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Heat transfer in the roll gap during hot rolling Hlady, Craig Ohrist 1994

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HEAT TRANSFER IN THE ROLL GAP DUPING HOT ROLLINGbyCRAIG OHRIST HLADYB. A. Sc., The University of British Columbia, 1991A THESIS SUBMITTED iN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFMASTER OF APPLIED SCIENCEinTHE FACULTY OF GRADUATE STUDIESDepartment of Metals and Materials EngineeringWe accept this thesis as conformingto the required standard..THE UNIVERSITY OF BRITISH COLUMBIAJanuary 1994© Craig Ohrist Hiady, 1994In presenting this thesis in partial fulfilment of the requirements for an advanceddegree at the University of British Columbia, I agree that the Library shall make itfreely available for reference and study. I further agree that permission for extensivecopying of this thesis for scholarly purposes may be granted by the head of mydepartment or by his or her representatives. It is understood that copying orpublication of this thesis for financial gain shall not be allowed without my writtenpermission.(Signature)Department of ( A1j&f iiLriThe University of British ColumbiaVancouver, CanadaDate 2, jqqiDE-6 (2/88)ABSTRACTA series of pilot mill hot-rolling tests involving AA5052, AA5 182, and coppersamples has been performed. These rolling tests encompassed a range of rollingpressures, rolling speeds, reductions and temperatures. In addition, two types oflubricants were employed in the hot rolling of the aluminum alloy samples.Instantaneous roll-gap heat-transfer coefficients (HTCs) have been calculated from roll-gap surface temperature measurements. These measurements were made from double-intrinsic thermocouples secured on the surface of the samples. Average roll-gap HTCshave been calculated from the bulk temperatures of the samples immediately before andafter rolling. The roll-gap HTC was calculated by an implicit, one-dimensional finite-difference technique. The resulting roll-gap HTCs of both the aluminum alloy and thecopper samples were compared to those obtained for steel rolling in a previous study.The roll-gap HTC has been proposed to be a function of the harmonic conductivity of thematerial being rolled and the roll, the ratio of the rolling pressure to the surface flowstress of the material being rolled, and the surface roughnesses of the roll and the materialbeing rolled.UTABLE OF CONTENTSPageABSTRACT iiLIST OF TABLES viiLIST OF ILLUSTRATIONS viiiNOMENCLATURE xACKNOWLEDGEMENTS xviCHAPTER 1. INTRODUCTION 1CHAPTER 2. LITERATURE REVIEW 42.1 Previous Estimates of the Roll-Gap Heat-Transfer Coefficient 42.2 Dependence of the Real Contact Area on Pressure 52.3 Dependence of the HTC on Pressure 72.3.1 Experimental Observations 72.3.2 Theoretical Treatment 82.4 Modelling Heat Transfer in the Roll Gap 122.4.1 Model Types and Solving Schemes 122.4.2 Surface Temperature Measurement Techniques in the Roll Bite 132.5 Friction in the Roll Gap 152.5.1 Characterization of Friction in the Roll Gap 152.5.2 Distribution of the Frictional Heat 162.6. Previous Considerations of Flow Stress Variation Through the Strip 16CHAPTER 3. SCOPE AN]) OBJECTIVES 18CHAPTER 4. EXPERIMENTAL DESIGN AND MEASUREMENTS 191114.1 Materials.194.1 .1 Homogenization Treatment 194.1.2 Flow Stress Characterization 194.2 Test Design 224.2.1 Test Facilities 224.2.2 Preparation of Test Samples 254.2.2.1 Aluminum Alloy Samples 254.2.2.2 Copper Samples 304.3 Test Procedure 304.3.1 Aluminum Test Schedule 304.3.2 Copper Test Schedule 334.4 Thermal Response 334.4.1 Thermal Response of Aluminum Samples 334.4.2 Thermal and Load Response of Copper Samples 38CHAPTER 5. HEAT TRANSFER MODEL DEVELOPMENT 395.1 Mathematical Formulation 395.2 Discretization of the Differential Equations 425.2.1 Nodes in the Strip 435.2.2 Nodes in the Roll 435.2.3 Solving Technique 445.3 Treatment of Heat Generation 455.3.1 Generation and Distribution of Frictional Heat 455.3.2 Bulk Heat due to Deformation 465.3.3 Depth of Heat Penetration into the Roll 475.4 Conductivities of Materials Used in this Study 48iv5.5 Model Verification .495.5.1 Validity of the l-D Model 495.5.2 Comparison with Analytical Solution 505.5.2.1 Verification of the Roll Finite-Difference Formulation 515.5.2.2 Verification of the Strip Finite-Difference Formulation 525.5.3 Convergence of the Model 53CHAPTER 6. ROLL-GAP HEAT-TRANSFER ANALYSIS 556.1 Measurement of Instantaneous Roll-Gap HTC 556.1.1 Aluminum Tests 556.1.2 Copper Tests 626.2 HTC Calculated from Initial and Final Sample Temperatures 636.2.1 Aluminum Tests 636.2.2 Copper Tests 68CHAPTER 7. RESULTS AND DISCUSSION 707.1 Effect of Rolling Parameters on the HTC 707.2 Friction in the Roll Bite 757.3 Comparison of the roll-gap HTC with Earlier Values for AluminumRolling 767.4 Generalized Correlation for the HTC 797.4.1 Underlying Assumptions 797.4.1.1 Dependence of the HTC on Pressure and Surface Hardness 797.4.1.2 The Dependence of the HTC on the Conductivity of the Workpiece andTool 807.4.2 Quantification of the Dependence of the HTC on Pressure and Conductivity837.4.2.1 Formulation of the General Equation 83V7.4.2.2 Modification of the Equation for Rolling Conditions 857.4.3 Numerical Solving Technique 877.5 Prediction of Roll-Gap HTCs Using the Developed Equation 897.5.1 Suitability of Conductivity and n as Equation Parameters 897.5.2 Effect of A on Equation Parameters 947.5.3 Significance of the General Constant C in the HTC-Prediction Equation 967.6 Maximum Theoretical HTC 987.7 Error in HTC Measurement due to Temperature Measurement Error 1007.8 HTC Measurement Error due to Roll Conductivity Error 1027.9 The Effect of the Heat-Transfer Coefficient on the Sample TemperatureProfile in the Roll Gap 104CHAPTER 8. SUMMARY AND CONCLUSION 1068.1 Summary of Results 1068.2 Recommendations for Further Study 1078.3 Concluding Remarks 108BIBLIOGRAPHY 109APPENDIX A 116Determination of Minimum HTC 116viLIST OF TABLESTable PageTable 4.1. Constants for constitutive steady-state stress equation 22Table 4.2. UBC pilot mill specifications 23Table 4.3. Conditions Employed in Aluminum Rolling Tests 32Table 4.4. Conditions Employed in Copper Rolling Tests 33Table 5.1. Peclet Numbers for Various Rolling Speeds and Reductions 50Table 6.1. FITCs Calculated from Bulk Aluminum Sample Temperatures 64Table 6.2. Statistical Comparison of HTCs Calculated from Bulk SampleTemperatures and Surface Temperatures in the Roll Bite 66Table 6.3. HTCs Calculated from Bulk Copper Sample Temperatures 68Table 7.1. Statistical Significance of Effect of Rolling Parameters on the HTC 72Table 7.2. Effect of Friction on the HTC 75Table 7.3. Value of C for Equation 7.15, 7.19a and 7.19b 90Table 7.4. Effect of A on Parameters of Equation (7.15) 93Table 7.5. Statistical Effect of A on Predictive Capability of Equation 7.15 94Table 7.6. Theoretical Maximum Roll-Gap HTCs 97viiLIST OF ILLUSTRATIONSFigure PageFigure 1.1 Contact between real surfaces 3Figure 2.1 Button model for contacting surfaces 10Figure 4.1 Schematic diagram of GLEEBLE apparatus 20Figure 4.2. Schematic diagram of the UBC pilot rolling mill 23Figure 4.3. Schematic diagram of roll lubrication system 25Figure 4.4. Design of aluminum sample 28Figure 4.5. Schematic diagram of aluminum test piece 29Figure 4.6. Close-up of surface thermocouple 29Figure 4.7. Schematic of copper test piece 30Figure 4.8. Thermocouple and load response for Test AL13 34Figure 4.9. Thermocouple and load response for Test AL15 35Figure 4.10. Thermocouple and load response for Test AL21 35Figure 4.11. Thermocouple and load response for Test AL24 36Figure 4.12. Thermocouple and load response for Test AL31 36Figure 4.13. Thermocouple and load response for Test CU7 38Figure 5.1. Discretization of roll and strip 42Figure 5.2. Flowchart of HTC-solving algorithm 45Figure 5.3. Sensitivity of numerical solution to mesh size 54Figure 6.1. Surface temperature and HTC for Test AL13 57Figure 6.2. Surface temperature and HTC for Test AL 15 57Figure 6.3. Surface temperature and HTC for Test AL21 58viiiFigure 6.4. Surface temperature and HTC for Test AL24 59Figure 6.5. Surface temperature and HTC of Test AL31 60Figure 6.6. Average HTC calculated from surface TCs vs. mean roll pressure 62Figure 6.7. Comparison of HTCs calculated from bulk sample temperatures andsurface temperatures in the roll bite 65Figure 6.8. Residual Errors from HTC Regression 67Figure 7.1. Effect of alloy type on the HTC 69Figure 7.2. Residual errors in comparison of AA5052 vs. AA5 182 70Figure 7.3. Temperature gradients between two asperities in contact 81Figure 7.4. Flowchart of algorithm used to predict the roll-gap HTC 86Figure 7.5. Comparison of experimental and predicted HTCs 87Figure 7.6. Comparison of predictive capabilities of Eq. 7.15, 7.19a and 7.19b 91Figure 7.7. Check of applicability of Equation 7.15 92Figure 7.8. Effect of A on predictive capability of Equation 7.15 94Figure 7.9. Effect of re-rolling on roll-gap HTCs 96Figure 7.10. Sensitivity of the HTC to the roll-gap exit temperature 98Figure 7.11. Error estimates of the HTC 99Figure 7.12. Effect of changing the model roll conductivity assumption on thecalculated HTC 101Figure 7.13. Sample temperature profile for an HTC of 378.5 kW/m2°C 103Figure 7.14. Sample temperature profile for an HTC of 37.85 kW/m2°C 103ixNOMENCLATUREa Radius of heat channel (m)A Constant used in steady-state stress constitutive equationAa Apparent contact area (m2)A Fractional contact areaAr Real contact area (m2)c Radius of button contact spot (m)C, C0 General constants (rn-i)C’ Inverse of C (m)C1 Constant used in Fourier equationC,,1 Specific heat of the material at node i (kJ/kg °C)Specific heat of the sample (id/kg °C)Cpr Specific heat of the roll (kJ/kg °C)db Constant used for Vickers microhardness correlationAE5 Change of energy of the sample due to rolling (kJ)f Fraction of the roll-bite contact time tcF Force (N)Fo Fourier numberh, h(t) Heat-transfer coefficient at the sample surface (kW/m2°C)H Surface hardness of the sample (kg/mm2)H(t) Sample thickness in roll bite (m)Havg Average thickness of the sample in the roll bite (m)Hb Bulk hardness of work piece (kg/mm2)xHf Exit thickness of the sample (m)H Entry thickness of the sample (m)Hj Thickness of the sample at time stepj (m)All Roll draft, H - Hf(m)km Harmonic mean conductivity (kW/m °C)Thermal conductivity of the slab (kW/m °C)kr Thermal conductivity of the roll (kW/m °C)kt Thermal conductivity of the tool (kW/m °C)Thermal conductivity of the workpiece (kW/m °C)L Projected arc length (m)m Interface friction factorn Constant used in steady-state constitutive equationn’ Number of contact spots per unit area (m2)Number of samples in population group 1Number of samples in population group 2N Angular velocity of the roll in rotations per minuteNr Number of nodes discretizing the roll surface layer 8N5 Number of nodes discretizing the sample half-thickness13a Apparent pressure (kg/mm2)1r Mean roll pressure (kg/mm2)Fe Peclet numberqf Frictional heat flux (kW/m2)qf, Frictional heat flux received by the roll (kW/m2)q f,s Frictional heat flux received by the strip (kW/m2)xiQ Activation energy (kJ)Qdef Heat generation due to deformation (kW)Q Heat flow (kW)r Radial thickness of roll (m)r1 Radial position of roll node i (m)Ar Radial thickness of roll node IR Gas constant (J/mole °C)R0 Roll radius (m)R* Radius of the roll interior not heated by the sample, R*=Ro (m)S Spread factor(S)2 Pooled varianceStandard deviation of population group 1Standard deviation of population group 2t Time (s)tc Contact time (s)At Change in time (s)T Temperature (°C, K)T Axial temperature of the roll (°C)Tf Bulk temperature of the sample after rolling (°C)T Bulk temperature of the sample before rolling (°C)Temperature of node i at time stepj (°C)Th Ambient temperature (°C)T5 Temperature of the sample (°C)T5(x t) Temperature of the sample at a position x at a time t (°C)Tr Temperature of the roll (°C)xliAT1 Component of ATc due to Material 1 (°C)AT2 Component of AT due to Material 2 (°C)AT Macroscopic difference in temperature between the surface temperaturesof two materials (°C)ATdef Increase in the sample temperature due to deformation (°C)Tr(X t) Temperature of the roll at a radius r at a time t (°C)ATr..s Average difference in the temperature between the sample and rollsurfaces in the roll bite (°C)AT Change in the sample temperature due to rolling (°C)tan 4 Mean of absolute slope of surface irregularities,tank +tan2t Absolute slope of tool surface asperitiestan Absolute slope of workpiece surface asperitiesV5 Half-volume of the sample in the roll bite (m3)W Width of the sample (m)W Width of the sample before rolling (m)Wf Width of the sample after rolling (m)x Through thickness direction (m)Ax Thickness of strip node i (m)y Length direction (m)Yield stress in pure tension (kg/mm2)z Transverse direction (m)Z Zener-Holloman parameterc Constant used in steady-state constitutive equation (MPa1)eff Harmonic thermal diffusivity (m2/s)xliiThermal diffusivity of the roll (m2/s)Thermal diffusivity of the sample (m2Is)1 Constant used in Fourier equation6 Depth of surface layer undergoing thermal cycle (m)Average asperity height of tool (m)Average asperity height of workpiece (m)61% Depth of internal point at which the difference between surfacetemperature and initial temperature of material is 1 pct. (m)A Cut-off time in the roll bite for HTC prediction (s)Mean strainMean strain rate (s-i)Strain rate (s-i)Efficiency of conversion of mechanical work to heat9 Root mean square of the roughness of the contacting surfaces (m),Standard deviation of profile height of tool asperities (m)Standard deviation of profile height of workpiece asperities (m)Coefficient of frictionMean of population group 1Mean of population group 2Vr Velocity of roll surface (mis)Velocity of sample (mis)(3 Angle of rotation of the roll (radians)Non-dimensional temperatureOpj Contact angle (radians)xivDensity of material at node i (kg/rn3)Pr Density of the roll (kg/rn3)p Density of the sample (kg/rn3)a Steady-state flow stress (MPa)a1 Steady-state flow stress of node i (MPa)aps Plane-strain steady-state flow stress (MPa)a Mean steady-state flow stress (MPa)-r Frictional shear stress (kg/mm2)Shear stress (kg/mm2)Angular speed of roll (radls)xvACKNOWLEDGEMENTSI would like to express my appreciation to Drs. I. V. Samarasekera, E. B. Hawboltand J. K. Brimacombe, for the welcome input and support they provided me these pasttwo years.Financial assistance from NSERC is grateftilly aknowledged. I would also like tothank ALCAN Ltd. for supply of materials and advice, and in particular, Dr. David Lloyddeserves special mention for his helpful support of this project.My gratitude also goes to fellow graduate student Wei Chang Chen; the advice andexpertise that he provided helped me greatly.xviCHAPTER 1INTRODUCTIONModem aluminum hot-rolling practice calls for the achievement of highproductivity coupled with the precise control of strip mechanical properties. These twingoals can only be attained through a thorough understanding of the complete hot rollingprocess. There is also a need for accurate, reliable and quick temperature measurementsof the strip during the hot-rolling process. Predictive and adaptive models used on-linecan then fully control the hot mill, responding to any situation to maintain quality andproduction standards.Of the total heat lost by aluminum strip during the hot-rolling process, three-quarters or more can be lost to the rolls [1]. Thus, accurate characterization of the roll-gap heat-transfer coefficient (HTC) is the single most important component of theknowledge base required for precise prediction and control of the strip temperature duringhot rolling.The importance of the (HTC) has been well realized [2] for steel hot-rolling. In thecase of aluminum rolling, however, there have been relatively few studies performed todate which measure the roll-gap HTC [3-6].Until recently, there have also been difficulties in applying the results of the studiesperformed in the laboratory to the industrial scale. The studies performed to date havebeen limited in scope and usefulness because of a lack of fundamental knowledge of themechanism of the roll-gap HTC. Therefore, there clearly exists a need to gain a12fundamental understanding of the physical basis behind the HTC, and thus also gain theability to characterize the HTC as a function not only of rolling parameters, but of thethermal and physical properties of the material being rolled as well.Even though the exact nature of the lubrication condition in the roll gap is notknown, there is a general agreement that some type of mixed lubrication conditionprevails [7, 8, 9]. This type of lubrication condition is also known as partialelastohydrodynamic lubrication; it describes the situation where the lubricant filmbetween the strip surface and the roll is partly interrupted by surface asperities of the rolland strip coming into direct contact with each other. The asperities of the strip in contactwith the roll then undergo deformation during the rolling process.Figure 1.1 shows a schematic of two surfaces contacting each other only at discretecontact points. As the normal pressure forcing the two surfaces together increases, theasperities deform and the contact area grows. The results of an increased contact area aretwofold. Firstly, the greater the contact area, the higher the friction force becomes as thelubrication layer is broken down. Secondly, since the majority of heat is transferredthrough the contact spots, the flux of heat between the two surfaces increases.23Figure 1.1 Contact between real surfaces (from Samarasekera [2])The concept that two nominally smooth surfaces are in fact rough on a micro-scale,and therefore, contact each other directly only at discrete spots is the basis of advancedfriction theory [10-12]. In addition, this concept has been used in studies characterizingthe HTC as a function of the true contact area, ie. the area of two contacting surfaces thatare in direct contact [13-17]. However, the potential of this concept as a means ofexplaining the nature of the roll-gap HTC in industrial rolling was only realized later bySamarasekera [2], who showed indirect evidence that the roll-gap HTC is stronglydependent on the real area of contact. Subsequent studies by Devadas et al. [18] and byChen et al. [191 have examined the relationship between the roll-gap HTC and the realarea of contact. The objective of this study, then, is to advance