Astm stp 1523 2010

KEYWORDS: boiling water heat transfer, inverse heat conduction, temperature measurement, thermocouple, water quench Wa-Cite as: Wells, M.. p Vector of coef cients for whole-domain regula

www.astm.orgISBN: 978-0-8031-7509-9Stock #: STP1523

JAI Guest Editors:

Lauralice de C.F Canale Michiharu Narazaki

Selected Technical Papers

STP 1523

Quenching and Cooling, Residual Stress and

Distortion Control

PhoTo CourTeSy of INduCToheaT, INC.,

aN INduCToTherm GrouP ComPaNy

Journal of ASTM International

Selected Technical Papers STP1523

Quenching and Cooling, Residual Stress and Distortion Control

JAI Guest Editors:

Lauralice C.F Canale Michiharu Narazaki

ASTM International

100 Barr Harbor Drive

PO Box C700West Conshohocken, PA 19428-2959

Printed in the U.S.A

ASTM Stock #: STP1523

Library of Congress Cataloging-in-Publication Data

Quenching and cooling, residual stress and distortion control / JAI guest editors,Lauralice C F Canale, Michiharu Narazaki

p cm (Journal of ASTM International selected technical papers; STP1523)Includes bibliographical references and index

ISBN: 978-0-8031-7509-9 (alk Paper)

1 Steel Quenching 2 Steel Defects I Canale, Lauralice de Campos Franceschini II.Narazaki, Michiharu

TN752.Q4Q456 2010

Copyright © 2010 ASTM INTERNATIONAL, West Conshohocken, PA All rightsreserved This material may not be reproduced or copied, in whole or in part, in any printed,mechanical, electronic, film, or other distribution and storage media, without thewritten consent of the publisher

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Citation of Papers

When citing papers from this publication, the appropriate citation includes the paperauthors,⬘⬘paper title’’, J ASTM Intl., volume and number, Paper doi, ASTM International,West Conshohocken, PA, Paper, year listed in the footnote of the paper A citation isprovided as a footnote on page one of each paper

Cover image illustrates a dual-spindle induction heating and quenching of steel shafts.Courtesy of Inductoheat Inc., An Inductotherm Group Company

Printed in Bridgeport, NJSeptember, 2010

THIS COMPILATION OF THE JOURNAL OF ASTM INTERNATIONAL (JAI), STP1523 on Quenching and Cooling, Residual Stress and

Distortion Control contains papers published in JAI highlighting the

impact of the quenching process on the heat treatment of metals This STP

is sponsored by ASTM Committee D-2 on Petroleum Products and

Lubricants

The JAI Guest Editors are Lauralice de C.F Canale, EESC Universidade

de S:o Paulo, Brazil, and Michiharu Narazaki, Utsunomiya University,Utsunomiya, Japan

M A Wells and K J Daun . 16 Heat Transfer During Quenching and Assessment of Quench Severity—A Review

K Narayan Prabhu and P Fernandes . 40 Formulation of a Guideline for the Determination of Heat Transfer Coefficient during Gas Quenching

T Luebben, M Lohrmann, S Segerberg, and P Sommer . 63 Determination of Surface Heat Transfer Coefficients of Cr12MoV Steel Cylinder during High-Speed Gas Quenching at Atmospheric Pressure

C Heming, L Jianyun, L Ziliang, H Lijun, and H Jie . 84 Enhancement and Local Regulation of Metal Quenching Using Atomized Sprays

U Alam, J Krol, E Specht, and J Schmidt . 91 The Effect of Agitation and Quenchant Temperature on the Heat Transfer Coefficients for 6061 Aluminum Alloy Quenched in Distilled Water

M Maniruzzaman, M Fontecchio, and R D Sisson, Jr. . 104

Modeling and Simulation

A Review on Modeling and Simulation of Quenching

Modeling of Quenching and Tempering Induced Phase Transformations in Steels

P T Rajeev, L Jin, T N Farris, and S Chandrasekar . 157 Improving Control of a Quenching Process by Coupling Analysis Methods

A Banka, J Franklin, Z Li, B L Ferguson, A Freborg, and M Aronov . 186 Generalized Equation for Cooling Time Evaluation and Its Verification by CFD

Analysis

P Krukovskyi, N Kobasko, and D Yurchenko . 205 Computer Simulation of Quenched Steel Working Stress

B Smoljan, S Smokvina Hanza, and D Iljkic´ . 228

A Novel Approach to Model Moving Heat Sources

F D Fischer, C Krempaszky, J Ocˇenášek, and E Werner . 245 Analysis of Stress Concentration around Inclusions due to Thermally Induced Strain to the Steel Matrix

M R Allazadeh, C I Garcia, A J DeArdo, and M R Lovell . 253 Considering the Body-Centered Cubic Lattice Parameter of␣-Fe Alloys versus

Concentrations of Solved Elements

V P Filippova . 269

L Li, S G Huang, L Wang, Y L He, J Vleugels, and O Van der Biest . 284 Model of Solid State Transformations of Ductile Cast Iron GJS-600

V Runser and V Schulze . 295

Distortion and Residual Stresses

Using Polyalkylene Glycol Quenchants to Effectively Control Distortion and Residual Stresses in Heat Treated Aluminum Alloys

T Croucher . 309 Minimizing Machining Distortion in Aluminum Alloys through Successful Application of Uphill Quenching—A Process Overview

T Croucher . 332 Explanation of the Origin of Distortion and Residual Stress in Carburized Ring Using Computer Simulation

K Arimoto, S Yamanaka, and K Funatani . 352 The Analysis and Control of Distortion in Carbonitrided and Nitrocarburized Thin-

Shelled Plain Carbon Steel Automotive Powertrain Components

V Campagna, D O Northwood, R Bowers, X Sun, and P Bauerle . 364 The Effect of Core and Carburized Surface Microstructural Stability on Residual Stress Evolution during Tempering

J Vatavuk, M Zicari di Monte, and A A Couto . 387 Explanation on the Origin of Distortion in Induction Hardened Ring Specimens by

Subroutines to Represent TTT and CCT Diagrams

E M Bortoleto, C F Lagatta, M G Di Vernieri Cuppari, I F Machado, and

R Martins de Souza . 436 Prediction of Distortion of Automotive Pinion Gears during Quenching Using CFD and FEA

D S MacKenzie, A Kumar, H Metwally, S Paingankar, Z Li, and B L Ferguson . 450

Property Predictions

Prediction of Quench-Hardness within the Whole Volume of Axially Symmetric

Workpieces of Any Shape

B Lišcˇic´, S Singer, and B Smoljan . 467 Correlation between Thermal and Mechanical Properties of the 10NiCr11

T Ghrib, M Bouhafs, and N Yacoubi . 489

An Efficient Numerical Algorithm for the Prediction of Thermal and Microstructure

Fields during Quenching of Steel Rods

S K Ali, M S Hamed, and M F Lightstone . 500

Quenchants and Quenching

Starch-Based Quenchants as an Eco-Friendly Alternative to Quenching Oil

S S Sahay . 525

Comparison of Structure and Quenching Performance of Vegetable Oils

E C de Souza, M R Fernandes, S C M Augustinho,

L de Campos Franceschini Canale, and G E Totten . 531 Severity of Quenching and Kinetics of Wetting of Nanofluids and Vegetable Oils

V Jagannath and K N Prabhu . 571

An Investigation on Quenching Performance of Hot Alkaline Bath

S Raygan, J Rassizadehghani, and M Askari . 584

A Feasibility Study of the Use of Bismuth Bath to Replace Lead Bath as the Quenching Media for Steel Heat-Treating

J Ru and Z Wang . 596 Spray Quenching in Induction Hardening Applications

V Rudnev . 609 Technology and Applications of Alternately Timed Quenching Technology

N Chen, X Zuo, S Zhou, and J Xu . 622 One More Discussion “What is Intensive Quenching Process?”

