Introduction

We experience the effects of force in everyday life and have an intuitive notion of force. For example, we exert a force on our body when we lift or push an object while we continuously (fortunately) feel the effect of gravitational forces, for instance while sitting, walking, etc. All parts of the human body in one way or the other are loaded by forces. Our bones provide rigidity to the body and can sustain high loads. The skin is resistant to force, simply pull on the skin to witness this. The cardiovascular system is continuously loaded dynamically due to the pulsating blood pressure. The bladder is loaded and stretched when it fills up. The intervertebral discs serve as flexible force transmitting media that give the spine its flexibility. Beside force we are using levers all the time in our daily life to increase the ‘ force ’ that we want to apply to some object, for example by opening doors with the latch, opening a bottle with a bottle-opener. We feel the effect of a lever arm when holding a weight close to our body instead of using a stretched arm. These experiences are the result of the moment that can be exerted by a force. Understanding the impact of force and moment on the human body requires us to formalize the intuitive notion of force and moment. That is the objective of this chapter.

Introduction

In the previous chapters the global behaviour of fibres was considered, without much attention to the detailed shape of these structures. Only the length change of the fibre played a role in the analysis. In the present chapter we address in a little bit more detail the deformation of long slender structures. These can be tendons, muscles, but also long bones. The aim is, to generalize the concepts introduced in previous chapters for discrete systems (i.e. springs) to continuous systems. To simplify matters the loading and deformation of a one-dimensional elastic bar is considered.

Equilibrium in a subsection of a slender structure

Consider a straight bar as visualized in Fig. 6.1(a), loaded by an external force F at x = L and fixed in space at x = 0. The figure shows the bar with the x-axis in the longitudinal or axial direction. It is assumed, that each cross section initially perpendicular to the axis of the bar remains perpendicular to the axis after loading. In fact it is assumed that all properties and displacements are a function of the x-coordinate only.

The objective is, to compute the displacement of each cross section of the bar due to the loading. The area of the cross section perpendicular to the central axis of the bar as well as the mechanical properties of the bar may be a function of the x-coordinate. Therefore the displacement may be a non-linear function of the axial coordinate.

Natural Convection Concepts

Introduction

Chapters 4 and 5 focus on forced convection problems in which the fluid motion is driven externally, for example by a fan or a pump. However, even in the limit of no externally driven fluid motion, a solid surrounded by a fluid may not reduce to a conduction problem because the fluid adjacent to a heated or cooled surface will usually not be stagnant. Natural (or free) convection refers to convection problems in which the fluid is not driven mechanically but rather thermally; that is, fluid motion is driven by density gradients that are induced in the fluid as it is heated or cooled. The velocities induced by these density gradients are typically small and therefore the absolute magnitude of natural convection heat transfer coefficients is also typically small.

The flow patterns induced by heating or cooling can be understood intuitively; hot fluid tends to have lower density and therefore rise (flow against gravity) while cold fluid with higher density tends to fall (flow with gravity). The existence of a temperature gradient does not guarantee fluid motion. Figure 6-1(a) illustrates fluid between two plates oriented horizontally (i.e., perpendicular to the gravity vector g ) where the lower plate is heated (to TH ) and the upper plate is cooled (to TC ). The heated fluid will tend to rise and the cooled fluid fall, resulting in the natural convection “cells” that are shown in Figure 6-1(a).

Introduction to EES

This extended section of the book can be found on the website www.cambridge.org/nellisandklein. EES (pronounced ‘ease’) is an acronym for Engineering Equation Solver. The basic function provided by EES is the numerical solution of non-linear algebraic and differential equations. EES is an equation-solver, rather than a programming language, since it does not require the user to enter instructions for iteratively solving non-linear equations. EES provides capability for unit checking of equations, parametric studies, optimization, uncertainty analyses, and high-quality plots. It provides array variables that can be used in finite-difference calculations. In addition, EES provides high-accuracy thermodynamic and transport property functions for many fluids and solid materials that can be integrated with the equations. The combination of these capabilities together with an extensive library of heat transfer functions, discussed throughout this text, makes EES a very powerful tool for solving heat transfer problems. This appendix provides a tutorial that will allow you to become familiar with EES.

Introduction to maple

This extended section of the book can be found on the website www.cambridge.org/nellisandklein. Maple is an application that can be used to solve algebraic and differential equations. Maple has the ability to do mathematics in symbolic form and therefore it can determine the analytical solution to algebraic and differential equations. Maple provides a very convenient mathematical reference; if, for example, you've forgotten that the derivative of sine is cosine, it is easy to use Maple to quickly provide this information.