research on this subject.3CHAPTER 2LITERATURE REVIEW2.1 Previous Estimates of the Roll-Gap Heat-Transfer CoefficientResearch on roll-gap heat transfer during the hot rolling of aluminum has laggedbehind that for steel. Thus, only a few estimates for the HTC specific to aluminum hotrolling have been published.Chen eta!. [3] reported values of 10 to 50 kW/m2 °C for the hot rolling of Al-5 pct.Mg alloy, using Type K thermocouples having a 1.0 mm wire diameter for thetemperature measurements. The roll-gap HTC was reported to increase continually alongthe roll gap. Furthermore, the HTC was found to be constant at any position in the rollgap relative to the entry point in three different tests which were conducted at threedifferent reductions. The studies were conducted with aluminum samples instrumentedwith four thermocouples. Two thermocouples were placed on the sample surface, onewas located 1.8 mm below the surface, and one was placed at the centreline of eachaluminum sample. Before rolling experiments were performed, the roll was heated to70°C. These workers expressed a belief that the HTC depends on a combination offactors such as rolling pressure, the nature of the oxide layer, position of the neutral pointand surface roughness; however, no evidence was provided to verify these claims.Semiatin et a!. [4] conducted high strain-rate ring upsetting tests of AA2024, analuminum-copper alloy, from which resulting HTCs were reported to lie between 15 and20 kW/m2 °C. Thermocouples placed at 0.15 mm and 0.91 mm from the die surface45recorded the subsurface die temperatures. Pressures attained during these tests were notreported.Timothy et al. [5] also employed subsurface thermocouples in the laboratory hotrolling of AA5083 to obtain a value of 15 kW/m2°C. These workers noted that the rolledaluminum sample regained a homogenous temperature 30°C lower than the initial rollingtemperature approximately 0.5 seconds after leaving the roll bite.Pietrzyck and Lenard [6] reported HTC’s between 18.5 and 21.5 kW/m2 °C in thewarm rolling (155°C to 210°C) of commercial pure aluminum, using extrinsicthermocouples embedded in the aluminum slab. These workers did not calculate anaverage roll-bite HTC from the instantaneous response of the surface thermocouples, butfrom the temperature drop incurred by the sample due to the rolling operation:h— 212AL (.)where Ps is the density of the aluminum, C,5 is the heat capacity of the aluminum, ATavgis the average temperature drop of the sample due to rolling, and a, E and Havg are theaverage flow stress, mean strain rate and average thickness of the sample, respectively, inthe roll gap. The term Vr is the rolling speed, ATrs is the average temperature differencebetween the sample and strip in the roll bite, and L is the projected arc length.2.2 Dependence of the Real Contact Area on PressureWilliamson and Hunt [57] conducted experiments in which a 12-mm diameter steelball was pressed into indium and aluminum samples with artificially-roughened surfaces.After the deformation of the sample, the surface profile of the deformed area was traced,6and a real contact area was measured. These experiments demonstrated that for localindentations, the real area of contact is proportional to the nominal contact area,independent of pressure. Furthermore, the experiments showed that the persistence ofasperities even at high pressures was not attributable to work-hardening mechanisms,friction, or the trapping of fluid within the valleys of the asperities. Evidence was alsogiven which suggested that for solids with homogeneous hardnesses, the real area ofcontact was equal to one-half the nominal area of contact; and that for solids withhardened surface layers, the real area of contact decreased to 25-3 5 pct. of the nominalarea of contact.In a related study, Pullen and Williamson [56] conducted experiments in whichaluminum samples with artificially-roughened surfaces were pressed by a flat, hardened-steel ram. Before being pressed, the aluminum samples, which were cylindrical in shape,were forced into tight-fitting holes in hardened-steel dies. The placement of thealuminum samples into the steel dies prevented any bulk deformation of the samples asthey were pressed by the ram. Thus, surface pressures of up to fifteen times the yieldstress of aluminum were attained in the study. From these experiments Pullen andWilliamson [56] established that the true area of contact can be characterized as follows:H (2.2)1+—-Hwhere 1a is the nominal pressure and H is the surface hardness of the material. The termAc is the fractional area of metal-metal contact; it is defined as7(2.3)where Aa is the apparent or bulk area of contact, and Ar is the ‘real’ area, the area ofmetal-metal contact. An important result obtained from Equation (2.2) is that, since foran unsupported material the ratio Pa/H can never be greater than unity, the maximumfractional contact area that can be obtained between two surfaces is 0.5. Experimentalsubstantiation of this calculated result is provided by Williamson and Hunt [57].2.3 Dependence of the HTC on Pressure2.3.1 Experimental ObservationsSemiatin et al. [4] noted that, in the absence of bulk deformation, the HTCincreases with applied interface pressure, attaining a constant value above a certainpressure. Furthermore, it was postulated that deformation must smooth asperities at theinterface, bringing the tool and workpiece into better thermal contact.Chen et aL [3] attempted to relate the measured variation of the HTC along the arcof contact with roll pressure and the change in true area of contact. However, themeasured HTC did not compare well with the calculated roll pressure variation.Samarasekera [21 more comprehensively explained the dependence of the roll-gapHTC at the interface between the workpiece and roll on the fractional area of contact. Itwas suggested that the HTC dependence on the fractional area of contact accounted forthe previously observed variation of the HTC on such parameters as rolling speed, degreeof reduction, gauge and lubrication. Experimental verification of the relationshipbetween HTC and real area of contact for the hot rolling of steel was provided by8Devadas et al. [18], and by Chen et al. [19], who established a linear relationship betweenthe roll-workpiece interface HTC and the roll pressure.2.3.2 Theoretical TreatmentA small body of research has been published in which the dependence of the HTCbetween two nominally flat but microscopically rough surfaces on contact pressure hasbeen considered. Cooper et aL [13] proposed the relationship7 / , \O.985--‘=1.45I--1 (2.4)kmfor heat conduction between the contacting surfaces of a tool and workpiece underpressure in a vacuum, where h is the HTC, a is the apparent or bulk pressure, and H isthe surface hardness of the workpiece. The term km is defined as the harmonic meanconductivity of the workpiece and tool:2kkkm= (2.5)I’t + Iwpwhere kt and k are the conductivities of the tool and workpiece, respectively. Theterms 9 and tan are surface roughness parameters; tan is the average of the absoluteslope of the surface irregularities of the two contacting surfaces,tan=jjtan24t +tan2 (2.6)9and 9 is the average standard deviation of the profile height of the surface asperities ofboth surfaces,(2.7)using a theory based on a Gaussian distribution of heights. The apparent pressure aacting on the two surfaces: is calculated from an applied load F acting on the apparentarea of contact Aa of the two surfaces:(2.8)Equation (2.4) was experimentally verified by a small set of tests conducted with Pa/Hratios of less than 0.1. The exponent 0.985 was experimentally derived from these tests.Fenech et al. [161, using a “button model” of two surfaces in contact, proposed thatl—A (2.9)ö O.47JA/n’+—- +k k kmneglecting heat transfer outside of direct metal-metal contact; ö and are the averagevoid height for the tool and workpiece surfaces, respectively. The term km refers to thehainionic mean conductivities of the tool and workpiece, as before, and n’ is the numberof contacts per unit area. The term A is the fractional area of metal-metal contact, asdefined in Equation (2.3). In the button model A is defined as the square of the radius of10the button contact spot c divided by the radius of the heat channel a, or Ac=(c/a)2. Figure2.1 shows how Fenech et al. [161 used the button model to simulate actual contact.(a) (b)6wp1Figure 2.1 Button model for contacting surfaces (from Fenech [16])(a) Actual contact(b) Button model(c) Heat channel for button modelMikic [17] presented variations of Equation (2.4) to take into account differentassumptions of the mode of deformation at the surface, ranging from pure plasticdeformation to pure elastic deformation. Considering plastic flow only, Mikic [17]proposed a modification of Equation (2.4):0.94h=ll3km1( i9 H+P00000000(c)(2.10)11As compared to Equation (2.4), the HTC as calculated by Equation (2.10) will be reducedat relatively high pressure-to-surface-hardness ratio, whereas at lower pressures the twoequations produce very similar values of h.Mikic also considered the case involving both plastic and elastic deformation. Thiscase involves a correction factor to the pure plastic deformation case. However, thecorrection assumes that the elastic displacement of each asperity can be consideredindependently. This assumption has been shown to become invalid at extremely smallPa/H ratios by Pullen and Williamson [56], who showed that the deformation zones ofeach asperity overlap and interfere with each other almost immediately under the slightestplastic deformation.Song and Yovanovich [20] developed an explicit equation relating the HTC to thebulk hardness of the material, rather than the surface microhardness:0.97—0.75h 1.13k1J 1’ (2.11)d25 Hb) tan)where Hb is the bulk hardness of the workpiece, as opposed to the surface hardness, anddb is a constant used for a Vickers microhardness correlation. This equation was shownto have good agreement with experimental data in the range i06 Pa/Hc 2.3x10,where H is defined as the contact microhardness, which is related to surface roughnesscharacteristics and Vickers microhardness test results.Chen [21] applied the work of Fenech [16] to the case of hot rolling of steel, andwith some suitable approximations and modifications, established a theoretical linearrelationship between the HTC and the apparent pressure:12h= 1 (2.12)0.47Jë/ii awhere C0 is a general constant.2.4 Modelling Heat Transfer in the Roll GapThere are many mathematical models in the literature which compute thetemperature distribution of the workpiece in the roll gap during hot rolling [5, 19, 22-33].Of these models, many assume that the contact resistance for the flow of heat at the roll-strip interface is negligible [22, 25, 27, 31, 33]. Other models assign a constant HTCthroughout the roll gap [5, 23, 26, 28, 29, 30, 32]. However, only Chen et al. [19] haverelated the HTC to rolling pressure.2.4.1 Model Types and Solving SchemesSmelser and Thompson [23] modelled hot rolling assuming purely viscous flowand employed a two-dimensional finite-element scheme to solve for forming loads andtemperature distribution. Timothy et al. [5] also used a 2-D finite element scheme intheir analysis, utilizing a general, non-linear code. Lenard and Pietzryck [26], on theother hand, adopted a rigid-plastic finite-element approach to model the rolling process.Dawson [34] used a 2-D finite-element formulation which employs deformationmechanism maps to evaluate different types of constitutive equations.However, in most studies which examined only temperature distribution in the stripand/or roll, the finite-difference method is most favoured. Both Lahoti et al. [25] andSellars [30] used a two-dimensional finite difference method in their modelling of therolling process. Tseng [27] also used a two-dimensional finite difference method, but13used a technique known as GFD-- generalized finite difference -- in which the strip isdiscretized by a non-orthogonal mesh which is compatible with the shape of the roll-stripinterface.However, the most widely-used technique for modelling temperature in the stripduring rolling is the one-dimensional, unsteady-state finite-difference formulation. [19,24, 29, 32, 33] These researchers all agree that heat conduction in the width and length ofthe strip is insignificant as compared to the heat conduction through the thickness of thestrip, ie. in the direction perpendicular to the roll-strip interface.2.4.2 Surface Temperature Measurement Techniques in the Roll BiteThe most novel roll-gap strip temperature measurement technique reported in theliterature was the use of surface temperature transducers by Kannel and Dow [35]. Thetransducers were made by vapour-deposition of titanium onto a steel roll. However, mostresearchers employed either surface or subsurface thermocouples embedded either in theroll or the strip, or a combination of the two, in their studies.Several researchers employed only subsurface thermocouples in their studies.Timothy et aL [5] used extrinsic chromel-alumel subsurface thermocouples embedded inan aluminum sample in their heat-transfer study of aluminum hot rolling. They locatedthe thermocouples 1.0 mm below the sample surface, at the quarter-thickness, and at thesample centreline. Semiatin et at. [4] utilized subsurface thermocouples in their two-dieexperiments. The thermocouples were located 0.15 and 0.91 mm from the die surface.Karagiozis and Lenard [36] employed extrinsic chromel-alumel thermocouples having awire diameter of 1.59 mm located 1.8 mm below the sample surface and at the samplecentreline in their study of heat transfer in the hot rolling of steel. In a study of coldrolling, Steindi and Rice [37] also used thermocouples embedded in the sample surface.14They employed 0.127 mm diameter copper-constantan thermocouples located 1.5 mmfrom the sample surface and at the sample centreline.There are disadvantages in utilizing subsurface thermocouples for measuring thethermal history of the strip during rolling. Timothy et aL [5] noted that the response timeof the thermocouples was a finite fraction of the total time the thermocouples spent in theroll gap. In follow-up analyses of the work performed by Karagiozis and Lenard [36],Lenard and Pietrzyk [26] and Pietrzyk et al. [38] determined that the response time of thesubsurface, extrinsic thermocouples was 0.4 seconds, which was too slow to accuratelyrecord temperature changes within the steel samples. Therefore, many researchers haveemployed surface thermocouples in their studies.Jeswiet and Rice [39] were concerned with the probability of large temperaturegradients in both the strip and roll close to the interface, and therefore used athermocouple that was embedded in the roll normal to the roll surface. The thermocouplewires were insulated from the roll and each other, and terminated at the roll surface.Therefore, as a strip sample was rolled, a double-intrinsic junction was established at thesurface of the sample surface. Devadas et aL [18] also used double-intrinsicthermocouples at the surface, but attached them to the sample rather than to the roll. Thethermocouples were Type K, having chromel-alumel wires of diameter 0.25 mm. Chen etal. [3] also used surface thermocouples as well as subsurface thermocouples in their studyof heat-transfer during the hot rolling of aluminum, but employed extrinsic chromelalumel thermocouples with wire diameters of 1.0 mm.The response time of small-gauge, intrinsic thermocouples has been shown to bevery fast, of the order of one millisecond [18]. Furthermore, the perturbation error of theintrinsic type of thermocouple (that is, the change in the local temperature caused by thepresence of the thermocouple) is the lowest of any thermocouple type [40]. And finally,15even severe deformation of the thermocouple wire has been found to have no significanteffect on the accuracy of the thermocouple response [411.2.5 Friction in the Roll Gap2.5.1 Characterization of Friction in the Roll GapAlmost no agreement was found in the literature on the nature of frictionalbehaviour at the roll-strip interface during hot rolling. Tseng et al. [311, for example,employed a value of 0.2 for the coefficient of friction in their study of hot rolling ofAA5052 alloy. Devadas and Samarasekera [32] employed a coefficient of friction whichwas dependent on the temperature of the steel strip. Chen et aL [42] employed a value of0.3 to approximate what they termed as ‘near sticking friction’.Dawson [35] states that, even in hot rolling, some type of sliding friction prevails,although no value of the coefficient of friction is offered. Despite this view, otherresearchers have instead adopted the concept of an interface friction factor, m:m=-- (2.13)towhere the frictional shear stress t is a fraction m of the shear stress ‘rj of the deformedmaterial. The interface friction factor, m, characterizes sticking friction, as opposed to theconventionally-employed coefficient of friction, i, which characterizes sliding friction:(2.14)16where the frictional shear stress is a fraction t of the normal load P being exerted uponthe deforming material.Male et aL [43] have shown that employing m rather than a in flow stresscalculations for ring compression tests improves the quantitative prediction of thefrictional component of the deformation load. Timothy et al. [5] set m=O.8 in theiranalysis, considering that value to be a typical one associated with sticking friction duringhot rolling. Semiatin et al. [4], also used a friction shear factor in their heat-transferanalysis of ring-upsetting tests.2.5.2 Distribution of the Frictional HeatTseng et al. [311 distributed the frictional heat uniformly between the roll and strip.In a more sophisticated approach, Haifa et al. [44] recognized that the thermalconductivity of steel decreases with temperature; therefore, in the case of steel hot rolling,since the roll surface is much cooler than the strip surface in the roll gap, the heatconductivity of the rolls is about two times that of the strip. Therefore, Haifa et a!. [44]distributed the heat generated due to friction more to the roll than to the strip. Wilson eta!. [24] distributed the frictional heat between the strip and rolls according to a ‘heatpartition coefficient’. However, no information was provided on how to calculate thevalue of this coefficient.2.6. Previous Considerations of Flow Stress Variation Through the StripSheppard and Wright [33] have previously noted