BOOKCOMP, Inc. — John Wiley & Sons / Page 1247 / 2nd Proofs / Heat Transfer Handbook / Bejan MACHINING PROCESSES: METAL CUTTING 1247 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 [1247], (17) Lines: 493 to 512 ——— -0.37898pt PgVar ——— Normal Page PgEnds: T E X [1247], (17) Uncut chip Cutting edge a o o ␥ c ␥ o ␣ c V c V c a c F f F n F s F c Chip yy x x ()a ()b f RЈ s n R c Figure 17.9 Chip formation during orthogonal metal cutting. (From DeVries, 1992.) is assumed to be continuous, at least in the vicinity of the cutting tool, characteristic of a ductile workpiece such as brass, low-carbon steel, or an aluminum alloy. Plastic shear is the principal mechanism of chip formation. There is a finite volume to the shear zone; however, at common cutting speeds, the angle between the planes, which defines the deformation zone, collapses, so that the deformation appears to occur in a single plane. For this reason, an infinitely thin zone of deformation is assumed in most models. Figure 17.9b shows the force components between the cutting tool and the workpiece that are needed to characterize the cutting process, including the heat generation and temperature distributions. DeVries (1992) has described the relationships among these forces and the geo- metrical features of chip formation, all based on the classical literature. Several the- ories are available to determine the shear plane angle φ o . One that minimizes the cutting power is φ o = π 4 − 0.5 β − γ o (17.21) where β is the friction angle, defined by tan −1 µ, and µ is the coefficient of friction between the chip and the tool, F f /F n . The resultant force at the shear plane must equal the resultant force at the tool–chip interface. R = F c + F s − R = F f + F n (17.22) BOOKCOMP, Inc. — John Wiley & Sons / Page 1248 / 2nd Proofs / Heat Transfer Handbook / Bejan 1248 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 [1248], (18) Lines: 512 to 545 ——— 2.80725pt PgVar ——— Normal Page * PgEnds: Eject [1248], (18) The shear force is then given by F s = (l s b)τ s = τ s ba sin φ 0 (17.23) where τ s is the plastic flow stress of the material, l s the length of the shear zone, a the depth of cut, and b the width of cut. Combining this information, one may obtain results for the force components F c and F n as F c = F s µ cos(φ 0 − γ 0 ) + sin(φ 0 − γ 0 ) cos(φ 0 − γ 0 ) − µ sin(φ 0 − γ 0 ) (17.24a) F n = F s 1 cos(φ 0 − γ 0 ) − µ sin(φ 0 − γ 0 ) (17.24b) The x component of the resultant force, F p , is also easily obtained. 17.4.2 Thermal Analysis Typically, the heat sources in orthogonal cutting can be regarded as being localized in three places in the cutting zone. Mechanical energy dissipated in the shear zone consists of that needed to cause plastic flow (a relatively small amount) and that converted into internal energy (heat). Usually, this is modeled as a planar heat source. The second significant source of heating is the rake face between the moving chip and the cutting tool face. The heat generated there by sliding friction is usually modeled as a uniformly distributed planar heat source. A tertiary source is frictional heating between the flank face of the cutting tool and the moving workpiece. For sharp tools, the contact area is very small, resulting in the neglect of this source in most models. The engineering information sought at the simplest analytical level is the temperature rise at the tool–chip interface, which is due to a combination of the shear plane heating of the chip and the frictional heating between the chip and the tool face. The focus in the remainder of this section is on these two processes and is based on the classical work of Trigger and Chao (1951) as described in DeVries (1992). Tool–Chip Interface Temperature Rise The (assumed) uniform heating of the chip as it passes through the shear zone is dealt with in the next section. Its temperature is assumed to be T c and it moves with a velocity V c as it passes over the tool surface through a frictional contact length l c . The total rake face heat flux is given by q r = F f V c bl c (17.25) Adapting the solution for the surface temperature rise due to a constant heat flux, eq. (17.16), we may determine the average temperature rise at the chip/tool interface to be BOOKCOMP, Inc. — John Wiley & Sons / Page 1249 / 2nd Proofs / Heat Transfer Handbook / Bejan MACHINING PROCESSES: METAL CUTTING 1249 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 [1249], (19) Lines: 545 to 602 ——— 0.12538pt PgVar ——— Normal Page