BOOKCOMP, Inc. — John Wiley & Sons / Page 1237 / 2nd Proofs / Heat Transfer Handbook / Bejan HEAT TRANSFER TO MOVING MATERIALS UNDERGOING THERMAL PROCESSING 1237 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [1237], (7) Lines: 248 to 276 ——— 2.26157pt PgVar ——— Long Page PgEnds: T E X [1237], (7) 1234567 0.0 0.2 0. 4 0.6 0.8 1.0 xxD * /4 = / LD/=4 LD/=7 LD/=2 LD/=1 Figure 17.3 Steady-state temperature distributions in a continuously moving cylindrical rod for different values of L/D(4 Pe = VD/α = 0.4;4Bi= hD/k = 0.2). Figure 17.3 shows the results for the temperature distribution in a circular rod (γ = D/4) for one particular case. Note that when L/D > 7 (for this case), the condition of an infinitely long rod is attained and the temperature at the end of the rod approaches asymptotically that of the environment. Under those conditions, it is easily shown that eq. (17.4) reduces to θ(x ∗ ) θ o = exp Pe − Pe 2 + 4Bi 2 x ∗ (17.6) Necessarily, eqs. (17.4) and (17.6) are valid only when the value of Bi is sufficiently small, to ensure that the temperature variation across the rod is negligible compared to that along its length. Furthermore, when Pe → 0, the problem reduces to that of a stationary extended surface. The problem depicted in Fig. 17.2 may be extended to include a more realis- tic boundary condition at x ∗ = 0 and L ∗ , taking into account the upstream and downstream thermal conditions. If θ(x ∗ →−∞) = θ o and θ must remain finite as x ∗ →+∞, the temperature distribution in the region exposed to the thermal environment is governed by the following three equations: d 2 θ 1 dx ∗2 − Pe dθ 1 dx ∗ = 0forx ∗ < 0 BOOKCOMP, Inc. — John Wiley & Sons / Page 1238 / 2nd Proofs / Heat Transfer Handbook / Bejan 1238 HEAT TRANSFER IN MANUFACTURING AND MATERIALS PROCESSING 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [1238], (8) Lines: 276 to 307 ——— 0.45232pt PgVar ——— Normal Page PgEnds: T E X [1238], (8) d 2 θ 2 dx ∗2 − Pe dθ 2 dx ∗ − Biθ = 0 for 0 <x ∗ <L ∗ (17.7) d 2 θ 3 dx ∗2 − Pe dθ 3 dx ∗ = 0forx ∗ >L ∗ for which Jaluria has provided the solutions as θ 1 = 1 + Pe m 2 e m 2 L ∗ − m 1 e m 1 L ∗ m 2 1 e m 1 L ∗ − m 2 2 e m 2 L ∗ e Pex ∗ θ 2 = 1 + Pe m 2 e m 2 L ∗ e m 1 x ∗ − m 1 e m 1 L ∗ e m 2 x ∗ m 2 1 e m 1 L ∗ − m 2 2 e m 2 L ∗ θ 3 = 1 + Pe ( m 2 − m 1 ) e PeL ∗ m 2 1 e m 1 L ∗ − m 2 2 e m 2 L ∗ (17.8) As in the more limited case, three physical parameters emerge from the analysis. The P ´ eclet number Pe represents a dimensionless speedoftheworkpiece and serves to compare the ability of the workpiece to advect energy through its motion to its ability to transfer energy along its length due to thermal conduction. The Biot number Bi may be thought of as a ratio of internal thermal resistance to external thermal resis- tance. These analytical results are limited to small values of Bi, because temperature gradients in the direction transverse to the workpiece motion have been neglected. Finally, L ∗ is the dimensionless length of the workpiece. When L ∗ is large enough (the value depends on Bi and Pe), the workpiece will approach thermal equilibrium with the thermal environment at T ∞ . Figure 17.4 shows some typical results. For increasing speed (Pe) there is a shorter time for heat loss up to given distance, so that the temperature decay is more gradual. Similarly, the thermal penetration of the temperature field upstream due to conduction is lower for increased Pe, because advection effects begin to dominate over thermal conduction. A large value of Bi implies more effective cooling by the external thermal environment, and the increased thermal gradient in the active heat transfer region leads to increased thermal penetration upstream. (It should be noted that the results in Fig. 17.4b are obtained for values of the Biot number for which the assumption of a one-dimensional temperature distribution is not fully justified. However, the analysis does illustrate effectively the physical significance of the dimensionless parameters, and this significance is repeated for many other MMPs.) Finally, when L ∗ is large enough (about 5 in Fig. 17.4b), the temperature