Analytical FiniteElementModelingandExperimentalspraycooling

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METALLURGICAL AND MATERIALS TRANSACTIONS A VOLUME 29A, MAY 19981485

Analytical/Finite-Element Modeling and ExperimentalVerification of Spray-Cooling Process in Steel

R. THOMAS, M. GANESA-PILLAI, P.B. ASWATH, K.L. LAWRENCE and A. HAJI-SHEIKH

An atomizer is a helpful tool that can be used to tailor the cooling rate of steel from the processingtemperature in order to get desired properties. It is important to determine the temperature distribution

in a specimen subjected to cooling by an atomized spray. A finite-element model for transient heattransfer and thermal-stress analysis is developed to determine the temperature and thermal-stressdistribution. The results of the finite-element heat-transfer model are compared with a finite-differencemodel. The heat-transfer model describes the heat-transfer processes in an AISI 4140 steel cylindersubjected to controlled atomized spray cooling from an initial temperature of 1273 K. The temper-ature fields predicted by the model are used both to predict the resulting microstructure using con-tinuous cooling transformation (CCT) diagrams and as an input for the thermal-stress model to predictthe occurrence of quench cracks. The thermal-stress model incorporates temperature-dependent ma-terial properties, heat generation due to phase changes, elastoplastic behavior of steel, and the vol-umetric expansion associated with the formation of martensite. The results of the finite-element modelare verified experimentally by recording temperature profiles, obtaining micrographs, and recordingthe occurrence of quench cracks.

I. INTRODUCTION

THE need for new materials has spurred innovation andresearch in metallurgical processing operations. In order toimprove and optimize productivity and quality, industriessuch as forging, welding, casting, and heat treatment areadopting process-modeling techniques to reduce defects,scrap, design lead time, and cost.[1,2] In most forge shops,the forged parts are piled together and allowed to cool inair. This process is very slow and a large number of partsare produced before the first part is cooled down and takenfor preliminary inspection. Moreover, the final productneeds to be heat treated to meet the customers specifica-

tion. Therefore, if specific properties can be engineered bymicrostructural control of alloys during the forming processitself, it would lead to substantial economic savings.

Accelerated cooling by quenching has been demonstratedas a possible method to get desired properties.[36] However,this method is limited to low-carbon and microalloyedsteels because of the possibility of quench cracks in me-dium- and high-carbon and alloy steels. Microalloyedsteels, by virtue of their low alloy content, are not veryhardenable and do not pose a significant challenge. How-ever, low-alloy steels contain larger amounts of alloyingadditions like Mo and Cr, which increase the hardenabilityof the steel and therefore increase the chances of quench-crack formation. Furthermore, the quenching process does

not facilitate the tailoring of cooling rates unless one usesa variety of quenching media and agitation.[7,8]

Water spray cooling using jets of water was found useful

R. THOMAS, Research and Development Engineer, is with Komag,San Jose, CA 95131. M. GANESA-PILLAI, Graduate Student, P.B.ASWATH, Associate Professor, and K.L. LAWRENCE and A. HAJI-SHEIKH, Professors, are with the Department of Mechanical andAerospace Engineering, University of Texas at Arlington, Arlington, TX76019.

Manuscript submitted March 23, 1995.

in controlling the cooling rate of hot-rolled medium-carbonsteel bars.[9] Accelerated cooling experiments in high-strength low-alloy (HSLA) steels using fog-jet nozzles in-dicate the possibilities of controlling the temperature gra-dient while cooling.[10] In an earlier work,[11] it wasexperimentally shown that an atomizer with a mixture ofair and water could be successfully employed to tailor thehardness and tensile strength of a microalloyed steel. Therelationships between composition, microstructure, and me-chanical properties have been studied by previous investi-gators.[1215] Campbell et al.[13,14,15] have developed amathematical model that incorporates the heat-transfer,transformation-kinetics, and property-structure-compositionrelationships to predict properties of plain carbon steels. Forthe analysis therein, the lumped heat-capacity method andthe one-dimensional (1-D) implicit finite-difference methodgave comparable results. The lumped heat-capacity modelassumes that the temperature gradient between the surfaceand the interior is negligible, which is indeed true in aircooling, but the model is valid only when the Biot number(Bi) is small (Bi 0.1).

This article deals with experiments and modeling ofspray cooling of AISI 4140, which is a typical alloy steel,from high temperatures by a two-component (air-water)spray. The experimental procedure is discussed briefly inSection II. Spray cooling is a new entrant in high-heat flux

cooling and the heat-transfer relations are not well estab-lished as they are in single-phase flow. The atomized sprayconsists of small liquid droplets in a conical jet of air. Byincreasing or decreasing the amount of liquid in the spray,one can increase or decrease the cooling rate at the surface.Therefore, the cooling rate can range from that of forced-convection air cooling to a quenching process.