PgEnds: T E X [1249], (19) (T r − T 0 ) avg = (T c − T 0 ) + 4 3 √ π B 3 q r ρ c c c V c Pe l c (17.26) where B 3 is the fraction of the frictional heat flux that is conducted into the chip, which is regarded as being infinitely thick for the short times associated with this process. Trigger and Chao (1951) provided an ad hoc theory for the value of B 3 ; however, a conservative estimate of the rake face temperature rise is obtained if B 3 is set equal to 1. Energy Generation at the Shear Plane The heat flux through the uncut chip cross section, bl s , the area of the shear plane, results in a temperature rise in the chip as it passes through the shear plane: q c = (T c − T 0 )c c ρ c V (17.27) In turn, q c is given by the following expression: q c = B 1 (F s V c/s − F p VB 2 ) bl s = B 1 [F p V(1 − B 2 ) − F f V c ] bl s (17.28) where B 1 is the fraction of the energy at the shear zone that goes into raising the temperature of the chip as opposed to plastic deformation energy (typical values: 0.85 <B 1 < 0.95). The quantity B 2 is the fraction of the power dissipated in chip formation that is conducted into the workpiece (typical values: 0.05 <B 2 < 0.15). In a more complete numerical study of the thermal field in the vicinity of the shear plane, Dawson and Malkin (1984) found that the average temperature rise of the chip as it crosses the shear plane may be expressed approximately as πkV (T c,avg − T o ) 2αq c = 3.11(1 − 0.22e −2.9φ o )e −0.7φ o · Pe 0.5e −3φ o l s (17.29) and the fraction of the shear plane energy removed by the chip in cutting is given by R = 1 − B 2 = V(2 sin φ o )(ρc)(T c,avg − T o ) 2q c (17.30) A plot of this result is given as Fig. 17.10. Assessment of Steady-State Metal Cutting Temperature Models This section concludes with a brief summary of a study by Stephenson (1991) which com- pared calculations from four steady-state metal cutting temperature models [Loewen– Shaw (Shaw, 1984); Boothroyd, 1975 (as modified by Tay et al., 1976); Wright et al., 1980; Venuvinod and Lau, 1986] with experimental results. Both tool–chip interface temperatures and deformation zone temperatures were considered. All the models assume that there are two heat sources, as discussed previously. Differences arise in the method in which heat generation is partitioned among the tool, chip, and workpiece, whether variations of thermal properties are considered, BOOKCOMP, Inc. — John Wiley & Sons / Page 1250 / 2nd Proofs / Heat Transfer Handbook / Bejan 1250 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 [1250], (20) Lines: 602 to 613 ——— -0.603pt PgVar ——— Long Page PgEnds: T E X [1250], (20) Figure 17.10 Fraction of shear plane energy that enters the chip during metal cutting. and how the rake face heat source is modeled. The Boothroyd and Wright models use an energy partition analysis due to Weiner (1955) and thus assume less heat flow into the workpiece than do the other two models, which use analysis based on Trigger and Chao (1951). The Loewen–Shaw and Venuvinod–Lau models take into consideration the variation of thermal properties, whereas the Boothroyd and Wright models do not. Finally, the Boothroyd and Loewen–Shaw models assume that the rake face source is a uniform heat source, while the Wright and Venuvinod–Lau models can be applied with more general source strength distributions. Stephenson (1991) found that the best results were obtained with the Loewen–Shaw and Venuvinod–Lau models, while the Boothroyd and Wright models overestimated both rake face and shear zone temperature rises by approximately a factor of 2 for a variety of materials and cutting conditions. All models failed when chips produced were discontinuous so that tool–chip contact length was not constant. Finally, all the models predicted shear plane temperatures that were too high, probably due to their common assumption of planar heat generation in the shear zone. 17.5 MACHINING PROCESSES: GRINDING The goals of this section are to review the mechanisms of heat generation in metal grinding and to provide an overview of some of the relevant modeling assumptions BOOKCOMP, Inc. — John Wiley & Sons / Page 1251 / 2nd Proofs / Heat Transfer Handbook / Bejan MACHINING PROCESSES: GRINDING 1251 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 [1251], (21) Lines: 613 to 625 ——— 0.25099pt PgVar ——— Long Page PgEnds: T E X [1251], (21) used in thermal analysis. A detailed exposition of mathematical modelingfor grinding is beyond the scope of this chapter; however, some key references to the literature are cited. 