htprof.qxp 9/3/2004 11:22 AM Page 1

PPRROOFFEESSSSOORR IINNDDUUCCTTIIOONN by Valery I. Rudnev • Inductoheat Group

Professor Induction welcomes comments, questions, and A common misassumption in suggestions for future columns. Since 1993, Dr. Valery Rudnev has been on the staff of induction Inductoheat Group, where he Ω µ currently serves as group henever someone is talking about the metal ( ·m), r is the relative mag- director – science and Winduction heating, reference is netic permeability, and F is the fre- technology. In the past, he was an associate often made to the phenomenon of skin quency (Hz), or (in inches) as: professor at several universities, where he 1 effect. is considered a fun- taught graduate and postgraduate courses. δ × ρ µ ½ damental property of induction = 3160 ( / rF) , (Eq. 3) His expertise is in materials science, , heating, representing a nonuniform applied electromagnetics, computer modeling, and distribution of an where electrical resistivity ρ is in units process development. He has 28 years of within the conductor cross section. This of Ω·in. experience in . effect will also be found in any electri- Thus, the value of penetration Credits include 15 patents and 118 scientific cally conductive body (workpiece) lo- depth varies with the square root of and engineering publications. cated inside an or in close electrical resistivity and inversely with proximity to the coil. According to this the square root of frequency and rel- Contact Dr. Rudnev at Inductoheat Group 32251 North Avis Drive phenomenon, eddy currents induced ative magnetic permeability. Mathe- Madison Heights, MI 48071 within the workpiece will primarily matically speaking, the penetration δ tel: 248/585-9393; fax: 248/589-1062 flow in the surface layer (the “skin”), depth, , in Eq. 1 is the distance from e-mail: [email protected] where 86% of all induced power will the surface of the conductor toward Web: www.inductoheat.com be concentrated. This layer is called the its core, at which the current decreases reference depth or current penetration exponentially to “1/exp” its value at depth, δ. The degree of skin effect de- the surface. The power density at this pends on the frequency and material distance will decrease to “1/exp2” its properties (electrical resistivity, ρ, and value at the surface. µ relative magnetic permeability, r) of Figure 1 illustrates the skin effect, the conductor.1 showing distribution of current Surface Core density from the workpiece sur- Traditional view of the skin effect face toward the core. At one It is often recommended to calcu- penetration depth from the sur- I = Isurface late the distribution of the current den- face (y = δ), the current will sity along the workpiece thickness equal 37% of its surface value. I = 0.368 Isurface (radius) using Bessel functions.2 How- However, the power density

ever, for electromagnetically “thick” will equal 14% of its surface Current density δ workpieces, the following simplified value. From this, we can con- equation is frequently used: clude that about 63% of the cur- rent and 86% of the induced Distance from workpiece surface –y/δ I = I0·e , (Eq. 1) power in the workpiece will be concentrated within a surface layer of Fig. 1 — Current density distribution due where I is the current density (in to the skin effect. (Ref. 1) thickness δ. A/m2) at distance y (m) from the Analysis of Equations 2 and 3 workpiece surface toward the core, I 0 shows that the penetration depth has is the current density at the surface different values for different materials 2 δ (A/m ), and is the current penetra- and is a function of frequency. tion depth (m). According to this equation, an density in- Selecting case depth, frequency duced within an inductively heated Surface hardening of and cast workpiece has its maximum value at represents the most popular ap- the surface and falls off exponentially. δ plication of induction heat treatment. Current penetration depth, , is de- The goal in surface hardening is to scribed (in meters) as: provide a martensitic layer on specific δ × ρ µ ½ areas of the workpiece to increase the = 503 ( / rF) , (Eq. 2) hardness, strength, and fatigue and where ρ is the electrical resistivity of wear resistance, while allowing the re- HEAT TREATING PROGRESS • SEPTEMBER/OCTOBER 2004 23 htprof.qxp 9/3/2004 11:22 AM Page 2