In metallurgical applications, water is the liquid of choicebecause of its low cost. The Biot number in such a processis very high, which means that there is a large gradient intemperature from the surface to the interior. For example,here the minimum conductivity (at the highest temperature)


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1486VOLUME 29A, MAY 1998 METALLURGICAL AND MATERIALS TRANSACTIONS A

Table I. Nominal Composition of AISI 4140

C Cr Mn Mo P (Max) S (Max) Si

0.38 to 0.43 pct 0.8 to 1.1 pct 0.75 to 1.0 pct 0.15 to 0.25 pct 0.035 pct 0.04 pct 0.15 to 0.30 pct

Fig. 1Schematic of the atomized spray-cooling setup used to cool thecylindrical specimens.

is 45 W/m/K, the radius is 0.0191 m, and the heat-transfer

coefficient is of the order of 105

W/m2

, which makes Bi 0.1. Therefore, the lumped parameter model (validonly when Bi 0.1) is inappropriate and the heat-conduc-tion equation needs to be solved. The heat-transfer coeffi-cients at the surface have to be prescribed as boundaryconditions. The relations that describe the surface heat-transfer coefficient during various phases of the spray-cool-ing process are presented in Section III. These relationswere obtained from experimental data and Monte Carlo in-verse heat-conduction analysis by Buckingham and Haji-Sheikh.[16,17] The internal temperatures are computed by theexplicit finite-difference formulation of the conductionequation in the cylindrical coordinate system.

The article also demonstrates the prediction of quench

cracks with commercial finite-element analysis (FEA) pack-ages using available information on a complex heat-transfer

phenomenon like spray cooling. The FEA method is cur-rently the most popular modeling technique, because of theflexibility in the definition of geometry, loading and bound-ary conditions. The tasks of building or updating a modeland analyzing the results have now been greatly simplified

by powerful FEA packages. The FEA package used in thiswork is ABAQUS, which provides an extensive library offunctions to define the complex phenomena in spray cool-ing.[18] The ABAQUS package allows detailed definition ofthe material, such as temperature-dependent properties,

phase-transformation effects, and spatial and temperaturevariation of heat-transfer coefficients, which are required to

describe the spray-cooling process. It is also very easy toincorporate new information as it becomes available in thefuture, in spray cooling or in phase-transformation kinetics,so that the predictions can be improved.

The finite-element modeling described in Section IV ofthis article consists of two steps. The first step describesthe heat-transfer behavior of the material during spray cool-ing by determining the transient temperature distributionwithin the material. The temperature profiles generated areused to predict the resulting microstructure of the material.Subsequently, these temperatures are used to calculate ther-mal stresses during cooling, which are used in turn to pre-

dict the occurrence of quench cracks. The data from thefinite-difference heat-transfer analysis and the finite-ele-ment predictions of temperatures are compared to measured

temperatures at a few selected sites after testing. This com-parison is for verifying the finite-element model. Samplesare removed from different locations of each test material,

polished, and used for metallographic studies. The tests areperformed to show microstructures and possible formationof cracks. A correlation is made between the predictedhardness of the steel using continuous cooling transforma-tion (CCT) diagrams available in the literature and the hard-ness measured by Jominy quench tests at various locations.In addition, a correlation is made between the actual mi-crostructure and the microstructure prediction based on thecooling rates. The results and conclusions are presented inSections V and VI, respectively.

II. EXPERIMENTAL APPARATUS ANDPROCEDURE

The material used for all experiments is AISI 4140 andits typical composition is shown in Table I. The experi-mental runs were conducted on an AISI 4140 steel bar,0.0381 m in diameter and 0.1016-m long. A schematic ofthe spray-cooling apparatus is shown in Figure 1. The ap-

paratus consists of a BETE 1/4XA-PR150-B atomizer, ob-tained from Bete Fog Nozzle Inc. (Greenfield, MA), thatinternally mixes compressed air and water in an annularregion within the nozzle to form a fine conical spray. Thewater is supplied by a water pump and its flow rate is con-trolled by a needle valve. The flow rate is measured by a

rotameter to aid manual flow control by the operator andby a flow meter, which sends the readings to a data acqui-sition system (DAS). The water pressure is monitored by a

pressure transducer, which sends data to the DAS, and alsoby a pressure gage. Compressed air is supplied by an aircompressor and is controlled by a ball valve. The air-flowrate is monitored, in the same manner as the water, by aflow meter, a pressure transducer, and a pressure gage. Theair-flow meter provides an electronic display of the flowrate and also sends these readings to the DAS. The waterand compressed-air lines meet at the atomizer, which usesthe energy of the compressed air to atomize the water, andthe mixture is expelled from the nozzle as a fine spray.More information on the nozzle characteristics is available

in a previous publication.[11]The specimen is first heated to a temperature of 1273 K

in a furnace and then quickly transferred into a fixture andexposed to the spray. During cooling, the specimen is lo-cated 0.41 m from the tip of the nozzle and in such a man-ner that it is upright and centered with respect to the spray.The particular distance was chosen because the spray wascharacterized only at that distance and the spray was fullydeveloped with a fairly uniform distribution of dropletsover a radial distance of 5 cm.[11] In actual practice, thespray completely engulfs the specimen. Specimens arecooled at four different cooling rates by varying the flow


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Table II. Experimental Variables

SpecimenNumber

Air Flow Rate(kg/s)

Water Flow Rate(kg/s)

1 as received as received 2 0.000732 0.00842 11.53 0.000698 0.00602 8.64 0.000778 0.00405 5.135 0.000819 0.00184 2.246 0.000909 0.00112 1.1

7 0.000997 0 0

Fig. 2Location of the thermocouples and coordinate conventions usedin the heat-transfer and finite-element models. Hardness was measuredalong the lines shown in the pictorial view, in the plane of thethermocouple locations.

