In this Unit we are looking in more detail than in Unit 1 at the ways in which heat is transferred. Whilst we have tried to confine the text to the basic physics, inevitably there is some discussion of wider issues and background to the calculation and simulation techniques for quantifying the thermal challenge that we will be expanding upon in later Units.

There are three distinct mechanisms by which heat can be transferred from one place to another, conduction, convection and radiation. In all three, the net flow of heat must be from high temperature to low temperature, as determined by the Second Law of Thermodynamics. Whilst we have to consider these as distinct mechanisms, and calculate their effects separately, in practice heat transfer considerations generally involve all three.

As a simple example of something from everyday life that involves these three mechanisms, think about a flask of coffee – the sort that has an inner plastic flask, and an outer plastic cover, with an air gap between. Forgetting the heat that flows through the stopper, what are the heat pathways by which energy is transferred from the hot coffee to the surrounding room?

Now see our comments.

Whilst radiation played some part in our model of the flask of coffee, in practical electronics cooling cases the contribution from radiation is small, especially where forced convection is used, so the situation is often simplified by ignoring radiation. This is very helpful when manual or spreadsheet calculations are used to scope the thermal issues.

In the sections that follow, we will be looking separately at conduction, convection and radiation, to show a little of how they work, and indicating how we might build models that help us calculate the heat flow.

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Conduction is the transfer of heat by molecular action through a substance, or between substances in contact with another. Its physical basis is the transfer of energy from more energetic particles to less energetic particles due to the interaction between them.

In the case of a **gas**, molecules at a higher temperature have a higher energy, which is stored as motion, rotation and vibration. As neighbouring molecules collide, energy is transferred from the more energetic to the less energetic, so that there is a net transfer of energy by a process of random motion that we refer to as energy diffusion.

To get a feel for how diffusion operates, go to this link and look at the animation of the diffusion process of mixing two gases. We recommend initiating at least the first experiment. Bear in mind that, although what you observe would be replicated in real life with vastly greater numbers of atoms, the real-life timescales for the two chambers to equilibrate is actually quite fast. And you see only diffusion, the result of the random movement of molecules, rather than convection.

In **liquids**, molecules are more closely spaced than in gases, and the molecular interactions are stronger and more frequent, so the conductivities are higher. For similar reasons, the thermal conductivity in a solid may be more than four orders of magnitude larger than that of a gas.

However, in a **solid**, whilst conduction still occurs by interaction between atoms, the atoms are fixed in position. The modern view ascribes energy transfer by conduction in a solid to ‘lattice waves’ induced by atomic motion. In a non-conductor, the energy transfer is exclusively by means of these lattice waves; in an electrical conductor, energy transfer also takes place through the translational motion of the free electrons^{1}. We make use of conduction in almost all electronic applications to remove heat from the junction region of a chip to its surface, and from here through the lead-frame to the board itself.

Consider a bar of material one end of which comes into contact with a heat source maintained at 100°C, whilst the other remains in contact with a heat sink maintained at 0°C. The bar itself is initially at 0°C, but heat will flow into the bar from the heat source, and the end of the bar near the source will increase in temperature. This heat will gradually travel along the bar, driven by the temperature gradient. In order for flow to take place, each section of the bar has to be at a slightly higher temperature than its neighbour further away from the heat source, however small the scale on which we are examining the bar.

Because there is always a source of heat at the hot end of the bar, and an infinite amount of heat is available, more and more heat flows along the bar, driven by the temperature gradient. However, at the cold end, heat must leave the bar so that the temperature remains 0°C. Eventually a steady state will be attained, where there is a constant flow of energy along the bar, and this condition will persist for as long as the heat source is in contact.

Figure 1 illustrates how the temperature along the bar will vary with time. The lowest curve represents an early time during the warming transient, and the uppermost line represents the eventual steady state, a straight line temperature distribution. Curves in between represent intermediate times in sequence.