PROFESSORPROFESSOR INDUCTIONINDUCTION mainder of the part to be unaffected ditional heating time is needed to some unique cases. For the great ma- by the process.1 The case depth, or allow heat to conduct to the desired jority of induction heating applica- hardness depth, is typically defined depth. Not only does this add unnec- tions, the current density (heat source) as the distance from the surface where essary time to the cycle, but there can distribution is not uniform and there the microstructure is at least 50% also be significant overheating of the always are thermal gradients within . Below this depth the hard- surface, which can lead to excessive the heated workpiece. These thermal ness begins to decrease drastically. grain growth. Overheating of the sur- gradients result in nonuniform distri- Power and frequency are two of the face can also cause butions of electrical resistivity and most important factors that affect case and excessive scaling. magnetic permeability within the depth. In surface hardening applica- If the chosen frequency is too low workpiece. This nonlinearity means tions, the frequency can range from as (Fig. 2, center), the heating is deeper that the classical definition of current high as 4000 kHz (used for special ap- than necessary. The result is a large penetration depth often does not fully plications such as hardening of thin heat-affected zone, additional work- apply. wire) to as low as line frequency (used piece distortion, and unnecessary New explanation: An assumption for hardening large rolls). waste of energy. In some cases, the of exponential current density distri- In many instances, it is possible to penetration depth can be so large, bution can be used for rough engi- achieve the same desired case depth by compared with the required case neering estimates for induction using different combinations of power depth, that it will not be possible to heating nonmagnetic materials (alu- density and frequency. For example, meet the pattern specification. minum and copper, for example) and when a shallow case is required it In general, the optimum frequency through heating of steels to might be possible to achieve the same will result in a current penetration forging temperature. results with a lower-than-optimal fre- depth that will be 1.2 to 2 times the re- However, in some applications, sur- quency in combination with a higher quired case depth. Maintaining this face hardening in particular, the power density applied for a shorter ratio compensates for the cooling/ power density distribution along the time. Conversely, if a deeper case is re- soaking effect of the workpiece’s cold radius/thickness has a unique “wave” quired with an existing system that uti- core. shape, which differs significantly from lizes a higher-than-optimal frequency, the commonly assumed, classical ex- then a lower power density in combi- Magnetic waves in hardening ponential distribution. Here, the nation with a longer heat time can be In most publications devoted to in- power density is maximum at the sur- used. Figure 2 compares a required duction heating and induction heat face, and decreases toward the core. case or hardness depth with the cur- treating, distributions of current den- But then, at a certain distance from the rent penetration depths obtained in hot sity and power density (heat source surface, the power density increases, using too high, too low, and op- distributions) along the workpiece reaching a maximum value before timal frequencies. thickness/radius are simplified, and again decreasing. If the frequency has been chosen described as exponentially decreasing This “magnetic-wave” phenom- correctly, the thickness of the non- from the surface into the workpiece enon was introduced by Davies and magnetic surface layer — the layer (see Eq. 1 and Fig. 1). It is important Simpson,2 and Losinskii.3 They intu- that is heated to above the Curie tem- to remember that this assumption is itively felt there should be situations perature — is somewhat less than the correct only for a solid body (work- where the power density (heat source) current penetration depth in hot steel piece) having constant electrical resis- distribution would differ from that of (Fig. 2, right). tivity and magnetic permeability. the traditionally accepted exponential If the frequency is too high for the Therefore, realistically speaking, this form. They provided a qualitative de- specified case depth (Fig. 2, left), ad- assumption can be made for only scription based on their intuition and understanding of the physics of the Current penetration depth in hot steel Required case depth Surface Core process. At the time, a quantitative descrip- tion of the phenomenon could not be developed due to limited computer power and the lack of software that could simulate the tightly coupled electrothermal phenomena of induc- tion heating processes. Of course, it Too high frequency Too low frequency Optimal frequency also was not possible to measure the Fig. 2 — How frequency affects current penetration depth in hot steel. If the frequency is too power/current density distribution high, left, surface overheating results, which can lead to excessive grain growth. If the frequency inside the solid body (workpiece). is too low, center, a higher power density and large heat-affected zone result, which can waste energy New software: Modern numerical and cause excessive distortion. The optimum frequency, right, results in a current penetration computation software, such as Induc- depth 1.2 to 2 times the required case depth. (Ref. 1) toheat’s ADVANCE, enables a quan- 24 HEAT TREATING PROGRESS • SEPTEMBER/OCTOBER 2004 htprof.qxp 9/3/2004 11:23 AM Page 3

