# Heat Transfer

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.

**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.

- 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.
- 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.
- 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.

**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:

- 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).
- 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).
- 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 (°C). 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).

**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.

**Convection**

Convection heat transfer occurs both due to molecular motion and bulk fluid motion. Convective heat transfer may be categorised into two forms according to the nature of the flow: natural Convection and forced convection.

In natural of ‘free’ convection, the fluid motion is driven by density differences associated with temperature changes generated by heating or cooling. In other words, fluid flow is induced by buoyancy forces. Thus the heat transfer itself generates the flow which conveys energy away from the point at which the transfer occurs.

In forced convection, the fluid motion is driven by some external influence. Examples are the flows of air induced by a fan, by the wind, or by the motion of a vehicle, and the flows of water within heating, cooling, supply and drainage systems. In all of these processes the moving fluid conveys energy, whether by design or inadvertently.

The left of Figure illustrates the process of natural convective heat transfer. Heat flows from the ‘radiator’ to the adjacent air, which then rises, being lighter than the general body of air in the room.

This air is replaced by cooler, somewhat denser air drawn along the floor towards the radiator. The rising air flows along the ceiling, to which it can transfer heat, and then back to the lower part of the room to be recirculated through the buoyancy-driven ‘cell’ of natural convection.

The word ‘radiator’ has been written above in that way because the heat transfer from such devices is not predominantly through radiation; convection is important as well. In fact, in a typical central heating radiator approximately half the heat transfer is by (free) convection.

The right part of Figure illustrates a process of forced convection. Air is forced by a fan carrying with it heat from the wall if the wall temperature is lower or giving heat to the wall if the wall temperature is lower than the air temperature.

If T1 is the temperature of the surface receiving or giving heat, and ∞ T is the average temperature of the stream of fluid adjacent to the surface, then the convective heat transfer Q is governed by Newton’s law:

**Q = h **_{c} A (T_{1} – T_{2}) or q = h_{c }(T_{1} – T_{2}) _{5}

**Q = h**

_{c}A (T_{1}– T_{2}) or q = h_{c }(T_{1}– T_{2})_{5}Another empirical quantity has been introduced to characterise the convective transfer mechanism. This is hc, the convective heat transfer coefficient, which has units [W/m2 K].

This quantity is also known as the convective conductance and as the film coefficient. The term film coefficient arises from a simple, but not entirely unrealistic, picture of the process of convective heat transfer at a surface. Heat is imagined to be conducted through a thin stagnant film of fluid at the surface, and then to be convected away by the moving fluid beyond. Since the fluid right against the wall must actually be at rest, this is a fairly reasonable model, and it explains why convective coefficients often depend quite strongly on the conductivity of the fluid.

The film coefficient is not a property of the fluid, although it does depend on a number of fluid properties: thermal conductivity, density, specific heat and viscosity. This single quantity subsumes a variety of features of the flow, as well as characteristics of the convecting fluid. Obviously, the velocity of the flow past the wall is significant, as is the fundamental nature of the motion, that is to say, whether it is turbulent or laminar. Generally speaking, the convective coefficient increases as the velocity increases.

A great deal of work has been done in measuring and predicting convective heat transfer coefficients. Nevertheless, for all but the simplest situations we must rely upon empirical data, although numerical methods based on computational fluid dynamics (CFD) are becoming increasingly used to compute the

heat transfer coefficient for complex situations.

Table above gives some typical values; the cases considered include many of the situations that arise within buildings and in equipment installed in buildings.

**Radiation**

While both conductive and convective transfers involve the flow of energy through a solid or fluid substance, no medium is required to achieve radiative heat transfer. Indeed, electromagnetic radiation travels most efficiently through a vacuum, though it is able to pass quite effectively through many gases, liquids and through some solids, in particular, relatively thin layers of glass and transparent plastics.

Above figure indicates the names applied to particular sections of the electromagnetic spectrum where the band of thermal radiation is also shown. This includes:

- the rather narrow band of visible light;
- the wider span of thermal radiation, extending well beyond the visible spectrum.

Our immediate interest is thermal radiation. It is of the same family as visible light and behaves in the same general fashion, being reflected, refracted and absorbed. These phenomena are of particular importance in the calculation of solar gains, the heat inputs to buildings from the sun and radiative heat

transfer within combustion chambers.

It is vital to realise that every body, unless at the absolute zero of temperature, both emits and absorbs energy by radiation. In many circumstances the inwards and outwards transfers nearly cancel out, because the body is at about the same temperature as its surroundings. This is your situation as you sit reading these words, continually exchanging energy with the surfaces surrounding you.

In 1884 Boltzmann put forward an expression for the net transfer from an idealised body (Black body) with surface area A1 at absolute temperature T1 to surroundings at uniform absolute temperature T2:

with σ the Stefan-Boltzmann constant, which has the value 5.67 × 10-8 W/m2 K4 and T [K] = T [°C] + 273 is the absolute temperature.

The bodies considered above are idealised, in that they perfectly absorb and emit radiation of all wavelengths. The situation is also idealised in that each of the bodies that exchange radiation has a uniform surface temperature. A development of Boltzmann’s law which allows for deviations from this pattern is

With ε the emissivity, or emittance, of the surface A1, a dimensionless factor in therange 0 to 1,

F12 is the view factor, or angle factor, giving the fraction of the radiation from A1 that falls on the area A2 at temperature T2, and therefore also in the range 0 to 1.

Another property of the surface is implicit in this relationship: its absorbtivity. This has been taken to be equal to the emissivity. This is not always realistic. For example, a surface receiving short-wave-length radiation from the sun may reject some of that energy by re-radiation in a lower band of wave-lengths, for which the emissivity is different from the absorbtivity for the wave-lengths received.

The case of solar radiation provides an interesting application of this equation. The view factor for the Sun, as seen from the Earth, is very small; despite this, the very high solar temperature (raised to the power 4) ensures that the radiative transfer is substantial. Of course, if two surfaces do not ‘see’ one another (as, for instance, when the Sun is on the other side of the Earth), the view factor is zero. Table below shows values of the emissivity of a variety of materials. Once again we find that a wide range of characteristics are available to the designer who seeks to control heat transfers.

The values quoted in the table are averages over a range of radiation wave-lengths. For most materials, considerable variations occur across the spectrum. Indeed, the surfaces used in solar collectors are chosen because they possess this characteristic to a marked degree. The emissivity depends also on temperature, with the consequence that the radiative heat transfer is not exactly proportional to T3.

An ideal emitter and absorber is referred to as a ‘black body’, while a surface with an emissivity less than unity is referred to as ‘grey’. These are somewhat misleading terms, for our interest here is in the infra-red spectrum rather than the visible part. The appearance of a surface to the eye may not tell us much about its heat-absorbing characteristics.