Most thermal situations will eventually reach a steady state, although they will only have a straight-line temperature distribution if the heat is being conducted through a material of constant cross-section, so that the heat transfer can be modelled in one dimension only. In real situations we have three dimensions to consider, and typically the sources of heat are transient rather than continuous and the distribution of heat varies with time. To analyse this we need a general equation of heat transfer that can be applied to a small portion of the heat-conducting medium.

Consider a small element, δ*x*, within the bar, as illustrated in Figure 2.

A temperature gradient exists across the element since the temperature is *T* on one side and *T + δT* on the other. The heat flux, *J*, across the element is related to the temperature gradient across it by

. . . . . . . Eqn. 1

where *T* represents temperature in Kelvin, *J* has the units Wm^{−2} and *k _{T}* is the thermal conductivity at temperature

- The minus sign is a consequence of the fact that heat is being transferred in the direction of decreasing temperature.

- The
*proportionality constant*, the thermal conductivity, is a characteristic not only of the material, but also of the temperature.

- This equation is for the rate of heat transfer
*per unit area*, so that the total heat conducted (units W) is the product of the heat flux and the area.

Fourier’s Law implies that the heat flux is a directional quantity, and at right-angles to the cross-sectional area. More generally, the direction of heat flow will always be normal to a surface of constant temperature, called an *isothermal surface*. Each flux being a vector quantity, we can write a more general statement of the conduction rate equation (Fourier’s Law) as:

. . . . . . . Eqn. 2

where is the three-dimensional gradient operator and *T*(x,y,z) is the scalar temperature field. It is implicit in Equation 2 that the heat flux of a vector is in a direction perpendicular to the isothermal surfaces, so an alternative form of the Law is:

. . . . . . . Eqn. 3

where *J _{n}* is the heat flux in direction n, which is normal to an isotherm, as shown for the two-dimensional case in Figure 3.

At this link we have a more extended analysis of the derivation of the heat diffusion equation.

Even though the real world is three-dimensional, a single-dimensional analysis will enable us to carry out many useful tasks, and can be sufficient for steady-state work. However, some challenges will need to be treated in two or three dimensions. For example, what happens if the surfaces are not plane? In many ways, this is something that is easier to approach with a sketch than by mathematical modelling – in Figure 4, we show a two-dimensional conduction problem, where we have drawn both the isotherms and lines of heat flow. Remember that, according to Fourier’s Law, the local heat flux is a vector that perpendicular to the lines of constant temperature.

Analytically the heat flux is a vector, the resultant of heat flux components in x and y directions. The heat equation for this becomes:

. . . . . . . Eqn. 4

Solving this equation may be done by a number of different analytical techniques, and also by graphical methods that involve constructing a heat flux plot as in Figure 4. This is a network of isotherms and heat flow lines from which the temperature distribution and heat flow may be inferred.

The procedure for constructing the flux plot is first to identify relevant lines of symmetry, so as to consider only part of the configuration, ensuring that there is no heat transfer in a direction perpendicular to the lines of symmetry. The next step is to identify known lines of constant temperature, and then sketch lines of constant temperature within the system. Note that these isotherms will always be perpendicular to the heat flow lines – as there is no heat flow perpendicular to these heat flow lines, they are sometimes referred to as ‘adiabats’. The heat flow lines are drawn with an eye to creating a network of curvilinear squares, with the heat flow lines and isotherms intersecting at right angles, and with all the sides approximately all the same length.

Whilst this approach does seem somewhat “artistic”, in fact it has in the past been used for much analytical work. It is also a useful aid to visualising the lines of heat flow.

The drawing below is of the heat flow within a semiconductor chip, from the junction towards the die mount. To what extent does this drawing follow the above guidelines?

Now look at our comments.

One we move away from a simple visualisation, and look for analytical solutions, we quickly move to the realm where simulation provides the only viable answers.

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We use the term ‘convection’ to describe energy transfer between a surface and a fluid moving over the surface. Convection is conceptually very different from conduction:

- the heat-transport medium is a fluid rather than a solid
- gross movement of the medium is involved
- the energy density in the medium is very much lower than with conduction.