that, in the rolling of aluminumslabs, due to the quenching effect of the roll on the surface of the aluminum slab, there isa flow-stress variation in the direction perpendicular to the roll-slab interface. Theseworkers observed the effect of the flow-stress variation on a structural difference between17the surface and centre of the aluminum slab, but did not consider any effect of the flowstress variation on heat transfer at the interface.CHAPTER 3SCOPE AND OBJECTIVESThe thesis of this study is that the roll-gap heat-transfer coefficient can becharacterized as a function of the mean rolling pressure and the thermal and physicalproperties of the material being rolled.The objectives of this project are three-fold:Firstly, to experimentally obtain heat-transfer coefficients in the roll gap during thehot rolling of the aluminum alloys AA5052 and AA5 182 and to establish a relationshipbetween the roll-gap heat-transfer coefficient and mean rolling pressure.Secondly, from the aluminum alloy rolling tests, additional copper rolling tests anddata from steel rolling tests from a previous study [211, to determine the relationship thatexists between the roll-gap heat-transfer coefficient and the thermal and physicalproperties of the material being rolled.And thirdly, ultimately to apply the heat-transfer coefficients developed in thisstudy to the prediction and control of the temperature profile of the strip being rolled, inorder to control and optimize the mechanical properties of the rolled strip.18CHAPTER 4EXPERIMENTAL DESIGN AND MEASUREMENTS4.1 MaterialsTwo aluminum-magnesium alloys and a commercial-pure copper were examined inthis study. The alumunum alloys studied were AA5052 (Al-2 pet. Mg) and AA5 182 (Al4.5 pet. Mg). They were supplied in the form of slices cut perpendicular to the verticalaxis of D.C. ingots for this study by the Kingston Research and Development Centre ofAlcan Ltd. The copper was provided from storage in-house in an annealed condition.4.1.1 Homogenization TreatmentSince the aluminum alloys were received in the as-cast condition, they weresubjected to a heat treatment in order to homogenize their physical properties. Thehomogenization treatment for each alloy was as follows. AAS 182 samples were heatedto 530°C and held at that temperature for one hour. AA5052 samples were heated to560°C and held at that temperature for two hours. The copper samples did not undergoany homogenization treatment.4.1.2 Flow Stress CharacterizationIn order to characterize the steady-state flow stress behaviour of both the aluminumalloys and the copper used in this study, 10 mm dia. x 15 mm long cylindrical samples ofthe materials were subjected to compression tests. All compression tests were performed1920by the Gleeble 1500 at UBC, a thermo-mechanical simulator which relies on resistanceheating to elevate and control the temperature of the specimens.Figure 4.1 shows a schematic diagram of the Gleeble apparatus. Tantalum orcarbon foil between the test specimen and the anvil prevented welding of the two togetherduring high temperature tests. A quartz L-strain device was used to measure the length ofthe test specimen, and a C-strain device measured the diameter of the test specimen.Providing that barrelling of the specimen does not occur to any large degree, thecombination of L-strain and C-strain measurements, in addition to the force required fordeformation of the specimen (recorded by the load cell), provided the informationnecessary to construct true stress-true strain data.Ta foilclampS.S. Jaw4anvilload cellFigure 4.1 Schematic diagram of GLEEBLE apparatusPrior to the actual deformation, the homogenized aluminum alloy test samples(both AA5052 and AA5 182) were heated at 5 °C/s to 530 °C, held at that temperature forthermocouple21one minute, cooled at 2 °C/s to the deformation temperature, and then held at thedeformation temperature for one more minute. This holding time ensured a uniformsample temperature. After deformation, the test samples were allowed to air cool. Eachalloy was tested at four different temperatures (300, 375, 430 and 520 °C), and at fivedifferent nominal strain rates (0.01, 0.5, 3, 7 and 10 s-1), for a total of twenty tests.Copper cylindrical test specimens were heated to 700 °C at 10 °C/s in the Gleeble,held at that temperature for one minute, then cooled to the deformation temperature at 5°C/s, and held at that temperature for one more minute. This ensured a uniform testingtemperature. After deformation, the specimens were allowed to air cool. The copper wastested at three different temperatures, 475, 575 and 675 °C, and at each temperature atthree different strain rates, 0.1, 1 and 10 s’, for a total of nine tests. The true stress-truestrain data obtained from the compression tests were then fitted to the hyperbolic-sineconstitutive equation:Z = e exp) = A sinh(cxa)” (4.1)In Equation 4.1, Z, the Zener-Holloman parameter, is the temperature-compensatedstrain rate; a is the strain rate; Q is the activation energy; R is the gas constant; T is theabsolute temperature; a is the steady-state flow stress and A, a. and n are constants. Theflow stress for the analysis was taken at a strain of 0.5. At this strain, the experimentalflow stresses were in a steady-state regime. The strain rate and sample temperature werecalculated over a strain range of 0.2 to 0.5. The equation parameters Q, a. , n, and in (A)were calculated using a method developed by Davies et aL [45], which allowed each22parameter to have an unconstrained value. The resulting parameters of the constitutiveequation are shown in Table 4.1.Table 4.1. Constants for constitutive steady-state stress equationParameter AA5 182 AA5052 CopperQ(kJ/mol) 185 189 173.2a(MPal) 0.0450 0.0317 0.0729n 1.818 3.536 1.257ln(A) 24.48 26.13 18.024.2 Test Design4.2.1 Test FacilitiesIn order to minimize the transfer time of the sample from the furnace to the rollstand, a square tube furnace was designed to butt against the rolls of the laboratory mill.The rolling mill used in this study is a two-high reversing mill with specifications shownin Table 4.2. The rolling mill was outfitted with a load cell to record the total separatingforce experienced by the mill during rolling, as well as a lubrication system that deliveredlubricant to the top roll of the mill and a guide located at the roll-gap exit to preventsamples from sticking to the top roll and curling upon exit from the roll gap. Figure 4.2shows a schematic diagram of the rolling mill and furnace.23Table 4.2. UBC pilot mill specificationsManufacturer STANATRolling Speed 34.3/68.5 rpmRoll Diameter 100 mmRoll Material Vanadium BB*Max. Roll Separating Force 200 kN*A proprietary alloy of VASCO Inc.-- similar to SAE 52100Rolling millFigure 4.2. Schematic diagram of the UBC pilot rolling millThe load cell, along with the three thermocouples attached to the instrumentedsample, was connected to a COMPAQ portable microcomputer, which was equipped witha DT2805 data acquisition board. During all rolling tests the data acquisition rate was1500 Hz.Load cell24The lubrication system consisted of a one-litre reservoir, attached to the top of themill, and a brush which spread the lubricant over the surface of the top roll. Lubricantpassed from the reservoir to the brush through a polypropylene hose at 200 mi/mm duringrolling. The bottom roll was lubricated by the excess lubricant that poured off the toproll. A container placed under the bottom roll collected the used lubricant.The exit guide, attached to the exit table of the rolling mill, formed a channel 10mm high through which the sample had to pass upon exit from the rolls. This preventedthe samples from curling up and therefore prevented the sample surface thermocouplesfrom being in contact with the top roll past the roll exit plane. Figure 4.3 schematicallyshows the lubrication system and the exit guide.25exit planeload celllubricant collectorFigure 4.3. Schematic diagram of roll lubrication system4.2.2 Preparation of Test Samples4.2.2.1 Aluminum Alloy SamplesAluminum samples (8.7 nmi x 50.8 mm x 127 mm) were machined as shown inFigure 4.4, with the axial direction of the original D.C. ingot shown with a dashed arrow.The width-thickness ratio of the samples was 5.84. Using an empirical spread-predictionequation proposed by Beese [461,lubricantreservoirbrushexit guideexit table26S 0.6l1’i exp—0. 321 ‘ (4.2)1jJ) 1?oAH)Jwhere H is the original thickness of the sample, W is the original width of the sample,R0 is the roll radius, AN is the draft, and S, the spread factor, is defined asln(W íw)5= (4.3)in(Ii IHf)The spread factor was calculated to be 0.0457. This corresponds to a 9.1 pct. deviationfrom a pure plane strain condition, since when S=0, the strain is plane, and when S=0.5,the strain is distributed equally in the width and length direction. This calculateddeviation from pure plane strain was judged to be small enough to not invalidate plane-strain assumptions used in calculations of flow stress.After a homogenization treatment (see Section 4.1.1 for details) each sample wasinstrumented with three thermocouples (1.6 mm dia. INCONEL-sheathed Type-Kthermocouples having Chromel-Alumel wires 0.25 mm in dia.); two of thethermocouples, Si and S2, were located on the sample surface and a third, Cl, at thecentre of the sample, as shown in Figure 4.5. Thermocouples Si and S2 were insertedinto horizontal holes extending halfway into the sample, and the exposed ChromelAlumel wires were brought to the surface through the vertical holes drilled from the topsurface, which intercepted the horizontal holes. The Chromel-Alumel wires were placedon the sample surface approximately one-half millimetre apart to establish a doubleintrinsic junction, and fastened to the sample surface by inserting the wires into a shallow270.6-mm dia. hole drilled into the sample surface, which was subsequently punched shut.Figure 4.6 shows a schematic view of the double-intrinsic junction. The thermocoupleholes were drilled oversize so that deformation of the sample would not deform thethermocouple sheathing; uncertainties involving the effect of thermocouple sheathdeformation on temperature measurement thus were avoided. For thermocouple Cl thethermocouple wires were spot-welded together to form an extrinsic junction. Electricalresistance checks ensured that the extrinsic junction made contact with the sample.283 mm rod (1 m long)22mmE:/2.4 mm dia. E22mm E3mmrod(lmlong)2.8 mm 4.35 mm 2.8 mm.9 1I I1EEEEEEE1&7rflm X50.8 mm2.4 mm diaFigure 4.4. Design of aluminum sample298.7 mmHandling rodFigure 4.5. Schematic diagram of aluminum test pieceChromel-Alumelwires (0.25 mm dia.)hole(0.6 mm dia.)(1.6 mm dia.)surface hole(2.4 mm dia.)AluminumSample4. //Oj127mmH— 50.8 mmBrass sheathocoupIe hole(2.4 mm dia.)INCON EL-sheathedType K thermocoupleFigure 4.6. Close-up of surface thermocouple304.2.2.2 Copper SamplesTwo copper samples were machined in a manner similar to the aluminum samples,with the omission of the centre thermocouple hole. After machining, the copper samples(6.4 mm x 50.8 mm x 114 mm) were instrumented with two surface thermocouples, asshown in Figure 4.7, using the same procedure that was developed for the aluminumsamples.Copper sampleThermocouplesFigure 4.7. Schematic of copper test piece4.3 Test Procedure4.3.1 Aluminum Test ScheduleIn general, each aluminum sample was rolled three times. Initially the sampleswere rolled without bulk plastic deformation. This first pass served to flatten thethermocouple wires into the sample, thereby assuring good electrical contact with thesample and establishing the double-intrinsic junction at the surface. The samples werethen returned to the furnace and reheated prior to being rerolled. For the second pass, theroll gap was narrowed so that a nominal bulk deformation of twenty percent was50.8 mm-HBrass sheath31achieved. The samples were then returned to the furnace a second time and reheated.The sample was then rolled once more, this time by a nominal reduction of ten percent.During the passes involving deformation, the rolls were continually lubricated by one oftwo oil (5 pct.) - water emulsions. These emulsions were prepared from two industrialrolling oils provided by Alean Inc. using an ultrasound mixing probe. During themajority of the aluminum rolling tests, the rolls were lubricated with a viscous, lowfriction lubricant, designated as Lubricant ‘A’. During the remainder of the aluminumrolling tests, the rolls were lubricated with a less-viscous, higher-friction lubricant,designated as Lubricant ‘B’. Table 4.3 presents the conditions of the aluminum tests.32Table 4.3. Conditions Employed in Aluminum Rolling TestsTest Material ReductionIDInitial Initial Rolling Strain Meanthickness Temp. Speed Rate Pressure(mm) (°C) (pct) (mis) (s’) (kg/mm2)AL1O 5182 7.04 470 5.1 0.358 5.21 14.26AL12 5052 8.70 515 19.8 0.358 9.83 10.41AL13 5052 7.01 510 11.2 0.180 3.93 10.09AL15* 5052 8.70 520 20.1 0.358 9.92 10.35AL16 5052 6.99 510 10.9 0.358 7.74 8.81AL18 5182 8.70 505 19.8 0.358 9.83 13.80AL19 5182 7.01 500 10.1 0.180 3.70 12.45AL21 5052 8.70 415 18.9 0.358 9.56 13.50AL22 5052 7.09 505 11.8 0.358 8.03 9.49AL24 5182 8.70 375 20.1 0.358 9.92 19.89AL27 5182 8.70 320 19.2 0.358 9.65 21.51AL28 5182 7.06 320 9.3 0.358 7.05 22.7AL3O 5052 8.70 320 20.3 0.358 9.99 16.46AL31 5052 6.96 320 10.9 0.358 7.76 17.27AL33 5052 8.70 370 20.1 0.358 9.92 14.75jJ,34* 5052 6.99 320 10.5 0.358 7.58 16.59J137* 5182 6.99 320 10.2 0.358 7.46 20.79AL39t 5182 8.70 300 8.4 0.358 6.01 22.57*These tests were performed with Lubricant ‘B’.Lubricant ‘A’ (except test AL39)tThis test was performed without lubrication.All other tests were performed with334.3.2 Copper Test ScheduleEach of the two copper samples was rolled three times in a similar manner as thealuminum samples (see 4.3.1). Table 4.4 presents the conditions of the copper tests.Table 4.4. Conditions Employed in Copper Rolling TestsTest Initial Initial Reduction Rolling Strain MeanID thickness Temp. Speed Rate Pressure(mm) (°C) (pct.) (mis) (s-i) (kg/mm2)CU4 6.35 700 13.2 0.358 9.05 11.20CU5 5.51 700 12.4 0.358 9.38 10.45CU7 6.35 650 13.6 0.358 9.21 11.10CU8 5.49 650 11.6 0.358 9.04 11.644.4 Thermal Response4.4.1 Thermal Response of Aluminum SamplesFigures 4.8-4.12 show the thermal response of some of the aluminum tests. Figure4.8 shows the thermal response of the three thermocouples during Test AL13, as well asthe output of the load cell. For this test, the output of all the thermocouples wasexceptionally smooth. The surface thermocouples showed good reproducibility. Thetemperature of each thermocouple decreased extremely rapidly from 505°C as the samplesurface was quenched by the roll to 3 00°C. As the sample at the thermocouple locationexited from the rolls, the sample surface temperature then quickly reheated toapproximately 425°. The temperature at the sample centre decreased smoothly from505°C to 425°C. The thermocouple response indicates that the sample temperature34became essentially homogeneous approximately 200 milliseconds after the rollingoperation was complete.In general, the apparent oscillation of the roll load is due to the electricalinterference from the 60 Hz A.C. power supply with the low-voltage (on the order of twoto three millivolts) signal of the load cell. The output signal from the load cell during thistest, however, exhibited greater noise than usual, possibly due to the power source.550 30000500 25000P 450 20000 : ::- 400 15000 c.2 TCC1350 10000 Roll load300 5000250 00.35 0.45 0.55 0.65 0.75 0.85 0.95Time (s)Figure 4.8. Thermocouple and load response for Test AL13353000025000200001500001000050000Time (s)Figure 4.9. Thermocouple and load response for Test AL15o TCSIo TCS2TCC1Roll Load0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5550500450a)400350I—30025045040000350.300Ea)I-2502005000040000- o TCS130000 -0 TCS2(U-2 TCC120000Roll Load1000000.4 0.5 0.6 0.7 0.8 0.9 1Time (s)Figure 4.10. Thermocouple and load response for Test AL2136400003200024000016000_____8000Figure 4.11. Thermocouple and load response for Test AL24Figure 4.12. Thermocouple and load response for Test AL31Figure 4.9 shows another test, Test AL 15, in which the reproducibility of the twosurface thermocouples is quite good, despite a small amount of noise in the output of48000o TCS1o TCS2. TCCIRoll Load0400360TestAL240_______P 320_______________I280G)240H2001600.4 0.5 0.6 0.7 0.8 0.9Time (s)360320, 280D240Ea)I-20016050000400003000020000100000co0cro TCSIo TCS2i. TCC1Roll Load0.2 0.3 0.4 0.5 0.6 0.7Time (s)37thermocouple S2. Figure 4.10, on the other hand, shows an instance (Test AL21) whereboth thermocouples appear to have failed. TC Si exhibited a questionable response whilein the roll bite, but seemed to recover after exiting the roll gap. TC S2, on the other hand,failed completely, sensing very erratic temperatures both in the roll bite and upon exitfrom the rolls. The reason for the erratic output of the thermocouples during this test isunknown.Figure 4.11 shows the results of Test AL24. The initial rolling temperature of thealuminum sample during this test was much lower than for Tests AL13, AL15, or AL21.Due to the higher flow stress of the aluminum sample during this low-temperature test,the roll load is seen to be significantly higher. Furthermore, the centreline thermocoupleTC Cl exhibits an initial temperature rise before decreasing to the lower, post-rollingtemperature. This temperature is due to the bulk heat of deformation, and is mostapparent at tests involving lower temperatures. The reason for this is that the temperatureincrease due to deformation is dependent on the flow stress of the aluminum sample,which is significantly higher at low rolling temperatures than at high temperatures.38300002500020000o TCS115000 o TCS20Roll load10000________________50000Figure 4.13. Thermocouple and load response for Test CU7Figure 4.13 shows the thermocouple and load output for Test CU7. The coppersamples did not contain centreline thermocouples. In general, the thermocouple outputresponse during the copper sample tests was not as well-behaved as the response of thethermocouples during the aluminum sample tests. In this case, the response of TC S2was anomalous in that it showed an initial temperature rise just prior to coming intocontact with the roll. Secondly, the output of TC S2 was generally erratic. The reasonfor the relatively poor response of the thermocouples during the copper tests is notknown.4.4.2 Thermal and Load Response of Copper Samples640560? 