induction-heated tool machining of elastomers part …

INDUCTION-HEATED TOOL MACHINING OF ELASTOMERS—PART 1:FINITE DIFFERENCE THERMAL MODELING AND EXPERIMENTALVALIDATION

Jie Luo, Hongtao Ding, and Albert J. Shih & Department of MechanicalEngineering, University of Michigan, Ann Arbor, Michigan, USA

& Experiments and finite difference thermal modeling of the induction-heated tool for endmilling of elastomers are investigated. Three sets of experiments are designed to calibrate the contactthermocouple for the tool tip temperature measurement, study the effect of tool rotational speedon induction heat generation and convective heat transfer, and measure the tool temperaturedistribution for finite difference inverse heat transfer solution and validation of modeling results.Experimental results indicate that effects of tool rotation on induction heat generation and con-vective heat transfer are negligible when the spindle speed is below 2000 rpm. A finite differencethermal model of the tool and insulator is developed to predict the distribution of tool temperature.The thermal model of a stationary tool can be expanded to predict the temperature distribution ofan induction-heated rotary tool within a specific spindle speed range. Experimental measurementsvalidate that the thermal model can accurately predict tool tip peak temperature.

Keywords Elastomer Machining, Finite Difference Modeling, Induction-Hated Tool

INTRODUCTION

Elastomers have the unique material properties of very low elasticmodulus and high percentage of elongation before fracture. These proper-ties make the effective material removal and precision machining of elasto-mers difficult. The early research in elastomer machining was conducted byJin and Murakawa (1). Benefits of the high speed and high tool helix andrake angles to enable the efficient machining of elastomers were demon-strated. Elastomer machining using sharp woodworking tools and the effectof cryogenically cooled elastomer workpiece were investigated by Shih et al.(2, 3). With proper selection of tool geometry, process parameters, andfixture stiffness, effective material removal and generation of clean groovesin end milling were demonstrated. The investigation showed that the

Address correspondence to Albert J. Shih, Department of Mechanical Engineering, University ofMichigan, Ann Arbor, MI, 48109, USA. E-mail: [email protected]

Machining Science and Technology, 9:547–565Copyright # 2005 Taylor & Francis LLCISSN: 1091-0344 print/1532-2483 onlineDOI: 10.1080/10910340500398225

down-cut tool, good fixturing, and cryogenic cooling of the workpiece arebeneficial for machining of elastomers.

A novel method using the induction heating of the rotary tool isdeveloped for machining of elastomers. This idea originates from thecurrent practice of manual carving the tread for prototype tire manufactur-ing. A thin, sharp, and resistance-heated grooving blade is used for thisoperation. Heating the tool is critical. If the tool does not reach a certaintemperature, carvers are not able to cut grooves on the tire. The heat soft-ens the elastomer materials ahead of the tool and allows it to penetrate theworkpiece efficiently. In this study, induction heating is used to raise thetemperature of a rotary tool. Experiments of heating the stationary androtary tool are performed to investigate the effect of rotation on tool tiptemperature for machining of elastomers. The finite difference thermalmodel, developed by Luo and Shih (4), is expanded to include the thermalinsulator and applied to predict the temperature of induction-heated tool.The concept of induction heated tool has broad potential applications infriction hole drilling (5) and friction spot welding (6). It can also beapplied in electrosurgery for cutting and coagulation of soft tissue usingheated tool (7).

The induction heating experiment setup and test procedure arepresented in the following section. Experimental results are summarizednext. The finite difference thermal modeling of the induction-heated tooland insulator follows. Finally, the modeling and experimental results arecompared.

EXPERIMENTAL SETUP AND TEST PROCEDURE

The configuration of induction-heated tool for machining of elastomersis shown in Figure 1(a). The non-contact induction heating provides a fast,consistent heating to a specific region adjacent to the cutting flute. As shownin Figure 1, a thermal insulator, made of low thermal conductivity Ti-6Al-4V,is used to reduce the heat transfer to the spindle.

Three sets of experiments, marked as Experiments A, B, and C, areperformed to calibrate the contact thermocouple for the tool tip tempera-ture measurement, study the effect of tool rotation on induction heatgeneration and convective heat transfer, and measure the tool temperaturedistribution for finite difference inverse heat transfer solution and vali-dation of modeling results, respectively.