17.5.1 Background Grinding is a precision machining process capable of delivering surface roughness 10 times lower than that achieved by metal cutting and with dimensional accuracy that is 10 times better. Grinding accounts for approximately 20% of all machining in the United States. One of the major differences between cutting and grinding relates to the num- ber and geometry of the cutting edges. Grinding uses an abrasive wheel with many randomly oriented cutting edges, while metal cutting uses a known number of cut- ting edges with a controlled geometry. Orthogonal metal cutting involves the use of positive or “moderately” negative rake angles. In grinding, the small abrasive grits with random orientation give rise to large negative rake angles. The chips or swarf produced in grinding are typically an order of magnitude smaller than that produced in metal cutting. These differences are illustrated in Fig. 17.11. Although surface grinding is not the most common type of production grinding, it is the simplest to model, and like orthogonal cutting, has been the focus of most modeling effort, much of which has occurred since 1990. Conventional grinding is characterized by small depths of cut (0.005 to 0.05 mm) and fast workpiece velocities (100 to 500 mm/s), whereas creep-feed grinding yields cut depths of 1 to 20 mm with very slow work- piece velocities (1 to 50 mm/s). In both cases the wheel velocity (typically, 20 to Figure 17.11 Schematic of the wheel–workpiece interface for a grinding process. (From DeVries, 1992.) BOOKCOMP, Inc. — John Wiley & Sons / Page 1252 / 2nd Proofs / Heat Transfer Handbook / Bejan 1252 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 [1252], (22) Lines: 625 to 656 ——— 1.80014pt PgVar ——— Long Page PgEnds: T E X [1252], (22) 80 m/s) is much larger than the workpiece velocity. Other types of grinding are exter- nal and internal cylindrical grinding, in which both the workpiece and the grinding wheel are cylindrical; face grinding, which uses the flat edges of a cylindrical wheel against a workpiece that moves perpendicular to the wheel motion; and abrasive cut- off, in which a thin cutting wheel slices through a workpiece. In every case, how- ever, the basic process involves the cutting action of individual abrasive grains acting against the workpiece. In grinding, more of the energy provided at the spindle is converted into heat, making the potential for thermal damage to the workpiece significant. In orthogonal metal cutting, the tool cutting edge experiences elevated temperatures. In the case of grinding, more of the heat generated goes into the workpiece, which can lead to elevated workpiece temperatures and metallurgical changes and subsurface damage called burning, as discussed by Malkin (1984) and Guo and Malkin (1992). Addi- tional review and recent advances in thermal modeling of grinding processes is con- tained in Jen and Lavine (1995), Guo and Malkin (1996), Zhang and Faghri (1996), and Ju et al. (1998). 17.5.2 Workpiece Temperatures during Grinding At the level of an individual grain, heat generation occurs in the shear plane, at the chip–grain interface due to friction there, and at the grain–workpiece interface. At this scale, the methods described in Section 17.4 may be used to estimate local temperatures, which may locally attain over 1000°C. Although these temperatures may be important with respect to wear of the abrasive, they are not usually indicative of the effects on workpiece quality, because they are so very localized (temporally and spatially) and since material experiencing such temperatures will quickly be removed by another grain. More relevant to workpiece damage is the average interference zone temperature that results from the overall effect of all the grains in the contact region and from the grinding fluid that is almost always present. In this way, Malkin (1984) determined an expression for the maximum grinding zone temperature rise to be T m − T o = βα 1/2 w εP k w bd 1/4 s a 1/4 V 1/2 w (17.31) where P is the grinding power, ε the fraction of the grinding power entering the workpiece as heat, a and b the depth and width of cut, V w the workpiece velocity, d s the wheel diameter, k w the thermal conductivity of the workpiece, α w the thermal diffusivity of the workpiece, and β a constant that depends on the heat source shape (1.13 for rectangular and 1.06 for triangular). Malkin used a semiempirical analysis to determine ε = u − 0.45u ch u (17.32) where