Even more significantly, boundaries are *opportunities* for heat transfer, rather than insurmountable barriers. So, for a board assembly, convection involving the surrounding air will be the major process by which heat leaves its surface. Although heat still has to enter the fluid by conduction^{2} between the fluid and the solid, the dominant mode of heat transfer is by large-scale movements within the fluid, with a small contribution from the random motion of fluid molecules (diffusion).

The amount of heat transferred by convection is given by:

. . . . . . Eqn. 5

where *A* is the area across which heat transfer takes place, *q* the heat flow in W, and *T _{solid}* and

Heat transfer coefficients are frequently temperature-dependent, and this is usually described in terms of their variation with absolute temperature, which is why the units of heat transfer coefficient are usually expressed in Kelvin. Otherwise, it is unimportant whether Kelvin or Centigrade scales are used, since it is the temperature *difference* that is of interest. However, it helps avoid error if, throughout a calculation, temperatures are specified consistently in only one of the two scales.

The challenge for heat transfer design lies in determining the value of *h*, as it is dependent upon many different factors. These include:

- The velocity of the air flow over the component/PCB surface

- The thermal conductivity of the cooling fluid, for example air or water

- The viscosity of the fluid

- The specific heat capacity of the fluid

- The type of fluid flow over the surface, whether turbulent or laminar

- The nature of the solid surface, for example its surface roughness

- The density of the fluid, bearing in mind that this can change with altitude.

Four of these properties, thermal conductivity, specific heat capacity, viscosity, and density are dependent upon the fluid. As the fluid is usually air, the designer often has little control over these factors. However, the designer does have more control over three of the remaining factors, the velocity and type of the air flow and the exposed component area, and may also be able to specify the nature of the surfaces.

A major influence on the heat transfer coefficient is the rate at which the fluid is flowing. A rapidly-moving fluid will remove more heat from a body than a more slowly-moving fluid at the same initial temperature, because more fluid is available to absorb the heat without itself experiencing a large rise in temperature. However, the rate of fluid flow also affects the nature of the interface between the solid and the fluid.

An example of the use of the heat transfer coefficient.

A chip has dimensions 5mm x 5mm x 3mm, and is at a temperature of 40°C. It is surrounded by air at a temperature of 20°C, and the heat transfer coefficient between the surface of the chip and the surrounding air is 10Wm

^{−2}K^{−1}.If the chip stops operating, so that no more heat is generated, calculate its average temperature after 3s. Assume that the density of silicon is 2,330kg·m

^{−3}and the specific heat capacity 705J·kg^{−1}K^{−1}.What would the average temperature be if fan cooling increased the heat transfer coefficient to 20Wm

^{−2}K^{−1}?Repeat the calculation for a cooling period of 3 minutes and a heat transfer coefficient of 10Wm

^{−2}K^{−1}.What assumptions have you made in your calculation?

Compare your answers with the ones given here

Heat transfer coefficients are also frequently used to describe the ease with which heat is transferred from one solid body to another. The concepts of thermal conduction or diffusion assume that the material is continuous and every part of it is in perfect contact with neighbouring material. However, if two pieces of material are initially separate and are brought together, then the junction will never be perfect and the material will never be thermally continuous. Because of this there will tend to be additional opposition to heat flow and a heat transfer coefficient can usefully be employed to represent the ease with which heat can move across the interface. Alternatively, the interface can be modelled as an additional thermal resistance, a concept that we will be introducing in Unit 8.

Convection divides into two broad categories, ‘natural’ (or ‘free’ ) convection and ‘forced’ convection, and there is a further division into whether the flow is laminar or turbulent. **Natural** **convection** describes the process in which the flow of the fluid is created by gravitational forces on the fluid as its density varies. As heat is transferred to the fluid in contact with a hot surface, the fluid temperature rises. A hot fluid being less dense^{3} than a cold one due to expansion, this temperature increase results in buoyancy forces that cause the warmed fluid to rise in relation to the surrounding fluids. Cooler fluid replaces the rising fluid, producing a continuous flow of fluid, but with a relatively low velocity. This ‘natural’ convection is therefore only applicable in systems where relatively small amounts of heat have to be removed.