Fig. 3Schematic of the heat-transfer profile showing different heat-

transfer regions during spray cooling using an atomizer (used to deriveEq. [6]).

rates of water and air. The flow rates and pressures recordedfor the four cooling rates are shown in Table II. The coolingrate increases with the increasing water-flow rate. A param-eter , defined as the ratio of the water-mass-flow rate tothe air-mass-flow rate, is used to represent each coolingrate. This parameter empirically relates the heat-transfer co-efficients with the air- and water-flow rates.

Four K-type thermocouples located as shown in Figure2 are used to record the temperatures at these locations

during cooling. To withstand the high temperatures, ther-mocouples sheathed in steel are used. The thermocouplesare fixed to the locations using a high-temperature PYRO-PUTTY,* which is a mixture of steel powder and ceramic

*PYRO-PUTTY is a trademark of Aremco Products, Ossining, NY.

compounds. Following the spray-cooling runs, the speci-mens are cut for microstructural studies, and quench cracks,if any, are recorded. In addition, Rockwell C hardness

measurements are made in a direction perpendicular to thespray and parallel to the spray at the midsection of thecylindrical specimen.

III. HEAT-TRANSFER ISSUES IN SPRAYCOOLING

The heat-transfer process in spray cooling is complex andwill not readily submit to numerical simulation. The factorsthat influence the local heat-transfer rate are droplet distri-

bution, thermal radiation, thermophysical properties, andkinematic quantities. The distribution of droplet size de-

pends on air pressure, water-flow rate, and nozzle charac-

teristics. The velocity field in the spray has axial and radialdependence and is usually accompanied by plume oscilla-tions. The thermophysical properties of the gas phase de-

pend on the temperature and mole fraction of theconstituents. The high-temperature nature of the heat-treat-ing process introduces additional factors that can influencethe prediction of temperature. The determination of tem-

perature- and location-dependent heat-transfer coefficientsfor the spray-cooling process is based on the results ofBuckingham and Haji-Sheikh.[16,17] This section outlines thederivation of these coefficients for the specimen geometryunder consideration.

The heat-transfer process during spray cooling can bedivided into three distinct regions: (1) a radiation-domi-

nated region where, depending on the amount of water inthe spray and the temperature of the surface, the waterdroplets will partially or totally vaporize before getting nearthe surface of the specimen and the air-water vapor-air mix-ture will cool the surface by convection; (2) a convection-dominated region, where, as the surface temperature de-creases, the water droplets begin to arrive near the surfaceand enter the boundary layer over the specimen and the rateof heat transfer will increase rapidly; and (3) a transitionregion, which links the previous two regions together. Inthe convection-dominated region, the heat-transfer mecha-nisms are more complex and the heat-transfer coefficient isvery large. However, when the Biot number is large, largeerrors in the value of the Biot number will have a small

influence on the computed internal temperature. A sche-matic form of the variation of heat flux with temperatureat the surface of a cylinder subject to spray cooling isshown in Figure 3. This schematic form adapted from theresults reported in Reference 17 is used in Section IIIC toderive the heat-transfer relations for the transition region.Since only the trends are important in the derivation, thevalues of the heat-transfer coefficient in Figure 3 do notmatter. Surface heat-flux profiles of a cylinder subjected tospray cooling are reported in the literature.[17,19] The peakheat flux is of the order of 106 W/m2.

The temperature Tmax is defined as the temperature at


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which the heat flux q at the stagnation point is maximum.The radiation-dominated heat-transfer process exists at tem-

peratures far above Tmax. In the radiation-dominated region,the liquid droplets will evaporate and heat transfer will beattributable to a homogeneous mixture of air and water va-

por; it is assumed that the mole fractions have no spatialdependence. The Tmax is also the temperature at which theconvective region starts.

The heat-transfer coefficient at the stagnation point (h0)

is calculated first, and then, using approximate spatial re-lationships, the heat-transfer coefficient at any point at that

particular time instant can be calculated as a function of h0.It is assumed that the gas is fully mixed and that there isno liquid in the spray for all locations with 90 deg.An accurate prediction of local values of the heat-transfercoefficient is a difficult task because the spatial variationsdepend on the distribution of droplet size, partial evapora-tion of liquid by radiation, and kinematic quantities. Forthis reason, only an estimation of the spatial variation ofthe heat-transfer coefficient is possible. The calculation ofh0 is outlined in the following sections for each of the heat-transfer regions.

In general, the local surface heat transfer is computed by

Eq. [1].

4 4q h(T T ) (T T ) [1]w a w

where is the emissivity of alloy steel and is the StefanBoltzmann constant. The local heat-transfer coefficient is h hR in the radiation-dominated region, and h hc whenthe phase-change process takes place in the boundary layer.

A. Radiative Region

The computations begin by using the heat-transfer rela-tion for cross flow over a cylinder when Tw Tmax. TheReynolds number is defined by Eq. [2].

u (1 ) D Ta 0 0 satRe [2]m

Tm a

where a is the density of the mixture at the adiabatic spraytemperature and m is the viscosity of the mixture calcu-lated at the mean bulk temperature. Some explanation isdue here for the special definition of the Reynolds number.In the radiation-dominated region, the assumption is that allthe liquid in the spray will evaporate before it arrives at thecylinder. In that case, the Nusselt-number data must exhibita behavior similar to cross flow of a single-phase fluid overa cylinder, where the Nusselt number depends on the Reyn-olds number and the Prandtl number. The process is very

complex, since the heat transfer is attributable to a mixtureof air and superheated water vapor. The water-vapor pro-duction contributes to an increase in the volume flow rateof the gas and a consequent widening of the jet. The factor(1 ) in the definition of Reynolds number accounts forthe vapor production in the air, assuming that the air andwater vapor are fully mixed. The quantity Ta/Tsat includesthe effect of thermal expansion of the gas before reachingthe boundary layer.