at the end of region 2 (at x ∗ = L ∗ )is very close to the ambient temperature, and any increase in L ∗ will not affect the heat transfer. Two-Dimensional Workpieces When temperature variations over the cross sec- tion of the workpiece are important relative to those in the direction of motion, a dif- ferential control volume of size dA c ×dx must be used to establish an energy balance BOOKCOMP, Inc. — John Wiley & Sons / Page 1239 / 2nd Proofs / Heat Transfer Handbook / Bejan HEAT TRANSFER TO MOVING MATERIALS UNDERGOING THERMAL PROCESSING 1239 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [1239], (9) Lines: 307 to 307 ——— -5.073pt PgVar ——— Normal Page PgEnds: T E X [1239], (9) Ϫ10 Ϫ8 Ϫ6 Ϫ5 Ϫ4 Ϫ2 0 0 5 42 10 86 0.88 0.88 0.9 0.9 0.92 0.92 0.94 0.94 0.96 0.96 1.02 1.02 1 1 0.98 0.98 x * x * Pe =10 L * = 0.5 L * =1 L * =2 L * =5 Pe =1 Pe = 0.1 Bi = 0.1 Bi = 0.4 L * =5 Pe =2 ()a ()b Figure 17.4 (a) Effect of varying Pe on the temperature distribution in a continuously moving solid ( Bi = 0.1;L ∗ = L/D = 0.2); (b) Effect of varying exposed rod length (L ∗ = L/D )on the temperature distribution ( Bi = 0.4;Pe = 2.0). (From Jaluria, 1993.) BOOKCOMP, Inc. — John Wiley & Sons / Page 1240 / 2nd Proofs / Heat Transfer Handbook / Bejan 1240 HEAT TRANSFER IN MANUFACTURING AND MATERIALS PROCESSING 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [1240], (10) Lines: 307 to 321 ——— 5.43802pt PgVar ——— Short Page * PgEnds: Eject [1240], (10) Figure 17.5 Isotherms for a long moving cylindrical rod: (a)Bi D = 10, Pe D = 0.4; (b) Bi D = 0.2, Pe D = 0.4. that will yield the temperature distribution. Now the coordinate system chosen must be more specific. For example, for a circular rod of diameter D, the steady-state con- duction equation becomes ∂θ ∂τ + Pe ∂θ ∂X = 1 R ∂ ∂R R ∂θ ∂R + ∂ 2 θ ∂X 2 (17.9) where τ = αt/D 2 ,X = x/D, and R = r/D. Initial and boundary conditions are as follows: BOOKCOMP, Inc. — John Wiley & Sons / Page 1241 / 2nd Proofs / Heat Transfer Handbook / Bejan HEAT TRANSFER TO MOVING MATERIALS UNDERGOING THERMAL PROCESSING 1241 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [1241], (11) Lines: 321 to 350 ——— 0.02715pt PgVar ——— Short Page PgEnds: T E X [1241], (11) τ > 0: at X = 0, θ = 1.0 for 0 ≤ R ≤ 0.5 at X = L(τ), ∂θ ∂X =−Bi · θ for 0 ≤ R ≤ 0.5 at R = 0, ∂θ ∂R = 0 for 0 ≤ X ≤ L(τ) at R = 0.5, ∂θ ∂R =−Bi · θ for 0 ≤ X ≤ L(τ) (17.10) In this case, the length of the rod is a function of time, L = V t , and convective heat loss is presumed to occur from the end of the rod. As L ∗ increases, a steady-state temperature distribution is attained and θ → 0asL ∗ →∞. Jaluria (1993) presented some typical results of a computational solution for a planar strip [for which eq. (17.9) must be modified slightly—D refers to the strip thickness in Bi D and Pe D ] and for a cylindrical rod. Figure 17.5 shows some typical isotherms for the latter case. Note that when Bi D = 0.2, the temperature variation through the rod is negligible. 17.2.2 Interaction between a Discrete Heat Source and a Continuously Moving Workpiece The situation in which a source of heat is confined to a small region on the surface of the workpiece is considered next. The one-dimensional case is discussed first, for a discrete and a distributed heat source. Then the two- and three-dimensional response of a solid to a moving heat source is also presented. These form the basis for understanding the behavior of a wide variety of manufacturing systems. Thin Plate or Rod with a Moving Planar Heat Source The situation is depicted in Fig. 17.6. The magnitude of the heat source is Q (in watts), and the surface is considered to transfer heat by convection to an environment at T ∞ with a heat transfer coefficient h. For a coordinate system fixed to the heat source moving Figure 17.6 Thin plate or rod with a moving planar heat source. BOOKCOMP, Inc. — John Wiley & Sons / Page 1242 / 2nd Proofs / Heat Transfer Handbook / Bejan 1242 HEAT TRANSFER IN MANUFACTURING AND MATERIALS PROCESSING 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [1242], (12) Lines: 350 to 404 ——— 0.26027pt PgVar ——— Normal Page * PgEnds: Eject [1242], (12) in the direction +x with velocity V (the workpiece is moving toward the source −x direction at a velocity of −V ), the quasi-steady temperature distribution in the workpiece is given by T(x ∗ ) − T ∞ = (T max − T ∞ ) exp + Pe 2 + 4Bi− Pe x ∗ 2 for x ∗ ≤ 0 (T max − T ∞ ) exp − Pe 2 + 4Bi+ Pe x ∗ 2 for x ∗ ≥ 0 (17.11) where the maximum temperature occurring in the workpiece is T max − T ∞ = Qγ k 1 Pe 2 + 4Bi (17.12) The maximum temperature is proportional to the magnitude of the heat source, de- creases with increasing workpiece speed, and is also lower for plates that are good conductors. The temperature drops quickly in the +x direction, and more slowly in the −x direction. Two limiting cases provide additional physical insight: When the workpiece speed is very large (Pe 2 4Bi), T(x ∗ ) − T ∞ = (T max − T ∞ ) for x ∗ ≤ 0 (T max − T ∞ ) exp ( −Pe · x ∗ ) for x ∗ ≥ 0 (17.13) T max − T ∞ = Q ρcA c V All the mass that passes by the source is heated to T max , which depends on the heat capacity of the material, ρc. For very low speeds (Pe 2 4Bi), the workpiece functions essentially as an infinitely long extended surface with a perfectly symmetric temperature profile: T −T ∞ = (T max − T ∞ ) exp − √ Bi · x ∗ T max − T ∞ = Qγ/k 2 √ Bi (17.14) Thin Plate with a Moving Line Heat Source Applications involving welding or heat treating with a localized source of energy (electron beam, laser beam, welding torch) provide an additional set of heat transfer problems. If the workpiece is very thin, temperature gradients through the thickness may be neglected compared to those in the direction of motion or transverse to it. When the surface loses heat to the surroundings at T ∞ with a heat transfer coefficient h, the temperature distribution is given by T −T ∞ = Q/H 2πk exp Pe H x ∗ 2 K o Pe H 2 2 + ( Bi H ) 2 r ∗ (17.15) BOOKCOMP, Inc. — John Wiley & Sons / Page 1243 / 2nd Proofs / Heat Transfer Handbook / Bejan HEAT TRANSFER TO MOVING MATERIALS UNDERGOING THERMAL PROCESSING 1243 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [1243], (13) Lines: 404 to 458 ——— 6.19666pt PgVar ——— Normal Page PgEnds: T E X [1243], (13) where H is the plate thickness, x ∗ = x/H is the dimensionless distance upstream of source, r ∗ = √ x ∗ + y ∗ ,y ∗ = y/H is the dimensionless distance transverse to the heat source, Pe H = VH/α, Bi H = hH/k, and K o is the modified Bessel function. When surface heat losses can be neglected, Bi H is set to zero in eq. (17.15). Semi-infinite Solid with a Moving Point Source The three-dimensional tem- perature distribution for the case of negligible surface heat losses is given by T −T ∞ = Q 2πkr exp − V(x + r) 2α (17.16) The symbols in eq. (17.16) are the same as used previously, with r = x 2 + y 2 + z 2 . There is not a convenient length scale because the plate is infinite in the x and y directions and semi-infinite in the z direction. If a finite plate thickness is introduced, eq. (17.16) may be rewritten in a dimensionless form as T −T ∞ = Q/H 2πkr ∗ exp − Pe H (x ∗ + r ∗ ) 2 (17.17) However, it must be remembered that H must be significantly larger than the thermal penetration in the z direction. Semi-infinite Plane with a Finite Size Moving Heat Source The quasi- steady state thermal response of a workpiece to frictional heating (e.g., in machining, extrusion, or drawing) or to heating from a moving finite source may be obtained from the solution for transient stationary heating of a semi-infinite solid (Incropera and DeWitt, 1996) with a constant surface heat flux. These results are applied to the case of a moving solid under a uniform heat flux q s , which acts over a distance x = 0 to x = l, with the remaining surface being insulated, to yield the following result for the dimensionless surface temperature rise: (T s − T ∞ )πkV 2αq s = π · Pe l √ x ∗ − √ x ∗ − 1 for x ∗ ≥ 1 π · Pe l · x ∗ for x ∗ ≤ 1 (17.18) The peak temperature occurs at x = l, with a drop in temperature downstream of the heat source. Equations (17.18) neglect any effects of heat conduction in the direction of motion. The more general results, cited by Carslaw