Can you think of any ways that the air flow velocity in natural convection can be improved?

In natural convection the driving force is the density difference in the fluid caused by heat conducted into it, so the direction of movement will be determined by gravity, and the heat flow will take the form of a ‘plume’ rising vertically above the hot surface.

Density differences have the same effect as very small pressure differences, and the flow created will be affected considerably by any obstacles in its path. For this reason, in a natural convection context we will need to use best practice in managing flow, and avoid any features with a high resistance to flow. For example, heat sinks for natural convection will have relatively widely-spaced fins, whereas those for forced convection may be packed closely together.

The amount of heat that can be transferred by natural convection is restricted, and it is common practice to force air over the surface to be cooled at higher than natural flow rates. Referred to as ‘**forced convection**’, the airflow is supplied externally, usually by some form of air handling system (AHS) either provided by the designer of the enclosure using fans to create a pressure difference and hence increased air flow, or applied directly to the heat-generating component, for example, as part of an integrated fan and heat sink.

In forced convection, as well as the actual amount of heat transferred, the pressure drop (*Δp*) becomes very important. Cooling fans are designed to move area from a low-pressure region to an area of higher pressure, but if the pressure in that higher-pressure area is too high, the fan may not operate as expected. There is obvious opportunity for a problem here if there is not enough clearance around the gas outlet, or if components and heat sinks are obstructing the flow path. Forced flow through a pin-fin array is a very effective way to increase heat transfer, but comes at the cost of a high pressure drop. There is also a possibility that, if the pin spacing is too dense, the air might flow *around* the heat sink instead of through it.

In both natural and forced convection the fluid velocity is reduced near its interface with the solid, which can be thought of as exerting ‘drag’ on the fluid. This layer of fluid with reduced velocity is known as the ‘boundary layer’, and its thickness (in the case of forced convection) is defined as the distance between the stationary surface to a distance where the moving fluid is less than 99% of the free-stream velocity. As heat must cross the layer in order to be removed from the system, the presence of a boundary layer will reduce the rate of heat transfer and depress the heat transfer coefficient.

The classic example for natural convection relates to a heated vertical place (Figure 5) immersed in an infinite medium (sufficient fluid for the bulk of the fluid to be at rest). The plate is hotter than the fluid, so buoyancy forces induce free convection in which the heated fluid rises, pulling with it fluid from the surroundings. However, the resultant velocity distribution is unlike that associated with forced convection: as well as being zero at the plate boundary, the velocity is also zero as we move some distance away from the plate.

The overall heat transfer may be expressed in terms of Newton’s Law of Cooling, with the usual provisos on the value of the convection heat transfer coefficient. However, the motion of the fluid is zero at the surface (a ‘no-slip’ condition is generally assumed to apply), so the heat transfer from the heated surface to the fluid in its immediate vicinity is by *conduction*. The heat transferred is therefore given by Fourier’s Law as:

. . . . . . Eqn. 6

In this case, the temperature gradient is the value in the fluid at the surface, and *k* is the thermal conductivity of the fluid, in this case usually air.

Convection heat transfer depends not only on how fast the fluid flows but on how well it conducts heat near the surface. The ratio between a fluid’s ability to conduct heat and then move it away is called its ‘Nusselt number’, given by the equation:

. . . . . . Eqn. 7

where *h _{c}* is the heat transfer coefficient,

In small fluid-filled spaces, the thermal boundary layer can be thick enough to prevent appreciable natural convection, so that heat transfer by conduction is dominant. This is why accurate simulation is important for a series of boards in a rack, particularly where these are spaced closely together.