When radiation dominates, the heat transfer is to a gas-eous mixture, and therefore the heat-transfer coefficient atthe stagnation point,hR,0, can be calculated using the for-

mula for single-phase flow over a cylinder. For the range104 Rem 2 10

5, the following Eq. [3][20] holds:

km 0.333h [82.81 0.003014 (Re Pr )R,0 m mD0

9 0.333 2 3.265 10 (Re Pr ) ] [3]m m

where hR,0 is the heat-transfer coefficient at the stagnationpoint for the radiation-dominated region. The thermophys-ical properties of air are taken from Incropera and DeWitt [21]

and the thermophysical properties of water vapor from Haaret al.[22] The method for obtaining the thermophysical prop-erties for a mixture of low-pressure gases is in Bird et al.[23]

B. Convective Region

When the stagnation-point temperature T0 is below Tmax,the convective region begins, and the water droplets reachthe surface of the specimen and boil away. In this region,the heat-transfer coefficient at the stagnation point hc,0 isgiven by Eq. [4].[17]

5 0 .8 4 0 .7 51.9 10 k (T T ) air w,0 ah [4]c,0

D0

where Tw,0 is the wall temperature at the stagnation point,Ta is the stream temperature under adiabatic conditions, is the mass-of-water to mass-of-air ratio, hc,0 is the heat-transfer coefficient at the stagnation point, D0 is the diam-eter of the cylinder, and kair is the thermal conductivity ofair computed at the mean film temperature.

C. Transition Region

The transition region is a poorly defined region that linksthe convective and radiative heat-transfer regions. The heat-transfer coefficient at the stagnation point h0 is approxi-mated by a linking function defined as follows:

h a b exp [(T T )] [5]0 w,0 max

where a, b, and are constants to be determined using thefollowing three conditions:

h h when T T ;0 max w,0 max

h h when T T is large; and0 R,0 w,0 max

dq/dt 0 or dq/dT 0 when T T .w,0 max

The form of the curve given by Eq. [5] is based on thetrend of the heat-flux variation with temperature (Figure 3).The first and the third conditions are based on the existenceof the maximum heat flux and hence the maximum heat-transfer coefficient when Tw,0 Tmax. The second conditionstipulates that when Tw,0 is much greater than Tmax, h0 should

tend to hR,0. The exponential term disappears when Tw,0 Tmax and therefore the form satisfies the requirements. Itshould be noticed that in this case, h0 is a function of hR,0,defined using radiative heat-transfer conditions. The valuesof a, b, and satisfying the conditions are as follows:

a h ,R,0

b h h , andmax R,0

hmax

[(h h )(T T )]max R,0 max a

Substituting these values in Eq. [5] we get the following:


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Fig. 4Variation of the heat-transfer coefficient with the angularcoordinate .[17]

Fig. 5Geometry of the axisymmetric finite-element model used todetermine the temperature distribution in spray cooling.

Fig. 6Schematic of the 2-D finite-element model used to determine thetemperature distribution.

T Tw,0 maxFACT

T Tmax ah h (h h ) e ,0 R,0 max 0 [6]

hmaxwhere FACT

h hmax R,0

D. Spatial Relationships

Empirical spatial relations relate the heat-transfer coef-ficient at any point to that at the stagnation point, h0. Thesurface of the cylinder facing the spray experiences both

radiative and convective heat transfer. The surface of thecylinder away from the spray is assumed to be in contactwith a mixture of water vapor and air only. The conventionfor the spatial variables and Z is shown in Figure 2. Itshould be noted that when 90 deg, h0 may be calculatedthrough the convective or the radiative relations, dependingon the temperature; i.e., if Tw Tmax, h0 hR,0, and if Tw Tmax, h0 hc,0. However, for 90 deg, h0 is calculatedusing the radiative relation. The spatial relations with

measured in radians are as follows:

h h 1 0.5 (1 B)0

1 4C cos 22 cos (1 0.18 Z ) [7]

4C 4C

where B 0.385 0.0426 0, C cos 0, 0 1.995 0.016 0, 0 when 4, and 0 4 when 4.

The angular dependence given previously is based on theheat-transfer data of Buckingham and Haji-Sheikh.[17] A

plot of the spatial variation of the heat-transfer coefficientaccording to Eq. [7] is shown in Figure 4. In the radiation-

dominated region, h0 approaches hR,0. Using the aforemen-tioned formulas the heat-transfer coefficient at any point onthe surface of the cylinder can be calculated, given the heat-transfer coefficient at the stagnation point.

IV. The Finite-Element Model

The FEA is divided into two phases: heat-transfer analysisand thermal-stress analysis. Three models are evaluated forthe heat-transfer analysis: an axisymmetric model, a two-dimensional (2-D) model, and a three-dimensional (3-D)model (Figures 5, 6, and 7, respectively). The axisymmetricmodel assumes that the heat-transfer coefficient is indepen-dent of the coordinate whereas the 2-D model assumesthat the heat-transfer coefficient is independent of the Zcoordinate. Physically, these assumptions mean that thereis no heat flow in the azimuthal direction for the axisym-metric case and in the Z direction for the 2-D case. It isshown subsequently that these approximations do not in-duce significant error in the solution. Hence, the thermal-stress analysis employs only an axisymmetric model. The

boundary conditions imposed for the three models areshown in Figure 8.