480400320I—2401600.6 0.65 0.7 0.75 0.8 0.85 0.9Time (s)CHAPTER 5HEAT TRANSFER MODEL DEVELOPMENT5.1 Mathematical FormulationIn order to convert the surface temperature response of the aluminum sample to aHTC, the general heat-conduction equation must be solved subject to initial and boundaryconditions. Several assumptions may be made to simplify the governing equation:1. The process is at steady state, so that at any fixed location in the roll bite thetemperature does not change with time;2. Since, for laboratory rolling, the Peclet number is high, on the order of 100, heatconductiàn along the length of the sample (y-direction) is assumed to benegligible compared to heat transfer by bulk motion;3. Heat transfer in the transverse (z) direction is assumed negligible due to the largewidth-to-thickness ratio;4. Frictional heat is generated along the arc of contact, and is distributed betweenthe sample and roll according to their respective thermal diffusivities.The two-dimensional, steady-state governing equation for heat conduction in thesample then can be written as=0 (5.1)3940Equation (5.1) can be expressed in a one-dimensional, unsteady-state form byemploying the transformation y where VS is the sample velocity, to becomeiks)+Qdef =p,C,,,- (5.2)where t is the time taken for an elemental volume to travel a distance y in the roll bite.Equation (5.2) is solved subject to an initial and two boundary conditions. Initially thetemperature of the sample, T5, is uniform:t=0, 0xLi-, 7=7(x,0) (5.3)Assuming symmetrical cooling of the sample about the centreline (x = 0), theboundary condition at this location ist0, x=0, —k,--=0 (5.4)The boundary condition at the sample surface (x = H(t)/2) can be expressed ast > o, x = H(t)— k, = hQ)(7’2— 7’Ro) (5.5)41where h(t) is the local heat-transfer coefficient at the roll-sample interface and 712 andyl?o are the temperatures of the sample and roll surfaces, respectively. The changingsample thickness is H(t).Since the sample and work rolls are coupled thermally, it is necessary to solve thegoverning equation of the sample simultaneously with that of the rolls. Neglecting axialheat conduction in the rolls and assuming that circumferential heat conduction isnegligible compared to bulk heat flow due to rotation of the rolls, conduction is confinedto the radial direction; and the governing heat conduction equation for the roll becomes181 6T’\ ÔT (5.6)I r I r prrôr’.. ãrjwhere t is the time taken for an elemental volume of the roll to rotate through an angle, 0,measured from a reference point.The temperature increase of the roll due to hot rolling is confined to a surfacelayer ö. Initially, the temperature of the roll is uniform, that is,t=O, R0—örR 7=7(r,O) (5.7)At R* = R0 - ö , the boundary condition istO, r=R*, (5.8)DrAt the roll surface (r = R0), the boundary condition is42t>O, r=R0, (59)5.2 Discretization of the Differential EquationsAn implicit finite difference method was used to solve the one-dimensionaltransient heat transfer equation developed in Section 5.1. The strip and roll werediscretized into three types of nodes; surface nodes, interior nodes, and adiabatic nodes,as shown in Figure 5.1. A heat balance was performed on each node to obtain generalequations.R0O.5H1O.5HfFigure 5.1. Discretization of roll and strip435.2.1 Nodes in the StripAt the surface of the strip, heat transfer to the environment is characterized by theheat transfer coefficient h(t). The heat balance on the surface node, Node 1, is then asfollows:r 2cL + h(t) + h(t)x, +--T (5.10)L& +Ax2 k cosO 2At] Ax1 +Ax2 kcosO 2Atwhere Node 1 is the strip surface node, a is the thermal diffusivity of the strip andj is the time step. The heat balance for an interior node i is:2 + 2c +-‘-lT’ = 2cL T+ 2cL, TH-’-T (5.11)L Ax11 + Ax1 Ax. + Ax._1 At ] ‘ Ax11 + Ax. 1+ Ax + Ax._1 ‘‘ AtFinally, the centreline node N5 has an adiabatic side, so the heat balance becomes:=+ AxN (5.12)AxN + AXN_I At]‘+ AxN,_l At5.2.2 Nodes in the RollThe roll node heat balances are similar to the strip heat balances. The heat balanceon the surface node of the roll is:442a (t;+-A,j r2(r+lAr=‘is 2 )Tj+1h(t)rT ‘ 2 ‘ TArj+At kr 2At(5.13)The heat balance for an interior roll node i is:2cj, +!Ar. 2dr.+!Ar 2cxI’r. +--Ar. 2dr+--z\r.‘‘ 2__‘J ‘ 2 2 ‘_‘T’+is’ 2+--T+ At; Ar, + & + At; ‘ Ar, + Ar, ‘ At(5.14)Finally, the heat balance for the adiabatic node Nr at a depth ö below the rollsurface is:2cxr(rN +—rN +—ATN —r, 2cT(rN +—rN J (rN +—ArN2 + 2 = 2 T’ + 2 T‘N, +ATN,._I 2AtN,ArN, +/.\rN,_iN,—I 2At N,(5.15)5.2.3 Solving TechniqueA Gauss-Siedel iteration method with successive over-relaxation was used in orderto solve the nodal equations. Figure 5.2 shows the general technique used to solve for theinstantaneous HTC in the roll gap.kr 2&45Figure 5.2. Flowchart of HTC-solving algorithm5.3 Treatment of Heat Generation5.3.1 Generation and Distribution of Frictional HeatThe frictional heat generation is given byqf_-vS—vrJ. P (5.16)46as formulated by Devadas and Samarasekera [32], in which I”s — Vrl, the relative velocitybetween the roll and strip, varies continuously along the arc of contact.Extending the approach of Hatta et aL [44], who distributed frictional heat based onrough estimates of the conductivities of the roll and strip, the frictional heat wasdistributed according to the thermal diffusivity ratio of the strip and roll:a(T)f\ ( \qf (5.17)+ rT)andqf,—qf—qf, (5.18)5.3.2 Bulk Heat due to DeformationPavlov’s equation [47] calculates the bulk heat of deformation in rolled strip as‘ef ln__LJ (5.19)pscps I-Jwhere a is the temperature-dependent flow-stress of the material being rolled, and H andHf are the entry and exit thicknesses of the strip, respectively. This equation assumes thatthe plastic deformation is uniform throughout the thickness of the strip. However, thisequation also assumes that all the mechanical work is converted into heat, and also onlyconsiders the total strain accrued in the strip as a result of deformation. Therefore, theequation was modified to address these deficiencies. The resulting equation is47Aef(t,l) = In1-J (5.20)p1C,,, k11)where r is the efficiency of the conversion of work to heat, set to 0.95 after Timothy etat. [5j. With this form of the equation, for a strip node i at time t, the increase intemperature due to deformation is a function of the strain rate-temperature dependentyield stress a, as well as the density and heat capacity of the strip at node i, and also ofthe strain accrued during the time step. Thus, Equation (5.20) takes into account thevarying amounts of strain that is accrued by the strip during each time step through theroll gap. Also, the increase in temperature of each node is dependent on the initialtemperature of that node. Therefore, the nodes closer to the surface of the strip, beingcolder than the nodes in the strip’s interior, and therefore also having a higher yield stress,increase more in temperature due to the bulk deformation than do the warmer, interiornodes.5.3.3 Depth of Heat Penetration into the RollDue to the extremely short contact time between the strip and roll, the thermalshock experienced by the roll only penetrates to a shallow surface layer of depth &Therefore, when modelling the temperature of the roll, it is only necessary to discretizethe roll to the depth , thus reducing the number of nodes in the roll by a considerableamount and saving computing time. Tseng [48] proposed an equation to calculate thedepth of the roll layer; however, the equation was developed for sustained rollingconditions as in an industrial situation. Therefore, this equation calculates a roll surface48layer that is much deeper than is required for the present study, as the test pieces werecompletely rolled within one roll revolution. Instead, a simple penetration depth wascalculated, assuming an infinite HTC (from [52]):61%—3.64 (5.21)where ö 1% is the roll depth at which the temperature changes by one percent of thedifference between the initial temperature of the roll and the ambient temperature, ar isthe thermal diffusivity of the roll, and t is the contact time. Taking tc to be 50milliseconds and ar to be 1.1 1x105 mis2, 81% was calculated to be 2.7 mm. SinceEquation (5.21) is strictly valid only for a semi-infinite slab, and taking into considerationthe approximate nature of the equation, for the purpose of this study, the thermal layerdepth 8 was set at 8 mm.5.4 Conductivities of Materials Used in this StudyThis study examined the HTC during rolling for three materials, AA5052, AA5 182,and copper. Furthermore, the results of this study were compared against those of Chen[21] for SS304. Finally, it was also necessary to know the thermal properties of the rollas well.The temperature-dependent thermal conductivity of both AA5052 and AA5 182 wasobtained from a study by Logunov and Zverev [49]. These researchers reported valuesfor an aluminum alloy known as AMG-3, which contains 3.2-3.8 pct. Mg, and for analuminum alloy known as AMG-5, which contains 4.8-5.8 pet. Mg. The experimentalthermal conductivities obtained in this study compared well with theoretical values.49Both the temperature-dependent heat capacity and thermal conductivity of thecopper were obtained from a report by Pehike et aL [50]. These workers have tabulatedvalues of heat capacity and thermal conductivity for copper for temperatures rangingfrom 273 to 2300 K.The material of the rolls used in this study was a proprietary Fe-alloy with achemical composition 1.0 pct. C, 0.32 pet. Mn, 0.25 pet. Si, 1.4 pet. Cr and 0.2 pet. V.Unfortunately, a literature search failed to find any published values of thermal propertiesfor this alloy. AISI 4140, an alloy containing 0.4 pet. C, 1 pet. Cr and 0.2 pet. Mo, has athermal conductivity of 42.7 W/m °C at 100°C [51]. An alloy containing 0.15 pet. C,0.57 pet. Mn, 0.26 pet. Si and 0.30 pet. V has a conductivity of 43.1 WIm °C at 100°C[611. To account for an observed slight lessening of thermal conductivity with increasedcarbon content [511, the roll conductivity was set at 41.0 W/m °C for the temperaturerange 25°C - 250°C. Finally, thermal properties for SS304 were obtained from Chen[211.5.5 Model Verification5.5.1 Validity of the 1-D ModelA nondimensional analysis was conducted to test the validity of the assumption thatheat flow in the length or rolling direction is negligible compared to heat flow in thethickness direction, ie. in the direction perpendicular to the roll-strip interface. The Peeletnumber, which is the ratio of heat flow due to bulk flow to heat flow due to conduction,defined asFe= 3&?0LC,,p, (5.22)50where ü is the angular roll speed, R0 is the roll radius, L is the length of the strip beingrolled in the roll bite, and and p are the heat capacity and density of the strip,respectively, is a useful parameter to evaluate this assumption. A Peclet number on theorder of 100 is generally considered to be the value below which heat transfer in thelength direction becomes non-negligible as compared to heat transfer in the thicknessdirection. The result of the Peclet number analysis for the UBC pilot mill is summarizedin Table 5.1.Table 5.1. Peclet Numbers for Various Rolling Speeds and ReductionsDeformation Angular Peclet Number(pct.) velocity(RPM)5 34.3 475 68.5 9310 34.3 6610 68.5 13120 34.3 9220 68.5 185The table reveals that at lower reductions and rolling speeds the assumptionbecomes less valid. It was therefore decided, in scheduling the rolling tests, to avoid fivepercent reductions. Ten percent was set as the minimum reduction, and 34.3 rpm as theminimum rolling speed, to maintain the validity of the 1-D heat transfer model.515.5.2 Comparison with Analytical SolutionEven though it was impossible to check the complete model against an analyticalsolution due to the complexity of the coupled roll-strip formulation, the model wasmodified in order to check the finite-difference solution of the strip and roll separatelyagainst analytical solutions.5.5.2.1 Verification of the Roll Finite-Difference FormulationThe model was modified so that the roll was uncoupled from the strip. The roll,initially at 24°C, was exposed to an environment of 400°C for 294 seconds. Therefore,the Fourier number (Fe,) for the roll in this situation becomesFo=-= (1.7.105m2/s)(294s) =2 (5.23)(O.05m)2where cr is the thermal conductivity of the roll, t is the time, and R0 is the roll radius.The HTC, h, was set to 1240 W1m2 °C, and the roll conductivity was fixed constant at 62W/m2 °C, in order to obtain a Biot (Bi) number of 1:Bi hR0 (1240W/m°C)(0.05m) = 1 (5.24)kr 62W/m°CSince Fo is greater than 0.2, it is possible to use a single term of the Fourier seriessolution to obtain an analytical solution of the roll axis temperature after 53 seconds[52]:520c=’7C1exp(—f3Fo) (5.25)where T is the roll axis temperature, Th is the ambient temperature, Tr is the initialtemperature of the roll, and C1 and 1i are constants. However, since the methods forsolving for the two constants are very tedious, for the roll a graphical technique was used.The Heissler-Gröber chart is a convenient method of determining the axial temperature ofa cylinder for precisely this type of problem. See [52], for example, for a completedescription of the H-B charts.The analytical solution of the axial temperature of the roll, obtained from the H-Bchart, was 38 1°C. The model, employing 200 nodes to discretize the roll, and using 50time steps, calculated the temperature of the roll axis to be 3 85°C. The values werejudged to be sufficiently close that the finite-difference solution was considered to beverified.5.5.2.2 Verification of the Strip Finite-Difference FormulationThe strip finite-difference solution was checked against an analytical solution in amanner very similar to the method employed for the roll. The strip, initially at 400°C,was exposed to an ambient temperature of 25°C for 1.331 seconds. For the purpose ofcomparing the model solution with the analytical solution, the thermal diffusivity a5 ofthe strip was fixed at 5.689x10 m2/s, and the conductivity k5 was fixed at 137 W/m °C.The Fourier number for a strip of thickness H = 8.7 mm is thenFo=______(5.689105m2/ s)(1.331s)= (5.26)(H2 (0.00435m)2)53The HTC was set to 31 264 W/m2 °C in order to set the Biot number to 1:Bi hL (31264W / rn2 °C)(0.00435m) = 1 (5.27)k 136W/rn °CFor this case, the Fourier number is again greater than 0.2, so a single term of theFourier series solution (Equation (5.24)) is accurate to greater than one percent.Furthermore, for the case of the strip geometry, tables are readily available which tabulatevalues of C1 and 1i. From [52.] C1 is equal to 1.1191 and 13i is equal to 0.8603.Equation (5.24) is then readily solved to yield a centreline temperature of the strip,46.7°C. The finite-difference solution, on the other hand, yielded a centrelinetemperature of 47.4°C. This value was arrived at using 75 nodes to discretize the stripand 1000 time steps. A relatively large number of time steps, compared to the number oftime steps used for verifying the roll finite-difference solution, was employed because ofthe high HTC. Since the difference between the model and analytical solution was lessthan one degree, the strip finite-difference formulation was considered to be verified.5.5.3 Convergence of the ModelThe implicit finite-difference method does not suffer stability problems as does theexplicit finite-difference method. Therefore, one would expect a continual improvementin the numerical solution (that is, the difference between the numerical solution and thetrue, analytical solution becomes less) as the mesh size and time step is decreased, to thepoint at which computer round off error becomes significant.54To test the convergence of the model solution, the mesh size and time step wasmodified for Test AL15, thermocouple Si. The base mesh used by the model is onehundred-fifty nodes to discretize a half-thickness of the strip, two hundred nodes todiscretize the surface layer, 8, of the roll, and fifty time steps to advance a slice of thestrip through the roll bite. As Figure 5.3 indicates, the HTC through the roll bite wascalculated to be essentially the same for all three mesh variations. Even when the meshwas increased to three hundred strip nodes and four hundred roll nodes, and one hundredtime steps were used, the calculated HTC changed only slightly. This confirms thestability of the finite-difference model and the convergence of the solution.3002505200I. 75 Ns, 100 Nr, 50 time steps150150 Ns, 200 Nr, 50 time steps0 100• 300Ns400Nr,lOOtimesteps50.10 I I I I I I I I0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.02Time in Roll Bite (seconds)Figure 5.3. Sensitivity of numerical solution to mesh size55CHAPTER 6ROLL-GAP HEAT-TRANSFER ANALYSIS6.1 Measurement of Instantaneous Roll-Gap HTC6.1.1 Aluminum TestsThe finite-difference model developed in Chapter 5 back-calculated theinstantaneous HTC as a function of time in the roll gap from the surface temperaturemeasurements. Figures 6.1-6.5 illustrate the calculated HTCs for the tests shown inFigures 4.8-4.12.According to general rolling theories, the normal pressure acting on the strip shouldincrease from the entry point to a maximum at the neutral point, the angle at which thevelocities of the strip and roll are equal. The pressure should then decrease from theneutral point to the exit point. This phenomenon is sometimes referred to as a ‘frictionhill’ [53]. Therefore, the roll-gap HTC, which is expected to be a function of the contactpressure, should follow the same trend; that is, to increase from the initial entry point to amaximum, then to decrease until the exit point is reached. Upon examination of Figures6.1-6.5, it can be seen that the calculated instantaneous HTCs show too much variabilityto qualitatively relate the HTC behaviour to a pressure variation within the roll gap.Often the HTC has been seen to increase again just before exit from the rolls, asseen in Figures 6.2 and 6.3. This could be evidence for a ‘double pressure peak’ whichhas been seen by other researchers investigating aluminum rolling [54, 55]. On the otherhand, the HTC as calculated by the model becomes unreliable in the latter stages of theroll bite. Figure 6.1, which shows the measured surface temperatures, calculated roll-gap56HTCs, and the calculated roll surface temperature based on the output of TC Si,illustrates this point. As the strip first enters the rolls, the temperature difference betweenthe strip and roll surface is large. However, as the strip proceeds through the roll gap, thestrip cools