N I Kobasko, M A Aronov, J A Powell, and G E Totten . 629 Energy Efficient and Eco-friendly Intensively Quenched Limited Hardenability Low

Alloy Steels

N I Kobasko . 644

An Overview of Technology and Equipment for Hardening of Large Steel Parts

L N Deyneko, N I Kobasko, V V Dobryvechir, and E I Litvinenko . 662 Accelerated Cooling of Steel Plates: The Time Has Come

A A Gorni and J H D da Silveira . 682 Overview of Pearlitic Rail Steel: Accelerated Cooling, Quenching, Microstructure, and Mechanical Properties

S S Sahay, G Mohapatra, and G E Totten . 692 Water and Polymer Quenching of Aluminum Alloys: A Review of the Effect of Surface Condition, Water Temperature, and Polymer Quenchant Concentration on the

Yield Strength of 7075-T6 Aluminum Plate

G S Sarmiento, C Bronzini, A C Canale, L C F Canale, and G E Totten . 728

Gas Quenching

Gas-Jet Quenching

P Stratton . 757 Quenching Homogeneity and Intensity Improvement in Batch Mode High Pressure Gas Quenching

R.-R Schmidt and U Fritsching . 776 Gas-Cooling of Multiple Short Inline Disks in Flow Along Their Axis

N Lior and D Papadopoulos . 795 Numerical Simulation of the Mechanical Properties of Cr12 Steel during Gas

A New Method to Study the Effect of Cooling Rate on the Decomposition of Austenite

in Advanced High Strength Sheet Steels

K Cho, C I Garcia, M Hua, J Lee, Y S Ahn, and A J DeArdo . 843

Quenchant Characterization by Cooling Curve Analysis

L C F Canale, X Luo, X Yao, and G E Totten . 861 Cooling Characteristic Test of Quenching Media

X Luo and J Li . 935 Correlation between Cooling Curves Obtained with a Silver Probe and Quenching

Properties of 5140 Steel Bars

R S Wang, Y Wang, and L P Su . 953 Measurement of the Cooling Power of Polyethylene Glycol Aqueous Solutions Used as Quenching Media

R Ikkene, Z Koudil, and M Mouzali . 977 Evolution of Quench Factor Analysis: A Review

P M Kavalco and L C F Canale . 991 Analysis of the Segerberg Hardening Power Equation

C Chun-huai and Z Jing-en .1021 The Influence of Surface Temperature on Rewetting Behavior During Immersion

Quenching of Hollow and Solid Cylinders

F Frerichs and Th Luebben .1032

Dilatometric Analysis

Dilatometric Analysis to Study Aging of Aluminum Alloys

R Gerosa, B Rivolta, and U Derudi .1055 Optimization of the Heat Treatment of a 17-4 PH Stainless Steel by Dilatometric

Technique

R Gerosa, B Rivolta, and A Sala .1066 Author Index .1077 Subject Index .1081

Quenching and distortion control continue to be of great concern to the als processing industry since they exhibit tremendous effects on quality andprofitability Recently, many papers on topics directly and indirectly related

met-to quenching and quenching processes have been published in the Journal ofASTM International (JAI) In view of the interest and importance of thesetopics to the thermal processing industry, a total of 59 JAI papers have beencollected together into this ASTM Special Technical Publication (STP) andthese papers have been organized into nine topical sections: Heat Transfer;Modeling and Simulation; Distortion and Residual Stresses; Property Pre-dictions; Quenchants and Quenching; Gas Quenching; Hardenability; Cool-ing Curve Analysis Methodologies; and Dilatometric Analysis

The Heat Transfer section includes papers describing experimental niques for measurements of heat transfer distribution at the spray cooledsurface and in gas quenching systems Also included are mathematical mod-els to promote accurate characterization of heat transfer throughout thequenching operation There are also papers which review the characteristics

tech-of various quench media, effects tech-of process parameters on quenching,mechanisms of thermal transport, and techniques for the estimation of heattransfer coefficients Heat transfer studies for atomized water sprays, liquidmedia, and gas quenching system are discussed

Modeling and Simulation are powerful tools in the design, optimization,and understanding of the quenching process and they are used increasingly

in component manufacture The development of simulation tools for ing is critical for improving process performance by minimizing distortionand maximizing service life Therefore, this section describes several ex-amples utilizing FEM (Finite Element Methods), a combination of methodssuch as CFD (Computational Fluid Dynamics) and FEM-based thermal pro-cess models, to provide efficient and effective solutions applicable to quench-ing and tempering processes Solid state transformations of ductile iron andnew design steels are also discussed

quench-One section is focused on Distortion and Residual Stresses Benefits sociated with polymer quenching and uphill quenching for aluminum alloysare discussed Carbonitriding and nitrocarburizing processes and their re-lationship with respect to size and shape distortion, retained austenite, andresidual stresses are also discussed Papers discussing tempering effects onas-quenched compressive residual stresses of carburized steel and the appli-cation of commercial codes including FLUENT, DANTE, CFD, ABAQUS,and Fortran subroutines to model and analyze residual strain, internal

as-ix

during steel heat treatment are included also.