Experiment A

The goal of Experiment A is to calibrate the temperatures measured bythe contact and cement-on thermocouples. The contact thermocouple is

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used to quickly touch and measure the tool tip temperature after stoppingthe rotating tool. The tool tip temperature is an important process para-meter for heated tool machining of elastomers. A slip ring can be usedto transmit the voltage signal of thermocouples attached to the tool tip(8). However, the thermocouple wire needs to route from the tool tip,through the heating coil (shown in Figures 2 and 4), to the slip ring nearthe spindle. Such configuration is not possible due to the heating ofthermocouple wire inside the induction heating coil. In this study, a contactthermocouple (Omega type KMQIN-062E-6) with a 0.7mm diameter minia-ture tip was selected. The discrepancy between the contact and cement-onthermocouple measurements does exist and requires a set of experimentsto identify the calibration curve.

Figure 2 shows the configurationofExperimentA.The toolwas stationary.A cement-on thermocouple was attached to the tool tip. The double-flute,down-cut tool (Onsrud 40-series) with the same cutting edge geometryas the router tool with good performance for effective material removal (2)was used. To adopt for induction heating, this new tool, as shown inFigures 1(b) and 2, has a longer overall length of 127mm and shorter flutelength of 31mm.

A type K cement-on thermocouple (Omega type CO1-K) was attachedto the tool tip. The close-up view of the tool tip is shown in Figure 3.Temperatures at the tip of the induction-heated stationary tool are mea-sured simultaneously by the cement-on and contact thermocouples.

FIGURE 1 (a) Induction-heated tool machining of elastomers and (b) the double-flute down-cut endmill and titanium thermal insulator.

Induction-Heated Tool Machining of Elastomers—Part 1 549

FIGURE 2 Configuration of end mill, thermal insulator, water-cooled induction heating coil, andthermocouple.

FIGURE 3 Tool tip and thermocouple locations: (a) view of tool tip with two symmetric Surfaces 1 and2 and (b) tool tip with the cement-on thermocouple attached to Surface 1 and the position for thecontact thermocouple.

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As shown in Figure 3(a), there are two symmetric surfaces, marked asSurfaces 1 and 2, at the tip of the doubt-flute tool. Due to the symmetry,temperatures on Surfaces 1 and 2 are expected to be the same. As shownin Figure 3(b), a cement-on thermocouple is attached to Surface 1, whilethe contact thermocouple touches Surface 2. Three tests with the same dur-ation of induction heating time, 10 s, and three induction heating powerlevels at 0.4, 0.5, and 0.7 kW were conducted to measure temperatures onSurfaces 1 and 2 simultaneously. The voltage signals from the cement-onand contact thermocouples are acquired using a PC-based data acquisitionsystem with the 250Hz sampling rate. The temperature data is averagedand recorded every 0.5 s. This temperature data-acquisition setup is alsoused for all tool heating experiments.

Experiment B

Experiment B studies the effect of tool rotation on heat transfer in theinduction-heated tool. The tool was vertically held to the spindle of a verti-cal end milling machine. Tests were performed at 0 (stationary), 1100,2000, and 2900 rpm to study the effect of spindle speed on the inductionheat generation and convective heat transfer. Low spindle speeds are usedfor elastomer machining to avoid chip burning (9). The induction heatingduration was 25 s and power level was about 0.7 kW for all tests. The induc-tion coil was positioned in the flutes section at the tool tip. As shownin Figure 2, a coordinate system with x-axis measuring from the tool tipin the direction toward the spindle is defined. The position of heatingregion in the tool or the location of the induction coil is denoted as xIC.In Experiment B, xIC ¼ 0mm.

Process parameters in Experiment B are summarized in Table 1. A base-line test was first performed when the tool is stationary. The contact ther-mocouple recorded the tool tip temperature after the 25 s inductionheating. Under the same setting of induction heating, three tests were con-ducted at 1100, 2000, and 2900 rpm to study the effects of tool rotation oninduction heat generation rate. The rotating tool was quickly stopped 25,50, and 100 s after the 25 s of induction heating. The contact thermocouplethen touched the tool tip and measured the temperature. The calibration

TABLE 1 Process Parameters for Experiment B

Spindle speed (rpm) Heating duration th(s) Duration of tool rotation(s)

0 (stationary) 25 01100 25 25, 50, 1002000 25 25, 50, 1002900 25 25, 50, 100


data in Experiment A was used to convert the measured temperature totool tip temperature.