and Jaeger (1959), may be expressed by (T s − T ∞ )πkV 2αq s = x ∗ (Pe l /2) (x ∗ −1)(Pe l /2) e u K o (|u|)du (17.19) The integral in eq. (17.19) has an analytical form with distinct results expressible for the three regions of interest, x ∗ ≤ 0, 0 ≤ x ∗ ≤ 1, and 1 ≤ x ∗ , and is shown BOOKCOMP, Inc. — John Wiley & Sons / Page 1244 / 2nd Proofs / Heat Transfer Handbook / Bejan 1244 HEAT TRANSFER IN MANUFACTURING AND MATERIALS PROCESSING 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [1244], (14) Lines: 458 to 458 ——— * 51.589pt PgVar ——— Normal Page PgEnds: T E X [1244], (14) Figure 17.7 Surface temperature distribution due to a moving heat source of length l(Pe = Vl/α). Figure 17.8 Peak surface temperature due to a moving heat source of length l, including effects of convective cooling. (From DesRuisseaux and Zerkle, 1970.) BOOKCOMP, Inc. — John Wiley & Sons / Page 1245 / 2nd Proofs / Heat Transfer Handbook / Bejan THERMAL ISSUES IN HEAT TREATMENT OF SOLIDS 1245 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [1245], (15) Lines: 458 to 467 ——— 0.0pt PgVar ——— Normal Page PgEnds: T E X [1245], (15) graphically in Fig. 17.7. When Pe l ≥ 20, the simpler result, eq. (17.18), is quite adequate and is usually the one cited when analyzing temperature rises in machining and grinding processes, which usually occur at speeds sufficiently high to yield large values of the P ´ eclet number. DesRuisseaux and Zerkle (1970) extended these results to account for convective heat losses from the surface. The dimensionless temperature at the trailing edge of the heating zone (x = l) is shown in Fig. 17.8 in terms of the heat transfer coefficient h, expressed dimensionlessly as 2 Bi l /Pe l = 2αh/kV . When the region between 0 ≤ x ≤ l is not convectively cooled, they suggest an approximation to the temperature in the heated zone obtained by increasing the heat source strength by an amount equal to the average convective flux that would have occurred in the heated zone. 17.3 THERMAL ISSUES IN HEAT TREATMENT OF SOLIDS Heat treatment is generally regarded as the controlled heating and cooling of metals for the purpose of altering their mechanical and physical properties. The material remains in a solid phase (usually below the eutectiod or the bulk melting temperature), and neither material removal nor significant alteration of shape occurs. Analysis of the heat transfer processes involved is straightforward, in principle, to obtain the appropriate temporal and spatial temperature response of the solid. However, achieving a desired result requires that the thermal behavior of a solid be carefully integrated with an understanding of the equilibrium properties, the kinetics of phase transformations and diffusion at various temperatures, and the relationship of the mechanical properties of a solid to its material structure. Consequently, study of this subject is usually confined to the expertise of metallurgists and materials specialists rather than heat transfer specialists. The interested reader is referred to Chapter 5 of DeGarmo et al. (1997) or Chapter 6 of Schey (2000) for an introductory discussion of the metallurgical issues, and to a large number of publications of the American Society of Metals (e.g., ASM, 1991) for more detailed information. A useful but oversimplified categorization of heat treatment processes divides them into bulk and surface heat treating. In the former the entire solid is maintained at an elevated temperature to obtain a metallurgical state indicated by the equilib- rium phase diagram. The key processing step is then the cooling process, in which the material goes through a nonequilibrium transformation. Under some conditions, a final stage of the process may occur at room temperature (e.g., natural aging), where diffusion occurs to convert an unstable supersaturated solution into a stable two-phase structure. An important tool in adapting such cooling processes is the time–temperature transformation (TTT) diagram, which characterizes the kinetics of solid–solid transformations at various temperatures. For example, thin specimens of a metal are