In the Property Predictions section, an empirical equation is proposed todeduce the Vickers hardness of carburized and quenched components Ther-mal and microstructure fields are predicted by numerical algorithms aswell

The Quenchants and Quenching section comprises of thirteen papers cussing a new ecofriendly starch-based quenchant and vegetable oil quen-chants Alumina-based nanofluids, hot alkaline bath, and bismuth bath arealso described as cooling media Accelerated quenching for steel plates isalso addressed and the resultant microstructure and mechanical propertiesare analyzed Intensive quenching and a timed quenching process are pre-sented as alternatives to conventional heat treating Spray quenching forinduction hardening applications and comparative results of water andpolymer quenchants for non-ferrous alloys are also covered

dis-Gas Quenching is considered a clean, non-toxic quenching medium thatleaves no residues to be removed after processing In this section, high pres-sure gas quenching processes exhibiting advantages including pure convec-tive heat transfer are discussed, as is an upstream gas flow profile of theload as a key factor determining heat transfer distribution from the mate-rial Finally, a simulation of the gas quenching processes is proposed as anefficient tool for investigation of mechanical properties, and their effects onquality are analyzed

A shorter section on the classic topic of Hardenability includes two papers

on the application of magnetic fields to enhance hardenability and an ration of the effect of cooling rate on austenite decomposition in highstrength sheet steels

explo-Quenching operations are a critical part of the heat treatment process.FEM simulations are used to optimize products and processes and are based

on boundary conditions expressed as heat transfer coefficients, which aremeasured using cooling curves Therefore, cooling curves are of great impor-tance for the evaluation of quenchants and quenching processes and thusthe section on Cooling Curve Analysis Methodologies is dedicated to thissubject A computational method for the prediction of cooling curves at thecenter of a steel bar based on a cylindrical silver probe is proposed An ex-periment to detect dynamic cooling and steel transformation behavior upondirect quenching is also discussed Test methods to characterize the coolingbehavior of quenching media, which was introduced in China, are criticallyanalyzed Review on quench factor analysis and the Segerberg hardeningpower equation is also provided

The final topic is Dilatometric Analysis Its use as a powerful test foroptimization of the age hardening parameters for aluminum alloys and for17-4 PH stainless steel is discussed

x

We believe that this Special Technical Publication of ASTM International

on QUENCHING AND COOLING, RESIDUAL STRESSES AND TION CONTROL will provide an important contribution to the heat treat-ment industry worldwide

DISTOR-Lauralice C.F CanaleSao Carlos, SP, BrazilMichiharu NarazakiUtsunomiya, Japan

xi

HEAT TRANSFER

M Pohanka,1 H Bellerova,1 and M Raudensky1

Experimental Technique for Heat Transfer Measurements on Fast Moving

Sprayed Surfaces

ABSTRACT:An experimental technique for the measurements of heat fer distribution at spray cooled surfaces is described The measurementswere done at moving surfaces up to a velocity of 12 m/s The samples ofdifferent cross sections 共flat, profile, rail, etc.兲 can move linearly Differentapproaches are used for the measurements of heat transfer coefficient共HTC兲 distribution or heat flux distribution at rotating cylindrical body Aninverse task for the computation of surface temperature, HTC, and heat fluxdistribution obtained from the measurements conducted for internal transienttemperature is described The paper describes necessary demands on ex-periment configuration and temperature measurements when studying highlytransient processes共fast moving objects under spraying nozzles or high cir-cumferential velocities of rotating surface兲 The results of HTC distribution forspray cooling are shown and are compared to water impingement density.Influence of water impingement density, water pressure, spray configuration,and surface velocity is studied Examples for water nozzles and for mistnozzles 共water-air兲 are given Emulsions and oils are beneficial for someindustrial applications of spray cooling The cooling experiments performedwith these liquids provide information about decrease of cooling intensityeven for a low concentration of the oils in water The results comparing thecooling properties of these liquids and their comparison to water are pre-sented

trans-KEYWORDS: heat transfer, spray cooling, coolants

Manuscript received April 17, 2008; accepted for publication February 11, 2009; lished online March 2009.

pub-1 Brno University of Technology, Faculty of Mechanical Engineering, Heat Transfer and Fluid Flow Laboratory, Technicka 2896/2, 616 69 Brno, Czech Republic

Cite as: Pohanka, M., Bellerova, H and Raudensky, M., ‘‘Experimental Technique for

Heat Transfer Measurements on Fast Moving Sprayed Surfaces,’’ J ASTM Intl., Vol 6,

No 4 doi:10.1520/JAI101801.

Reprinted from JAI, Vol 6, No 4

doi:10.1520/JAI101801 Available online at www.astm.org/JAI

Copyright © 2009 by ASTM International, 100 Barr Harbor Drive, PO Box C700, West Conshohocken, PA 19428-2959.

3

Methods of measurement of heat transfer at sprayed surfaces are discussed inthis paper It should be specified first what parameters can influence the results.The first factor is a device producing the spray The most frequent in industry is

an application of cluster of nozzles Alternatives are single nozzle, slots, andholes producing a free coolant jet The most typical coolant is water but anyliquid can be found in technologies Coolant itself is the second factor Use ofoil and emulsions is discussed in this paper The third factor is the cooledobject, its shape, velocity, and especially the surface temperature Boiling ofcoolant at a hot surface makes the task strongly temperature dependent Astable vapor layer can be formed at the hot sprayed surface A vapor layerprotects the surface from a direct contact with the coolant and the cooling is of

a low intensity Stability of the vapor layer is coupled to the surface ture When the temperature decreases and the vapor layer collapses, the cool-ing instantly increases The cooling intensity can be ten times higher in the lowtemperature region in comparison to the intensity in the high temperatureregion The border共dividing兲 temperature between these two temperature areas

tempera-is the Leidenfrost temperature关1,2兴 If the Leidenfrost temperature for a givennozzle and spray parameters is inside the operating ranges of a device, rapidchanges of cooling could be expected during operation This is typical for spraycooling in continuous casting

There are in principal two possibilities of how to measure heat transfer athot surfaces covered by sprays The first method uses heated elements wheretheir temperature is held constant and the second method is based on transienttemperature measurements during cooling of test samples

A small heated element can be used for scanning the impact area by ing the element and measuring the local values of heat transfer coefficient Agood precision is usually obtained only for the nozzle axis area The source ofdifferences is obvious: jet droplets spraying on a large hot surface evaporate,steam is generated, and the character of flow and conditions for the impact ofthe droplets on the hot surface is much different than the flow on the cold platewith the heated spot

mov-A full scale test sample or test samples larger than jet impact are necessaryfor elimination of the errors caused by neglecting real fluid flow conditions andevaporation when using small heated sensors Overlapping of jets is a typicalexample where the size of a test plate has to cover at least the impact area oftwo neighboring nozzles

It was found that the stationary tests with no relative motion between thenozzle and the cooled surface do not provide reliable results for technologieswhere the cooled object moves Even low velocities order of metres per minute共typical for continuous casting兲 can influence heat transfer

Experimental Technique

The principle of the experiment is simple: a heated test sample moving underthe spray is cooled and the temperature history inside the sample is recordedusing a sampling frequency up to 360 Hz The temperature history is used for

the computation of cooling intensity The most difficult part is precise ments of fast temperature changes inside the body The presence of any sensordisturbs the temperature field in the body A detailed numerical model of thebody with sensor compensates impingement of the presence of the sensor atthe temperature field.

measure-Configuration of the test for the cooling study of rotating roll is describedhere The goal of the experiment is to obtain the distribution of heat transfercoefficient共HTC兲 at the roll surface for a given spray configuration and spraycondition The term “distribution” means the HTC in both longitudinal andcircumferential directions of the cylinder and a function of surface tempera-ture

The test bench is schematically shown in Fig 1 The basic part is an mented roll共1兲 Nozzles are placed around the roll in an arbitrary configuration共2兲 A mechanically opened deflector is placed between the nozzles and the rollsurface with the purpose of reflecting sprayed coolant during the initial stages

instru-of the experiment The roll is driven by a frequency control motor allowing thesetting of circumferential velocity The test roll is hollow and the surface isformed by a steel shell A part of the test roll surface is formed by a test platecast from stainless steel Special thermocouple-based sensors, individually cali-brated, are built into the test plate The question of sensor parameters is de-scribed in detail in Ref关3兴 The distance between the cooled surface and ther-mocouple 共measurement point兲 placed inside the roll is 0.5 mm Shieldedungrounded thermocouples of diameter 0.5 mm are used Eight sensors areplaced on one line in longitudinal directions of the roll The pitch between twoneighboring sensors is50 mm The result of calibration is used in the numeri-cal model of the test plate with sensors This model is the basic part of the

FIG 1—Scheme of test bench, 共1兲 test roll, 共2兲 spray nozzles, 共3兲 velocity control unit, 共4兲 data logger.