The effect of tool rotation on convective heat transfer is studied by com-paring the tool tip temperature at different time durations (25, 50, and100 s) of rotation. The induction heating duration remains unchanged at25 s. The change in temperature qualitatively reveals the increase in heatconvection coefficient of a rotating tool.

Experiment C

The temperature distribution of the induction-heated stationary toolwas measured in Experiment C. The experiment setup and configurationare shown in Figures 4 and 5. Two type K thermocouples, marked as Ther-mocouples 1 and 2 with location x1 and x2 from the tool tip, respectively,were cemented on the tool. In this study, x1 ¼ 79mm and x2 ¼ 3mm.The induction heating duration was 10 s.

During the induction heating, the thermocouple temperature measure-ment was not possible due to the interference of electromagnetic field gen-erated by the induction heating coil. The peak temperature of athermocouple was used as the input for the inverse heat transfer solutionbased on the finite difference thermal modeling to find the induction heat

FIGURE 4 Setup of tool, water-cooled induction heating coil, and thermocouples in Experiment C.

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generation rate. Experimental results were also used to validate tempera-ture distribution estimated by the finite difference modeling.

RESULTS OF TOOL INDUCTION HEATING EXPERIMENTS

Results of Experiment A—Calibration of Contact Thermocouple

Figure 6 shows the calibration results of Experiment A. For all threetests, represented by different symbols, the contact thermocouple tempera-ture, Th, is lower than the cement-on thermocouple temperature, Tc. Alldata points show a linear trend. The regression analysis is used to find astraight line to model the relationship between Th and Tc.

FIGURE 5 Configuration of the induction heating in Experiment C.


Tc ¼ 1:70Th � 0:97 ð1ÞThe Tc is about 1.7 times the Th. The discrepancy between Th and Tc is likelydue to the combination of the following three reasons.

1. Thermal contact resistance: Thermal contact resistance (10) is determ-ined by the contact area between the thermocouple tip and the tool.For the cement-on thermocouple, the 2.5mm by 3mm area of the tipfully contacts with the tool. For the contact thermocouple, only a smallregion of the 0.7mm diameter spherical tip touches the tool. The con-tact area of the contact thermocouple tip is much smaller and thethermal contact resistance is higher.

2. Heat loss from the thermocouple tip to the ambient air: As shown inFigure 3(b), the cement-on thermocouple tip is covered by a layer ofcement with thermal conductivity of 1.15W=mK. The contact thermo-couple tip is exposed directly to the ambient air and results highertemperature gradient and more heat loss.

3. Uncertainty of contact position between the thermocouple tip andthe tool: The variation of contact point on Surface 2, as shown inFigure 3, during the experiment also contributes to the temperaturevariation.

Equation 1 is used to convert measured temperatures in Experiments B.

FIGURE 6 Calibration for the contact thermocouple probe based on cement-on thermocouple.

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Results of Experiment B—Effect of Tool Rotation

Results of Experiment B are shown in Figure 7. The measured tempera-ture of the stationary tool is the baseline data. During the 25 s heatingduration, the electromagnetic interference makes the thermocouple tem-perature measurement impossible. The measured temperatures are shownfor t after 30 s. The time, t, is measured from the start of induction heating.It takes about 5 s from the end of induction heating for the contact thermo-couple to touch the tool tip for temperature measurement.

For the stationary tool, as shown in Figure 7, the temperature rapidlyrises, in about 3 to 5 s, to 420�C shortly after the contact thermocoupletouched the tool tip at t ¼ 30 s.

At 1100 rpm, measured temperatures of the tool with the 25 s rotationaltime match closely with the stationary tool heated under the same induc-tion heating setup. This indicates that the influence of tool rotation oninduction heat generation and convective heat transfer is not significant.The lag time to reach the peak temperature is longer for tests with therotating tool than the stationary tool. It is due to the time required to stopthe tool and apply the contact thermocouple to touch the tool tip.

