Cover Page

Energy Engineering Set

coordinated by
Abdelhanine Benallou

Volume 2

Energy Transfers by Conduction

Abdelhanine Benallou



“Never regret the time that was needed for doing good.”
Joseph Joubert,
“De la sagesse [Wisdom]”,
Pensées, essais et maximes [Thoughts, Essays and Maxims],
Chapter IX, 1866

For several years, I have cherished the wish of devoting enough time to the writing of a series of books on energy engineering. The reason is simple: for having practiced for years teaching as well as consulting in different areas ranging from energy planning to rational use of energy and renewable energies, I have always noted the lack of formal documentation in these fields to constitute a complete and coherent source of reference, both as a tool for teaching to be used by engineering professors and as a source of information summarizing, for engineering students and practicing engineers, the basic principles and the founding mechanisms of energy and mass transfers leading to calculation methods and design techniques.

But between the teaching and research tasks (first as a teaching assistant at the University of California and later as a professor at the École des mines de Rabat, Morocco) and the consulting and management endeavors conducted in the private and in the public sectors, this wish remained for more than twenty years in my long list of priorities, without having the possibility to make its way up to the top. Only providence was able to unleash the constraints and provide enough time to achieve a lifetime objective.

This led to a series consisting of nine volumes:

The present book is the second volume of this series. It concerns the study of conduction heat transfer.

As we will see, the calculation methods established in this book present multiple applications in engineering: heat exchanger design and sizing, energy conservation, thermal insulation, metal tempering, etc.

A series of exercises is presented at the end of the book, aimed at enabling students to implement new concepts as rapidly as possible. These exercises are designed to correspond as closely as possible to real-life situations occurring in industrial practice or everyday life.

Abdelhanine BENALLOU
June 2018


Conduction is one of the most important heat exchange mechanisms. It occurs by contact-continuity between systems. Energy is transferred, step by step through a medium, by transmission of the excitations of an atom to the neighboring atoms.

The present volume from the Energy Engineering set is devoted to studying heat conduction in detail. It highlights the equations that govern the phenomenon, and then applies them to a number of practical situations. In this context, special attention is paid to calculation methods used in analyses aiming at energy conservation. This involves, for example, calculating energy fluxes dissipated through the inner walls of industrial furnaces, financial assessment of heat losses and the choice of the economic thickness for thermal insulation.

Within the same context, great importance is placed in calculations allowing the implementation of regulations of thermal installations in buildings which define building energy efficiency requirements that are mandated by different countries. Implementation of these regulations usually involves determining the heat losses passing through the walls of heated or air-conditioned buildings.

Such applications are not only important to observe applicable laws and regulations, but they generally prove to be instrumental in researching energy savings in furnaces, steam pipes, reactors, freezing circuits, etc.

Chapter 1 of this volume is dedicated to establishing general equations translating energy balances in conduction, in different coordinate systems (plane, cylindrical and spherical). Applying a balance approach, conduction equations are put in place for the most general situations: systems in transient or stationary state, with or without internal energy generation.

Subsequently (Chapter 2), these equations are used to solve a number of physical problems of interest to the engineer: heat transfer through a simple, plane wall, loss through a multi-layer wall composed of several inner walls and transfer through cylindrical jackets (steam pipes, for example) or spherical containers (storage reservoirs). The choice of the different application examples presented in this chapter was made based on problems usually encountered in industrial practices or in sustainable-development approaches.

Chapter 3 of this volume is reserved for thermal insulation. Within this context, special attention has been devoted to heat lagging materials, given their importance in actions aiming at reducing energy loss. The appendix provided at the end of this document presents a set of data that offers guidance in choosing which insulation to use. Moreover, this data enables the reader to perform the calculations required in order to validate a lagging material for a given project. Indeed, the technical and economic analyses carried out in this chapter show that thermal insulation can lead to important reductions in overall production costs.

Chapter 4 is reserved for thermal analyses of buildings where the study of heat losses through the walls (which may or not be glazed) of a building leads to the estimation of the annual costs of these energy losses. These analyses are conducted bearing in mind that, in different countries, new thermal regulations impose rigorous constraints on the permitted heat losses of new constructions.

Next, Chapter 5 examines conduction problems in the presence of energy generation. They correspond to practical situations that are encountered in exothermic reactors or in fuel tubes of nuclear reactors, or in radiators used to dissipate surplus heat resulting from electronic circuit operation.

Chapter 6 details the different methods for solving conduction problems in transient state. They provide answers to common engineering problems, particularly in the field of metal tempering (Cengel, 1998).

The last chapter of this volume is devoted to presenting a number of exercises and their solutions. Note that when choosing these exercises, our aim was to stay as close as possible to the reality of industrial operations and to practical energy engineering problems.