The value of heat transfer coefficient depends on the characteristics of flow. As well as being divided into free (natural) and forced convection, convection is also characterised as being ‘laminar’ or ‘turbulent’. When flow is laminar, the bulk of the fluid is moving at a common speed, although this will reduce at the solid boundary. Compared to laminar flow, turbulent flow gives more opportunities for fluid molecules to impact on the heated surfaces and with each other. So turbulent flows have a greater capacity to carry heat, in the same way that forced convection has a higher capability of removing heat than natural convection. Figure 6 shows a heated surface at temperature T_{s}, surrounded by a cooler moving fluid at a bulk temperature of T_{∞} and free-stream velocity of U_{∞}. Note that, as with natural convection, the velocity in the fluid reduces as the surface is approached, and that the fluid at the interface is stationary.

This shows the laminar sub-layer and the buffer layer that separates the laminar and turbulent layers

In the laminar boundary layer, fluid motion is highly ordered, and it is possible to identify ‘streamlines’ along which particles move. As well as a velocity component in the direction of flow, there will be some velocity component normal to the surface, leading to growth of the boundary layer. By contrast, in the turbulent boundary layer, fluid motion is highly irregular, with velocity fluctuations. These increase both the surface friction and convection transfer rate. Fluid mixing resulting from the fluctuations makes boundary layer thicknesses larger in turbulent conditions and boundary layer profiles of speed, temperature and concentration flatter than with laminar flow.

A useful concept is that of the Reynolds number, a dimensionless grouping of variables that is used to predict whether or not flow will be turbulent. If we think of the transition as occurring at a location x_{c, }where, for a flat plate, the characteristic length is the distance from the leading edge, the Reynolds number is given by:

. . . . . . Eqn. 8

The *critical* Reynolds number is the value of Re_{x} for which transition begins. Its value depends on the surface roughness and the turbulence of the free stream. The figures for the critical Reynolds number vary, but in the case of flow in a tube, a Reynolds number of up to 2,200 corresponds to laminar flow, and a value of above 10,000 to the kind of fully turbulent flow shown in Figure 7. Note that in turbulent flow there is a complex formation of eddies, with fluctuations in velocity, pressure, density and temperature. At intermediate values, the probability increases that the flow will be turbulent. Because of the unpredictability of flow in this transition range, calculations for the heat transfer coefficient may have more than a 50% margin of error, so it is best to design a system that does not operate in this region.

(a) shows the three layers of flow; (b) shows the profile of the velocity within the tube

Natural convection does not necessarily imply laminar flow. A typical transition from laminar to turbulent flow is shown in Figure 8. If the plate is not perfectly flat, or if we string a wire across the initiation edge, the onset of turbulence can occur sooner.

Turbulent conditions characterise many flows of practical interest, and can be advantageous in providing increased transfer rates. The onset of turbulence depends firstly on whether disturbances originate from the free stream, are the effect of surface roughness, or are induced deliberately by interrupting the flow by trip wire or component, and secondly on whether these disturbances are amplified or attenuated in the direction of fluid flow. This in turn depends on the ratio of the inertia of the fluid to the viscous forces, which is what the Reynolds number actually measures: if the Reynolds number is small, inertial forces are small, so they are dissipated and the flow remains laminar; for a large Reynolds number, the inertial forces are sufficient to amplify disturbances, so a transition to turbulence occurs.

The effectiveness of the contact between fluid and surface is highly dependent on the type of flow; in laminar or ‘streamline’ flow, the direction of flow is parallel to the surface, and a boundary layer exists between fluid and surface. This boundary layer presents a barrier to heat, so the effective value of the heat transfer coefficient is reduced. By contrast, no such boundary layer persists in turbulent flow conditions, so the heat transfer coefficient is higher. The transition between laminar and turbulent flow depends both on the velocity of the fluid and any disturbances within it – at a sufficiently high velocity, the flow will always be turbulent; at very low velocities the flow will be laminar; at intermediate velocities, the type of flow will be indeterminate, depending on local conditions. A good visual illustration of this from the natural world is the flow of a river, where the flow can be deduced from observation of floating objects. In the central section of the river, the flow is relatively smooth and linear; towards the edges, small eddies and vortexes can be seen, where the swirling current comes in contact with the bank. Similarly with the flow in a gas, much turbulence is caused by friction between surfaces and the gas flowing past.