POHANKA ET AL., doi:10.1520/JAI101801 5

inverse heat conduction task used for the data evaluation Temperature data arestored together with instant roll position data in a data logger 共4兲 rotatingwithin the roll.

The cooling experiment is transient The preparation of an experimentstarts by heating the test plate with an electric heater Only the test plate itself

is heated, the rest of the tested roll remains at room temperature The roll isstationary during test plate heating

The experiment starts when the temperature of the test plate reaches auniform start temperature The heater is then removed, rotation starts, thepump is switched on, and the pressure共or flow rate兲 is set The closed deflectorprotects the cylinder surface from spray

The computer controls the movement of the deflector 共the deflector ismoved by pneumatic units兲 The movement of the deflector is linked with theposition of the test plate The deflector is opened at the moment when therotating test plate is in the opposite position—out of water impact This mecha-nism ensures that the deflector opening and closing is at precisely measuredtiming for all experiments in the same way

Temperatures are not measured at the surface but at a measured pointlocated under the surface An example of the measured temperature history atone point is in Fig 2, the solid line representing measured temperature Tem-perature decreases shown on the curve represent the runs under the spray.Each drop and increase on the curve represents one revolution of the roll.Variation of the surface temperature is higher than variation of the measured

FIG 2—Example of measured temperature history 共solid line兲 and computed surface

temperature 共dashed line兲.

temperature The computed surface temperature is plotted in Fig 2 by adashed line.

The procedure of the cooling experiment at linearly moving objects is lar to the experiments with rotating surface The linear test bench共see Fig 3兲carries a test plate or a test piece 共for example, rail兲 with embedded thermalsensors The experiment starts by heating the test sample The initial tempera-tures vary with respect to the studied problem Maximum used temperature is1250° C when cooling in continuous casting is studied The hot test samplemoves under the spray located in the middle of the long test bench

simi-Evaluation of Measured Data

Temperature history recorded during the experiment 共see Fig 2兲 is used forcomputing time dependent boundary conditions关4兴 共surface heat flux, surfacetemperature, and heat transfer coefficients兲 The boundary conditions are com-puted using the sequential estimation of the time varying boundary conditionsproposed by Beck关5兴

The method uses sequential estimation of the time varying boundary ditions and uses future time steps data to stabilize the ill-posed inverse problem关3兴 Future time for our measurements was 0.01 s To determine the unknownsurface heat flux at the current time t m, the measured temperature responses

con-T i *,mare compared with the computed temperatureT j mfrom the forward solverusingnffuture times steps

FIG 3—Principal scheme of the linear test bench.

POHANKA ET AL., doi:10.1520/JAI101801 7

properties Using the linear minimization theory, the value of the surface heatflux that minimizes Eq 1 is

qˆ m=兺f=m+1 m+n f 兺j n ⬅i;i=1 T*

共T i *,f−兩T j f兩q m=0兲␨i f

兺f=m+1 m+n f 兺i=1 n T*

where兩Tj f兩q m=0 are the temperatures at the temperature sensor locations puted from the forward solver using all the previously computed heat fluxes,but without the current oneq m The␨i fis the sensitivity of the ithtemperaturesensor at timet f to the heat flux pulse at timet m These sensitivity coefficientsare mathematically the partial derivatives of the computed temperature field tothe heat flux pulse, but in this case they physically represent the rise in tem-perature at the temperature sensor location for a unit heat flux at the surface.The sensitivity coefficient of our interest is defined as

com-␨j m=⳵T j m

Once the heat flux is found for the timet m, the corresponding surface ture T surf m may be computed using the forward solver When the surface heatfluxq mand surface temperatureT surf m are known, the heat transfer coefficient iscomputed from

at the “present” time is computed, the time indexm is incremented by one, and

the procedure is repeated for the next time step

Results

For each kind of experiment, the most representative one was repeated Thevariation in results is generally less than 5 % Future time used for inverseevaluation was0.01 s which corresponds to the distance of the installed ther-mocouple from the investigated surface关3兴 Depending on the moving velocitythis future time corresponds to distance from0.1 to 120 mm

Heat Transfer and Parameters of Spray

Measurements with mist water-air nozzles on the linear test bench are showingthat computation of heat transfer from water impingement density is in somecases impossible Mist water-air nozzles are used for modern continuous cast-ing plants The nozzles produce fine droplets and the main cooling effect iscaused by evaporation

The estimation of cooling intensity obtained from spray cooling density共L/m2· s兲 is frequently used for simulation purposes 关6兴 The heat transfer testsshowed that there is no explicit link between coolant flow and cooling intensity

In other words, the identical nozzle can have a different cooling intensity forthe same water flow when changing air flow Heat transfer is not influencedonly by water flow but also by droplet size and velocity Simple estimation ofthe heat transfer coefficient from water impingement density can be affected bysignificant errors.

Two heat transfer tests were done with the following spray parameters Themist nozzle with a spray angle of 80° was located at a spray height of300 mm.The cooled surface velocity was 1 m / min Initial temperature of the cooledsteel plate was1100° C Both experiments used a water flow of 10 L / min Theonly parameter that was different in these two tests was air flow The firstexperiment uses30 m3/ h, the second experiment uses 15 m3/ h The change ofthis parameter caused the change of droplet size and velocity Water and airflows were set by the combination of water and air pressures

The results of the tests are in Fig 4 There are two curves for heat transfercoefficient共HTC兲 for surface temperature in an interval from 900 to 1100°C Itcan be seen that the cooling intensity is significantly different even if the water

FIG 4—Heat transfer coefficient 共HTC兲 distribution in the sprayed area, two

experi-ments with identical mist water-air nozzle, water flow 10 L / min, identical spray tern, and different droplet sizes and velocities caused by different air flow 共30 and

pat-15 m3/ h兲.

POHANKA ET AL., doi:10.1520/JAI101801 9

flow is the same in both cases The average value of heat transfer coefficient forExperiment 1 is860 W / m2· K and for Experiment 2 is 510 W / m2· K The dif-ference is 40 %.

