The P´eclet number Pe represents a dimensionless speed ofthe workpiece and serves to compare the ability ofthe workpiece to advect energy through its motion to its ability to transfer en
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0.0 0.2
0 4 0.6 0.8 1.0
x */4 = /x D
L D/ = 4
L D/ = 7
L D/ = 2
L D/ = 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; 4 Bi = 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 ofthe rod approaches asymptotically that ofthe environment Under those conditions, it is easily shown that eq (17.4) reduces to
θ(x∗)
θo = exp
Pe−Pe2+ 4 Bi
(17.6)
Necessarily, eqs (17.4) and (17.6) are valid only when the value ofBi 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 ofa
stationary extended surface
The problem depicted in Fig 17.2 may be extended to include a more realis-tic boundary condition atx∗ = 0 and L∗, taking into account the upstream and
downstream thermal conditions Ifθ(x∗ → −∞) = θ o andθ must remain finite
as x∗ → +∞, the temperature distribution in the region exposed to the thermal
environment is governed by the following three equations:
d2θ1
dx∗2− Pedθ1
dx∗ = 0 f orx∗< 0
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d2θ2
dx∗2 − Pedθ2
dx∗ − Biθ = 0 for 0< x∗< L∗ (17.7)
d2θ3
dx∗2 − Pedθ3
dx∗ = 0 f orx∗> L∗
for which Jaluria has provided the solutions as
θ1=
1+Pe
m2e m2L∗
− m1e m1L∗
m2e m1L∗− m2e m2L∗
e P ex∗
θ2= 1 + Pe
m2e m2L∗
e m1x∗
− m1e m1L∗
e m2x∗
m2
1e m1L∗− m2
2e m2L∗
θ3= 1 + Pe(m2− m1) e P e L∗
m2
1e m1L∗− m2
2e m2L∗
(17.8)
As in the more limited case, three physical parameters emerge from the analysis
The P´eclet number Pe represents a dimensionless speed ofthe workpiece and serves to compare the ability ofthe 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 ofas a ratio ofinternal thermal resistance to external thermal resis-tance These analytical results are limited to small values ofBi, because temperature gradients in the direction transverse to the workpiece motion have been neglected
Finally,L∗ is the dimensionless length ofthe workpiece WhenL∗ is large enough
(the value depends on Bi and Pe), the workpiece will approach thermal equilibrium with the thermal environment atT∞
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 ofthe temperature field upstream due to conduction
is lower for increased Pe, because advection effects begin to dominate over thermal conduction A large value ofBi 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, whenL∗ is large
enough (about 5 in Fig 17.4b), the temperature at the end ofregion 2 (at x∗= L∗) is
very close to the ambient temperature, and any increase inL∗will not affect the heat
transfer
sec-tion ofthe workpiece are important relative to those in the direcsec-tion ofmosec-tion, a dif-ferential control volume of sizedA c × dx must be used to establish an energy balance
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⫺10
⫺5
0
0
5
4 2
10
8 6
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.)
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Figure 17.5 Isotherms for a long moving cylindrical rod: (a) Bi D = 10, Pe D = 0.4; (b)
BiD = 0.2, PeD = 0.4.
that will yield the temperature distribution Now the coordinate system chosen must
be more specific For example, for a circular rod of diameterD, the steady-state
con-duction equation becomes
∂θ
∂τ + Pe
∂θ
∂X =
1
R
∂
∂R
R ∂θ
∂R
+ ∂2θ
whereτ = αt/D2, X = x/D, and R = r/D Initial and boundary conditions are as
follows:
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τ > 0: atX = 0, θ = 1.0 for 0≤ R ≤ 0.5
atX = L(τ), ∂θ
∂X = − Bi · θ for 0≤ R ≤ 0.5
∂R = 0 for 0≤ X ≤ L(τ)
atR = 0.5, ∂R ∂θ = − Bi · θ for 0≤ X ≤ L(τ)
(17.10)
In this case, the length ofthe rod is a function oftime,L = V t, and convective heat loss is presumed to occur from the end of the rod AsL∗ increases, a steady-state
temperature distribution is attained andθ → 0 as L∗→ ∞ 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 BiD and PeD] and for
a cylindrical rod Figure 17.5 shows some typical isotherms for the latter case Note that when BiD = 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 ofheat is confined to a small region on the surface ofthe 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 ofa solid to a moving heat source is also presented These form the basis for understanding the behavior ofa wide variety ofmanufacturing systems
depicted in Fig 17.6 The magnitude ofthe heat source is Q (in watts), and the
surface is considered to transfer heat by convection to an environment atT∞with
a heat transfer coefficienth For a coordinate system fixed to the heat source moving
Figure 17.6 Thin plate or rod with a moving planar heat source
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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∞=
(Tmax− T∞) exp
+
Pe2+ 4 Bi − Pex∗
2
forx∗≤ 0
(Tmax− T∞) exp
−
Pe2+ 4 Bi + Pex∗
2
forx∗≥ 0
(17.11)
where the maximum temperature occurring in the workpiece is
Tmax− T∞=Qγ k 1
The maximum temperature is proportional to the magnitude ofthe 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(Pe2 4 Bi),
T (x∗) − T∞=
(Tmax− T∞) exp (−Pe · x∗) forx∗≥ 0 (17.13)
Tmax− T∞= ρcA Q
c V
All the mass that passes by the source is heated to Tmax, which depends on the heat capacity ofthe material,ρc For very low speeds (Pe2 4 Bi), the workpiece
functions essentially as an infinitely long extended surface with a perfectly symmetric temperature profile:
T − T∞= (Tmax− T∞) exp−√Bi· x∗
Tmax− T∞= Qγ/k
2√
Bi
(17.14)