the specific grinding energy u is equal to the grinding power divided by the vol- umetric removal rate, and u ch is the chip formation component of this energy, which BOOKCOMP, Inc. — John Wiley & Sons / Page 1253 / 2nd Proofs / Heat Transfer Handbook / Bejan MACHINING PROCESSES: GRINDING 1253 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 [1253], (23) Lines: 656 to 682 ——— 0.46103pt PgVar ——— Long Page PgEnds: T E X [1253], (23) may be regarded as a material property (about 13.8 J/mm 3 for ferrous materials). Typical values of ε determined in this way are 0.7 to 0.9. Equation (17.31) neglects heat transfer to the wheel (through the grains) and the grinding fluid, which may restrict its validity to dry grinding or to the occurrence of film boiling in the grinding fluid. However, it does permit determination of the onset of workpiece burn (critical grinding zone temperature) in terms of a measured level of the grinding power. If a grinding fluid is present, a very approximate analysis yields the following for the fraction ε: ε = 1 1 + (V s /V w ) 1/2 [(kρc) c /(kρc) w ] 1/2 (17.33) where k c and (ρc) c in the product (kρc) c are weighted volumetric averages of the thermal properties of the grain, grinding fluid, and air (porosity), and V s is the wheel velocity. When applied to creep-feed grinding conditions, the energy partition to the workpiece is determined to be a small fraction of that in conventional grinding. Equation (17.33) may be modified for cubic boron nitride (CBN) grinding wheels to account for the much higher thermal conductivity of the CBN abrasive grains than that for aluminum oxide, by neglecting the effects of the grinding fluid (valid for conventional grinding), to yield (kρc) c = (1 − φ) 2 (kρc) g (17.34) for insertion into eq. (17.33). The average wheel surface porosity φ is difficult to estimate reliably; however, ad hoc estimates based on experimental measurements imply that ε ≈ 0.2 and φ ≈ 0.9 for typical conditions with ferrous workpieces. More recent attempts to predict temperature rise during grinding have sought to eliminate the need to estimate a partition of the grinding energy or to determine effec- tive fluid/grinding wheel thermal properties. The analytical details are too involved to present here. However, the general approach, indicated schematically in Fig. 17.12, is as follows. Separate analytical solutions are determined analytically for the con- duction heat transfer to an individual abrasive grain, in terms of a heat flux q g ;for conduction into the grinding fluid from the workpiece, in terms of a heat flux q f ; Figure 17.12 Heat flow paths in the vicinity of an abrasive grain–workpiece interface. (After Lavine and Jen, 1991.) BOOKCOMP, Inc. — John Wiley & Sons / Page 1254 / 2nd Proofs / Heat Transfer Handbook / Bejan 1254 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 [1254], (24) Lines: 682 to 703 ——— -0.03pt PgVar ——— Normal Page PgEnds: T E X [1254], (24) for the background heat transfer into the workpiece from the entire grinding zone, in terms of a heat flux q wb ; and for the heat transfer into the workpiece from an individual grain, in terms of a heat flux q wg . Then an appropriate coupling of these solutions is established using continuity of temperature and conservation of energy applied in a mathematically consistent way to each interface. The reader is referred to Jen and Lavine (1995), Lavine and Jen (1991), and Ju et al. (1998) for additional details. 17.6 THERMAL-FLUID EFFECTS IN CONTINUOUS METAL FORMING PROCESSES In this section we review briefly some of the important thermal and fluid consider- ations in continuous deformation processes such as drawing, rolling, and extrusion. Metal forming processes exploit the property of metals that allows them to flow plasti- cally in the solid state. By simply moving the material to the shape desired, as opposed to removing unwanted regions, there is little or no waste. In general, a temperature increase in the workpiece brings about a decrease in material strength, an increase in ductility, and a decrease in the rate of strain hardening (all of these effects tend to promote the ease of deformation). 