A. Heat-Transfer Module of the Finite-Element Model

The heat-transfer model defines the process of spraycooling and the thermal behavior of the material over thetemperature range under consideration. The governingequation for heat transfer in the cylinder for all the threecases analyzed is as follows:

T k T q "' C [8]p

t

The initial condition for all cases is as follows: at t 0,T Tin 1273 K. The term q is the heat generated perunit volume. This term is invoked when there is an exo-thermic or endothermic phase transformation. For the axi-symmetric case, the following holds:


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Fig. 7Schematic of the 3-D finite-element model used to determine thetemperature distribution.

Fig. 8Boundary conditions used in the finite-element models.

Fig. 9Temperature dependence of the thermal conductivity and specificheat of AISI 4140.[24]

1 T T k T kr k

r r r z z

Tk h(T T ) at r 0.01905 m, t 0w a

r

Tk 0 at r 0, t 0 [9]

r

Tk 0 at z 0, t 0

z

Tk 0 at z 0.0508 m, t 0

z

For the 2-D case, the following holds:

1 T 1 T k T kr k 2r r r r

Tk h(T T ) at r 0.01905 m, t 0w a

r

Tk 0 at r 0, t 0 [10]

r

Tk 0 at 0, t 0

Tk 0 at , t 0

For the 3-D case, the following holds:

1 T 1 T T k T kr k k 2r r r r z z

Tk 0 at z 0 and z 0.0508 m, t 0

z

Tk h (T T ) at r 0.01905 m, t 0w a

r

Tk 0 at r 0, t 0

r

Tk 0 at 0 and , t 0

[11]

The thermal behavior of materials is defined by speci-fying the temperature dependence of thermal conductivityand specific-heat capacity[24] and the latent heat effects dueto phase transformations (Figure 9). The heat generated bythe austenite-to-ferrite/pearlite phase transformation resultsin a sharp peak for the specific-heat curve. The quantity ofheat generated is specified by calculating the area under the

peak of the dotted line in the Cp vs T curve. This latent

heat effect is suppressed when the cooling rate is too highfor the transformation to take place and hence is incorpo-rated in the model only for the lower cooling rates. In thecases where the phase transformation does occur, a latentheat of 106 J/kg is defined between 1013 and 1053 K. Theregion within which the heat generation is specified is ob-tained by determining the temperature range in which theinflection occurs in the experimental cooling curve.[24]

The spray-cooling process is characterized by specifyinglocation- and temperature-dependent heat-transfer coeffi-cients, as defined previously. This definition is straightfor-ward for the 3-D case because both the spatial coordinates

are explicitly defined in the model geometry. For the 2-Dcase, the heat-transfer coefficients are defined by specifyingthat Z 0 in the spatial relationships. This is justified be-cause the heat-transfer coefficient varies by less than 10 pctin the range of Z considered (Figure 4.). The variation of


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Fig. 10Temperature dependence of the tensile strength and elasticmodulus of AISI 4140 steel.[24]

Fig. 11Thermal expansion coefficient with and without phasetransformation in AISI 4140 steel.

heat-transfer coefficient with is significant and hencechoosing an arbitrary value for would lead to large errors.Therefore, for the axisymmetric case, the average of theheat-transfer coefficients over the range 0 is cal-culated by integrating Eq. [7] from 0 to and dividing

by , and this average value is specified over the circum-ference. The temperature profiles at different locations forthese experiments, calculated by the finite-differencemethod, are reported in an earlier publication.[19]

B. The Thermal-Stress Module

This module calculates the thermal-stress distributionwithin the specimen during cooling. The model imports thetemperatures from the heat-transfer analysis to calculatestresses induced due to thermal gradients and phase trans-formations, which in turn are used to predict the occurrenceof quench cracks in the specimen. This analysis is con-ducted using only the axisymmetric model, as mentioned

previously. Elastoplastic-material behavior is specified bydefining temperature-dependent material properties such aselastic modulus and tensile strength (Figure 10). Informa-tion on these properties is not available for the range of

temperatures considered in this article. In the calculations,the elastic modulus and tensile strength are considered to

be the same as those at 800 K, for all higher temperatures.This is shown to be a conservative estimate. The elasticmodulus decreases with increasing temperature. The tensilestress is calculated by the product of the elastic modulusand the thermal strain. If the calculated stress is more thanthe ultimate tensile strength, the model predicts a crack.Since an elastic modulus higher than the true value is used,

the stress is being overestimated. It is true that the tensilestrength to which the calculated stress is compared is alsohigher than the true value. However, tensile strength at 860K is already very small and keeping it the same at highertemperatures will not alter the predictions at all. Therefore,the predictions are conservative and satisfactory. Phase-transformation effects are modeled by appropriately modi-fying the temperature dependence of the thermal-expansioncoefficients, as explained in the following sections.

Martensitic transformation

The 6 pct volumetric expansion accompanied by the mar-tensitic transformation has a significant effect on the ther-

mal-stress distribution. The reaction takes place at 615 Kin AISI 4140 when the cooling rate is high. At temperatureshigher than 615 K, the thermal-expansion coefficients areas shown by the solid line in Figure 11.[24] From Eq. [12],it can be seen that a 6 pct volumetric expansion is equiv-alent to a 2 pct linear expansion.