and the roll surface heats up, causing the difference in temperatures betweenthe two surfaces to lessen. In the case of Test AL13, for example (Figure 6.1), thedifference in temperatures between the strip temperature and the roll temperature (ascalculated from the output of TC Si) becomes very small near the end of the roll bite.The HTC, h, is defined as the heat flux from the strip, q, divided by the difference in thestrip and roll surface temperatures,q (6.1)As T5- Tr becomes smaller, any error in the calculation of Tr affects the calculatedHTC to a larger and larger degree. For example, a ten degree error in the calculation ofTr when the difference in roll and strip surface temperatures is only twenty degrees wouldcause 100 pet. error in the calculated HTC. Therefore, in the latter part of the roll bite,where the strip and roll surface temperatures are relatively close together, the calculatedHTC is subject to large errors. This problem is exacerbated in the case of metals whichare rolled at lower temperatures. In the hot rolling of steel, for example, the temperaturedifference between the strip and roll surfaces remains considerable throughout the rollgap, and therefore any error in the calculation of the roll surface temperature through theuse of a heat-transfer model affects the calculated HTC to a lesser extent than in the hotrolling of aluminum.57600500__C-)400300C-)I—200 z100300250_C.)200150 .C-)I—100 z50700o TCS1TCS2Roll Temp.HTC-S1HTC - S20 0.01 0.02 0.03 0.04 0.05Time from entry into the roll bite (s)0Figure 6.1. Surface temperature and HTC for Test AL135505005’ 450400Dw 350EI—2502005505003’ 450400G) 350aE_____ci)I-_ _______250200 00 0.01 0.02 0.03 0.04 0.05Time from entry into the roll bite (s)Figure 6.2. Surface temperature and HTC for Test AL15Figure 6.2 shows the calculated HTC for Test AL15. This test illustrates the highsensitivity of the roll-gap HTC to slight fluctuations in temperature. The two350TCS1TCS2HTC - SIHTC - S258thermocouple responses during this test were nearly identical. However, atapproximately fifteen milliseconds into the roll gap, the surface temperature as recordedby TC S2 increases just slightly -- approximately ten degrees. This small increase intemperature caused the instantaneous HTC to drop from 475 kW/m2 °C to 370 kW/m2 °C.The high sensitivity of the roll-gap HTC to small variations in surface temperaturearises because of the magnitude of the HTC. A large HTC corresponds to a lowresistance to heat flux at the roll-strip interface. Therefore, even small changes in surfacetemperature would be a result of a large change in the magnitude of the HTC.Figure 6.3 shows the calculated instantaneous HTC for Test AL2 1. Again, thesensitivity of the calculated HTC to variations in measured surface temperature is seen.For this test, it is apparent that TC Si failed while in the roll bite for some reason, thenrecovered as it emerged from the roll bite. On the other hand, TC S2 continued tofunction satisfactorily throughout the test.450400C.)350o. 300E1)I—250200500400C.)C300E200I10000 0.01 0.02 0.03 0.04 0.05Time from entry into the roll bite (s)Figure 6.3. Surface temperature and HTC for Test AL210 TCS10 TCS2HTC-S1HTC-S259Figure 6.4 shows, in addition to the measured surface temperatures and theresulting HTCs, the measured centre temperature as recorded by TC Cl, and thecalculated centre temperature (C.C.T.). In this case, the model had difficulties incalculating the roll-gap HTC for either surface temperature. In both cases, the calculatedHTC quickly reaches the cut-off value of the model (arbitrarily set at 1x106 kW/m2 °C) atwhich point contact resistance is essentially zero, and remains there until the end of theroll bite. This indicates that the model was incapable of calculating a HTC that wouldaccount for the measured drop in surface temperature of the sample.Figure 6.4 also shows the measured and calculated temperature of the interior of thesample. The interior temperature of the sample at first rises due to the heat ofdeformation, then decreases as the heat is conducted to the sample surface. A comparisonof the two shows good agreement between the recorded temperature rise of the sampleC.)0a)aEa)I-40036032028024020010008006004002000C)aNE0I—zo TCS1o TCS2- TCC1• C.C.T.HTC-S1HTC- S20 0.02 0.04 0.06 0.08Time from entry into the roll bite (s)Figure 6.4. Surface temperature and HTC0.1for Test AL24interior due to the bulk heat of deformation, and the temperature rise as calculated by themodel.0 0.01 0.02 0.03 0.04 0.05Time from entry into the roll bite (s)Figure 6.5. Surface temperature and HTC of Test AL3 16000a,1..CuICEa)F-360320280240200160500400C-)300200I1000o TCS1o TCS2HTC-S1HTC - S2Figure 6.5 shows the results of Test AL3 1. In this test, even though both surfacethermocouples seemed to be responding properly, there is a large difference in theirvalues throughout the roll gap. This corresponds to a large difference in the calculatedHTC from TC Si and TC S2.Since there was no means available to measure the normal pressure variationthrough the roll bite, it was decided to characterize the average HTC as a function of themean roll pressure. The mean roll pressure r was defined asF (6.2)wJiii61where F is the total roll load, R0 is the radius of the roll, and All and Ware the differencebetween the entry and exit thickness, and the width of the sample, respectively. Thesquare root of the product of the roll radius and draft is the projected length of the rollbite. In order to make a comparison between tests meaningful, the time in the roll bitewas nondimensionalized as a fractionf of the total time in the roll bite, or f=t/tc. Theaverage HTC was calculated from the instantaneous HTC by employing a simplenumerical integration method. Figure 6.6 shows the average HTC at O.5t plotted againstthe mean roll pressure for all aluminum tests performed in this study. Ideally, one wouldlike to show the average HTC for the entire time in the roll bite. However, as statedbefore, in the latter part of the roll bite, the calculated HTC tended to increase to themodel-imposed maximum. Whenever this occurred, the calculated average value overthe entire roll bite became meaningless. Therefore, O.5t was chosen, because at thispoint in the roll bite, the calculated HTC for most tests was still relatively well-behaved.The scatter in the data is seen to be large. The data ranges from 60 to 1200kW/m20C. It is difficult, given the scatter, to establish any trend or dependence of theHTC on the mean roll pressure by considering the average HTC calculated from theresponse of the surface thermocouples in the roll bite.6214001200_-.. 10000800600(_)400 a200 a aa a a aaI0 I I I I I I8 10 12 14 16 18 20 22 24Mean Roll Pressure (kg/mm2)Figure 6.6. Average HTC calculated from surface TCs vs. mean roll pressure6.1.2 Copper TestsThe model experienced difficulties in calculating the instantaneous HTC versustime in the roll bite for all the copper tests. The calculated HTC in each case quicklyreached the cutoff value imposed by the model. These situations were similar to theproblem encountered in trying to calculate the HTC for Test AL24 (Figure 6.4). Thereason for the problems encountered with the copper rolling tests may be the high thermalconductivity of copper. The Peclet number for the copper rolling tests is on the order of60 because of the high thermal conductivity of copper. This Peclet number is lower than100, the value usually considered to be the minimum necessary for conduction along therolling direction to be considered negligible. In cases where the roll-gap HTC is high,such as in aluminum and copper hot-rolling, even a small amount of conduction along thelength direction would increase the HTC calculated by a model that assumes conductiononly in the thickness direction.636.2 HTC Calculated from Initial and Final Sample Temperatures6.2.1 Aluminum TestsIt was mentioned before (see Section 4.4.1) that within 250 milliseconds of exitingthe roll bite, the aluminum samples reattained a homogeneous temperature, which wastypically thirty to sixty degrees cooler than the initial bulk temperature. The finite-difference model was modified so that it was capable of guessing an initial roll-gap HTC,predicting a bulk temperature from the initial guess of HTC, and continually modifyingthe FITC until the predicted and measured final bulk temperatures of the sample agreed towithin one degree. This provided an alternative method of determining an average rollgap HTC. Table 6.1 shows the HTCs calculated by this method for each test.64Table 6.1. HTCs Calculated from Bulk Aluminum Sample TemperaturesTest ID Mean Roll Initial Rolling Temperature Mean HTCPressure Temperature After Rolling(kg/mm2) (°C) (°C) (kW/m2°C)AL12 10.41 513 454 264.5AL13 10.09 505 424 273.0AL15 10.35 516 457 258.6AL16 8.81 510 452 221.3ALI8 13.80 505 450 240.5AL19 12.45 497 419 276.2AL21 13.50 415 370 293.3AL22 9.49 505 447 401.0AL24 19.89 377 338 378.5AL27 21.51 321 295 278.4AL28 22.70 327 295 464.0AL3O 16.46 329 298 273.2AL31 17.27 323 289 387.5AL33 14.75 371 333 276.0AL34 16.59 321 287 424.6AL37 20.79 321 290 463.0AL39 22.57 274 267 65.1Figure 6.7 shows the results of Table 6.1, and compares the HTCs calculated bythe difference in bulk temperatures with the HTCs calculated from the surfacethermocouple responses. Upon comparing the two different methods of calculating the65roll-gap HTC with the aid of Table 6.2 and Figure 6.8, it is evident that the HTCscalculated from the difference in sample temperatures before and after rolling is thesuperior method. The data calculated from the bulk sample temperature difference showsconsiderably less scatter than the data calculated from the surface thermocouple responses(the coefficient of determination for the HTCs calculated from the bulk sampletemperatures is seven times as high as for the HTCs calculated from the surfacetemperature responses). In addition, the regressions of the HTCs calculated from the twodifferent techniques are very similar, which supports the view that both techniques arecalculating the same parameter. That is, the sample surface thermocouples are measuringthe true surface temperature of the sample, and not some temperature intermediatebetween the surface temperature of the sample and of the roll. If the surfacethermocouples had been measuring a temperature lower than the actual surfacetemperature, the mean HTC as calculated from the surface thermocouples would havebeen higher than the mean HTC calculated from the difference in sample temperaturesbefore and after rolling.661400 _J c SurfaceT-Data__________________________________Regression1200a Bulk T-Data1000 Regression8000 0600C.) 0400200 o0 j C 00 I I8 12 16 20 24Mean Roll Pressure (kg/mm2)Figure 6.7. Comparison of HTCs calculated from bulk sample temperatures and surfacetemperatures in the roll bite67Table 6.2. Statistical Comparison of HTCs Calculated from Bulk Sample Temperaturesand Surface Temperatures in the Roll BiteHTCs from BulkSample TemperatureHTCs from SurfaceTemperature ResponseAnother reason the method of calculating the HTC from the sample bulktemperature is superior to the method of calculating the HTC from the surfacetemperature response, is that the Significance F -- calculated from an analysis of variance-- is much lower than for the HTCs calculated from the surface temperature response,even though the slope of the linear regression is less. The Significance F is a statisticalmeasure of the hypothesis that the roll-gap HTC increases with pressure, versus the nullhypothesis, that the roll-gap HTC does not increase with pressure. In this case, from theHTCs calculated from bulk sample temperatures, the confidence that the HTC rises withpressure is (100 pct.)(1-0.0009)=99.91 pet., versus only 74.7 pet. from the HTCscalculated from the surface temperature response.Slope of Linear Regression 11.4 14.6Coefficient of Determination (r2) 0.3 94 0.056Significance F 0.0009 0.253681000_______________________o From surface T - DataC)800 IFromBuIkT-Data600400 020: .. : I12 °‘ 16 20 • 24-200 0 0 00 0-400Mean Roil Pressure (kg/mm2)Figure 6.8. Residual Errors from HTC RegressionFigure 6.8 shows the residual errors (defined as the regression value at a pressuresubtracted from the data point value) from the linear regressions shown in Figure 6.7. Itshows that, for the HTCs calculated from the sample surface temperature responses in theroll gap, not only are the residual errors larger in general, but are not evenly spread out --relatively few, high-magnitude positive errors cancel out the more numerous, but lower-magnitude, negative errors. The residual errors resulting from the regression of the HTCscalculated from the sample bulk temperatures, on the other hand, are smaller inmagnitude, and are distributed more evenly. This indicates that the linear regression ismore suited to the bulk sample temperature HTC data than to the sample surfacetemperature response HTC data.6.2.2 Copper TestsCopper has a higher thermal conductivity than aluminum. Therefore, the coppersamples after rolling regained a uniform, bulk temperature after rolling more quickly than69the aluminum samples. Consequently, the same technique involving the difference inbulk sample temperature before and after rolling used in calculating a mean HTC throughthe roll gap for alumimun rolling can be easily utilized in the case of copper rolling aswell. The HTCs in the roll gap calculated by this method for the copper rolling samplesare tabulated in Table 6.3.Table 6.3. HTCs Calculated from Bulk Copper Sample TemperaturesTest ID Mean Roll Initial Temperature Mean HTCPressure Rolling After(kg/mm2) Temperature Rolling (kW/m2°C)(°C) (°C)CU4 11.64 574 528 341.2CU5 11.10 578 523 427.2CU7 10.44 602 551 347.0CU8 11.20 523 481 388.2Despite the fact that the mean roll pressure was quite similar for each copperrolling test, the calculated HTC again shows substantial variation from test to test. Again,this is due to the extreme sensitivity of the HTC to small changes in temperature at highHTC values.CHAPTER 7RESULTS AND DISCUSSION7.1 Effect of Rolling Parameters on the HTCFigure 7.1 shows selected HTCs calculated from the sample bulk temperaturesversus rolling pressure for the two aluminum alloys, AA5 052 and AA5 182, examined inthis study. HTCs were chosen only from tests that were conducted at 20 pet. deformationat 68.5 rpm using Lubricant ‘A’, so that the effect of alloy type was isolated.4000300— AC.) £ A 0AE A200A A’\5052100o AA518208 10 12 14 16 18 20 22 24Mean Roll Pressure (kg/mm2)Figure 7.1. Effect of alloy type on the HTCAccording to the theory (see Section 7.3), there should be a difference in alloybehaviour because of the difference in thermal conductivities, as well as the difference in7071flow stress behaviour between the two alloys. Since AA5052 has a slightly higherthermal conductivity than AA5 182 (an average of 152 W/m °C for AA5052 versus 136W/m°C for AA5 182), one would expect to see a marginally higher HTC vs. rollingpressure for the AA5052. In order to test whether or not this effect is statisticallyobservable, the following approach was taken. Firstly, a line of regression was calculatedfrom the AA5052 HTC data. Secondly, residual errors for both AA5052 and AA5 182were calculated from the regression analysis of the AA5052 data, Figure 7.2. Finally, aconfidence interval for the difference of the means of the residual errors of the AA5 052and AA5 182 HTC data was calculated.80_______________U60 A M5052j PA51824O20A AwC 0— A A I ID12 16 20 24-20A-40! 0Mean Roll Pressure (kg/mm2)Figure 7.2. Residual errors in comparison of AA5052 vs. AA5 182The confidence interval of the difference between two means determines a rangethat the difference between the means of two population samples lies within at a certainlevel of confidence. In this case, the mean of the residuals of the AA5052 data points iszero. However, the mean of the AA5 182 data points is not zero, because these data72points were not included in the regression. If the mean of the AA5 182 residual errors issignificantly different than zero, then it is reasonable to conclude that the roll-gap HTC isaffected by alloy type. Hogg and Ledolter [58] provide an equation for the confidenceinterval for the difference of two means, used for situations where the variance isunknown and the sample sizes are small:(a ‘\ Ii 1—+— (7.1)jv’’where iii and 112 are the means of the two populations, n andn2 are the sizes of the twopopulations, t is the Student t-distribution at a confidence level a and having r degreesof freedom, and Si,, called the pooled variance, combines the variance estimates from thetwo samples in proportion to their degrees of freedom:= (7.2)° n1+n2—2where S1 and 2 are the variances of the two samples.When applied to a graph of residual errors, the confidence interval of the differenceof the two means establishes whether or not there is a statistically significant deviation ofdata values of one group from the line of regression calculated from the other data group.Table 7.1 shows the results of the confidence interval tests at the 90 pct. confidencelevel that were performed on the residual errors shown in Figure 7.2. It also shows theresults of the same tests for comparisons of rolling speed, lubrication type and the effectof re-rolling.73Table 7.1. Statistical Significance of Effect of Rolling Parameters on the HTCEffect of: Group 1 Group 2 l’1-I’2 ± 90%C.I.Alloytype AA5052 AA5182 0.9±40.0Re-rolling 20 pct. 10 pct. -69.4 ± 65.5Rolling speed 68.5 rpm 34.3 rpm 86.5 ± 79.4Lubrication type ‘A’ ‘B’ -52.8 ± 67.9The results of the confidence interval test reveal that there is no statistical basis forconcluding that the type of alloy has an effect on the roll-gap HTC behaviour. Thedifference in means between the residual errors of the HTCs of the two alloys lies wellwithin the 90 pet. confidence interval. Another way to consider the result of theconfidence interval would be to consider the set [I-1-i-2 - 90 pet. C.I., !