Whether the flow is laminar or turbulent has a considerable effect on the heat transfer coefficient, so has been the subject of significant attention. At what point does flow become turbulent? One answer lies in the Reynolds number . . .

From time to time our discussion has skirted around this topic . . .

If you read the literature, you will come across a seemingly endless number of so-called “dimension-less groups”. These are numbers used to describe the physical phenomena in heat transfer, and have been named after outstanding researchers in the field. You have already come across Nusselt’s and Reynolds’ contribution, and names like Rayleigh and Prandtl will delight! Fortunately, computer simulations work from first principles, and generally you do not need to understand the detail.

For more about Reynolds, Nusselt and the others, read our short commentary on *Dimensionless numbers* and *Dimensionless Numbers in Heat Transfer* in the CoolingZone tutorial section.

Fortunately, in real situations, surfaces are rarely flat and smooth, but have a number of sharp features from the components on the boards, and this reduces the flow rate at which the transition to turbulent flow occurs. And in cases such as ‘impingement’ cooling, where the flow of air is directed normal to the surface being cooled, turbulent flow is often deliberately instigated in order to obtain the most effective cooling regime.

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In our discussion of convection and flow, although we have generally used the term ‘fluid’, the use of air as the coolant will probably have been inferred even when not stated explicitly. Indeed, because it is by far the most common approach, our main focus for the remainder of the module will be on air cooling. However, all the factors involved, and the modelling approaches used, apply equally to the flow of a liquid such as water. Or to more exotic ways of cooling high-power assemblies that may involve refrigerant liquids, inert fluorocarbon fluids, or gases such as carbon dioxide. The only change is that the calculations need to be carried out with appropriate values of the physical constants.

So what factors are involved in determining how much heat is taken from a surface by fluid flow across it? Think about this, and make your own list before reading further.

The amount of heat transferred depends on the *mass* of fluid transported past the hot surface, and the *temperature rise* of the fluid. The second of these will reflect the conductivity of the fluid, the temperature difference between the hot surface and the fluid, the length of time during which contact takes place, and the specific heat of the fluid; the mass flow rate will depend on the density, and the rate of flow. Whilst most of these parameters are defined by the physical constants of the fluid being used, both contact time and flow rate are determined by the rate at which the fluid flows, and it is this aspect that we seek to control.

Fluid flow is a complex topic whose detailed consideration is found in many text books. Fortunately detailed knowledge of this subject, and the ability to make hand estimates of fluid flow, is no longer needed, because the simulation software builds in the necessary physics. However, you do need to understand the basic factors involved in flow, if only to be able to understand the literature.

At its simplest, fluid flow is determined by three factors:

- The ‘moving force’, which is the pressure difference across the body of fluid being considered, created either by differences in density in a natural convection situation, or by pressure applied by an air handling system.
- The
*internal*resistance to flow of the fluid, expressed as its viscosity. - The
*external*resistance to flow provided by physical features of the environment, both deliberate obstructions as in the case of filters, or aspects such as high-profile components that impede flow.

The pressure differences needed to generate flow vary substantially between different fluids: extruding a viscous polymer requires considerable pressure, of the order of 20 bar; pumping water can be achieved with only a modest ‘head’ of 30 feet, corresponding to 1 bar, although mains pressures are typically 6 bar to be able to provide water at the top of the hill! For moving gases, which have very much lower viscosities, the issue is more one of handling volume than creating a pressure, and the pressures involved are relatively low. If you think of the metrological situation, relatively few millibars of pressure difference can create substantial air movement. So gas pressure differences are often quoted in non-SI units, such as ‘inches water gauge’.

See *Expressing values of pressure* for more information on the ways in which pressure is measured.

