This chapter discusses the basics of conduction and induction heating. It presents simple electrical and electrothermal equations for the conductors, and provide a basis for a practical contact.

When AC flows in a conductor, the problem of inductance is superimposed on simple resistive flow. Its own alternating magnetic field links with the rest of the conductor to concentrate the current in the surface layers. This is the phenomenon of skin-effect, which has to be carefully considered when using AC. There is no way of dealing with the subject non-mathematically, so it has to be expressed in mathematical language, but we shall try to give a physical interpretation of the theory wherever possible. Since we are dealing with what is happening at every point in the conductor and skin-effect makes the electromagnetic quantities different at every depth, circuit concepts are inadequate. These field quantities are converted to the more-familiar circuit values at the end.

If we move from the semi-infinite slab, with its mathematically simple solutions but abstract realisation, to a slab of finite thickness, where the current can flow in both sides at the same time, then we get slightly more complex solutions but more practical conditions. The flat bus-bar which is much wider than its thickness is a good approximation to this; although small surface currents will flow in the two narrow edges, these can be ignored in the first instance.

The circular cross-section problem is the problem that is normally solved in the literature, both because of its inherent importance and because it is an excellent example of the use of Bessel functions. Most of these solutions are aimed at the circuit viewpoint, i.e. resistance, reactance and impedance, rather than power loss, which is the purpose of direct resistance heating (DRH). A.G. Warren (“Mathematics Applied to Electrical Engineering”, Chapman and Hall, London, 1949) has an excellent section (pp. 243ff.) on current in a circular conductor but the book has long been out-of-print; also, it uses unrationalised CGS concepts, so we shall start again from first principles. For the reader not conversant with Bessel functions, we do not make a lot of fuss about them here: suffice it to say that they are the equivalent in a cylindrical world of the sinusoids and exponentials of the previous chapters and, like them, are described by infinite series. We find that there is a close similarity between the solutions for the slab and those for the cylinder. Just as, in the thin slab, we found that the two sides interacted to give hyperbolic functions, Bessel functions are needed because every point in the section of a circular conductor is affected by the currents flowing in the rest of the conductor.

It is obviously easier to heat round tubes (or rectangular tubes) by DRH than the equivalent solid section. If the tube wall is thin enough, regardless of diameter, the current will occupy the cross-section substantially uniformly and the problem becomes simple. In this chapter we shall examine the equations already derived, but with new boundary conditions, to try to arrive at the rules for 'thick' and 'thin' tubes.

In many industrial metallurgical processes, the material needs to be heated as part of a moving system. Examples are rolling, intermediate annealing, tempering, etc., where the material starts off on one reel and, after going through various processes, including DRH, ends up on another reel. This chapter is not concerned with the process in general, only the part of the run where the work is heated. For present purposes, since the material is usually flexible - wire, small rod or thin narrow strip - we can ignore the problems of current distribution across the section and assume uniform current density and heating.

We started this book with the conduction heating of metals because this is nearer to our everyday experience. We moved from the simple passage of DC in the metal to the complications that arise from the use of alternating current and, during the derivations, saw that it is normally preferable to use DC, whenever possible. Another reason for starting with conduction was that it is a very effective way to heat metals if the shape is right. It allowed us to develop the theory from first principles in a way that closely matches the way we were taught, starting with Ohm's law and with the current passing from end to end of the material between the electrodes. The understanding that has been de veloped in those chapters can now be extended into the realm of induction heating.

The analysis of Chapter 3 can be used, with different boundary conditions, for the induction heating of a slab surrounded by a coil. The width w is assumed to be large compared with the thickness 2b. This assumption of w≫2b implies negligible end-effects at the extremities of w. If 2b is greater than a certain thickness, to be discussed later, the two surfaces will be independent and the solution will be twice the value given in Chapter 7. If the slab is thin, i.e. b is small, the current distribution will be somewhat like that shown in Fig. 8.1, where the H on each side of the slab induces some current in the other half, tending to oppose the stronger current induced by the nearer magnetic field. Note that the full exponential fall-off is not completed for either current. We also define the direction of the exciting field H_x, which implies that the current density is in the z direction. It is convenient to make the problem symmetrical by moving the y axis to the centre-line.

This chapter presents the induction heating of cylinders using similar theory to that discussed previously.

This chapter tackles about the basic theory of induction heating of tubes, but with different boundary conditions at the inner diameter. The solution is akin to that of conduction heating of tubes, but there is an added complication due to the flux inside the tube. We derive it to bring out the principles.

This chapter discusses the induction heating of hollow cylinder from inside. This topic has been included because of its possible practical importance in the preheating of the 'containers' of extrusion presses. These very thick hollow cylinders have to be brought to starting temperature before work commences or kept at temperature during breaks; at present, this is done very ineffectively using surface heaters on the external surface of the cylinder, so that the heat has to find its way through the thick metal. This could be done much more efficient ly, using an induction coil temporarily suspended from the ram to heat the active surface, where it is needed.

In this chapter, permeability μ and resistivity ρ are treated as numbers in the depth-of-penetration formula δ=√(2ρ/μω) and the consequences of changing them are noted, using the theory that has been developed in earlier chapters, μ and ρ both change during heating, so this is important. However, before we consider them combined, we examine simple changes of each for the semi-infinite slab. Throughout this chapter, we only treat the constant-E condition for conduction (where E is the electric field strength).