The same trend can be observed at 2000 rpm. The tool rotation doesnot significantly affect the induction heat generation and convective heattransfer.

At 2900 rpm, as shown in Figure 7, the effect of the tool rotation onconvective heat transfer can be recognized. A larger discrepancy of tem-peratures between the stationary tool and rotating tool can be seen. Suchdiscrepancy is contributed due to the higher convection coefficient causedby a faster tool rotational speed.

The effect of the duration of tool rotation on increasing the tempera-ture discrepancy relative to the stationary tool can also be identified inFigure 7.

Results of Experiment C—Induction Heating of a Stationary Tool

Figure 8 shows measured temperatures at Thermocouples 1 and 2from 0 to 200 s in Experiment C. The duration of induction heating is10 s. The heat generated by induction heating transfers like a thermal wavefrom the heating region toward both sides of the tool. Thermocouple 2 iscloser to the induction heating region than Thermocouple 1. Therefore,Thermocouple 2 has shorter time (83 s) to achieve higher peak tempera-ture (168�C), marked as point P in Figure 8. The peak temperature ofThermocouple 1, denoted as point S (100 s, 86�C), is used as the inputfor the inverse heat transfer solution of the heat flux generated by induc-tion heating. The results of inverse heat transfer solution and analysis ofthe temperature distribution of the tool will be discussed later.


FIGURE 7 Effect of tool rotation on the tool temperature (25 s duration of induction heating).

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FINITE DIFFERENCE THERMAL MODELING OF AN INDUCTIONHEATED TOOL

Explicit finite difference method is applied to model the tool tempera-ture distribution with the thermal insulator and to estimate the surfaceheat flux, q 00ind , generated by induction heating. The finite difference ther-mal model, modeling results of a stationary tool, and comparison of theexperimental and modeling results are discussed in the following sections.

Finite Difference Thermal Model

The explicit finite difference formulation of an inverse heat transfermodelto calculate the induction heat flux for a workpiece with the cross-section Biot

FIGURE 8 Comparison of the measured and calculated temperature for heating duration as 10 s(P: peak temperature of Thermocouple 2 and S: peak temperature of Thermocouple 1).


number less than 0.1 has been developed by Luo and Shih (4). Undersmall Biot number, the temperature in the cross-section is almostuniform. A single value can be used to represent the temperature of thecross-section in the finite difference formulation to solve the inductionheat flux using the inverse heat transfer solution and to analyze the tem-perature distribution during heating and cooling. As shown in Figure 5,the radius of the tool, r, is 6.35mm. The width of the thermal insulator,w, is 4.75mm. The tool material is M7 high speed steel with thermal con-ductivity (k) of 19.0W=mK (Tarney, private communication, 2004). TheTi-6Al-4V thermal insulator has thermal conductivity (ks) of 6.6W=mK(11). The Biot numbers of the tool and thermal insulator cross-sectionsare 0.0045 and 0.0049, respectively, under the convection coefficient of6.8W=m2K (4). These Biot numbers are both much smaller than 0.1. Inthis study, the finite difference method is further expanded to investigatethe heat transfer in the tool with a thermal insulator.

As shown in Figure 5, the tool is divided to three types of regions: insu-lation, heating, and cooling. The insulation region is represented by line abwith length Lins. The heating region is represented by line cd with length Lh.Four cooling regions, marked as 1, 2, 3, and 4, have length Lc1, Lc2, Lc3, andLc4, respectively. The cylindrical section of the tool, line ae, has a constantcross-section area Ab and perimeter Pb. The flutes section of the tool, line eg,is further divided into the cooling region 3 (line ef) and cooling region 4(line fg). The cooling region 4 has the constant cross-section area At andperimeter Pt. Cooling region 3 is a geometrical transition section. Thecross-section area and perimeter are assumed to vary linearly from Ab andPb to At and Pt, respectively.