heated to obtain an equilibrium condition, followed by rapid quenching to a specified temperature. The transformation to a new stable metallurgical state is then observed as a function of time, and the points where a transformation begins and ends are noted. The locus of these points usually takes the shape of a pair of “C” BOOKCOMP, Inc. — John Wiley & Sons / Page 1246 / 2nd Proofs / Heat Transfer Handbook / Bejan 1246 HEAT TRANSFER IN MANUFACTURING AND MATERIALS PROCESSING 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [1246], (16) Lines: 467 to 493 ——— 0.0pt PgVar ——— Normal Page PgEnds: T E X [1246], (16) curves. The nose of the “C” is at an intermediate temperature, which is where the transformation occurs most rapidly, a compromise between the equilibrium driving force for the transformation and the species diffusion rates. Real quenching processes occur under a continuous cooling condition, and a modification of the TTT diagram known as the continuous cooling transformation (CCT) curve is often overlaid on the TTT diagram. Then an actual temperature vs. time plot provides the metallurgist with information on the materials characteristics that will result from a particular cooling path. Achieving a uniform cooling rate and a resulting uniform structure is rarely possible for large workpieces, because quenching or cooling is naturally a nonhomo- geneous process. Viskanta and Bergman (2000) have discussed in more detail some of the heat transfer issues related to quenching of metals. An example of surface heat treatment is a local annealing and quenching process to achieve a hard, wear-resistant surface coupled to a tough, fracture resistance core. Common heating techniques include flame heating, induction heating, laser beam heating, and electron beam heating. An important issue is that of control of the motion of the heat source and of the workpiece, because only a portion of the surface may be exposed at a time (see Section 17.8). 17.4 MACHINING PROCESSES: METAL CUTTING The goals of this section are to review the mechanisms of heat generation in metal cutting, discuss the relevant modeling assumptions used in thermal analysis, and describe the limitations of these models. 17.4.1 Background The process of chip formation during machining consumes a great deal of power. The overall cutting power is given by P = VF p (17.20) where F p is the power component of the cutting force, which is parallel with the workpiece velocity, and V is the velocity of the workpiece material. Most of this power is converted into heat and then partitioned to the workpiece chip and cutting tool. Some of the detrimental effects of high temperatures include (1) increased tool wear, and hence shortened tool life (high temperature at the cuttingedge is the primary factor associated with accelerated wear); (2) decreased process efficiency (escalation of temperature limits the rate of material removal, specifically the cutting speed); and (3) decreased surface quality as a result of residual stresses and thermal distortion. The simplest case of metal cutting, which has been the focus of most modeling effort regarding thermal behavior, is orthogonal cutting. A single cutting edge is oriented normal to the direction of motion, and the chip flows up the surface of the wedge-shaped cutting edge, with a velocity V c , in the same plane as the velocity of the workpiece. Figure 17.9a illustrates the chip, tool, and workpiece geometry. The chip . core. Common heating techniques include flame heating, induction heating, laser beam heating, and electron beam heating. An important issue is that of control of the motion of the heat source and. 0forx ∗ < 0 BOOKCOMP, Inc. — John Wiley & Sons / Page 1238 / 2nd Proofs / Heat Transfer Handbook / Bejan 1238 HEAT TRANSFER IN MANUFACTURING AND MATERIALS PROCESSING 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [1238],. energy balance BOOKCOMP, Inc. — John Wiley & Sons / Page 1239 / 2nd Proofs / Heat Transfer Handbook / Bejan HEAT TRANSFER TO MOVING MATERIALS UNDERGOING THERMAL PROCESSING 1239 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [1239],