Chapter 1 discussed the analytical and numerical solution of 1-D, steady-state problems. These are problems where the temperature within the material is independent of time and varies in only one spatial dimension (e.g., x). Examples of such problems are the plane wall studied in Section 1.2, which is truly a 1-D problem, and the constant cross section fin studied in Section 1.6, which is approximately 1-D. The governing differential equation for these problems is an ordinary differential equation and the mathematics required to solve the problem are straightforward.

In this chapter, more complex, 2-D steady-state conduction problems are considered where the temperature varies in multiple spatial dimensions (e.g., x and y). These can be problems where the temperature actually varies in only two coordinates or approximately varies in only two coordinates (e.g., the temperature gradient in the third direction is negligible, as justified by an appropriate Biot number). The governing differential equation is a partial differential equation and therefore the mathematics required to analytically solve these problems are more advanced and the bookkeeping required to solve these problems numerically is more cumbersome. However, many of the concepts that were covered in the context of 1-D problems continue to apply.

Shape factors

There are many 2-D and 3-D conduction problems involving heat transfer between two well-defined surfaces (surface 1 and surface 2) that commonly appear in heat transfer applications and have previously been solved analytically and/or numerically.

Introduction to Heat Exchangers

Introduction

A heat exchanger is a device that is designed to transfer thermal energy from one fluid to another. The term “heat exchanger” like “heat transfer” is inconsistent with the thermodynamic definition of heat; these devices would be more appropriately called thermal energy exchangers. However, the term “heat exchanger” is ubiquitous. Heat exchangers are also ubiquitous; nearly all thermal systems employ at least one and usually several heat exchangers.

The background on conduction and convection, presented in Chapters 1 through 7, is required to analyze and design heat exchangers. This section reviews the applications and types of heat exchangers that are commonly encountered. Subsequent sections provide the theory and tools required to determine the performance of these devices.

Applications of Heat Exchangers

You may be unaware of just how common heat exchangers are in both residential and industrial applications. For example, you live in a residence that is heated to a comfortable temperature in winter and possibly cooled in the summer. Heating is usually accomplished by combusting a fuel (e.g., natural gas, propane, wood, or oil) that provides the desired thermal energy, but also produces combustion gases that can be harmful. Therefore, your furnace includes a heat exchanger that transfers thermal energy from the combustion gases to an air stream that can be safely circulated through the building.

Conduction Heat Transfer

Introduction

Thermodynamics defines heat as a transfer of energy across the boundary of a system as a result of a temperature difference. According to this definition, heat by itself is an energy transfer process and it is therefore redundant to use the expression ‘heat transfer’. Heat has no option but to transfer and the expression ‘heat transfer’ reinforces the incorrect concept that heat is a property of a system that can be ‘transferred’ to another system. This concept was originally proposed in the 1800's as the caloric theory (Keenan, 1958); heat was believed to be an invisible substance (having mass) that transferred from one system to another as a result of a temperature difference. Although the caloric theory has been disproved, it is still common to refer to ‘heat transfer’.

Heat is the transfer of energy due to a temperature gradient. This transfer process can occur by two very different mechanisms, referred to as conduction and radiation. Conduction heat transfer occurs due to the interactions of molecular (or smaller) scale energy carriers within a material. Radiation heat transfer is energy transferred as electromagnetic waves. In a flowing fluid, conduction heat transfer occurs in the presence of energy transfer due to bulk motion (which is not a heat transfer) and this leads to a substantially more complex situation that is referred to as convection.

The single objective of this book is to provide engineers with the capability, tools, and confidence to solve real-world heat transfer problems. This objective has resulted in a textbook that differs from existing heat transfer textbooks in several ways. First, this textbook includes many topics that are typically not covered in undergraduate heat transfer textbooks. Examples are the detailed presentations of mathematical solution methods such as Bessel functions, Laplace transforms, separation of variables, Duhamel's theorem, and Monte Carlo methods as well as high order explicit and implicit numerical integration algorithms. These analytical and numerical solution methods are applied to advanced topics that are ordinarily not considered in a heat transfer textbook.

Judged by its content, this textbook should be considered as a graduate text. There is sufficient material for two-semester courses in heat transfer. However, the presentation does not presume previous knowledge or expertise. This book can be (and has been) successfully used in a single-semester undergraduate heat transfer course by appropriately selecting from the available topics. Our recommendations on what topics can be included in a first heat transfer course are provided in the suggested syllabus. The reason that this book can be used for a first course (despite its expanded content) and the reason it is also an effective graduate-level textbook is that all concepts and methods are presented in detail, starting at the beginning. The derivation of important results is presented completely, without skipping steps, in order to improve readability, reduce student frustration, and improve retention.