or heat treating with a localized source ofenergy (electron beam, laser beam, welding torch) provide an additional set ofheat transfer problems Ifthe workpiece is very thin, temperature gradients through the thickness may be neglected compared to those
in the direction ofmotion or transverse to it When the surface loses heat to the surroundings atT∞ with a heat transfer coefficienth, the temperature distribution
is given by
T − T∞=Q/H
2πk exp
PeH x∗
2
K o
PeH 2
2
+ ( Bi H )2r∗
(17.15)
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whereH 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, PeH = VH/α, Bi H = hH/k, and K ois the modified Bessel function
When surface heat losses can be neglected, BiH is set to zero in eq (17.15)
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, withr =x2+ y2+ z2 There is not a convenient length scale because the plate is infinite in thex and y
directions and semi-infinite in thez direction Ifa finite plate thickness is introduced,
eq (17.16) may be rewritten in a dimensionless form as
T − T∞= Q/H
2πkr∗exp
−PeH (x∗+ r∗)
2
(17.17)
However, it must be remembered thatH must be significantly larger than the thermal
penetration in thez direction.
quasi-steady state thermal response ofa 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 ofa moving solid under a uniform heat fluxq
s, which acts over a distancex = 0
tox = l, with the remaining surface being insulated, to yield the following result for
the dimensionless surface temperature rise:
(Ts − T∞)πkV
2αq
π · Pel√
x∗−√x∗− 1 forx∗≥ 1
(17.18)
The peak temperature occurs atx = l, with a drop in temperature downstream ofthe
heat source Equations (17.18) neglect any effects of heat conduction in the direction ofmotion The more general results, cited by Carslaw and Jaeger (1959), may be expressed by
(Ts − T∞)πkV
2αq
x∗(Pel/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
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Figure 17.7 Surface temperature distribution due to a moving heat source of lengthl (Pe =
V l/α).
Figure 17.8 Peak surface temperature due to a moving heat source of lengthl, including
effects of convective cooling (From DesRuisseaux and Zerkle, 1970.)
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graphically in Fig 17.7 When Pel ≥ 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 ofthe 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 ofthe heating zone(x = l) is shown in Fig 17.8 in terms ofthe heat transfer
coefficienth, expressed dimensionlessly as 2 Bil/Pel = 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 ofmetals
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 ofshape occurs Analysis
of the heat transfer processes involved is straightforward, in principle, to obtain the appropriate temporal and spatial temperature response ofthe solid However, achieving a desired result requires that the thermal behavior ofa solid be carefully integrated with an understanding ofthe equilibrium properties, the kinetics ofphase transformations and diffusion at various temperatures, and the relationship of the mechanical properties ofa solid to its material structure Consequently, study ofthis subject is usually confined to the expertise ofmetallurgists and materials specialists rather than heat transfer specialists The interested reader is referred to Chapter 5 of DeGarmo et al (1997) or Chapter 6 ofSchey (2000) for an introductory discussion ofthe metallurgical issues, and to a large number ofpublications ofthe 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 ofthe 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
ofsolid–solid transformations at various temperatures For example, thin specimens ofa 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 ofthese points usually takes the shape ofa pair of“C”
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curves The nose ofthe “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 ofthe 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 ofthe heat transfer issues related to quenching ofmetals
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 ofcontrol ofthe motion ofthe heat source and ofthe workpiece, because only a portion ofthe surface may
be exposed at a time (see Section 17.8)
The goals ofthis section are to review the mechanisms ofheat generation in metal cutting, discuss the relevant modeling assumptions used in thermal analysis, and describe the limitations ofthese models
The process ofchip formation during machining consumes a great deal ofpower The overall cutting power is given by
whereF p is the power component ofthe cutting force, which is parallel with the workpiece velocity, andV is the velocity ofthe workpiece material Most ofthis
power is converted into heat and then partitioned to the workpiece chip and cutting tool Some ofthe detrimental effects ofhigh temperatures include (1) increased tool wear, and hence shortened tool life (high temperature at the cutting edge is the primary factor associated with accelerated wear); (2) decreased process efficiency (escalation oftemperature limits the rate ofmaterial removal, specifically the cutting speed); and (3) decreased surface quality as a result of residual stresses and thermal distortion
The simplest case ofmetal cutting, which has been the focus ofmost modeling
effort regarding thermal behavior, is orthogonal cutting A single cutting edge is
oriented normal to the direction ofmotion, and the chip flows up the surface ofthe wedge-shaped cutting edge, with a velocityVc, in the same plane as the velocity ofthe
workpiece Figure 17.9a illustrates the chip, tool, and workpiece geometry The chip