3 3 3 3L LV (L L) L 3Lf i i i 6 pct [12]

3 3V L L Li i

Considering a 1-D case of expansion, the thermal-expan-sion coefficient is related to the initial and final lengths byEq. [13] as follows:

f iL L (1 T) [13]0 0

where is the final length without transformation, Li is thefL0initial length, 0 is the thermal-expansion coefficient with-out considering transformation, and T is the change intemperature from the initial condition. In Eq. [13], 0 refersto the thermal-expansion coefficients given by the solid linein Figure 11, which do not consider the effect of expansiondue to the martensitic transformation. For temperatures be-low 615 K, 0 values are modified to account for the ex-

pansion in the following way. If represents the finalfLTlength, taking into account the transformation, is relatedfLTto by Eq. [14].fL0

f fL L 1.02 [14]T 0

This arises because of the 2 pct linear expansion due to the

martensitic transformation. Hence, the following holdf f iL L 1.02 L (1 T) [15]T 0 T

where T is the thermal-expansion coefficient consideringthe expansion due to the martensitic transformation, whichgives the following:

i iL (1 T) 1.02 L (1 T) [16]0 T

Simplifying, we get the following:

0.02 1.02 T0 [17]T

T


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Fig. 12Comparison of experimental cooling curves with the predictionsfrom the heat-transfer model for the thermocouple at location 2.

Fig. 13Comparison of experimental cooling curves with the predictionsfrom the axisymmetric finite-element model for the thermocouple locatedat position 2.

Fig. 14Comparison of the experimental cooling curves with predictionsfrom the 2-D finite-element model for the thermocouple located at location 2.

This relation is used to calculate the modified thermal-expansion coefficients, and these values are compared tothe original thermal-expansion coefficients in Figure 11.

V. RESULTS AND DISCUSSION

The heat-transfer module of ABAQUS predicts the tem-perature distribution within the specimen as a function oftime at the various cooling rates. These temperature profiles

are used to predict the resulting microstructure at the lo-cation under consideration and serve as an input for thethermal-stress module. The predictions made by FEA arecompared with experimental results in the following sec-tions.

A. Temperature Profiles

1. Finite-difference heat-transfer-analysis predictionsThe temperatures at various locations were calculated us-

ing the explicit finite-difference scheme and compared withthe experimental data. The temperature profiles and surfaceheat-flux history are reported in detail by Thomas et al.[19]

A cooling curve at location 2 (the center of the cylinder)

is shown in Figure 12 for comparison and discussion. De-spite the complications involved in modeling the heat trans-fer in an atomized spray, the predictions match theexperiments reasonably well. The cooling rate, as expected,is highest for 11.5, which has the largest water-flowrate. At 11.5 there is a direct transformation of aus-tenite to martensite and bainite, and hence there is no bumpin the cooling curve. However, for 2.24 and 1.1, thematch between the experiments and the heat-transfer mod-els is not accurate at temperatures below 750 K. In Figure12, the experimental temperature profiles at the lower cool-ing rates have a tendency to level off at around 750 K. Thecooling rate is slowed down here because at this tempera-ture the formation of pearlite is favored, and the latent heat

due to the austenite-to-pearlite phase transformation is re-leased. The finite-difference heat-transfer model was notable to predict the temperature profile well, because thelatent heat was not included in the calculations. The latentheat was not included because of the dependence of thetransformation temperature on the cooling rate and chemi-cal composition. At the higher cooling rates, there is a di-rect transformation from austenite to martensite and bainite.A latent heat corresponding to this transformation will bereleased, but because the cooling rates are very high, thelatent heat is quickly transferred to the external flow. There-fore, there is no response to this heat generation in thetemperature profile, whereas there is a clear slowing downof the cooling process for the lower cooling rates, wherein

the formation of pearlite is favored. A sharp increase incooling rate at approximately 615 K in the heat-transfermodel is also seen in the experimental data. This is asso-ciated with the start of the convective region when dropletsof water are able to arrive at the boundary layer withoutcompletely evaporating.

2. Finite-element predictionsTemperature profiles are generated by the axisymmetric,

2-D, and 3-D models and are compared with experimentalresults in Figures 13, 14, and 15, respectively. For the ax-isymmetric model shown in Figure 13, 2.24 and 1.1


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Fig. 15Comparison of the experimental cooling curves with predictionsfrom the 3-D finite-element model for the thermocouple located at position 2.

Fig. 16Temperature history at location 2 calculated using theaxisymmetric finite-element model for 1.1 and 2.24superimposed on the CCT diagram for AISI 4140 steel.

Fig. 17Temperature history at location 2 calculated using theaxisymmetric finite-element model for 5.13 and 11.5superimposed on the CCT diagram for AISI 4140 steel.

have an inflection in the cooling curve due to heat gener-ated by the austenite-pearlite transformation. This latent

heat effect is accurately reproduced by the model. At thehigher cooling rates, i.e., for 5.13 and 11.5, the aus-tenite-pearlite transformation does not occur and hence theassociated inflection is absent. The 2-D and 3-D model pre-dictions are shown in Figures 14 and 15, respectively. The2-D model assumes that there is no heat flux in the Z di-rection of the specimen, whereas the 3-D model does notmake this assumption. Since the 2-D and 3-D models pre-dict almost identical results, it can be concluded that thereis negligible heat flux in the Z direction.