‘1-l’2 + 90 pet.C.I.], and if zero is contained in the set, there is no evidence at the 90 pet. confidencelevel that the means of the residual errors of the two alloy types are different.In a similar fashion, the effect of re-rolling, roll speed and lubrication type wasdetermined by the use of a residual error plot and the confidence interval test, and theresults are shown in Table 7.1. In each case, the tests chosen for comparison wereselected so that no parameter was varied except for the one being tested.A small enhancement of the HTC behaviour of the aluminum samples rolled to 10pet. might be expected as compared to the aluminum samples rolled to 20 pet. if surfaceroughness has an effect on the HTC, since the samples rolled to 10 pet. deformation hadall been previously rolled. This could have had the effect of smoothening the samplesurface, so that in the second (10 pet.) deformation, a greater metal-metal contact would74exist between the roll and the sample, and therefore the HTC would be increased. Theconfidence interval test performed on two data sets isolating the effect of re-rolling thesamples on the HTC show that the mean of the residuals for the 10 pct. tests is negative(which implies that the HTCs of the 10 pct. reduction tests are higher than the 20 pct.HTCs) and lies slightly outside the confidence interval. Thus, there is some evidence tosupport the conclusion that the initial surface roughness has a small effect on the roll-gapHTC. However, the surface roughness of the samples or roll was not characterized in thisstudy, so the effect can not be quantified.The confidence interval test performed on the data comparing the tests conducted at34.3 rpm and the tests conducted at 68.5 rpm shows that the HTCs for the 34.3 rpm testsare slightly lower than the HTCs of the 68.5 rpm tests. This effect was unexpected and isdifficult to explain. It may be that, since at lower rolling speeds the Peclet number islower, some conduction may be taking place in the rolling direction. If this were thecase, the sample surface would be slightly warmed by heat transferring from the entrydirection of the sample to the exit sample, thus decreasing the apparent roll-gap HTC. Inthis case, however, the effect is only a minor one, and the one-dimensional approach isstill adequate even for the 38.3 rpm tests.The confidence interval tests performed for the comparison of the two lubricantsshow that the difference of the means of the HTCs obtained using Lubricant ‘A’ andLubricant ‘B’ lies within the 90 pct. confidence interval. Thus, there is no evidence toindicate that the use of Lubricant ‘B’, the less-viscous lubricant of the two, enhances theroll-gap HTC. This corroborates the findings of at least two other studies. Chen et al. [3]concluded that, for the hot rolling of aluminum, the roll-gap HTC was independent of thepresence or absence of lubrication. Furthermore, Williamson and Hunt [57] presented75evidence that the trapping of lubrication does not affect asperity behaviour, which in turnimplies that the presence of a lubricant should have no effect on the contact HTC.7.2 Friction in the Roll BiteEven though the validity of the use of a constant coefficient of friction throughoutthe roll bite has been questioned (see Section 2.1.1), it is a simple way to include theeffect of friction in the roll bite and easy to incorporate into the heat-transfer model. Thecoefficient of friction can be calculated by a variety of methods, but for the present studythe following equation (from Schey [59]) is useful:(7.3)where ji is the coefficient of friction, All is the draft and R0 is the roll radius. From thefact that 20 pct. reductions were achieved in this study with 8.7 mm thick specimens, itfollows that the coefficient of friction was at least 0.19. A value somewhat higher thanthis, 0.3, was assigned to the model for all tests. This value is also the same that was usedby Chen et a!. [42].In order to evaluate the effect this assumption had on the calculated HTC, thecoefficient of friction was varied for two aluminum tests, AL15 and AL37. Table 7.1shows the sensitivity of the calculated HTC on the coefficient of friction.76Table 7.2. Effect of Friction on the HTCFriction HTC (kW/m2°C)Coefficient Test AL 15 Test AL370.1 230 4290.3 231 4290.5 231 429It is evident, upon examination of Table 7.2, that the choice of the coefficient offriction has an insignificant effect on the calculated HTC. It is possible that, wheneverHTCs are very high, such as in aluminum and copper rolling, the quench of the surface ofthe material being rolled is so severe that the heat flux at the interface generated due tofriction is insignificant as compared to the heat flux at the interface that is due to the highHTC.7.3 Comparison of the Roll-Gap HTC with Earlier Values for Aluminum RollingPrevious estimates of the roll-gap HTC for aluminum rolling are an order ofmagnitude less than the values determined in this study. One of the deficiencies of theprevious works has been the lack of consideration paid to the roll-gap I-{TC as a functionof the rolling pressure. This may help explain why figures reported for laboratory rollingexperiments have usually been an order of magnitude less than those reported for actualmill conditions. For example, Lenard and Pietzryck [6] reported values of 4.8 W1m2 °Cfor the hot rolling of steel on a laboratory scale rolling mill at a rolling speed of only 4.0rpm. These workers went on to point out that successful modelling of temperatureprofiles of steel strip rolled on production mills requires the use of a HTC in the range of23.3 to 81.0 kW/m2 °C. These researchers did not attempt to explain the discrepancy77between the laboratory- and production-scale HTCs. However, it is probable that thepressures experienced by the steel strip under production conditions are much greaterthan under laboratory scale predictions. This would result in a larger true area of contactbetween the roll and steel strip under production conditions, and hence a higher HTC.Another defiency of previous work has been the lack of consideration paid to theexperimental setup. Chen et al. [3] reported roll-gap HTCs of between 10 and 60 kW/m2°C. These workers observed the HTC to continually increase almost linearly in the rollgap to a maximum at the exit point. A comparison of three different reductions revealedno change in the magnitude of the HTC. It is believed that the results by these workersare consistent with temperatures measured by thermocouples with response times that areslow compared to the time the samples spent in the roll gap. These workers reportedusing 1.0 mm diameter intrinsic surface thermocouples in their study; it is believed thatthis type of thermocouple would not have a sufficient response time. If this were thecase, then as long as the actual decrease in temperature at the sample surface as it enteredthe roll bite was quicker than the response time of the thermocouples, the thermocoupleswould continually cool throughout the roll bite in an attempt to ‘catch up’ with the actualsample surface temperature. Therefore, no effect of pressure on the HTC would beobserved, because the thermocouples would already be cooling at their maximum rates.In the Appendix an equation (Equation A.9) has been developed that, given themeasured bulk temperature of an aluminum sample before and after rolling, calculates a‘minimum value’ HTC. By assuming that the roll and sample surface temperaturesremain fixed while in the roll gap, Equation (A.9) calculates the lowest theoreticallypossible roll-gap HTC that can account for the temperature drop of the sample due to heatloss to the rolls. When Equation (A.9) is applied to data from this study, it calculatesminimum value HTCs that are on the order of one-fifth to one-eighth the HTCs calculated78by the finite-difference model (see Appendix for a sample calculation). In other words,Equation (A.9) has been found to calculate roll-gap HTCs that are five to eight timeslower than the true HTCs measured in this study. The reason for the discrepancy betweenthe values calculated by the finite-difference model and Equation (A.9) is that thedifference in the roll and sample temperatures in the roll gap does not remain at its initialvalue, but decreases continuously as the roll surface heats up and the sample surfacecools. Thus, given a specific heat flux, the calculated HTC increases in order tocompensate for the lower driving force for the flow of heat.In a rolling test of AA5083 Timothy eta!. [5] reported a HTC of 15 kW/m2 °C. Incomparison, the minimum HTC as calculated by Equation (A.9) using the data providedby these workers is 14.2 kW/m °C. The similarity of the estimate provided by Timothyet aL [5] and the lowest possible HTC suggests that the estimate by these workers ismuch too low.Pietrzyk and Lenard [6] reported HTCs between 18.5 and 21.5 kW/m2 °C in warmrolling of commercial pure aluminum. The HTC as calculated by Equation (A.9), on theother hand, is 18.4 kW/m2 °C (see Appendix for sample calculation). Again, the fact thatthe roll-gap HTC reported by these workers is similar to the minimum HTC suggests thatthe value reported is too low. Since the calculation by Pietrzyk and Lenard of the roll-gap HTC was performed using Equation (2.1), which is an equation similar in form toEquation (A.9), and the HTC calculated by Pietrzyk and Lenard [6] and by Equation(A.9) are very similar, the possibility arises that these workers, due to the slow responsetime of their surface thermocouple [36], overestimated the value for LTrs, the averagedifference in temperature between the roll and sample surface in the roll bite. If this werethe case, then their value for ATr.s would be similar to the term T5 - Tr used in Equation79(A.9), and they would in fact have calculated a value close to the minimum-valuesolution.7.4 Generalized Correlation for the HTCThe theoretical treatments of the dependence of the HTC on contact pressurediscussed in Chapter 2 have two major flaws that have to be overcome before anysuccessful application of the theoretical equations to rolling. First of all, the equationshave generally been validated for low apparent pressure-to-microhardness(1a’1) ratios(0.01 to 0.1) -- their applicability to high (a/) ratios that occur in bulk formingprocesses such as hot rolling (0.2- 0.3 and perhaps higher) has not been proven.Secondly, the fact that a material’s surface microhardness is temperature-dependent, andtherefore is dependent on the HTC, has not been taken into account. Thus, an explicitformulation of the HTC is not possible.7.4.1 Underlying Assumptions7.4.1.1 Dependence of the HTC on Pressure and Surface HardnessIn developing an equation for the prediction of the HTC in rolling, the followingline of reasoning was taken. Firstly, both surfaces, that of the roll and of the workpiece,have an initial surface roughness profile. For any commercial rolling operation, theroughness of the workpiece can be assumed to be greater than the roughness of the roll;therefore, the workpiece surface becomes smoother as a result of the rolling operation.Some evidence of this phenomenon has been presented by Chen [211. It is a difficultmatter, then, to incorporate the roughness of the workpiece as part of a HTC-predictionequation because the workpiece profile changes significantly throughout the roll bite.80Therefore, it was chosen to include surface roughness parameters of the workpiece intothe HTC-prediction equation as a general constant.Secondly, when the hot workpiece first comes into contact with the roll, as metal-metal contact is established at the asperity tips, there is an immediate quench of theworkpiece surface. The greater the extent of metal-metal contact, the higher the resultingHTC and the quicker the quench. However, the lowering of the workpiece surfacetemperature leads to hardening of the workpiece asperities. The plastic deformation ofasperities effectively ceases when the material surface becomes hard enough and metal-metal contact extensive enough that the true pressure acting on the surface of theworkpiece -- the roll force divided by the true area of metal-metal contact-- becomesequal to the surface flow stress of the workpiece material. Therefore, the HTC isdependent not only on the pressure being applied to the two surfaces in contact, but alsoon the surface hardness of the material.7.4.1.2 The Dependence of the HTC on the Conductivity of the Workpiece and ToolConductivity must also be a factor in determining the behaviour of the roll-gapHTC. The thermal conductivity affects the HTC in two ways. Firstly, it plays a role indetermining the temperature-dependent yield stress of the asperities. At an appliedapparent pressure, the higher the conductivity of the sample, the quicker the heatextracted from the sample surface is replaced from the sample interior. Therefore, thesurface temperature of the sample quenches slower than if the sample had a lowerconductivity. The HTC thus tends to increase, since because the surface temperature ofthe sample remains higher, the surface of the sample stays softer. Thus, the surfaceasperities would deform more at an applied pressure, and therefore, the HTC would beexpected to be enhanced with increased sample conductivities.81Secondly, at the points of direct metal-metal contact between two materials, theconductivity of each material determines the relative flow of heat from one material to theother. Therefore, the HTC is expected to be proportional to an effective conductivity,which is a function of the conductivities of the two materials, at the points of contactbetween the surfaces of the two materials. Superficially, it might seem that the effectiveconductivity, keff, might be expressed as the sum of the conductivities of the twocontacting materials, as shown in Equation (7.4a). However, a more appropriateparameter is the ‘harmonictconductivity of the two materials, which is the inverse of thesum of the inverses of the conductivities of both materials, as shown in Equation (7.4b).keff = k1 + k2 (7.4a)(7.4b)A full proof of the applicability of Equation (7.4b) in an HTC-prediction equation isprovided by Cooper et al. [13]; however, the reason why the harmonic thermalconductivity and not the average thermal conductivity should be included in the HTCprediction equation can be demonstrated with the aid of Figure 7.3.Figure 7.3 shows the temperature profile between two materials of differing thermalconductivities, k1 > k2, in direct contact with each other at the meeting point of twoopposing asperites. Since at the contact plane of the two asperities there is no interfacialgap between Material 1 and Material 2, the temperatures of the two materials at thecontact plane are the same, T0. The term ATe is the macroscopic difference intemperature between the surface temperatures of Material 1 and Material 2, and AT1 and82AT2 are the components of ATc due to the conductivity of Material 1 and Material 2,respectively. Because Material 1 has a higher conductivity than Material 2, AT1 is lessthan AT2.Now, two limiting cases can be examined; firstly, the case where k1 approachesinfinity, and secondly, the case where k2 approaches zero. In the first case, as k1approaches infinity, AT1 approaches zero. The macroscopic-temperature gap ATE, and byinference, the HTC, is then entirely a function of the conductivity of Material 2, k2. Theeffective thermal conductivity, keff, is then equal to k2, as calculated correctly byEquation (7.4b); whereas in Equation (7.4a), keff is calculated to be infinity. In thesecond case, as k2 approaches zero, the plane of contact becomes an insulated boundary,and the temperature gradient becomes zero at the plane of contact. Since there is notemperature gradient, there is no flow of heat across the plane of contact and the effectiveconductivity becomes zero. Again, Equation (7.4b) calculates this result correctly,whereas Equation (7.4) calculates keff to be equal to k1.83AT2Figure 7.3. Temperature gradients between two asperities in contact7.4.2 Quantification of the Dependence of the HTC on Pressure and Conductivity7.4.2.1 Formulation of the General EquationReflecting this approach, an equation of the general form, patterned after Cooper eta!. [13], is formulated as:h=CkI 1 (75)l-A )where C is a general constant that replaces the surface roughness terms that are found inthe equation of Cooper eta!. [13]. The term k is defined askktWpk +AT1A TC0Material 1 C-) Material 2(7.6)84which is similar in form to Equation (2.5) and is equivalent to Equation (7.4b). Finally,nthe term—A— characterizes a family of equations that approaches 0 as A (thereal contact area Ar/Aa) approaches 0, and approaches infinity as Ac approaches 1.Employing the relation of contact area to normal load proposed by Pullen andWilliamson [56] (shown in a slightly different form than in Equation (2.2)),a (77)H+Fwhere-1’a is the apparent pressure (the force acting over the total area), and substitutingthis relation into Equation (7.5) yields/ n \lih=CkI-I (7.8)HJwhere C is a general constant with units m1. The roll-gap HTC is now characterized as afunction of the applied pressure and the surface hardness of the material being deformed.Equation (7.8) is, in effect, a generalized version of the equation proposed by Cooper etaL [13]. In local indentation tests, it has been found that full plastic deformation at thesurface occurs when the applied pressure is approximately three times the yield stress [17,57]. Therefore, the surface hardness H is calculated as:(7.9)85where Y0 is the bulk yield stress of the material being deformed. By incorporating thetemperature-strain rate dependent nature of the yield stress at the sample surface,7uriace,E) (7.10)and substituting Equation (7.9) and (7.10) into Equation (7.8), the following equation isobtained:11(7.11)Equation (7.11) is inherently an implicit equation, because the roll-gap HTC ischaracterized as a function of the surface temperature of the workpiece, which in turn isdependent on the HTC.7.4.2.2 Modification of the Equation for Rolling ConditionsEquation (7.11) describes the relationship between the HTC and apparent pressurebetween two surfaces in metal-metal contact. However, this surface hardness-yield stressrelation has not been verified for the case of rolling. In addition, Equation (7.8) containsthe general constant, C, which replaces any surface roughness parameters and can alsotake into account any constant multiple of the yield stress; in this case the samplehardness is simply taken to be the flow stress at the sample surface:86H=a(1uacee) (7.12)where cr is the flow stress in free tension, Tsurface is the surface temperature of thesample, and C is the strain rate at the sample surface.In the case of hot rolling, because of the large temperature gradient that existsbetween the surface of the sample and the sample interior while the sample is in the rollbite, the strain rate at the sample surface is not accurately known. Therefore, a meanstrain rate, C, is used:• E (7.13)Cwhere t is the contact time and E, the mean strain of the sample caused by the rollingoperation, is defined as:2(H’ (7.14)VHf)where H and Hf are the entry and exit thicknesses of the sample, respectively.By substituting Equation (7.12) into Equation (7.8), and Equation (7.14) and (7.13)into (7.12), Equation (7.11) is thus modified for the rolling case:87(7.15)where r is defined in Equation (6.2).7.4.3 Numerical Solving TechniqueThe program that was used to calculate the HTC in the roll gap based on the samplesurface temperature was modified to predict the roll-gap HTC, employing an iterativetechnique, as shown in Figure 7.4.88Figure 7.4. Flowchart of algorithm used to predict the roll-gap HTCIn order to determine values for the equation parameters C and n, the equation wasfit to the experimental data obtained by Chen [21] for stainless steel rolling tests. Therewere two reasons that the stainless steel data, rather than the data for aluminum alloyobtained in the present study, were used. Firstly, the data from Chen [21] exhibits lessscatter than the data obtained in this study. Therefore, the coefficients for the equationcould be determined from the stainless steel data with more precision. Secondly, thepressure range was greater in the study by Chen [21] than in the present study. Thus,fitting the equation parameters to the stainless steel data enables the equation to beapplied to the aluminum and copper rolling experiments in this study without need forextrapolation.897.5 Prediction of Roll-Gap HTCs Using the Developed EquationIt was found that for A 0.015 seconds, C 15 800 m1 and n = 1.4, Equation(7.15) provided predictions of the HTC that fitted the line of regression through theobserved steel rolling data. Using the same parameters, the equation was then applied toselected aluminum rolling tests and to all the copper rolling tests. Figure 7.5 shows thatthe roll-gap HTCs predicted by Equation (7.15) for aluminum hot rolling lie in themidrange of the experimental data, while the predicted HTCs for copper rolling lie at theupper range of the experimental data.500.