17.6.1 Background Deformation processes tend to be classified as hot working (recrystallization occurs simultaneously; T initial ≥ 0.6T melt ), cold working (T initial = 0.3T melt ), or warm work- ing. However, because the general principles governing deformation at different tem- peratures are basically the same, classification according to specific input and output geometries and material and production rate conditions is often more useful. There- fore, one may broadly characterize forming operations under the headings of forging, sheet metal forming, drawing, extrusion, and rolling, the latter three being examples of continuous processing. An understanding of thermal effects in such systems is integrally coupled to a characterization of the metal flow, stresses, lubrication, and material handling and design of the forming equipment, so that a simple thermal analysis even for generic types of systems may not characterize a process completely. Among classical references, Altan et al. (1983) and Schey (1983) are detailed and complete, albeit with limited consideration of thermal and heat transfer effects. Yang (1992) and Tseng et al. (1990) also provide useful discussions of thermal effects in extrusion and drawing and in rolling, respectively. The focus of the present section is on factors associated with the temperature rise of the workpiece and the die (or roll). 17.6.2 Considerations for Thermal–Fluid Modeling in Extrusion and Drawing Figure 17.13 includes schematic illustrations of generic continuous deformation pro- cesses. (Although extrusion is a batch process, it is often studied as a quasi-steady- state process.) The principal distinctions among these are the delivery of the force to BOOKCOMP, Inc. — John Wiley & Sons / Page 1255 / 2nd Proofs / Heat Transfer Handbook / Bejan THERMAL-FLUID EFFECTS IN CONTINUOUS METAL FORMING PROCESSES 1255 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 [1255], (25) Lines: 703 to 716 ——— 0.42099pt PgVar ——— Normal Page PgEnds: T E X [1255], (25) Figure 17.13 Schematic diagrams of rolling, drawing, and extrusion processes. (After Altan et al., 1983.) the workpiece, the reduction in the cross section of the workpiece, and the speed. Heat generation arises from two sources: plastic deformation heating of the workpiece and frictional heating between the workpiece and the die (Fig. 17.14). Deformation Heating Considerations In wire drawing and often in rolling, the deformation and hence the heating of the workpiece due to the plastic flow of the material may be considered to be nearly uniform (Wright, 1976). Under these conditions, the uniform temperature increase of the wire is given by BOOKCOMP, Inc. — John Wiley & Sons / Page 1256 / 2nd Proofs / Heat Transfer Handbook / Bejan 1256 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 [1256], (26) Lines: 716 to 735 ——— -0.97295pt PgVar ——— Normal Page PgEnds: T E X [1256], (26) Figure 17.14 Surface and core temperature distributions during a drawing process. ∆T DH = W D ρ w c w τ s ln(A o /A 1 ) ρ w c w (17.35) where W D is the deformation work per unit volume, τ s the flow stress of the work- piece, A o and A 1 the initial and final cross-sectional areas, and ρ w c w the volumetric specific heat of the workpiece. Frictional Heating Considerations Frictional heating is concentrated near the wire–die interface, resulting in severe temperature gradients. In the simplest physical model, the latter is regarded as arising from a friction coefficient (assumed known and constant) and the normal stress on the workpiece, which is usually assumed to be the yield stress of the workpiece. Apparent friction coefficients can vary from 0.01 (hydrodynamic lubrication) up to 0.5 (boundary lubrication). Then if a suitable partition coefficient for the transfer of heat between the workpiece and die can be discerned, an estimate of temperature levels may be obtained. Even though uniform friction and heat partition coefficients may not be realistic, many modeling efforts are based on their use (Snidle, 1977; El-Domiaty and Kassab, 1998). Figure 17.15 shows results for a calculation of two-dimensional temperature distributions in a drawn steel wire using a similar model for the die–workpiece interface; Fig. 17.16 shows a similar result for a hot aluminum extrusion process. . background heat transfer into the workpiece from the entire grinding zone, in terms of a heat flux q wb ; and for the heat transfer into the workpiece from an individual grain, in terms of a heat. considered, BOOKCOMP, Inc. — John Wiley & Sons / Page 1250 / 2nd Proofs / Heat Transfer Handbook / Bejan 1250 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 [1250],. (From DeVries, 1992.) BOOKCOMP, Inc. — John Wiley & Sons / Page 1252 / 2nd Proofs / Heat Transfer Handbook / Bejan 1252 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 [1252],