Note that flow is related to pressure *difference*, rather than to absolute pressure, so that systems that don’t involve any ‘non-return’ component will respond equally well to fluid being pulled as well as pushed. The analogy to this is with the early days of flight, where some designs had propellers at the rear – convenient, because the forward-facing machine gun didn’t destroy the propeller! [It was only when machine-gun and propeller could be synchronised that the conventional propeller-in-front configuration was adopted] For the purposes of electronic air movement, we can arrange fans to blow or suck, and both arrangements will be seen, although blowing is the most common, because the airflow patterns created are more easily defined and filtration is easier to implement.

The viscosity of a fluid is determined by the interaction of the molecules, so is affected by the temperature. It is also affected by the *density* of the fluid, which is a function of temperature; for a gas, the density is also a function of pressure, and so will change with altitude.

We have already mentioned filters as an example of deliberate obstruction of airflow; finger guards and grills have the same effect, but are less likely to become clogged, so are more constant with time. But these are only *some* of the elements in path resistance – in a typical assembly, resistance will be provided by surface effects, by adjacent surfaces (such as parallel boards and parallel fins), and by changes in airflow direction.

This last might be straightforward to calculate, as in the case of bends and constrictions in a tube, but is usually more difficult to estimate, as in a rack-mounted unit, where the air flow is constrained to follow a particular path by features such as back-planes.

Of course, some internal resistant elements may be deliberately introduced. An example is where a baffle is used to divert cooling air away from a low-impedance channel to an area that requires greater cooling, but happens to have a higher impedance to flow.

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Radiation is a second natural process that helps in many cooling situations. Although a lower-order effect, in a number of practical cases radiation’s contribution can be similar to that of natural convection.

If you put your hand near (but not on!) a light bulb, you will feel heat from that bulb. You could argue that the bulb surface is hot and that the hot glass surface heats the air next to it, which then transfers its heat via natural convection. However, even if there were a vacuum between your hand and the bulb, you would feel almost as much heat, transferred to your hand by radiation.

Because component-emitted radiation only becomes a significant heat transfer mechanism at high temperatures, it is usually not necessary to consider radiation effects between components. However, it is possible to predict the effect of radiation by using the Stefan-Boltzmann Law, which relates the energy emitted from a surface to its temperature:

. . . . . . . Eqn. 9

where *s* is the Stefan-Boltzmann constant and has the numerical value 56.7 x 10^{−9}Wm^{−2}K^{−4}, *T* is the absolute temperature of the radiating surface in K, and *A* is the surface area of the emitting surface.

Since the value of the Stefan-Boltzmann constant is low, radiation tends to become a dominant mechanism of cooling only at very high temperatures. Figure 9 is a plot of radiant heat versus absolute temperature for a 1cm perfectly emitting surface. It is important to notice the marked non-linearity of the graph.

Figure 9 assumed that the surface was a one of the hypothetical perfect radiators that are known as ‘black bodies’. A black body is considered to absorb all incident radiation from all directions at all wavelengths without reflecting, transmitting, or scattering it. For a given temperature and wavelength, no other body can emit more radiation than a black body. The radiation emitted by a black body at any temperature *T* is the maximum possible emission at that temperature.

Most surfaces are *not* perfect radiators. And, although surfaces that *look* black are generally better radiators than other surfaces, they may still be far from perfect. To account for the fact that radiating surfaces are not perfect we use a quantity known as the emissivity, ε, which is defined by the equation:

. . . . . . . Eqn. 10

where *q* is the energy emitted by the particular surface and *q _{b}* the energy emitted by the equivalent black body at the same temperature. A perfect black body radiator has an emissivity of unity, so all other surfaces have emissivity between zero and one. Some typical values of emissivity are given in Table 1.

Surface | Emissivity |
---|---|

polished metal | less than 0.1 |

oxidized metal | 0.5 |

non-metals | more than 0.5 |

painted surfaces | more than 0.7 |

Emissivity varies with other surface properties such as roughness and is generally temperature-dependent, because the radiation properties of a surface depend on the wavelength of radiation. However, in radiation calculations, it is not practical to consider heat transfer at each wavelength, and it is common to simplify the analysis by assuming a uniform emissivity over the entire wavelength spectrum. This *grey body approximation* is frequently used in engineering applications.

It is important to realize that bodies do not just *cool* by radiation, but may be *heated* by the radiation emitted by another body. After all, that is how the Earth is heated by the sun! In fact, for most of the time bodies *exchange* radiation; in the simplified case where the receiving body absorbs all the radiation, and none is reflected, the net flow of heat is given by:

. . . . . . . Eqn. 11

where *e* is the emissivity of the radiating surface, *A _{projected}* is the area of the absorbing body that is “seen” by the radiating body, and

If we take the case of radiation exchange between a small surface and a much larger surrounding isothermal surface, such as the walls of a room or furnace, all the radiated power will be collected, and the difference between the thermal energy that is emitted and that which is gained, expressed per unit area of surface, is given by:

. . . . . . . Eqn. 12

where *ε* is the emittivity, *E* the emitted radiation, *α* the absorptivity and *G* the absorbed radiation. For a body in equilibrium with the enclosure, there can be no net gain in energy, so the radiation emitted must be equal to the energy absorbed, and Equation 12 reduces to:

. . . . . . . Eqn. 13

where *(T)* indicates that the factor is a function of temperature. This is a special case of the relationship between the absorptivity and emissivity of a body that is usually referred to as * Kirchhoff’s law in thermodynamics*, to distinguish it from its analogue in Kirchhoff’s laws of circuits. The form of Kirchhoff’s law that involves no restrictions is given in Equation 14:

. . . . . . . Eqn. 14

That is, the *emissivity* of a surface at a specified wavelength, direction and temperature is equal to the *absorptivity* of the surface at the same wavelength, direction and temperature. Although we have to exercise care in generalizing this result to the average values of emissivity and absorptivity over the entire wavelength spectrum, it is frequently useful to do so. However, bear in mind that, whilst Equation 13 is always valid, the equality of Equation 14 is applicable *only* where the radiation properties of the body are independent of the wavelength, or when the incident and the emitted radiation have the same spectral distribution.

It is often more convenient to express radiation heat exchange in the form of Equation 15, where the radiation heat transfer coefficient is given by Equation 16:

. . . . . . Eqn. 15

. . . . . . Eqn. 16

Here we have modelled the radiation in a manner similar to convection, and linearised the radiation rate equation. However, the radiation heat transfer coefficient used is strongly dependent on temperature.

In a relatively few pages it hasn’t been possible to do more than scratch the surface of the detail of how heat is transferred. If this is an aspect you wish to follow up, there are many books on thermodynamic and related principles, but most are unnecessarily mathematical, and far from clear in their explanations. One of the few we found reasonably approachable was* Fundamentals of Heat and Mass Transfer* by Incropera and DeWitt, currently at the fifth edition, but available from a number of sources at widely-differing prices. If you like problems, there are plenty of these; there is also a linked CD/book with an equation-solving software package. Like many such books, it starts with pages of symbols. Although off-putting, this has some utility, because (unfortunately) different books use slightly different symbols and definitions. Definitely an aspect to beware of!

Title | Author | ISBN | Publisher | Date |
---|---|---|---|---|

Fundamentals of heat and mass transfer | Frank P. Incropera and David P. DeWitt | 0471386502 | John Wiley & Sons | 5th edition, 2002 |

One of the advantages of Incropera’s book is that it is firmly rooted in the metric system. Although the authors are Americans, they have resisted the use of the British units still preferred by many workers in the US.

Each of these lists is in the order in which the material is referenced in the Unit text. However, note that links to SAQ answers are *not* included!

- Virtual laboratory:
*Thermodynamic equilibrium*, University of Oregon

- Supplementary note:
*The heat diffusion equation* - Supplementary note:
*Expressing values of pressure*

- Supplementary note:
*Dimensionless numbers* - Web page:
*Dimensionless Numbers in Heat Transfer*,*CoolingZone*tutorial

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