There is no information available in the literature for thelatent heat of formation for the phase transformation fromaustenite to martensite for AISI 4140 steel. Therefore, thefinite-element calculations were initially carried out includ-

ing a fictitious but large value for the latent heat of for-mation for the austenite-to-martensite transformation. Therewas no change in the temperature profiles when comparedwith the calculations when the latent heat was not included.Therefore, in the subsequent finite-element calculations, thelatent heat was included for the austenite-to-pearlite trans-formations observed at the lower cooling rates and ne-glected for the austenite-to-martensite transformationobserved at higher cooling rates.

It is observed that despite the large variation of the heat-transfer coefficient with , the axisymmetric model givesthe best match and is therefore used to predict the micro-structures. The 3-D model does not give any specific ad-vantage and therefore is not used in subsequent

calculations. This is not surprising, considering the com-plexity of the heat-transfer phenomenon in spray cooling.

B. Microstructures

Once the cooling curves are known, the resulting micro-structure can be predicted by superimposing these coolingcurves on a CCT diagram of AISI 4140 steel. [25] When acooling curve is superimposed on the CCT diagram, the

phases that form can be determined by the regions that thecurve passes through. The steel is initially made entirely ofaustenite (A). As the austenite cools it transforms to other

phases, namely, pearlite (P), ferrite (F), bainite (B), andmartensite (M). The labeled solid lines represent the startof the various transformations, namely, austenite-to-ferrite(Fs), austenite-to-pearlite (Ps), austenite-to-bainite (Bs), andaustenite-to-martensite (Ms). The labeled regions show the

phases that exist with it. The dashed lines represent thefraction of the total austenite that has transformed. Com-

bining these two, it is possible to find out which phases andhow much of each phase will form at a particular coolingrate.

1. 1.1At this cooling rate, the curve enters the region of the

CCT diagram (Figure 16). The curve first crosses Fs intothe ferrite-forming region and then crosses Ps into the pearl-ite-forming region. At this point, only 1 pct of the austenitehas transformed and hence the fraction of ferrite formed isstill negligible. However, as discussed previously, ferritecontinues to form along with pearlite between the Ps and

Bs lines. When the curve crosses Bs into the bainitic region,


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(a) (b)

(c) (d)

Fig. 18Optical micrographs of the microstructure of the cylindrical specimens cooled at ( a) 1.1 showing 90 pct pearlite and 10 pct bainite, (b) 2.24 showing 50 pct pearlite and 50 pct bainite, (c) 5.13 showing bainite and martensite, and (d) 11.5 showing martensite and bainite.

Table III. Summary of Microstructure Prediction Results

Experimentally Observed

MicrostructurePredicted Microstructure Based on

Heat-Transfer Model

0 P F P F1.1 90 pct P 10 pct B 80 pct (F P) 20 pct B2.24 50 pct P 50 pct B 50 pct (F P) 50 pct B5.13 B M 15 pct F 60 pct B 25 pct M8.6 B M 10 pct F 40 pct B 50 pct M

11.5 B M 10 pct B 90 pct MQuench 100 pct M 100 pct M

about 80 pct of the austenite has transformed into ferrite pearlite. The remaining austenite then transforms to bainite.The predicted microstructure is therefore 80 pct ferrite

pearlite and 20 pct bainite. The actual microstructure isshown in Figure 18(a). The microstructure is predominantlydark-etched fine-grained pearlite with a few light-etched

patches of bainite. The characteristic needles of bainite canbe seen within the light etching.

2. 2.24Figure 16 shows the curve first crossing Fs, and then,

after 1 pct transformation, across Ps. After 50 pct of the

austenite has transformed into ferrite pearlite, the re-maining austenite transforms to bainite. The predicted mi-crostructure is therefore 50 pct ferrite pearlite and 50 pct

bainite. The observed microstructure (Figure 18(b)) con-firms these predictions very clearly. The dark etching showsvery fine pearlite and the light etching shows the needlelike

bainitic structure. The proportion of each phase is approx-imately 50 pct.

3. 5.13As shown in Figure 17, after about 15 pct of the austenite

transforms to ferrite, bainite starts to form. When the curve


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Fig. 19Schematic diagram of the formation of radial cracks due tomartensitic transformation.

Fig. 20Maximum thermal stress and ultimate tensile stress calculatedusing the axisymmetric finite element against the correspondingtemperature during cooling for 5.13.

crosses Ms, 75 pct of the austenite has transformed, whichmeans that 60 pct of the transformed product is bainite. Theremaining austenite transforms to martensite and hence thefinal predicted microstructure is 15 pct ferrite, 60 pct bain-ite, and 25 pct martensite. The observed microstructure(Figure 18(c)) is almost entirely needle like, being a com-

bination of bainite and martensite. Here, it is difficult todifferentiate between the martensitic needles and the bain-

itic needles and hence no estimates on percentages aremade.

4. 11.5Figure 17 shows that after 1 pct of the austenite trans-

forms into ferrite, bainite begins to form. After 10 pct ofthe austenite has transformed into bainite, the remainingaustenite forms martensite. The predicted final microstruc-ture is therefore 10 pct bainite and 90 pct martensite. Theactual microstructure (Figure 18(d)) shows a predominantlymartensitic microstructure. A summary of the microstruc-ture predictions is shown in Table III.