Nodes in the Heating Region. Based on the energy balance method,explicit finite difference equations are developed for nodes in the heatingregion. As shown in Figure 9(a), the tool is modeled as a line of series ofnodes. At node m, the control volume, cross-section area, and perimeterof the tool are Vm, Am, and Pm, respectively. The energy exchange in controlvolume Vm is determined by the conduction between the node m and adjac-ent nodes m – 1 and m þ 1 in the axial direction and the convection, radi-ation, and induction heat generation on the peripheral surface PmDx,where Dx is the width of the control volume. The conduction heat fluxfrom node m – 1 to m and from node mþ 1 to m in the axial direction isrepresented by q 00ca;m�1 and q 00ca;mþ1, respectively. The symbol a in q 00ca;m�1and q 00ca;mþ1 indicates the heat flux in the axial direction. The convectionand radiation heat flux on the peripheral surface of the control volumeat node m is q 00conv and q 00rad , respectively. q

00conv and q 00rad can be calculated by

the formulas developed by Luo and Shih (4). During induction heating,a uniform heat flux q 00ind is applied on the peripheral surface of the heating

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region. The energy balance equation of the control volume at node m is:

ðq 00ca;m�1 þ q 00ca;mþ1Þ � Am þ ðq 00ind � q 00conv � q 00radÞ � Pm � Dx

¼ qc � AmDx �T

pþ1m � T

pm

Dtð2Þ

where q and c are the density and specific heat of the tool material, respect-ively, Dt is the time increment, and T

pm is the nodal temperature at node m

and time step p.

Nodes in the Cooling Region. Equation 2 with q 00ind ¼ 0 is the energy bal-ance equation for nodes in the cooling region.

Nodes in the Insulation Region. As shown in Figures 9(b) and 9(c), theheat transfer in the insulation region is conducted in both axial and radialdirections. The energy balance analysis is performed for node m in the tooland node ms in the thermal insulator.

At node m in the tool, the energy exchange in control volume Vm, asshown in Figure 9(b), is determined by the conduction between node mand adjacent nodes m� 1, mþ 1, and ms, marked as q 00ca;m�1, q

00ca;mþ1, and

q 00cr ;m , respectively. q00cr ;m is conducted in the radial direction, from node m

to ms, through the contact area Pm Dx between the tool and the thermalinsulator. The q 00cr ;m can be expressed as (10):

q 00cr ;m ¼ 2 � ðTpm � T

pmsÞ � p � Dx

ðln 2=k þ lnððPm þ PmsÞ=ð2PmÞÞ=ksÞð3Þ

where Tpms is the nodal temperature at node ms and time step p.

FIGURE 9 Heat exchange between adjacent nodes in the tool and thermal insulator: (a) at node m inthe heating region, (b) at node m in the insulation region, and (c) at node ms in the insulation region.


The energy balance of the control volume Vm at node m is:

ðq 00ca;m�1 þ q 00ca;mþ1Þ � Am � q 00cr ;m � Pm � Dx ¼ qc � AmDx �T

pþ1m � T

pm

Dtð4Þ

At node ms in the thermal insulator, as shown in Figure 9(c), the energyexchange in the control volume Vms is determined by the conductionbetween the node ms and adjacent nodes ms � 1, msþ 1, and m, markedas q 00ca;ms�1, q 00ca;msþ1, and q 00cr ;m , respectively, and the conduction betweenthe node ms and surrounding surface, marked as q 00cr ;ms. For q 00cr ;m andq 00cr ;ms, the symbol r indicates the heat flux in the radial direction. The heatflux q 00cr ;ms is conducted through the contact area Pms Dx between the thermalinsulator and the surrounding surface, where Pms is the outer perimeter andDx is the width of the control volume Vms at nodems. The q

00cr ;ms is determined

by the boundary condition on the contact area Pms Dx.The boundary condition on the interface between tool and insulator is

difficult to clearly define. Two boundary conditions, representing two extremecases, are studied. One boundary condition assumes constant temperatureand the other hypothesizes no heat flux (adiabatic) across the boundary.

If the contact surface remains at its initial temperature, Tsp , during theheating and cooling stage, the heat flux q 00cr ;ms is:

q 00cr ;ms ¼2p � ðTp

ms � TspÞ � Dx � kslnðð2PmsÞ=ðPm þ PmsÞÞ

ð5Þ

For the adiabatic boundary condition, which assumes that the thermalinsulator is a perfect insulator, q 00cr ;ms ¼ 0.