Interaction of Sprays

Early experiments completed in the 90s were motivated by an effort to studyheat transfer caused by single nozzle as a base for more complex spray con-figurations Finding the methods for heat transfer computation for complexproblems using basic experiments was not successful An example is given forcooling using a combination of spray bars

Three rows of nozzles with different sizes were used for a study of mutualinteraction of sprays共see Fig 5兲 All experiments were done using a roll diam-eter of 600 mm, circumferential velocity of 3 m / s, and pressure of 0.5 MPa.Spray Bar A used the largest nozzles giving15 L / s per 1 metre of roll length.Spray Bar B used mid-sized nozzles with a flow rate of5 L / s / m and Spray Bar

C used a flow rate of1 L / s / m The positions of the axes of impact area on theroll were at distances of200 mm measured at the roll circumference

Experiments with a single bar showed that heat transfer in the sprayed area

is not proportional to the coolant flow rate The portion of flow rates for SprayBars A, B, and C is15+ 5 + 1 while the portion of HTC is 5 + 3 + 1, respectively.The program of experiments was continued by experimenting using two rows

of nozzles The results in Fig 6 show the HTC distributions for the experiments

A + B, A + C, and C + D共see Fig 5兲 Note that the spray pattern is not overlappingthe spray pattern of the neighboring bar The position axis in Fig 6 is con-nected with the roll circumference Entrance side is on the right共positive axis兲and the axes positions of impact areas of Spray Bars A, B, and C are in 200, 0,and 200 mm, respectively The rise of HTC at 400 mm was caused by an in-stalled wiper on top of the roll The most important finding is that the datausage for a single row of nozzles, where multiple rows of nozzles have been

FIG 5—Configuration of spray nozzles.

used, is only possible in cases when the impact areas are significantly far awayfrom one another CaseA + B共see the solid line in Fig 6兲 is most enlightening.When the surface point receives Spray A, the less intensive Spray B can holdHTC at a constant level! A similar conclusion can be formulated when studyingthe experimentB + C in Fig 6 The comparison of experiments showed that agiven combination of sprays cannot be used for any superposition.

Surface Temperature

The surface temperature determines the mechanism of heat transfer Spraycooling is influenced by the presence of boiling Significant points on the boil-ing curve known from pool boiling can be found in spray cooling The mostimportant point is the Leidenfrost temperature Spray cooling efficiencystrongly depends on surface temperature A stable vapor layer can be formed atthe cooled surface The stable vapor layer protects the surface from a directcontact with the coolant and the cooling becomes of lower intensity Stability ofthe vapor layer is coupled to the surface temperature When temperature de-creases and the vapor layer collapses the cooling instantly grows The coolingintensity can be ten times higher in a low temperature region in comparison tothe intensity in a high temperature region The border between these two tem-perature areas is the Leidenfrost temperature discussed by Raudensky andHorsky关7兴 and by Bernardin and Mudawar 关8兴

Spray cooling experiments starting at surface temperatures of about300° Care in the area below the Leidenfrost temperature This is not to say that heattransfer is temperature-independent in this low temperature region

FIG 6—HTC distribution for tests using two spraying rows of nozzles.

POHANKA ET AL., doi:10.1520/JAI101801 11

A simple experiment using two rows of nozzles shows the influence ofsurface temperature Figure 7 shows heat transfer coefficient distribution onthe cylinder surface The diameter of the cylinder is600 mm and circumferen-tial velocity 1 m / s Horizontal axis in Fig 7 is for a position on the cylindercircumference where zero represents the surface point on the level of roll axis.The position of the first row of nozzles is 100 mm and the position of thesecond row is 500 mm 共measured at the roll circumference兲 Figure 7 showsthat the highest intensity of heat transfer was obtained for low surface tempera-tures, below 100° C Temperature intervals of 100– 200° C provide lower HTCvalues but the difference is not significant Increasing surface temperaturemakes heat transfer less intensive.

The above can be explained by negative role of vapor formation Vaporforming by the contact of a hot surface with water prevents the surface fromactual direct contact with the coolant The second effect is that impacting watersticks well to the cold surface and supports cooling outside the direct impact ofdroplets Water in the mixture with vapor is easily sprayed out due to centrifu-gal forces acting on the rotating surface

Cooling by Emulsions and Oils

The cooling intensity of five types of coolants was determined The pure waterwas taken as the referential value Additions cause change of specific capacity,thermal conductivity, and surface tension, which leads to change of coolingintensity Five measurements were done for five different water pressures vary-ing from 0.1 MPa to 1 MPa Pure oil was examined next These experiments

FIG 7—Study of the influence of surface temperature on heat transfer.

were done for the concentration of 100 % with pressure changing from0.1 MPa to 0.5 MPa Experiments were done for constant pressure and flowrate was measured together Oil volume flow was 18.7 % higher than water, andoil density is799 kg/ m3 Emulsions and salt dilutions were examined for dif-ferent concentrations of these substances in water at a liquid pressure of0.5 MPa Two types of emulsions were tested First is metal-working liquid,concentration 2 %, 4 % of mineral oil in water The second emulsion is withLubrodal共mixture of water with polymers and organic salts, dynamic viscosity

at20° C is 400 mPas, density 1110 kg/ m3兲 Concentrations of 2 %, 5 %, and 10

% of Lubrodal 192 were exanimate No measurable differences in flow rate ofwater and emulsions for identical pressures were observed

The experiments were stationary 共no movement兲 with the tested samplemade from austenitic steel plate The initial temperature was 250° C Thenozzle with a spray angle of 60° placed vertically at a distance of150 mm fromthe cooled surface was used

The results were compared with the experiment for pure water with sure0.5 MPa named PW5, shown in Fig 8 The reference experiment PW5 uses

pres-a wpres-ater impingement density of12.75 L · m−2s−1 The HTC is computed as anaverage value for surface temperatures below 100° C The HTC of emulsionwith mineral oil reaches 80–84 % of the HTC of pure water, the HTC of emul-sion with polymers reaches 84–87 % The pure oil cooling intensity reaches13–35 % The water with hardness of16.55 mmol/ L reaches 101 % of the HTC

of pure water, the water with hardness of 30.1 mmol/ L reaches 96 % of theHTC of pure water The Emulsion 2 cools better in comparison with the Emul-sion 1 with mineral oil 共metal-working liquid兲 using the same concentration.The pure oil cooling is not very effective The water hardness influences coolingintensity very little The best coolant is pure water

The heat transfer coefficient is decreasing with an increasing amount ofadmixtures in water共2, 5, 10 % of emulsion Lubrodal→87, 86, 84 % of HTC; 2,

4 % of mineraloil→84, 80 %兲 The pressure of coolant influences cooling more

than the amount of admixtures共see Fig 8兲 The higher pressure means higher

FIG 8—Heat transfer coefficient for all types of examined coolants; PW–Pure water 共pressure 0.1, 0.3, 0.5, and 1 MPa兲, O—Pure oil 共pressure 0.1, 0.3, 0.5 MPa兲; MWL–

Metal-working liquid 共concentration 2 %, 4 %兲; L–Lubricant 共concentration 2, 5, and

10 % 兲; SD–Salt dilution 共hardness 16.55 and 30.10 mmol/L兲.

POHANKA ET AL., doi:10.1520/JAI101801 13

cooling intensity 共water pressure 0.1 MPa→53 %, water pressure 0.5 MPa

→100 %, water pressure 1 MPa→143 %, oil pressure 0.1 MPa→13 %,

0.3 MPa→18 %, 0.5 MPa→35 %兲 The increasing hardness of water declines

cooling intensity, but its influence is minimal

Conclusions

Spray cooling of moving surfaces is a difficult heat transfer task which can beprecisely solved only when using experimental techniques A large number ofspray parameters influence heat transfer intensity An experimental techniqueproviding boundary conditions for numerical models was described by Horsky

et al 关6兴 Heat transfer in the sprayed area is significantly dependent on thesurface temperature The Leidenfrost temperature for spray cooling is typicallyhigher than500° C and for intensive sprays exceeds 1100° C Even in the tem-perature interval below the Leidenfrost temperature there was found a seriousdependence of the heat transfer coefficient on surface temperature The experi-ments proved that decreasing values of heat transfer coefficient occurred withincreasing temperature of the sprayed moving surface

The study of the velocity proved a decrease in heat transfer intensity on arotating cylinder with the speed of rotation Inclination of coolant jets againstand along the direction of movement proved the surprising fact that higherHTC values are obtained when the jets follow the sprayed cylindrical surface.Any superposition cannot be used and only the experiment with experimentalinvestigation into the mutual interference of several sprays can be used.The experiments made using a combination of several sprays proved theimpossibility of finding a simple method to apply the results from simple con-figurations 共for example, experiments with single nozzle or single rows ofnozzles兲 to those using complex spray configurations

Acknowledgment

This investigation was supported by the Czech Science Foundation No 106/06/0709

References

关1兴 Bernardin, J D and Mudawar, I., “A Cavity Activation and Bubble Growth Model

of the Leidenfrost Point,” ASME Trans J Heat Transfer, Vol 124, No 5, 2002, pp.

864–874.

关2兴 Bernardin, J D and Mudawar, I., “A Leidenfrost Point Model for Impinging

Drop-lets and Sprays,” ASME Trans J Heat Transfer, Vol 126, No 2, 2004, pp 272–278.

关3兴 Pohanka, M., “Limitation of Thermal Inverse Algorithm and Boundary Conditions

Reconstruction for Very Fast Changes on Boundary.” Engineering Mechanics 2007,

Svratka, Czech Republic, 2007, pp 229–230 ISBN 978–80–87012–06–2.

关4兴 Raudensky, M., “Heat Transfer Coefficient Estimation by Inverse Conduction

Al-gorithm,” Int J Numer Methods Heat Fluid Flow, Vol 3, 1993, pp 257–266.

关5兴 Beck, J V., Blackwell, B., and Charles, R C., Inverse Heat Conduction: Ill-Posed

Problems, Wiley, New York, 1985 ISBN 0–471–08319–4.

关6兴 Horsky, J., Raudensky, M., and Zela, L., “Experimental Study of Heat Transfer

with Reference to Numerical Simulations in Hot Rolling.” 7th International

Con-ference on Steel Rolling, Makuhari, Chiba, Japan, Proceeding published by The

Iron and Steel Institute of Japan, 1998, pp 216–220.

关7兴 Raudensky, M and Horsky, J., “Secondary Cooling in Continuous Casting and

Leidenfrost Temperature,” Ironmaking Steelmaking, Vol 32, 2005, pp 159–164.

关8兴 Bernardin, J D and Mudawar, I., “The Leidenfrost Point: Experimental Study and

Assessment of Existing Models.” ASME Trans J Heat Transfer, 121, 1999, pp 894–

903.

POHANKA ET AL., doi:10.1520/JAI101801 15

M A Wells1 and K J Daun1

Accurate Determination of Surface Heat

Fluxes During Quenching Characterized by Boiling Water Heat Transfer

ABSTRACT:Mathematical models play a critical role in design, optimization,and control of water quenching treatment in metals processing Modelingthese processes correctly is predicated on accurately characterizing the heattransfer at the product surface throughout the quench operation, usually with

a boiling curve These curves are most often generated from time-resolvedsubsurface temperature measurements carried out during quenching, al-though data analysis is complicated by the ill-posed nature of the underlyingproblem This paper reviews the best experimental and analytical practicesthat have been developed for recovering the boiling water curve through aninverse heat conduction analysis

KEYWORDS: boiling water heat transfer, inverse heat conduction,

temperature measurement, thermocouple, water quench

Wa-Cite as: Wells, M A and Daun, K J., Accurate Determination of Surface Heat Fluxes

During Quenching Characterized by Boiling Water Heat Transfer, J ASTM Intl., Vol 6,

No 1 doi:10.1520/JAI101818.

Copyright © 2009 by ASTM International, 100 Barr Harbor Drive, PO Box C700, West Conshohocken, PA 19428-2959.

16

p Vector of coef cients for whole-domain regularization, Eq 17

P j Coef cient for whole-domain regularization, Eq 17

qs Surface heat ßux,W ám−2

r Thermocouple cavity radius, m

T Temperature, K

T0 Initial body temperature, K

Td Temperature at a depthd, K

Ts True surface temperature, K

TsIHC Surface temperature returned by inverse heat conduction

algorithm, K

t Time, s

x Distance from surface, m

x Vector of unknown heat ßuxes, Eq 10

of the surface heat ßux versus surface temperature throughout the quenchingprocess During the quenching process, rapid nonlinear changes in the heat ßuxoccur as the surface cools and the boiling regime progresses from lm boiling,through transition and nucleate boiling, and ultimately into convective coolingwithout boiling

The most common way of estimating the transient surface heat ßux duringquenching is to perform an inverse heat conduction 共IHC兲 analysis 关5 9兴 ontime-resolved temperature measurements carried out at a known depth withinthe sample Invariably, however, the measured temperatures contain someerror due to the thermocouple instillation in the sample as well as the inherentthermocouple characteristics, like measurement noise caused by EMF interfer-ence and heat conduction from the junction through the wires These errors are

WELLS AND DAUN, doi:10.1520/JAI101818 17

exacerbated by the extreme heat ßux conditions typical of water quenching,which makes the inverse problem especially ill-posed This paper describes thebest experimental practices that have been developed to mitigate these errors,followed by a review of the analytical techniques used to stabilize the ill-posedinverse heat conduction problem Finally, the sensitivities of the quenchingproblem to uncertainties in the thermophysical parameters are addressed andquanti ed.

Experimental Temperature Measurement

Time-resolved temperature measurements in quenching experiments are mostoften done using a thermocouple, which consists of two dissimilar metalsjoined together at two ends called junctions.ÓWhen the junctions are at dif-ferent temperatures, a voltage is produced that can be used to infer the tem-perature difference, and the temperature of one junction if the temperature ofthe other junction is known One junction is formed in the voltmeter that mea-sures the thermocouple voltage, while the other junction can be made either bywelding the two thermocouple wires together into a bead or by welding eachindividual wire directly to the product being quenched, so that an intrinsicjunction is formed These two types of junctions are illustrated in Fig 1 Theintrinsic junction necessitates a two-point thermocouple measurement, whichprovides the average temperature of the thermocouple wire at the product sur-face Instrumentation of the thermocouples using an intrinsic junction is pre-ferred during quench situations since it results in less mass at the junction andconsequently a more rapid sensor response with changing surface heat ßux.Thermocouples can either be attached directly onto the quenched surface

or instrumented within the sample being quenched Welding the thermocoupledirectly to the surface of the quenched product may at rst appear attractive asthere is no time lag in the measured surface thermal history Unfortunately, thisoption is usually precluded by the severe perturbing effect the thermocouplehas on the surface temperature, since the thermocouple wires act as pin nsÓ

in the presence of the quenching medium and conduct heat away from thesurface 关10,11兴 Moreover, the thermocouple probe and wires also interferewith the ßow of the quenched medium over the surface

A better alternative is to weld or position the thermocouple at the bottom of

a cavity at a known interior depth relative to the quenched surface Typicallythis is done very close to the surface共⬃1 mm兲 to minimize the ill-posedness ofthe inverse conduction problem that must be solved to recover the surface heatßux, as described later in the paper Figure 1 shows a schematic of a typicalthermocouple used to measure the temperature history in a sample duringquenching The thermocouple wires are separated by an insulating materialsuch as mullite or MgO, and the entire assembly is then covered using a metalsheath The presence of a submerged thermocouple inside a quenched solidstill disturbs the temperature eld, although to a much lesser extent than would

be the case for a surface-mounted probe In situations where the material rounding the thermocouple has a much higher thermal conductivity than thecavity, which is normally the case in metal quenching, the measured junction

sur-temperature is lower than the sur-temperature at the same depth but far away fromthe thermocouple This effect is caused by the low thermal conductivity of thecavity relative to the surrounding material, which prevents heat from ßowingfrom within the bulk material to the quenched surface, as shown schematically

in Fig 2 The thermal shadowÓ of the thermocouple cavity causes the therms to curve around the thermocouple tip as shown Fig 3; this effect usu-ally extends about two diameters on either side of the cavity The temperaturedisturbance grows as the surface heat ßux and cavity diameter increase, and as

iso-Thermocouplesheath

Junction

(a)

MgO InsulationChromel wire

Alumel wire

(b)

weld

(c)

FIG 1 Sketch of K-type thermocouple showing 共a兲 the outward appearance of

ther-mocouple and 共b兲 cross-sectional view.

WELLS AND DAUN, doi:10.1520/JAI101818 19

q s (t)

d

2rFIG 2 Schematic showing the effect of the thermocouple hole on heat flow within the quenched surface.

FIG 3 Disturbance in the temperature field due to the presence of a thermocouple hole oriented parallel to the heat flow 共r=1.0 mm, d=2.0 mm, k=100 W/máK, qs

= 0.5 MW/ m2兲 关13兴.

the conductivity of the material decreases 关12兴 The greatest thermal bances are experienced when the thermocouple cavity is parallel to the heatßow共perpendicular to the quenched surface兲, while this effect is minimized byaligning the cavity parallel to the surface or perpendicular to the heat ßow关12兴.

distur-In industrial practice, however, this orientation is often precluded by the etry of the product being quenched

geom-The extent of the thermal perturbation caused by the presence of an rior thermocouple can be estimated from the Biot number, de ned by

inte-Bi =hLc

whereh is the heat transfer coef cient at the surface of the quenched sample,

Lcis the characteristic dimension of the material being cooled, in this case theradius of the thermocouple cavity, r and k is the thermal conductivity of the

material being quenched If the Biot number is smaller than 0.1, heat ducted laterally at the thermocouple tip compensates for the thermal resistance

con-of the cavity, so the presence con-of the thermocouple will not signi cantly disturbthe temperature eld around it On the other hand, a Biot number higher than0.1 means that the conduction resistance of the quenched object is signi cantlyhigher than the resistance associated with convection heat transfer at the sur-face, and consequently the thermocouple cavity must be included in the heattransfer model Figure 4 shows the quench conditions for different materialswhere the thermocouple cavity must be included in the heat transfer model toobtain accurate results; each line corresponds toBi= 0.1 for different TC radii.Experimental conditions corresponding to points above the line indicate Bi

⬎0.1, in which case the presence of the cavity can be ignored, while pointsbelow the line denoteBi 0.1, which shows that the cavity has a non-negligibleeffect on the temperature eld For steel, which has a thermal conductivity of

20 W ám−1áK−1, the thermocouple cavity must be taken into consideration inthe analysis when h is greater than 1000 W ám−2áK−1 for r = 2 mm, or

2000 W ám−2áK−1 for r = 0.5 mm On the other hand, the higher thermal

con-ductivity of the aluminum alloy 共160 W/máK兲 means that the thermocouplecavity can be ignored for convection coef cients below 16, 000 W / m2áK 共r

= 2 mm兲 or 8000 W/m2áK共r=4 mm兲.

Thermocouple Compensation Technique

One way of dealing with perturbing effect of the thermocouple cavity on thetemperature eld is based on observing that the temperature measured by thethermocouple located at a depthd from the quenched surface corresponds to a

smaller depth,dein an unperturbed temperature eld This equivalent depthÓ

turbed surface, but closer to the surface than the actual thermocouple position

d A negative equivalent depth means that the thermocouple is too close to the

surface and the disturbance of the temperature eld cannot be corrected lytically

ana-The factor 1.1 was rst recommended by Beck and Hurwicz关14兴 for to-radius ratios close to unity These authors developed a null-calorimeterÓtechnique for designing quenching experiments in which Eq 2 is used to calcu-late the sensor depthd that would make de= 0, so that the temperature mea-sured by the thermocouple corresponds to the surface temperature of the un-perturbed temperature eld共i.e., in the absence of the thermocouple hole兲 Inquenching experiments, however, it is rarely possible to position the thermo-couple with suf cient precision, and the sensor depth is instead determined bysectioning the sample or through non-destructive techniques such as X-rayphotography and ultrasound In this case, the measured cavity depth and ra-dius is substituted into Eq 2 to retrieve the equivalent depthde, which in turn isused in the inverse conduction algorithm described later in the paper Theequivalent depth method is also useful when the disturbance of the surfacetemperature by the thermocouple cavity 共e.g., hot/cold spots兲 must be mini-mized, since this technique allows a greater distance between the thermocoupleand the surface than the null calorimeter technique

depth-By supplying the equivalent depthd in place of the actual depthd of the

FIG 4 Effect of thermal conductivity k, heat transfer coefficient h, and thermocouple hole radius r on the Biot number Under quench conditions where the Biot number is greater than or equal to 0.1 the thermocouple hole will need to be compensated for in the IHC 关12兴.