C. Thermal Stresses

The thermal-stress model generates the stress distributionwithin the material during cooling. These stress distribu-tions, along with the maximum-stress failure criterion, areused to predict the occurrence of quench cracks. The max-imum principal stresses are plotted against their associatedtemperatures and are shown in Figures 20 and 21 for 5.13 and 11.5, respectively. These maximum stresses arecompared with the tensile strength of AISI 4140, which isalso a function of temperature.[24] The maximum-stress cri-terion states that if any of the three principal stresses in thematerial exceed the tensile strength, failure will occur.Since the thermal stresses at lower cooling rates are lowerand no quench cracks are predicted, the stress distributionsare not shown for 1.1 and 2.24.

The effect of the martensitic transformation on thermal-stress and radial-crack formation is shown schematically inFigure 19. When the material cools at a high cooling rate,there is a large thermal gradient between the core and thesurface of the cylinder. The surface reaches the martensitictransformation temperature first and forms martensite. Mar-tensite is a hard and brittle phase and its formation is ac-companied by a 6 pct volumetric expansion. When thesurface transforms to martensite, the resulting expansion isnot resisted because the core is at a higher temperature andtherefore is soft. At a later time, the core of the cylinderstarts to form martensite. In this case, the expansion of thecore is resisted by the hard surface, which is already mar-tensite. This resistance by the surface induces a tensile hoopstress on the surface that can cause a radial quench crackto form. It is important to note that the formation of mar-tensite must be accompanied by large thermal gradients be-tween the core and the surface for the transformation toinduce thermal stresses. If martensite forms all over thecylinder at the same time, the thermal stresses that are in-duced will be much smaller. The steel must also possess

sufficient hardenability to form martensite at the surface aswell as the core, and AISI 4140 does have high hardena-bility.

1. 5.13Figure 20 shows the variation of maximum stress during

cooling, and it is compared with the tensile strength. InTable III, it is shown that martensite does form at this cool-ing rate. The thermal-expansion coefficients are thereforeappropriately modified for this model. However, this doesnot have any effect on the maximum stresses shown in thefigure, which indicate that the cooling rate for this case isnot high enough for the martensitic transformation to havean effect on the thermal-stress distribution. The hoop stressand longitudinal stress are comparable and larger than the

radial stress. Because the maximum stresses do not ap-proach the tensile strength at any point, the model predictsthat the material will not fail. No quench cracks were ob-served during experiments at this cooling rate.

2. 11.5In Figure 21, the maximum stress is plotted against the

temperature at the location where the maximum stress oc-curred. This plot shows the effect of the martensitic trans-formation on thermal-stress distribution. Around thetemperature at which martensite forms, which is 615 K, allthree maximum stresses shoot up to very close to the tensile


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Fig. 21Maximum thermal stress and ultimate tensile stress calculatedusing the axisymmetric finite element against the correspondingtemperature during cooling for 11.5.

Fig. 22Typical radial crack found in specimens for 11.5

Fig. 23Rockwell hardness profiles along the diameter perpendicular tothe direction of the spray for different values of .

strength. The maximum hoop stress is the largest of thethree stresses and hence a radial crack is predicted in thematerial. This is consistent with the earlier discussion and

is confirmed by Figure 22, which is a picture of a typicalradial crack that occurred in specimens at this cooling rate.

D. Rockwell Hardness

Hardness profiles of the spray-cooled specimens, takenin the direction perpendicular to the spray, are shown inFigure 23. The hardness increases with from Rc 30 forthe as-received material to about Rc 60 for water quench-

ing. There is little variation in hardness between the surfaceand the core of the cylinder for all cooling rates. A jumpin hardness is observed between 2.24 and 5.13. Thisis because, as shown in previous sections, martensite beginsto form at 5.13, leading to an increase in hardness.

Table IV lists the measured cooling rate and that pre-dicted by the heat-transfer analysis at 973 K, and the hard-ness at room temperature. For a given cooling rate, theRockwell C hardness can be found from a standard Jominyend-quench table for 4140 steels. Comparison of the datain Table IV indicates that there is good match between themeasured hardness and the hardness determined from aknowledge of the cooling rates determined by the heat-transfer models. Therefore, the finite-element model, along

with the available charts, is a convenient method of deter-mining the microstructure and hardness at any locationwithin the specimen, without the need for measurement oftemperature and cooling rate.

VI. CONCLUSIONS

The following statements summarize the analysis of thespray-cooling process and the scheme proposed to modelthe process.

1. The air/water atomizer can be used to tailor the coolingrate from the forging temperature in order to obtain de-sired properties in a one-step process from the forging

operation without any post heat treatment.2. The proposed heat-transfer model provides a reasonable

estimate of the temperature profile in a AISI 4140 steelbar during cooling by an atomized spray of air and wa-ter.

3. The temperature distribution in spray-cooled steel canbe predicted to reasonably good accuracy using finite-element modeling. The latent heat effects of the phasetransformations and temperature dependence of material

properties can be effectively incorporated in the model.4. Resulting microstructures in spray-cooled steel can be

predicted by means of the CCT diagram in combinationwith predicted temperature profiles.

5. Hardness at any location in spray-cooled steel can bepredicted by means of Jominy end-quench diagrams incombination with predicted temperature profiles.

6. The thermal-stress model accurately predicts the occur-rence of quench cracking.

7. