conditions at x0 and x6 , 0= at these two boundaries. For j = 0, Eq. 5.16:

100~1 0 =T j = 0 Eq. 5.25

For j = 1, Eq. 5.14:

+=

++

2004.0

2002.0101000~

004.05.0~

004.05.0

002.05.0~

002.05.0 3

210 TTT j = 1 Eq. 5.26

For j = 2, Eq. 5.14:

+=

++

2004.0

2004.0101000~

004.05.0~

004.05.0

004.05.0~

004.05.0 3

321 TTT j = 2 Eq. 5.27



For j = 3, Eq. 5.14:

+=

++

2004.0

2004.0101000~

004.05.0~

004.05.0

004.05.0~

004.05.0 3

432 TTT j = 3 Eq. 5.28

For j = 4, Eq. 5.14:

+=

++

2004.0

2004.0101000~

004.05.0~

004.05.0

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