PROFESSORPROFESSOR INDUCTIONINDUCTION titative estimation of the magnetic- 1000 wave phenomenon (also known as the (1832) CD 1 “dual-properties” phenomenon ) 800 based on a coupled approach of (1472) solving electromagnetic and thermal 600 problems. (1112) An example is given in Fig. 3, which 400 (752) shows the temperature profile, left, Temperature, °C (°F) 200 CD

and power density distribution, right, (392) Power density (heat source) along the radius of a 36 mm (1.42 in.) in diameter carbon steel shaft at the 10 (0.39) 12 (0.47) 14 (0.55) 16 (0.63) 18 (0.71) 10 (0.39) 12 (0.47) 14 (0.55) 16 (0.63) 18 (0.71) final stage of heating using a fre- Radius, mm (in.) Radius, mm (in.) quency of 10 kHz. For comparison, the Internal area Surface Internal area Surface dotted curve in the power density dis- the to above it. Fig. 3 — Actual temperature profile and tribution represents the classical ex- In other applications, the impor- power density distribution for induction sur- ponential distribution. tance of this complex phenomenon face hardening of carbon steel shafts using a As can be seen, a power density may differ. In applications such as sur- frequency of 10 kHz. Case depth (CD) is 2 mm (0.08 in.). The dashed line in the graph at right maximum is always located at the sur- face hardening, the magnetic-wave is the commonly assumed power density dis- phenomenon plays a very important face of the cylinder, and in the sub- tribution. (Ref. 1) surface area, the density decreases to- role in the prediction of final temper- ward the core. However, at a certain ature profile and case depth. On the Core Surface distance from the surface, the density other hand, in applications such as starts to increase again, creating a through hardening or induction “magnetic wave.” The cause of this heating of steel products prior to hot Cold stage phenomenon is the remaining mag- forming, the duration of the transition netic properties of the heated steel at stage is much shorter compared with this distance. both the cold stage and, in particular, Note that in some applications, due the hot stage. For example, the hot Power density to this phenomenon, the maximum heating stage in a heating-for-forging value of heat sources can be located in application is usually 65 to 70% of the an internal layer of the workpiece and total heating time (which also includes not at its surface. A discussion of how the cold and transition stages). Here, this can occur at higher frequencies (25 and in other applications like it, the Transition stage and 100 kHz) can be found in Ref. 1. magnetic-wave phenomenon has an Consideration of the wavelike dis- insignificant effect on the final tem- tribution of power density (heat perature distribution and often can be Power density source) will make a significant differ- ignored. ence in the frequency chosen for in- Note that the magnetic-wave phe- duction surface hardening. Conclu- nomenon can also play an important sion: frequency selection is not as easy role in some low-temperature induc- a task as it appears to be at first glance. tion heating applications, such as A detailed evaluation of the entire coating and plating. For example, if a Hot stage heating process using modern com- nonmagnetic, electrically conductive puter modeling software is required. coating is applied to a carbon steel part,

the phenomenon will occur and can be Power density Magnetic waves in bars and billets quite pronounced, depending on the frequency and thickness of the The electromagnetic wave phe- 0.5 (13) 1.0 (25) 1.5 (38) nomenon just described is always pre- nonmagnetic surface deposit. Radius, in. (mm) sent when the final temperature of a Fig. 4 — Power density profiles at different magnetic workpiece exceeds the Curie References stages of in-line induction heating of 75 mm 1. Handbook of Induction Heating, by V. (3 in.) in diameter carbon steel bars. (Ref. 1) point. Figure 4 shows power density Rudnev, D. Loveless, R. Cook, and M. profiles at different stages during in- Black: Marcel Dekker Inc., New York, 2003, line induction heating of 75 mm (3 in.) 800 p. in diameter carbon steel bars. The 2. Induction Heating Handbook, by J. Davies magnetic-wave effect takes place and P. Simpson: McGraw-Hill Inc., New York, 1979, 426 p. during the transition from the cold to 3. Industrial Applications of Induction the hot heating stage, when the work- Heating, by M.G. Losinskii: Pergamon piece temperature rises from below Press plc, Oxford, 1969, 460 p. HEAT TREATING PROGRESS • SEPTEMBER/OCTOBER 2004 25