Fundamental Equations of Conduction

1.1. Introduction

In this chapter we explore the general equations that reflect energy balances in conduction. These equations will then be used to solve a number of physical problems of interest to the engineer.

Equations governing conduction are extremely useful, both for quantifying energy losses in industrial installations (steam pipes, furnaces, reactors, etc.) and for the application of thermal building regulations. In each case, a comprehensive implementation of these equations makes it possible to calculate the fluxes of energy losses if no action was taken. The analysis methods presented in the following sections offer the possibility of quantifying energy flows, as well as indicating the actions to implement in order to limit energy loss.

For each of the situations that are explored, special attention is paid to the practical applications of the equations established. Thus, problems relating to the thermal insulation of industrial installations are approached both technically, by assessing the fluxes of energy losses, and with regard to the profitability of the investments to be made in order to achieve thermal insulation. For each of the situations explored, special attention is paid to the practical applications of the equations established. Thus, thermal insulation problems are concretely dealt with by means of technical and economic analyses, which demonstrate the benefit in terms of reducing production costs.

In this chapter, the problems relating to conduction in the presence of energy generation are also examined. They correspond to practical situations encountered in exothermic reactors or in nuclear-reactor fuel bars.

1.2. General equations of conduction

Consider a solid, of arbitrary shape, in which we assume that there is conduction heat transfer.

In order for the development to be general, we will assume that there is heat generation in the solid, and that P is the generation power per unit volume.

Let us establish an energy balance on a microscopic element of this solid with sides dx, dy and dz (see Figure 1.1).

The general balance equation is as follows:


which can be written simply as:


Figure 1.1. Differential energy balance

1.2.1. Expressing the term (I O)

In the most general case, energy flows in the three directions, ox, oy and oz. Consequently, its flux entering or exiting the solid element considered will have three components: one component for each direction of propagation. These three components will be indicated as follows:

ϕx: flux in direction x
ϕy: flux in direction y
ϕz: flux in direction z

The flux vector is then written:image. Expressing ϕx

Conduction heat input is the energy flux in the direction x, expressed at abscissa x. It is given by: image.

Similarly, the output corresponds to the energy flux in the direction x, expressed at abscissa x + dx: image.

Hence: image.

Therefore: imagedxdydz. Expressing ϕy

Using a similar development we obtain: image dxdydz. In direction z

We have: image dxdydz. For the three directions

When we consider the flow of energy in the three propagation directions, the term (I – O) is then given by: I − O = (I − O)x + (I − O)y + (I − O)z.

Therefore: image.

1.2.2. The term “generation”

P being the volumetric generation power, the energy generated in the volume element dxdydz is given by: ϕG = P dx dy dz.

1.2.3. The term “accumulation”

If m designates the mass of the volume element considered and if Cp is the sensible heat of the material constituting it, the energy accumulated in this element is given by: image

The mass, m, can be expressed as follows: m = ρ dx dy dz.

Therefore: image.

1.2.4. Energy balance equation

The balance equation can then be written as follows for isotropic materials:


This energy balance equation is often referred to as the general conduction equation.

1.3. Equations of conduction in different coordinate systems

Anisotropic or orthotropic materials (variable thermal conductivity λ depending on the direction) are sometimes encountered in engineering calculations. This is particularly true when energy is called upon to propagate in wood or in salts of nonhomogeneous crystalline structures. For such situations, the forms of the conduction equation that take account of this variability in thermal conductivity are to be used.

Yet in practice, we mainly encounter isotropic materials, that is, materials whose thermal conductivity, λ, is the same in all directions. Indeed, this is often the case for metals and alloys usually encountered in industrial equipment and devices. However, even in the isotropic case, λ is not necessarily constant. It is identical in all directions, but can vary with the temperature, for example. For this reason, this section is dedicated to presenting equations of conduction for the isotropic case, with either variable or constant thermal conductivities.

Moreover, the geometry of the solid determines which coordinate system to use. Thus, when the solid studied is a parallelepiped shape, for example, it makes sense to choose the Cartesian coordinates for analysis.


Figure 1.2. The coordinate system depends on the shape of the solid considered

Likewise, when the system studied is a pipe or a ball, we will opt for cylindrical or spherical coordinates, respectively.

The following sections present the equations of conduction in different coordinate systems for isotropic cases, but whose thermal conductivities can be either variable or constant.