thermocouple to the inverse heat conduction analysis, the algorithm returnsthe surface heat ßuxqsand temperatureTs,IHCthat correspond to the unper-turbed temperature eld The surface heat ßux predicted using this method isthe rst step to generating the boiling curve for this quench condition Onefurther correction is needed to complete the boiling curve and this involvescorrecting the surface temperatureTs,IHC predicted by the inverse conductionalgorithm As shown in Fig 3, the disturbance of the thermocouple hole on thethermal eld perpetuates to the surface of the material, such that the surfacetemperature directly above the thermocouple is slightly colder than the surfacetemperature in a region unperturbed by the presence of the thermocouple.Hence, a further correction is needed to infer the actual surface temperatureabove the thermocouple, which corresponds to the surface heat ßux calculatedthere using the IHC algorithm and is given as

The correction is a function of the cavity radius and depth,r and d, respectively,

and the thermal conductivity of the surrounding material,k:

⌬Ts=Aqs共Br2+ Cr兲exp共Dd兲

whereA, B, C, and D are all tting constants are given in Table 1关15兴 Figure 5shows a comparison of the predicted boiling curves using these correction tech-niques against the actual measured one

Inverse Heat Conduction Models

Once the time-resolved thermocouple data has been collected, it is then used toinfer the transient surface heat ßux by carrying out an inverse heat conductionanalysis Figure 6 shows a schematic of how measured thermocouple data isintegrated with an IHC model to calculate the surface heat ßux history duringthe quench operation

Heat Conduction Equation

Depending on the geometry of the product, the heat conduction equations erning the ßow of heat in the quenched material can be one, two, or threedimensional For a one-dimensional, transient conduction problem, the heattransfer and temperature eld are related by

gov-TABLE 1 Fitting constants 关15兴.

Inverse algorithm

Model

FIG 6 Schematic of methodology to determine transient heat flux boundary condition data during a quench operation based on internal temperature measurements.

When performing an inverse heat conduction analysis to predictqs共t兲 from

a set of time-resolved thermocouple data, the accuracy of the calculated results

is limited by the uncertainty in the material thermophysical properties such asthermal conductivity and volumetric speci c heat, and the precision of thethermocouple location Additionally, measurement noise in the thermocoupledata caused by EMF interference can substantially impact the accuracy of therecovered heat ßux due to the ill-posedness of the problem, as described below

Inverse Algorithm

The process of determining the transient surface heat ßux from time-resolveddata measured by a subsurface thermocouple is equivalent to solving a bound-ary inverse conduction problem that is mathematically ill-posed To understandthe challenges inherent in solving this type of problem, it is useful to reduce theproblem to the one shown in Fig 7, where the solid is modeled as semi-in nite,the thermal conductivity and thermal diffusivity are independent of tempera-ture, and the thermocouple measurement corresponds to the temperature at apoint at a depthd below the surface of the medium.共The perturbing effect ofthe thermocouple cavity on the temperature eld can be accommodated usingthe equivalent depth technique described above.兲

In this simpli ed problem the temperature measurementT d 共t兲 and the

sur-face heat ßuxqs 共t兲 are related by an integral equation

In inverse conduction problems Eq 8 is called Duhamel s integral equation.

Ifqs共t兲 were known a priori, determining the temperature response Td共t兲 would

be done simply by carrying out the integral; this is the forward problem The object of this analysis, however, is to solve the inverse problem, i.e., to infer qs共t兲

fromTd共t兲 values measured at a discrete set of times, which is mathematically

ill-posed Ill-posed problems were rst de ned by the mathematician JacquesHadamard关17兴, who stated that for problems to be amenable to mathematicalanalysis, they must:共i兲 have a solution, that is, 共ii兲 unique and 共iii兲 numericallystable, meaning that small perturbations to the input data should cause equallysmall errors in the recovered solution Any problem that does not satisfy each

of these criteria is mathematically ill-posed Inverse heat conduction problems

like the one described here satisfy Hadamard s rst and second criteria: theremust exist some heat ßux at the boundary that is responsible for experimentallyobserved temperature data; furthermore, in the limit of in nitesimal time steps

it can be reasoned that in nitesimal temperature changesdT correspond to a

unique increment in the surface heat ßuxdqsstarting fromt = 0.

However, inverse heat conduction problems violate Hadamard s third rion, that small errors in the thermocouple measurements, unavoidable in anexperimental setting, are ampli ed by deconvolution of Eq 8 into much largererrors in the recovered surface heat ßux history This behavior arises from twosources关18兴: rst, the inßuence of the heat input at t=0 on Td共t兲 is both lagged

crite-and damped with respect to the surface temperature; crite-and secondly, the term effect of a variation inqs共t*兲, say ⌬qs共t*兲, causes an extremely small change

long-inTd共t兲, ⌬Td共t兲, when t−t*becomes large due to the smoothing properties of thekernelk 共t,t*兲 When the problem is written in its inverse form to nd qs共t兲 from

Td共t兲, the increment in thermocouple temperature ⌬Td共t兲 that indicates the

existence of ⌬qs共t*兲 is easily overwhelmed by the measurement noise in thethermocouple reading

The extent of ill-posedness can be better understood and quanti ed by writing Eq 8 in matrix form:

where b and x correspond to Td共t兲 and qs共t兲, respectively, and A is a

lower-triangular matrix In experimental analysisb contains thermocouple

measure-ments sampled at uniformly spaced time intervals of⌬t, x contains parameters

that specifyqs共t兲, and A is derived from the kernel function if it is available, or

a numerical approximation ofk 共t,t*兲 otherwise The simplest model for q is to

assume a uniform heat ßux between sampling times as shown in Fig 8, atechnique rst used by Stolz关20兴 to analyze data from a quenching experiment.For this problem,xi = q si,t 僆共t i−1 , t i 兲, b i = k / 兵dá关Td共t i 兲−T0兴其 and the elements of

ierfc共u兲 =冑1␲exp共− u

When the problem is written this way, A depends only of the Fourier number,

⌬Fo=␣⌬t/d2, where⌬t is the sampling increment between thermocouple

read-ings

Because A is lower-triangular x can be found ef ciently by

back-substitution, which was rst done by Stolz关20兴 共Consequently, this procedure

is sometimes called the Stolz methodÓand A is the Stolz matrixÓ关19兴.兲 Inreality, however,b = bexact+␦b, where␦b contains measurement noise, and as a

result the predicted heat ßuxes are also contaminated with errors, x = xexact

deconvolution can be estimated from关18,21兴

FIG 8 Heat flux at the boundary is modeled by uniformly-spaced “steps.”

WELLS AND DAUN, doi:10.1520/JAI101818 27