This chapter discusses the nonlinear relationship between flux density and the magnetic field intensity which is described by the magnetisation curve for ferromagnetic materials as applied in conduction and induction heating. Much effort has been spent on theories to take this nonlinearity into account.

Heating in scanning and single-shot hardening applications is usually caused by heavy currents flowing in single conductors near the surface of the part. Because the conductor is near the surface (compared with the radius of the work), it is usually valid to treat this as a line current at a distance from a flat plane.

In this chapter, we examine further the simple storage of heat, and then present introductory ideas on heat flow. The idealised solutions with all the heat entering at the surface, which we have seen to be a reasonable first assumption, are derived for slabs and cylinders.

We showed in Chapter 15 that the temperature difference between centre and surface is often too high for the workpiece to be used immediately, so that a short soaking period is needed to reduce the differential. Since we start this chapter with the end results of Chapter 15, surface heating is implicit here: this is modified in Chapter 18, where we take into account the fact that the heat is not produced at the surface.

The radiation loss 𝒫_R is given by the Stefan-Boltzmann law: 𝒫_R = 5.67×10^-8εT_s ⁴ W/m², where ε is the emissivity coefficient of the surface (dimensionless) and T_s is the absolute surface temperature (K). The constant 5.67×10^-8 W/m²K⁴ is known as Stefan's constant.

In this chapter, we modify the previous assumptions to allow for the two practical conditions: (1) the actual heat distribution in the metal; (2) the heat lost by radiation from the surface. The loss equations can be combined with the heat-flow equations to give an accurate solution. To deal with this, we expand the heat-flow equation to include the volumetric production of heat.

In surface hardening, the work surface is brought rapidly to temperature by a high power-density applied at the surface. The aim is to heat a shallow layer without affecting the rest of the workpiece. These two features the high intensity and the thin layer mean that different criteria from billet heating apply. The skin depth is small, there is no need to use cylindrical coordinates or thin slab ideas; this problem can be treated as a semi-infinite slab. In practice, during hardening, the surface will be heated under one of two conditions: (1) a variable temperature θ(t) applied to the surface (2) a variable power-density P(j) applied to the surface. These are not really separate, since one implies the other; it is a question of which quantity is known. Before discussing these cases, it is helpful to consider the problem of a semi-infinite slab suddenly placed in a new temperature environment. This is not an artificial problem, as it is what happens when a piece to be hardened is heated in an oven, but it is not the induction surface-hardening problem; it is used here to introduce the concepts.

Water is widely used in induction-heating applications to cool hollow tubular conductors. Because of the inherent importance of the topic, a great deal of experimental work has been done to verify the theory, and the results can be used with confidence.

In the literature, it is claimed as an advantage of induction billet heating that the coils can be arranged to give a 'tapered' temperature distribution, which is advantageous for extrusion. It is, therefore, interesting to see how such an axial temperature inequality evens out with time.

Induction heating is a convenient method for bulk-heating metals to a set temperature. It replaces furnaces, which tend to be large and which have the disadvantage of long start-up and shut-down times, so that their effectiveness is low. By contrast, the induction heater is relatively small in size and is immediate ly available for use. It is clean and relatively efficient. The power goes directly into the workpiece. The heating times are usually short a few minutes (except when a large thermal mass is being heated) and the process fits well into automated production methods. Although electricity costs are higher per unit of energy, this is offset by higher efficiencies. In Chapter 7, it was shown that induction heating takes place within about one skin depth (say 9 mm for iron, 10 mm for copper at 50 Hz, cold). The heat has to reach the rest of the billet or slab by conduction. Since non-ferrous metals have high heat conductivities this presents few problems, but for steel and other low-conductivity metals we have to avoid surface overheating, or perhaps even melting. Heat flow is an important part of the study of through-heating.

In the last chapter, induction was used for heating large masses of metal, where the aim was a uniform temperature throughout. There, the possible high in tensity of induction heating was limited by the rate at which the heat could be conducted away from the surface. Surface hardening makes use of this 'disadvantage'. This application of induction heating is very important. Whilst the basic principles are obviously the same, the use is very different.

This chapter discusses applications of induction heating, including: soldering and brazing; tube welding; heating of resin kettles and other vessels; paint drying; induction heating in plastic working; annealing and stress relieving; longitudinal flux induction heating; transverse-flux heating; semiconductor processing; and travelling-wave heaters.

This chapter discusses induction melting installations. There is no basic difference between the use of a pulsating field to heat a solid billet or to melt metal for, say, casting. The principles of induction heating still apply; the pulsating field at the surface of the melt induces voltages in the material, which acts as a short-circuited secondary, causing currents to flow and heat the metal. All the ideas on skin depth can be used without modification for melted material. The relative permeability is obviously unity, except when steel scrap is being heated up to Curie point.

This chapter is necessarily vaguer than Chapters 22-25 because there are fewer established uses of direct resistance heating (DRH). This is sad because, as shown in Chapters 1-6, there are great advantages, especially if DC is used. Before looking at DC, let us discuss some reasons for the neglect. Direct resistance heating is ideally suited to heating steel pieces having a uniform cross-section which are long compared to their other dimensions. Very rapid heating is possible, avoiding scale formation and surface decarburisation, and giving high production rates. Efficiencies higher than 90% have been obtained. Heating times of less than a minute can be expected.