________________________0 •AA-Data0 0n AA.- Prediction300E 0 • Cu-DataA AAA Cu - Prediction200A A Steel-DataZ A10:0 10 20A30 50Steel - PredictionMean Rolling Pressure (kg/mm2)Figure 7.5. Comparison of experimental and predicted HTCs7.5.1 Suitability of Conductivity and n as Equation ParametersEven though all the HTC-prediction equations in the literature include thermalconductivity as a parameter [13, 15, 16, 17, 201, one might expect the thermal diffusivity90to be a more appropriate parameter to include because of the transient nature of theproblem. However, the flow of heat is only transient at the macroscopic level.Upon examining only two asperities in contact with each other (referring onceagain to Figure 7.3 where Material 1 is the sample material and Material 2 is the rollmaterial), there is no resistance to heat flow from one asperity to the other at the plane ofcontact between the two asperities. When the two asperities first contact each other,immediately the temperature at the plane of contact becomes T0, which is determined by(from [60])7’ p (p p ‘ Ps ps10— ‘S mit— Yr mit ‘s,init)p5C + pC13where Tr,jit and Ts,init are the initial temperatures of the roll and sample, respectively, pr and Ps are the densities of the roll and sample, respectively, and Cpr and are thespecific heats of the roll and sample, respectively.Treating each asperity as a control volume, and assuming that for the short contacttime T0 remains constant and the interior temperatures of the roll and sample remainlargely unaffected, any inflow of heat into an asperity through the contact plane must bebalanced by a corresponding oufflow (and vice-versa). Therefore, considering eachasperity contact plane individually, the heat flow from a sample asperity to a roll asperityis a steady-state and not a transient event. The mathematical problem is then describedby the general equationV2T=O (7.17)91or, considering heat flow to be one-dimensional only, and defining the z-direction asperpendicular to the contact plane, with z=O at the contact plane,_Ikfl=pC =O (7.18)ãzk ôzJ ãtIt can be seen that, since the right-hand side of Equation (7.18) is equal to zero, thethermal conductivity, and not the thermal diffusivity, becomes the relevantthermophysical material property.In order to verify experimentally the validity of the preceeding argument, thepredictive ability of Equation (7.15) was compared with that of two modified equations,h = Caeff(-) (7.19a)and/ n \flh=CI-I (7.19b)HJwhere His defined in Equation (7.12), and eff is defined as——=-i--+-i- (7.20)eff r as92to determine whether the thermal conductivities or the thermal diffusivities of the roll andsample were more appropriate equation parameters, or if a parameter involving materialthermal properties was required at all.Each equation had the parameter C adjusted to predict the HTC values for stainlesssteel rolling tests, as determined by Chen [211. Table 7.3 shows the resulting value of Cfor each equation.Table 7.3. Value of C for Equation 7.15, 7.19a and 7.19bEquation Value of Ch = Ck(P1.1H)”4 15 800 (m1)h= CcLeff(Pr/II)1.4 9.7x1 010 (JIm4 °C)h= C(PrIH)1.4 2.54x10 (W1m2 °C)The equations were then applied to predict the HTC for aluminum and copperrolling tests. Figure 7.6 shows that the equation using ‘eff as the relevant materialthermal property (Equation 7.1 9a), overpredicts the HTC for aluminum rolling, while theequation that doesn’t incorporate any thermal property parameter (Equation (7.1 9b)),underpredicts the HTC for aluminum rolling. This provides experimental confirmationthat the harmonic conductivity is a more appropriate parameter to include in Equation(7.8) than the thermal diffusivity.93550450350250C-)_____________I150150 250 350 450 550HTC (kW/m2 °C) - RegressionFigure 7.6. Comparison of predictive capabilities of Eq. 7.15, 7.19a and 7.19bFigure 7.7 presents the logarithm of the HTC-harmonic conductivity ratio plottedagainst the logarithm of the rolling pressure-surface flow stress ratio. The dotted linerepresents the relationship between the HTC-harmonic conductivity and pressure-surfacehardness ratios as characterized in Equation (7.15), using n1.4 and C15 800 m4. Thesolid line represents the line of best fit through the data, and the shaded lines represent the+1- 95 pct. confidence intervals for the line of best fit. The slope of the line of best fitthrough the data is 0.92, which is close to the values for n proposed by other investigators[13, 20, 21]. However, due to the scatter of the thta, at the 95 pct. confidence level thetrue slope of the regressed line (and therefore the value of n) lies somewhere between0.44 to 1.4. The value used for n in this study, 1.4, thus lies on the extreme edge of theconfidence interval of the data.Least SquaresRegressiona Experimentn h=Ck(P/H)A1.4h=Ca(P/H)A1.40 h=C(P/Hyl.49432.50I—0.5-1.2 -1 -0.8 -0.6 -0.4 -0.2 0In (Pr/I-I). AA. CopperA S.S.Prediction-n=1.4Best fit-—— -95 pct C.I.+95 pct. C.I.Figure 7.7. Check of applicability of Equation 7.157.5.2 Effect of A on Equation ParametersA difficulty in using Equation (7.15) involves the choice of A, the length of time inthe roll bite, to use as the point at which the workpiece surface flow stress is calculatedfrom the surface temperature. Obviously, as A is changed, the surface temperature of theworkpiece changes as well. This leads to the conclusion that the parameters C and n ofEquation (7.15) are not unique, but rather are dependent on the choice of A.To determine whether the choice of A affects the prediction capabilities of Equation(7.15), three different times, A=O.005, 0.010, and 0.015 seconds were chosen forcomparison. Table 7.4 shows the values of the parameters C and n for the differentvalues of A. It was found that the choice of A changes the values of C and n necessary tocalibrate the equation to predict the HTC values for the stainless steel data.A•• •• I•95Table 7.4. Effect of A on Parameters of Equation (7.15)A(s) C(m1) n0.005 24 500 2.20.010 19600 1.70.015 15800 1.4Even though n appears to decrease as A increases, its value at A=0.015 s is stillsignificantly higher than the values proposed by Cooper et a!. [13] (0.985), Song andYovanovich [20] (0.97) or Chen [21] (1.0). Equations (7.15) was then applied topredicting the HTC for the aluminum rolling tests using the three different values of A, Cand n shown in Table 7.4; the results are presented in Figure 7.8. In addition, thecomparison of the mean of the residual errors of the predicted HTCs versus the regressionthrough the experimental data (the same technique as developed in Section 7.1) ispresented and compared to the 95 pet. confidence interval of the experimental data inTable 7.5. It can be seen that only the roll-gap HTCs calculated using the equationparameters A=0.015 s, n=1.4 and C=15 800 m1 adequately predict the experimentalHTCs of the aluminum rolling tests.9650000 0 004000D. Experiment0C=19600, n=t7, t0.OlOs0.o 0 0 C=24500, n=2.2, t=0.005sI— 300C15800,n=1.4t=0.015s. U_________________.U200 I I8 13 18 23Mean Rolling Pressure (kg/mm2)Figure 7.8. Effect of / on predictive capability of Equation 7.15Table 7.5. Statistical Effect of z on Predictive Capability of Equation 7.15Equation Parameters Itpredict.ilexp. ± 95%C.I.A(s) n C(m1)0.05 2.2 24 500 85.0 ± 35.60.10 1.7 19600 63.5±35.60.15 1.4 15800 27.9±35.67.5.3 Significance of the General Constant C in the HTC-Prediction EquationEquation (7.8) may also be expressed in a dimensionless form:(7.21)k iH}97where C’ is equal to C1, and therefore has units of length (m). The left-hand side ofEquation (7.21) has the same form as both the Nusselt and Biot numbers. However, theNusselt number is usually defined as the ratio of convection heat transfer to fluidconvection heat transfer; C’ refers to a fluid layer width and k is the conductivity of thefluid [52]. This definition of the left-hand side of Equation (7.21) is clearlyinappropriate. Instead, considering it as a form of Biot number may be more suitable.The term k would then be the conductivity of the solid, and C’ would refer to acharacteristic length, in this case a roughness parameter.Therefore, by utilizing Equation (7.15) to predict roll-gap HTCs for aluminum andcopper rolling tests, employing the value of C developed from steel rolling data, theimplicit assumption involved is that the initial roughness parameters of the samples of thedifferent rolling materials are the same, or unimportant. However, it is been establishedthat there is statistical evidence that re-rolling enhances the roll-gap HTC (see Table 7.1)in the hot rolling of the aluminum samples. Figure 7.9 shows the logarithm of the HTCharmonic conductivity ratio plotted against the logarithm of the rolling pressure-surfacehardness ratio for the aluminum tests, comparing the roll-gap HTCs measured from thefirst and second passes. By once again employing the statistical techmique developed inSection 7.1, and taking a regression line through the first pass (20 pct. reduction) data andcomparing the means of the residual errors of the first pass and second pass data, it wasfound that the 95 pct. confidence interval of the difference of the means of the residualerrors of the first and second pass data is 0.166, whereas the mean of the residuals errorsof the second pass is 0.239. Therefore, the difference of the means of the residual errorswas within the interval 0.239 +1- 0.166 at the 95 pct. confidence level. Since zero is notpart of the interval, this suggests that because of the first rolling pass, the surfaces of the98aluminum samples were smoothened, causing the roll-gap HTCs to be enhanced duringthe second pass. This HTC-enhancement can be characterized in Equation (7.8) and(7.15) by employing a larger value of C, or in Equation (7.21) by employing a lowervalue of C’, which implies a connection between the general parameter C or C’ and theinitial surface roughness of the sample being rolled.3a 1st PassA A A 2ndPass2.5- A___• • A()F— 2.25= aI aA A— 2 I• I •A A1.751.5 I-0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0In (Pr/H)Figure 7.9. Effect of re-rolling on roll-gap HTCs7.6 Maximum Theoretical HTCWhen deforming an unsupported material, it is obvious that the applied pressurecannot exceed the yield stress of the material being deformed. Therefore, the pressure-surface flow stress ratio cannot exceed unity, and therefore a theoretical maximum roll-gap HTC is obtained:hCk (7.21a)99or, expressed in a dimensionless form,k (7.21b)where k is calculated from Equation (7.6).Table 7.6 shows the calculated maximum theoretical roll-gap HTC for each of thefour materials studied, using the value of C (15 800 m) developed from the data fromChen [21].Table 7.6. Theoretical Maximum Roll-Gap HTCsMaterial Ic (W/m °C) Max. HTC (kW/m2°C)AA5052 32.3 510AA5182 31.5 498Copper 37.0 585Stainless steel 15.5 245From Figure 7.5, it is seen that none of the measured copper and aluminum roll-gapHTCs exceed the theoretical maximum HTCs shown in Table 7.6. One data point fromthe steel series of tests exceeds 245 kW/m2 °C, but not by an amount that exceeds theerror limit at that level of HTC (see next section).1007.7 Error in HTC Measurement due to Temperature Measurement ErrorThe scatter of data for the hot rolling of aluminum has been seen to be considerable.However, this is expected due to the magnitude of the HTCs that were measured in thisstudy. As explained earlier (see Section 6.1), at large values of the HTC, the HTCmeasured from the surface thermocouple responses is highly sensitive to smallfluctuations in surface temperature. Similarly, the HTC measured from the bulk sampletemperatures before and after roilling also becomes increasingly sensitive to small errorsin temperature measurement. As an example of the increase in uncertainty in roll-gapHTC measurement at larger HTCs, consider Figure 7.10. Using Test AL24 as a base,HTCs were calculated based on hypothetical roll-gap exit temperatures. The roll-gapentry temperature for Test AL24 was 376.7 °C, and the true roll-gap exit temperature was338 °C. The finite-difference model calculated roll-gap HTCs based on hypothetical roll-gap exit temperatures in the range 334 °C - 370 °C.900 —800700C) 60000IE 500400Ci_ 300200.1000 I I I I I330 335 340 345 350 355 360 365 370Sample Temperature After Rolling (°C)Figure 7.10. Sensitivity of the HTC to the roll-gap exit temperature101Figure 7.10 shows that the roll-gap HTC exponentially approaches infinity as thehypothetical roll-gap exit temperature decreases. From this graph, error estimates ofcalculated roll-gap HTC values can be established. Figure 7.11 shows the level of errorexpected at different levels of the calculated roll-gap HTC, assuming that the post-rollingbulk sample temperatures measured by the thermocouples were accurate to within ± 1 or2 °C of the true sample temperatures.Figure 7.11 shows that the order of the error increases very rapidly as the HTCapproaches higher values. In the present study, HTCs in the range of 200 - 450 kW/m2were measured. Assuming errors of temperature measurement in the range of 1 to 2°C, pet. errors in HTC measurement range from 10 - 20 pet. at the lower HTC level to upto 15- 40 pet. at the upper HTC level. Due to the fact that the HTCs measured by ChenC1w1008060402000 100 200 300 400 500 600Calculated HTC (kW/m2°C)Figure 7.11. Error estimates of the HTC700102[21] ranged from 10 - 250 kW/m2 °C, the errors associated with the measured HTCs weremuch less, and therefore less scatter was exhibited.However, keeping in mind that the ultimate aim of measuring the roll-gap HTC isto predict the strip temperature, the large scatter in the measured HTC is not verysignificant. At such large HTCs as were measured in this study, the calculated striptemperature is not very sensitive to large variations in the HTC. At larger HTCs, theresistance to heat flow due to the contact resistance at the strip-roll interface becomesonly a small proportion of the total resistance to heat flow. For example, at a roll-gapHTC of 200 kW/m2 °C, the resistance to heat flow due to conduction through thealuminum sample and through the roll layer, 6, is five and thirty times greater,respectively, than that due to contact resistance at the roll-sample interface. In fact,another interpretation of Figure 7.11 is considering it a graph of acceptable error in theassumed roll-gap HTC if a 1°C or 2°C error in strip temperature prediction is acceptable.Then it is easily seen that as the roll-gap HTC increases, a larger percentage error in thereported roll-gap HTC becomes acceptable.7.8 HTC Measurement Error due to Roll Conductivity ErrorBecause the chief resistance to heat flow is the roll, the calculated HTC is sensitiveto the choice of roll conductivity. Figure 7.12 shows the effect of changing the value forroll conductivity, as assumed by the model on the average HTC calculated from thesurface thermocouple responses. By using a value for roll conductivity that is consideredhigh by 50 pct. (62 W/m°C), the calculated HTC has dropped by a factor ofapproximately two, but their is no appreciable effect on the sensitivity of the HTC to rollpressure.103800700600_____________________kroll=41 W/m°Cg4 500Regression400• kroll=62 W/m°C0 300I— a Regression200a ___aa a a100 a aI0 I I8 12 16 20 24Mean Roll Pressure (kg/mm2)Figure 7.12. Effect of changing the model roll conductivity assumption on the calculatedHTCThe reason for the decrease in calculated HTC is that, at higher roll conductivities,heat that transfers from the sample surface to the roll surface is extracted away from theroll surface into the roll interior quicker than at lower roll conductivities. Therefore, theroll surface temperature, as calculated by the model, remains cooler. Thus, the averagedifference in roll and sample surface temperatures throughout the roll gap remainsgreater, which has the effect of reducing the calculated roll-gap HTC. It is felt that, basedon the conductivities for various steels with similar compositions to the roll alloy used inthis study (see Section 5.4), that the roll thermal conductivity value (41.0 W/m °C) isaccurate to within 2 W/m °C. Based on Figure 7.12, the corresponding effect oncalculated HTC values is about 7.5 pct. Thus, the uncertainty introduced into the HTCmeasurement due to uncertainty in the value of roll conductivity is relatively small ascompared to the error introduced into the HTC due to uncertainties in temperaturemeasurement.1047.9 The Effect of the Heat-Transfer Coefficient on the Sample