Note for Heat Transfer - HT by Ravichandran Rao

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1. Introduction Energy is defined as the capacity of a substance to do work. It is a property of the substance and it can be transferred by interaction of a system and its surroundings. The student would have encountered these interactions during the study of Thermodynamics. However, Thermodynamics deals with the end states of the processes and provides no information on the physical mechanisms that caused the process to take place. Heat Transfer is an example of such a process. A convenient definition of heat transfer is energy in transition due to temperature differences. Heat transfer extends the Thermodynamic analysis by studying the fundamental processes and modes of heat transfer through the development of relations used to calculate its rate. The aim of this chapter is to console existing understanding and to familiarise the student with the standard of notation and terminology used in this book. It will also introduce the necessary units. 1.1 Heat Transfer Modes The different types of heat transfer are usually referred to as ‘modes of heat transfer’. There are three of these: conduction, convection and radiation. x Conduction: This occurs at molecular level when a temperature gradient exists in a medium, which can be solid or fluid. Heat is transferred along that temperature gradient by conduction. x Convection: Happens in fluids in one of two mechanisms: random molecular motion which is termed diffusion or the bulk motion of a fluid carries energy from place to place. Convection can be either forced through for example pushing the flow along the surface or natural as that which happens due to buoyancy forces. x Radiation: Occurs where heat energy is transferred by electromagnetic phenomenon, of which the sun is a particularly important source. It happens between surfaces at different temperatures even if there is no medium between them as long as they face each other. In many practical problems, these three mechanisms combine to generate the total energy flow, but it is convenient to consider them separately at this introductory stage. We need to describe each process symbolically in an equation of reasonably simple form, which will provide the basis for subsequent calculations. We must also identify the properties of materials, and other system characteristics, that influence the transfer of heat. 1

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1.2 System of Units Before looking at the three distinct modes of transfer, it is appropriate to introduce some terms and units that apply to all of them. It is worth mentioning that we will be using the SI units throughout this book: x The rate of heat flow will be denoted by the symbol Q . It is measured in Watts (W) and multiples such as (kW) and (MW). x It is often convenient to specify the flow of energy as the heat flow per unit area which is also known as heat flux. This is denoted by q . Note that, q Q / A where A is the area through which the heat flows, and that the units of heat flux are (W/m2). x Naturally, temperatures play a major part in the study of heat transfer. The symbol T will be used for temperature. In SI units, temperature is measured in Kelvin or Celsius: (K) and (qC). Sometimes the symbol t is used for temperature, but this is not appropriate in the context of transient heat transfer, where it is convenient to use that symbol for time. Temperature difference is denoted in Kelvin (K). The following three subsections describe the above mentioned three modes of heat flow in more detail. Further details of conduction, convection and radiation will be presented in Chapters 2, 3 and 4 respectively. Chapter 5 gives a brief overview of Heat Exchangers theory and application which draws on the work from the previous Chapters. 1.3 Conduction The conductive transfer is of immediate interest through solid materials. However, conduction within fluids is also important as it is one of the mechanisms by which heat reaches and leaves the surface of a solid. Moreover, the tiny voids within some solid materials contain gases that conduct heat, albeit not very effectively unless they are replaced by liquids, an event which is not uncommon. Provided that a fluid is still or very slowly moving, the following analysis for solids is also applicable to conductive heat flow through a fluid. 2

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Figure 1.1 shows, in schematic form, a process of conductive heat transfer and identifies the key quantities to be considered: Figure 1-1: One dimensional conduction Q : the heat flow by conduction in the xdirection (W) A : the area through which the heat flows, normal to the x-direction (m2) 3

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dT dx : the temperature gradient in the x-direction (K/m) These quantities are related by Fourier's Law, a model proposed as early as 1822: Q = -k A dT dx or q = -k dT dx 1 (1.1) A significant feature of this equation is the negative sign. This recognises that the natural direction for the flow of heat is from high temperature to low temperature, and hence down the temperature gradient. The additional quantity that appears in this relationship is k , the thermal conductivity (W/m K) of the material through which the heat flows. This is a property of the particular heat-conducting substance and, like other properties, depends on the state of the material, which is usually specified by its temperature and pressure. The dependence on temperature is of particular importance. Moreover, some materials such as those used in building construction are capable of absorbing water, either in finite pores or at the molecular level, and the moisture content also influences the thermal conductivity. The units of thermal conductivity have been determined from the requirement that Fourier's law must be dimensionally consistent. Considering the finite slab of material shown in Figure 1.1, we see that for one-dimensional conduction the temperature gradient is: T - T1 dT = 2 dx L 2 Hence for this situation the transfer law can also be written T -T Q = kA 1 2 L D = or q = k T1 - T2 L 3 (1.2) k U C (1.3)4 Table 1.1 gives the values of thermal conductivity of some representative solid materials, for conditions of normal temperature and pressure. Also shown are values of another property characterising the flow of heat through materials, thermal diffusivity, which is related to the conductivity by: 3 Where U is the density in kg / m of the material and C its specific heat capacity in J / kg K . 4