1.3.1. When λ is not constant

  • – Rectangular coordinates:
  • – Cylindrical coordinates (r, θ, z):
  • – Spherical coordinates (r, θ, ψ):

1.3.2. When λ is constant

  • – Rectangular coordinates:
  • – Cylindrical coordinates (r, θ, z):
  • – Spherical coordinates (r, θ, ψ):

1.3.3. Simplified cylindrical and spherical coordinates

Very often, temperature variations are confined to only two directions, and sometimes even only one. The latter cases are frequently encountered when the heat propagates from the center of a pipe to the outside under the action of a large temperature gradient in the r direction, while only a small gradient exists in the longitudinal direction. This also happens in the case of a sphere that is in a medium or bath of homogeneous temperature: it is obvious that, in this context, the existence of a temperature gradient between the center of the sphere and the bath will result in heat propagation along r only; due to the symmetry of the problem and the homogeneity of the bath, the other directions will not be involved in the transfer.

In these situations, simplified coordinates r and z are used for cylindrical systems, and r and θ, for spherical systems.

  • – Simplified cylindrical coordinates (r, z):
  • – Simplified spherical coordinates (r, θ):

1.3.4. One-dimensional conduction

In most problems of interest to the engineer, using a single heat-propagation direction is sufficient to be able to describe fairly accurately what happens in practice (Jakob and Hawkins, 1957).

Thus, the equations of conduction presented previously in Cartesian, cylindrical and spherical coordinates can be further simplified when a single propagation direction is retained. This direction is generally x for Cartesian coordinates and r for cylindrical and spherical coordinates, in which case we will speak of one-dimensional or one-directional problems. This type of problem is commonly encountered as a simplification in engineering calculations. In this case, the equations are:

  • – In rectangular coordinates: image
  • – In simplified cylindrical coordinates: image
  • – In simplified spherical coordinates: image

1.4. Reading: metal tempering

Tempering is a heat treatment used in metallurgy to improve the mechanical properties of metals. It enables the hardness or the overall resistance of a given metal to be increased. This treatment generally consists of heating the metal to raise its temperature, then cooling it at a given speed. The thermal stresses thus imposed lead to changes in the metal’s crystalline structure.

As a result, with each metal having a crystalline form within which atoms of a so-called “interstitial” compound can be inserted, the solubility of these atoms in the mesh depends on the structure of the latter and its size. If a metal’s temperature variations are conducted so that the phase change point is approached, they can then lead to changes in the crystalline structure, in particular with respect to its interstitial distances. We will thus pass from a mesh having important sites to a tighter mesh imprisoning interstitial atoms in the new crystalline structure.

Bearing in mind that the mechanical properties are closely related to the geometries of the crystalline meshes, it follows that the different temperature variations induce a stress in these meshes, and thus a change in the mechanical properties of the metal considered.

The origin of metal tempering dates back to the early 20th Century, when blacksmiths subjected the sharpened parts of a tool to this type of heat treatment, in order to increase their mechanical strength. The metal was heated to red before being immersed in water.

The temperature to which the metal is heated before cooling is significant. It is called the “phase change temperature”. It must be chosen in such a way that it allows the “suspension” of the interstitial chemical compounds dissolved in the metal’s crystalline structure.

The heating time, or the time during which the metal is subjected to high temperatures, is also significant. It must be long enough to ensure the transformation of the entire heated mass, but it must not be too long either.

The cooling time, for its part, is also a decisive parameter. Indeed, the latter must allow cooling of the entire mass considered at a great enough speed to lead to the imprisonment of interstitial chemical elements having diffused into the crystalline structure of the solid during the heating phase. This cooling operation is generally performed by soaking the metal part in a cooling fluid. The fluids used are either water or salt water to generate faster cooling.

As we will see in Chapter 6, the cooling speed will depend on the following three elements:

  • – The heat transfer in the metal considered, and therefore its thermal conductivity.
  • – The nature of the fluid used.
  • – The heat transfer between the tempering fluid and the solid, and therefore the convection heat transfer coefficient, h.

In Chapter 6 of this volume we will see how the conduction equations are used to accurately calculate the time required for a certain degree of heat penetration into a solid.

Knowledge of this time period will permit heating the material to a sufficiently high temperature for just long enough and, next, cooling it at an appropriate speed.

In this manner, chemical elements will be imprisoned in the crystalline structure creating, after cooling, tensions in the meshes which will contribute to improving the mechanical resistance of the tempered part.

Although mechanical resistance of metals can be augmented through tempering, it should be noted, however, that this thermal treatment often causes metals to lose their elasticity. Indeed, the modifications induced by tempering in the meshes of the crystalline structure constrain movements of the crystalline irregularities (vacancy defects or dislocations) which are important for elasticity. Thus, tempering results in a loss of the plastic deformation mechanism that allows a metal to return to its resting state following a deformation. Actually, tempering brings a metal’s elastic resistance closer to its failure resistance.