The energy balance of the control volume Vms at node ms is:

ðq 00ca;ms�1 þ q 00ca;msþ1Þ � Ams þ ðq 00cr ;m � Pm � q 00cr ;ms � PmsÞ � Dx

¼ qscs � AmsDx �T

pþ1ms � T

pms

Dtð6Þ

where qs is the density and cs is specific heat of the thermal insulator andAms is the cross-section area at node ms.

Modeling Results of the Stationary Tool

The induction heat flux q 00ind is solved by minimizing an objective func-tion, which is the square of the discrepancy between the experimentalmeasurement (point S in Figure 8) and modeling result (4). In the experi-mental configuration shown in Figure 5, the position of heating region

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xIC ¼ 33.0mm. The length of region, Lc1 ¼ 47.7mm, Lc2 ¼ 2.0mm, Lc3

¼ 17.0 mm, Lc4 ¼ 14.0mm, Lh ¼ 11.3mm, and Lins ¼ 35.0mm. The gridwith 24, 1, 9, 7, 6, and 19 nodes is assigned in cooling regions 1, 2, 3,and 4, heating region, and insulation region, respectively. The grid spacingis about 2mm in all regions. For the tool used in this study, Pb ¼ 40mm,Pt ¼ 34mm, Ab ¼ 127mm2, and At ¼ 71mm2.

The experimentally measured temperature, thermocouple location, andmaterial properties of the tool and thermal insulator are required inputs forthe finite differencemodeling. For theM7 high speed steel used for the tool,q ¼ 7950 kg=m3 and c ¼ 460 J=kg�K (Tarney, private communication, 2004).For the Ti-6Al-4V used for the sleeve, qs ¼ 4428 kg=m3 and cs ¼ 580 J=kg�K(11). The peak temperature at Thermocouple 1, point S in Figure 8 (100 s,86.4�C), and the location of thermocouple 1, x1 ¼ 79mm, are used as theinput for the finite difference inverse heat transfer solution. The calculatedq 00ind is 1.508MW=m2 under the constant temperature boundary conditionand 1.442MW=m2under the adiabatic boundary condition. These two valuesare the upper and lower bounds for the estimated induction heat flux corre-sponding to two extreme boundary conditions on the contact area betweenthe thermal insulator and surrounding surface. The adiabatic boundarycondition assumes no heat loss through the boundary and results in a highertool temperature of two boundary conditions with the same induction heatflux in the heating region. Therefore, with the same temperature at Thermo-couple 1 as the input for the inverse heat transfer solution, the constanttemperature boundary condition generates a larger estimated q 00ind than theadiabatic boundary condition.

Comparison of the Experimental and Modeling Results

Based on two estimated values of induction heat flux q 00ind , two sets oftemporal and spatial distribution of temperature on the tool are calculated.Results of finite difference modeling and experimental measurements atThermocouples 1 and 2 are illustrated in Figure 8. As expected, the experi-mental measurement and finite difference modeling temperatures matchat point S, the input for the finite difference inverse heat transfer solution.The finite difference modeling is able to predict the temperature duringthe induction heating period (0 < t < 10 s). In this period, the tempera-ture at thermocouple cannot be experimentally measured due to the elec-tromagnetic interference. When 0 < t < 40 s, the discrepancy between thefinite difference modeling results under adiabatic and constant surfacetemperature boundary conditions is negligible. The heat flux across theinsulation region is still small. Therefore, the effect of boundary conditionon the temperatures at Thermocouples 1 and 2 is not significant. Whent > 40 s, the effect of the boundary condition becomes more apparent.


At Thermocouple 1, when 40 < t < 100 s, the estimated temperatureunder constant surface temperature boundary condition is higher thanthat of adiabatic boundary condition due to the larger estimated inductionheat flux q 00ind in the heating region. q 00ind is the input for the temperaturedistribution analysis of the finite difference modeling. However, whent > 100 s, the estimated temperature under constant surface temperatureboundary condition is lower than that under the adiabatic boundary con-dition. This is due to the effect of continuous heat loss by conductionbetween thermal insulator and surrounding surface surpasses the influenceof higher estimated induction heat flux.

At Thermocouple 2, when t > 40 s, the estimated temperature underconstant surface temperature boundary condition is always higher thanthat under the adiabatic boundary condition. This is due to the larger esti-mated q 00ind for constant surface temperature boundary condition. BecauseThermocouple 1 is much closer to the insulation region (Figure 5), theeffect of boundary conditions on the temperature at Thermocouple 2 isless significant than at Thermocouple 1.