This extended chapter can be found on the website www.cambridge.org/nellisandklein. Mass transfer occurs whenever fluid flows; that is, some mass is transferred from one place to another. However, the focus in this chapter is on the transport of one chemical species (or component) within a mixture of chemical species that occurs as a direct result of a concentration gradient, independent of a pressure gradient. This type of mass transfer is called diffusion. Mass transfer, like momentum transfer, plays an important role in many important heat exchange processes and devices. For example, mass transfer is critical to the operation of cooling coils, cooling towers, and evaporative coolers and condensers that are commonly used in refrigeration and power systems. The energy transfer that occurs as a result of mass transfer can significantly improve the performance of these heat transfer devices. The processes of heat and mass transfer are analogous. The governing equations for heat and mass transfer are similar and therefore many of the relations and solution techniques that have been developed for heat transfer can be directly applied to mass transfer processes.

Chapter 9: Mass transfer

The website associated with this book www.cambridge.org/nellisandklein provides many more problems.

Introduction

Chapters 4 through 6 discuss convection involving single-phase fluids. The thermodynamic state of single-phase fluids is sufficiently far from their vapor dome so that even though temperature variations may be present, only one phase exists (vapor or liquid). In this chapter, two-phase convection processes are examined. Two-phase processes occur when the fluid is experiencing heat transfer near the vapor dome so that vapor and liquid are simultaneously present. If the fluid is being transformed from liquid to vapor through heat addition, then the process is referred to as boiling or evaporation. If vapor is being transformed to liquid by heat removal, then the process is referred to as condensation.

Chapter 6 showed that temperature-induced density variations in a single-phase fluid may have a substantial impact on a heat transfer problem because they drive buoyancy induced fluid motion. However, the temperature-induced density gradients that are present in a typical single-phase fluid are small and so the resulting buoyancy-induced fluid velocity is also small. As a result, the heat transfer coefficients that characterize natural convection processes are usually much lower than those encountered in forced convection processes. The density difference between a vapor and a liquid is typically quite large. For example, saturated liquid water at 1 atm has a density of 960 kg/m3 while saturated water vapor at 1 atm has a density of 0.60 kg/m3. Large differences in density lead to correspondingly large buoyancy-induced fluid velocities and heat transfer coefficients.

Internal flow concepts

Introduction

Chapter 4 discusses the behavior of the momentum and thermal boundary layers associated with an external flow. An external flow is broadly defined as one where the boundary layer can grow without bound; for the flat plate considered in Section 4.1, the boundary layer was never confined by the presence of another object. An internal flow is defined as one where the growth of the boundary layer is confined. Internal flows are often encountered in engineering applications (e.g., the flow through tubes or ducts) and this section discusses the qualitative behavior of internal flows. Many of the concepts that are discussed in Section 4.1 for an external flow can also be applied to internal flows in order to provide a physical understanding of their behavior.

Momentum considerations

Figure 5-1(a) illustrates laminar external flow over a plate and shows, qualitatively, the momentum boundary layer and velocity distribution that results. Figure 5-1(b) illustrates laminar flow through a passage that is formed between two parallel plates; notice that at some location, the momentum boundary layer becomes bounded.

The momentum boundary layers growing from the upper and lower plates in Figure 5-1(b) meet at some distance from the inlet; this distance is referred to as the hydrodynamic entry length, xfd, h . The momentum boundary layer thickness will remain constant as the fluid moves further down the flow passage (i.e., for x > xfd, h ).

Introduction to laminar boundary layers

Introduction

Chapters 1 through 3 consider conduction heat transfer in a stationary medium. Energy transport within the material of interest occurs entirely by conduction and is governed by Fourier's law. Convection is considered only as a boundary condition for the relatively simple ordinary or partial differential equations that govern conduction problems. Convection is the transfer of energy in a moving medium, most often a liquid or gas flowing through a duct or over an object. The transfer of energy in a flowing fluid is not only due to conduction (i.e., the interactions between micro-scale energy carriers) but also due to the enthalpy carried by the macro-scale flow. Enthalpy is the sum of the internal energy of the fluid and the product of its pressure and volume. The pressure-volume product is related to the work required to move the fluid across a boundary. You were likely introduced to this term in a thermodynamics course in the context of an energy balance on a system that includes flow across its boundary. The additional terms in the energy balance related to the fluid flow complicate convection problems substantially and link the heat transfer problem with an underlying fluid dynamics problem. The complete solution to many convection problems therefore requires sophisticated computational fluid dynamic (CFD) tools that are beyond the scope of this book.