Temperature Profilein the Roll GapFigures 7.13 and 7.14 show the effect the HTC has on the temperature profile of theworkpiece. Figure 7.13 shows the calculated temperature profile for a sample, initially ata temperature of 3 77°C, rolled to 20 pct. deformation with an HTC of 378 kW/m2 °C. Atthis HTC, the sample surface cools from the initial temperature to less than 254°C withinthe first 10 pct. of the roll bite length and then remains at almost a constant temperaturethrough the remainder of the roll bite. In the interior of the sample, the temperature hasincreased due to the heat of deformation.Figure 7.14 shows the calculated temperature profile for a sample with an assumedHTC of 37.8 kW/m2 °C, which is ten times less than the HTC assumed in Figure 7.13.As compared to Figure 7.13, the surface temperature in Figure 7.14 cools much moregradually through the length of the roll bite. Also, the interior zone of the sample, whichwas heated to above the initial rolling temperature due to the heat of deformation, isenlarged because of the reduced temperature gradient..0.05100E>-0.000105Figure 7.13. Sample tejuperat1 profile for an HTC of 378.5 kWIifl2°CFigure 7.14. Sample mperaW profile for anIITC of 37.80.000.0.0500284324•354-364-o053.0.008 -0.006 .0.004X - dir (metres).0.002-0.0500.0.0530.0.008 -0.006X - dir (metres)kWIm2°CCHAPTER 8SUMMARY AND CONCLUSION8.1 Summary of ResultsThe roll-gap HTC has been found to be a function of the harmonic conductivity ofthe roll and the material being rolled, the ratio of the mean rolling pressure to the surfaceflow stress being rolled, and the surface roughness. This equation is expressed in theform(8.1)where C is a parameter probably related to initial surface roughness, k is the harmonicconductivity of the roll and sample, r is the applied roll pressure acting on a sample anda[Tsurface EJ is the flow stress of a sample at the roll-sample interface. The physicalmechanism relating the dependence of the HTC to rolling pressure is the deformation ofasperities at the workpiece surface. As pressure increases, the asperities of the workpiecedeform, thus increasing direct metal-metal contact at the interface of the roll andworkpiece. Since the main flow of heat from the workpiece to the roll is through the106107points of metal-metal contact, as the area of metal-metal contact increases, contactresistance to heat flow decreases and the HTC increases.The roll-gap HTC of samples which had been previously rolled has been shown tobe higher than samples which had not been previously rolled. This is possibly due to thesurface of the samples being smoothened by the first pass through the rolling mill, thusincreasing metal-metal contact between the sample and roll during the second pass.Furthermore, it was found that the choice of lubricant did not have a statisticallyobservable effect on the behaviour of the roll-gap HTC. In addition, the HTCs calculatedfrom the bulk temperature of the samples before and after rolling proved superior to thosecalculated from the surface thermocouple responses, because the scatter of the data wasreduced.The HTCs obtained in this study are about an order of magnitude greater than thosereported in previous studies of aluminum rolling. The usefulness of the earlier studieshas been limited because of inadequate consideration of measurement techniques, faultymethods of analysis, and difficulties in translating laboratory test results to an industrialscale due to a lack of understanding of the importance of rolling pressure incharacterizing the HTC.8.2 Recommendations for Further StudyThe attempt to relate the roll-gap HTC to rolling and material parameters has beenhampered by the large amount of scatter in the measured data. It has been shown that thescatter in the measured HTCs is due, to a large extent, to the magnitude of the HTC.Therefore, a further series of rolling experiments involving lower HTCs would lessen theerrors involved in the measurement of the HTC, thus allowing for a more precisecomparison of experimental and theoretical values.108In addition, it has been suggested in this study that the initial surface roughness ofthe samples has an effect on the roll-gap HTC. It was not possible to quantify this effect,however, because the surface profiles of the samples were not recorded. A series of testsinvolving samples of differing roughnesses would help clarify the relationship betweenthe value C in Equation (8.1) and the roughness parameters.8.3 Concluding RemarksSellars writes,“Use of a computer model with ‘typical’ values of [roll-gap HTCJprovides a more reliable way of determining temperatures than attempting tomeasure them directly.” [30]This statement is paradoxical, because the determination of these ‘typical’ values ofthe HTC requires reliable strip temperature measuring techniques! Furthermore, these‘typical’ values of the roll-gap HTC could only be obtained by experiments that had to beperformed for each different rolling operation. It is hoped that the information presentedin this study will provide a more rational basis for determining roll-gap HTCs for rollingoperations than has previously been available.BIBLIOGRAPHY[1.1 P. J. S. Brooks, “Advances in Hot Rolling Technology”, AluminumTechnology ‘86, T. Sheppard, Ed., The Institute of Metals, 1986[2.] I. V. Samarasekera, “The Importance of Characterizing Heat Transfer in HotRolling of Steel Strip”, Proceedings of the International Symposium onMathematical Modelling of Hot Rolling of Steel, Aug. 1990, 148-167[3.] B. K. Chen, P. F. Thomson and S. K. Choi, “Temperature Distribution in theRoll-Gap during Hot Flat Rolling”, Journal of Materials ProcessingTechnology, Vol. 30, 1992, 115-130[4.] S. L. Semiatin, E. W. Collings, V. E. Wood and T. Altan, “Determination of theInterface Heat Transfer Coefficient for Non-Isothermal Bulk-FormingProcesses”, Journal of Engineering for Industry, Feb. 1987, Vol. 109, 49-57[5.] S. P. Timothy, H. L. Yiu, J. M. Fine and R. A. Ricks, “Simulation of SinglePass of Hot Rolling Deformation of Aluminium Alloy by Plane StrainCompression”, Materials Science and Technology, Mar. 1991, Vol. 7, 195-201[6.] M. Pietrzyk and J. G. Lenard, “A Study of Boundary Conditions in HotJColdFlat Rolling”, Second International Conference on Computational Plasticity, D.R. J. Owen, E. Hinton and E. Ofiate, Eds., Barcelona, Spain, 1989, 947-958[7.] K. T. Lang, “Lubrication and Thermal Effects in Metal Processing”,[8.] I. Yarita, “Problems of Friction, Lubrication, and Materials for Rolls in theRolling Technology”, Transactions ISIJ, Vol. 24, 1984, 1014-1035109110[9.] C. L. Wandrei, “Review of Hot Rolling Lubricant Technology for Steel”, ASLESpecial Publication SP-17, 1984[10.] W. R. D. Wilson and S. Sheu, “Real Area of Contact and Boundary Friction inMetal Forming”, International Journal of Mechanical Science, Vol. 30, No. 7,1988, 475-489[11.1 M. P. F. Sutcliffe, “Surface Asperity Deformation in Metal Forming Processes”,International Journal of Mechanical Science, Vol. 30, No. 11, 1988, 847-868[12.1 N. Bay, “Friction Stress and Normal Stress in Bulk Metal-FormingOperations”, Journal of Mechanical Working Technology, Vol. 14, 1987,203-223[13.] M. G. Cooper, B. B. Mikic and M. M. Yovanovich, “Thermal ContactResistance”, International Journal of Heat and Mass Transfer, Vol. 12, 1969,279-300[14.] 3. V. Beck, “Determination of Optimum, Transient Experiments for ThermalContact Resistance”, International Journal of Heat and Mass Transfer, Vol. 12,1969, 621-633[15.] B. B. Mikic, “Thermal Constriction Resistance due to Non-Uniform SurfaceConditions; Contact Resistance at Non-Uniform Interface Pressure”,International Journal of Heat and Mass Transfer, Vol. 13, 1970, 1497-1500[16.] H. Fenech, J. J. Henry and W. M. Rohsenow, “Thermal Contact Resistance”, in‘Developments in Heat Transfer’, W. M. Rohsenow, Ed., The M.I.T. Press,Cambridge, Massachusetts, 1964[17.] B. B. Mikic, “Thermal Contact Conductance; Theoretical Considerations”,International Journal of Heat and Mass Transfer, Vol. 17, 1974, 205-214111[18.1 C. Devadas, I. V. Samarasekera and E. B. Hawbolt, “The Thermal andMetallurgical State of Steel Strip during Hot Rolling: Part I. Characterization ofHeat Transfer”, Metallurgical Transactions A, Vol. 22A, Feb. 1991, 307-3 19[19.] W. C. Chen, I. V. Samarasekera and E. B. Hawbolt, “Characterization of theThermal Field during Rolling of Microalloyed Steels”, 33rd MWSP ConferenceProceedings, Vol. XXIX, ISS-AIME, 1992, 349-357[20.] S. Song and M. M. Yovanovich, “Relative Contact Pressure: Dependence onSurface Roughness and Vickers Microhardness”, Journal of Thermophysics,Vol. 2, No. 1, Jan. 1988, 43-47[21.1 W. C. Chen, “Thermomechanical Phenomena during Rough Rolling of SteelSlab”, M.A.Sc. Thesis, The University of British Columbia, 1991[22.] B. F. Bradley, W. A. Cockett and D. A. Peel, “Transient TemperatureBehaviour of Aluminium during Rolling and Extrusion”, Iron and SteelInstitute International Conference on Mathematical Models in MetallurgicalProcess Development, London, Feb. 1969, 79-92[23.] R. E. Smelser and E. G. Thompson, “Validation of Flow Formulation forProcess Modelling”, Advances in Inelastic Analysis, Winter Annual Meeting ofthe ASME, Boston, Massachusetts, 1987, 273-282[24.] W. R. D. Wilson, C. T. Chang and C. Y. Sa, “Interface Temperatures in ColdRolling”, Journal of Materials Shaping Technology, Vol. 6, 1989, 229-240[25.] G. D. Lahoti, S. N. Shah and T. Altan, “Computer-Aided Analysis of theDeformations and Temperatures in Strip Rolling”, Journal of Engineering forIndustry, Vol. 100, May 1978, 159-166[26.] J. G. Lenard and M. Pietrzyk, “The Predictive Capabilities of a Thermal Modelof Flat Rolling”, Steel Research, Vol. 60, No. 9, 1989, 403-406112[27.] A. A. Tseng, “A Numerical Heat Transfer Analysis of Strip Rolling”,Transactions of the ASME: Journal of Heat Transfer, Vol. 106, Aug. 1984,512-517[28.] J. Pavlossoglou, “Mathematical Model of the Thermal Field in Continuous HotRolling of Strip and Simulation of the Process”, Arch. Eisenhuttenwes., Vol.52, No.4, 1981, 153-158[29.] A. Laasroui and J. J. Jonas, “Prediction of Temperature Distribution, FlowStress and Microstructure during the Multipass Hot Rolling of Steel Plate andStrip”, ISIJ International, Vol. 31, No. 1, 1991, 95-105[30.] C. M. Sellars, “Computer Modelling of Hot-Working Processes”, MaterialsScience and Technology, Vol. 1, Apr. 1985, 325-332[31.1 A. A. Tseng, S. X. Tong, S. H. Maslen and J. J. Mills, “Thermal Behavior ofAluminum Rolling”, Transactions of the ASME: Journal of Heat Transfer, Vol.112, May 1990, 301-308[32.] C. Devadas and I. V. Samarasekera, “Heat Transfer during Hot Rolling of SteelStrip”, Ironmaking and Steelmaking, Vol. 13, No. 6, 1986, 3 11-322[33.] T. Sheppard and D. S. Wright, “Structural and Temperature Variations duringRolling of Aluminium Slabs”, Metals Technology, July 1980, 274-28 1[34.1 P. R. Dawson, “A Model for the Hot or Warm Forming of Metals with SpecialUse of Deformation Mechanism Maps”, International Journal of MechanicalSciences, Vol. 26, No. 4, 1984, 227-244[35.] J. W. Kannel and T. A. Dow, “The Evolution of Surface Pressure andTemperature Measurement Techniques for Use in the Study of Lubrication inMetal Rolling”, Transactions of the ASME: Journal of LubricationTechnology, Oct. 1974, 61 1-616113[36.] A. N. Karagiozis and J. G. Lenard, “Temperature Distribution in a Slab DuringHot Rolling”, Transactions of the ASME: Journal of Engineering Materials andTechnology, Vol. 110, Jan. 1988, 17-21[37.] S. I. Steindi and W. B. Rice, “Measurement of Temperature in the Roll Gapduring Cold Rolling”, Annals of the CIRP, Vol. 22, No. 1, 1973, 89-90[38.] M. Pietrzyk, J. G. Lenard and A. C. M. Sousas, “A Study of TemperatureDistribution in Strips during Cold Rolling”, Heat and Technology, Vol. 7, No.2, 1989, 12-25[39.] J. Jeswiet and W. B. Rice, “Measurement of Strip Temperature in the Roll Gapduring Cold Rolling”, Annals of the CIRP, Vol. 24, No. 1, 1975, 153-156[40.] Manual on the Use of Thermocouples in Temperature Measurement, ASTMCommittee E-20 on Temperature Measurement and Subcommittee E20.04 onThermocouples, ASTM, 1981[41.] N. P. Bailey, “The Measurement of Surface Temperatures, AccuraciesObtainable with Thermocouples”, Mechanical Engineering, Vol. 54, Aug. 1932,553-556[42.] B. K. Chen, P. F. Thomson and S. K. Choi, “Computer Modelling ofMicrostructure during Hot Flat Rolling of Aluminum”, Materials Science andTechnology, Vol. 8, No. 1, Jan. 1992, 72-77[43.] A. T. Male, “The Relative Validity of the Concepts of Coefficient of Frictionand Interface Friction Shear Factor for Use in Metal Deformation Studies”,ASLE Transactions, Vol. 16, No. 3, 177-184114[44.] N. Haifa, J. Kokado, H. Nishimura and K. Nishimura, “Analysis of SlabTemperature Change and Rolling Mill Line Length in Quasi Continuous HotStrip M:iIl Equipped with Two Roughing Mills and Six Finishing Mills”,Journal of the Japan Society for Technology of Plasticity, Vol. 21, 1980, 230-236[45.] C. H. J. Davies, I. S. Geitser, E. B. Hawbolt, J. K. Brimacombe and I. V.Samarasekera, “Modelling the Hot Deformation of 6061/Alumina Composites”,Symposium on Light Metals Processing and Applications, CIM Conference ofMetallurgists, 1993, Quebec City[46.] J. G. Beese, “Ratio of Lateral Strain to Thickness Strain during Hot Rolling ofSteel Slabs”, Journal of the Iron and Steel Institute, June 1972, 433-436[47.] Z. Wusatowski, ‘Fundamentals of Rolling’, 1969, Oxford, Pergamon Press[48.] A. A. Tseng, “Roll cooling and Its Relationship to the Roll Life”, MetallurgicalTransactions A, Vol. 20A, Nov., 1938-2305[49.] A. V. Logunov and A. F. Zverov, “Investigating the Thermal Conductivity andElectrical Resistance of Aluminum and a Group of Aluminum Alloys”, Journalof Engineering Physics, Vol. 15, No. 1, 1256-1260[50.] R. D. Pehike, A. Jeyarajan and H. Wada, “Summary of Thermal Properties forCasting Alloys and Mold Materials”, University of Michigan, Dec. 1982,NSRINEA-82028[51.1 Metals Handbook, Desk Edition, H. E. Boyer and T. L. Gall, Eds., ASM, 1985[52.] F. M. White, ‘Heat and Mass Transfer’, 1988, Addison-Wesley[53.] E. Orowan, “A Simple Method of Calculating Roll Pressure and PowerConsumption in Hot Flat Rolling”, Special Reports: Iron and Steel Institute,First Report of the Rolling Mill Research Committee, Sec. V, 1946, 124-126115[54.1 F. A. R. Al-Salehi, T. C. Firbank and P. R. Lancaster, “An ExperimentalDetennination of the Roll Pressure Distributions in Cold Rolling”, InternationalJournal of Mechanical Science, Vol. 15, 1973, 693-710[55.] L. Lai-Seng and J. G. Lenard, “Study of Friction in Cold Strip Rolling”, Journalof Engineering Materials and Technology, Vol. 106, April 1984, 139-146[56.] J. Pullen and J. B. P. Williamson, “On the Plastic Contact of Rough Surfaces”,Proceedings of the Royal Society of London, Vol. 327A, 1972, 159-173[57.] J. B. P. Williamson and R. T. Hunt, “Asperity Persistence and the Real Area ofContact Between Rough Surfaces”, Proceedings of the Royal Society ofLondon, Vol. 327A, 1972, 147-157[58.] R. V. Hogg and J. Ledolter, ‘Engineering Statistics’, MacMillan Publishing Co.,1987[59.] J. A. Schey, ‘Metal Deformation Processes: Friction and Lubrication’, MarcelDekker Inc., 1970[60.] E. Osinski, “Hot Rolling - Coefficient of Heat Transfer between Rolls andStrip”, Internal Report, Nov. 1993[61.1 tHandbook of Thermophysical Properties of Solid Materials’, MacMillan Co.,1961APPENDIX ADetermination of Minimum HTCIt is possible to calculate, without need of an iterative solution, a minimum solutionfor the HTC. This is accomplished by using the assumption that, while in the roll bite,the surface temperature of the roll and strip do not change from their initial temperatures.The roll-gap HTC then simply becomes(A.1)where QI Ac is the average heat flux in W/m2,and T5 and Tr is the surface temperature ofthe strip and roll, respectively, throughout the roll gap. These temperatures are set to theinitial bulk temperature of the strip and roll. The heat flux is equivalent to the net energylost by the sample, AE, while in the roll bite, divided by the contact time, t:(A2)CThe energy loss is related to the temperature drop, t\T, of the sample through thevolume V, the density Ps’ and the heat capacity of the sample:AE pCJ”M (A.3)116117The volume V5 is taken to be a half-volume of the sample in the roll bite.Furthermore, it is approximated to be the product of the sample width, the contact length,and the average of the entry and exit thickness:= !WJR0AH(I + Hf) (A.4)where W is the width of the sample, R0 the radius of the roll and All the draft. Thesquare root term is the projected length of the contact arc. In addition, Hi is the entrythickness of the sample, and Hf 1S the exit or fmal thickness of the sample.The area of the sample in contact with the roll is characterized as= WJR0AH (A.5)Substituting Equation (A.4) into Equation (A.3), (A.3) into (A.2), and (A.2) and(A.5) into Equation (A.1), the following equation is obtained:h= p,c,,,(H +Hf)AT (A.6)4t(7-7)The term AT has to include not only the initial and final temperature of the strip, Tand Tf, but the heat of deformation ATdef as well:AT=7—7;+A7ef (A.7)118where N is the angular velocity of the roll in rpm.Substituting Equation (A.7) and (A.8) into (A.6), the equation for a lower-boundroll-gap HTC is obtained:Finally, the contact time, tc, can be calculated as:tc =60(A.8)— p[N( + Hf)(1 — + ef)--(I -Hf)120cos 2 (2;—7;)Applying this equation to Test AL24, for example, yields:7t(2636kg I m3 )(900J I kg0C)(68.5rpm)(O.0087m + O.00695m)(377° C — 338°C + 10°c)20Im C0.05m—!(0.0087m— 0.00695m)120cos 20.05m119The estimate of the HTC from Equation (A.9) for Test AL24 is 49.3 kW/m2 °C, ascompared to 378.5 kW/m2 °C, as calculated in Section 6.2.1. The HTC as calculated byEquation (A.9) is thus eight times lower than that calculated by the finite-differenceprogram. The discrepancy is due to the assumption that the roll and sample remain attheir initial temperature, whereas in fact the temperature differential decreases withincreasing time in the roll gap.When Equation (A.9) is applied to the data supplied by Timothy et a!. [5], thefollowing HTC is obtained:h= it(26ookgI m)(i 120J / kg°C)(lOrpm)(0.03m + 0.0159m)(460° C — 430° C + 18°c)0.184m — !(0.03m— 0.0159m)l2Ocos’ 2 (46o°c—24°c)0.0184m= 14200W/m°CyieldsAnd similarly, the calculation for the test performed by Pietrzyk and Lenard [6]= 18400W/m2°C7t(2700kg / m)(9OOJ / kg°C)(7.3rpm)(0.013m + 0.0106m)(210°C— 175° C + 8°c)0.127m1210°C—24°C)

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