At Thermocouple 2, the discrepancy between the experimental mea-surements and finite difference modeling is generally larger than at Ther-mocouple 1. The experiment shows the peak temperature at the tool tip(Thermocouple 2) occurs at 83 s. The finite difference modeling predictsthe peak temperature at about 105 to 110 s. The geometry approximationof the complex flutes section and the use of constant thermal propertiesboth contribute to this discrepancy. However, the experimentally mea-sured peak temperature at Thermocouple 2, 168�C, is still close to thepeak temperature range (162�C, 167�C) estimated by the finite differencemodeling.

Experiment C described earlier is conducted several times with differ-ent power settings. For each time, the induction heat flux is estimated bythe inverse heat transfer solution and the temperature distribution and his-tory are analyzed by the finite difference modeling. The estimated peaktemperature range at Thermocouple 2 matches the experimental tempera-ture measurement, which demonstrates the capability of the finite differ-ence thermal modeling to predict tool tip peak temperature. This isimportant to design the heating strategy for induction-heated tool machin-ing of elastomers.

CONCLUSIONS

Experiments and finite difference modeling of the induction-heatedtool was described in this paper. Experiments were conducted to study theheat generation and transfer in a rotating tool. A calibration line between

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temperatures measured by the cement-on and contact thermocouples wasestablished. Tool rotation was illustrated to have insignificant effect oninduction heat generation and convective heat transfer at spindle speedlower than 2000 rpm. In the following study (9) the low spindle speedwas adopted in the elastomer end milling experiments to avoid burningand smoking. Based on these observations, the temperatures of an induc-tion-heated rotary tool can be estimated by the finite difference modelingof a stationary tool. The complete finite difference thermal modeling of thetool and insulator is developed to predict the tool temperature. Comparedto the experimental measurements, the accuracy of predicted tool tip peaktemperature was demonstrated.

More accurate finite difference modeling results can be achieved byapplying better geometrical representation of the tool and using moreaccurate temperature-dependent properties of the tool material. Basedon the finite difference thermal modeling developed in this study, the toolinduction heating strategy can be developed for elastomer machining.Experimental results for elastomer machining tests are presented in thefollow-up study (9).

NOMENCLATURE

Ab area of the cross-section in the insulation region and coolingregion 1 and 2

Am area of the cross-section at node m in the toolAms area of the cross-section at node ms in the thermal insulatorAt area of the cross-section in the cooling region 4c specific heat of the tool materialcs specific heat of the thermal insulatork thermal conductivity of the tool materialks thermal conductivity of the thermal insulatorLcj width of the cooling region j of the toolLh width of the heating region in finite difference modelingLins width of the insulation region of the toolm node number in the toolms node number in the thermal insulatorp time stepPb perimeter of the cross-section in the insulation region and

cooling region 1 and 2Pm perimeter of the cross-section at node m in the toolPms outer perimeter of the cross-section at node ms in the thermal

insulatorPt perimeter of the cross-section in the cooling region 4


q 00ca;m�1 conduction rate from adjacent node m � 1 to node m along axialdirection

q 00ca;ms�1 conduction rate from adjacent node ms � 1 to node ms along axialdirection

q 00cr ;m conduction rate from node m to adjacent node ms along radialdirection

q 00cr ;ms conduction rate from node ms to surrounding environment alongradial direction

q 00conv convective heat fluxq 00ind surface heat flux generated by induction in the heating regionq 00rad radiation heat fluxr radius of the toolTc reading of cement-on thermocoupleTh reading of contact thermocoupleT

pm nodal temperature at node m and time step p

Tpms nodal temperature at node ms and time step p

Tsp initial temperature of the machine spindlet timeDt time incrementVm volume of the control volume at node mVms volume of the control volume at node ms

w thickness of the thermal insulatorxIC location of the induction coil from the tip of the toolxj location of thermocouple j from the tip of the toolDx width of the control volumeq density of the tool materialqs density of the thermal insulator

ACKNOWLEDGMENTS

This research is sponsored by the National Science Foundation (DMIGrant #0099829) and Michelin Americas R&D Corp.

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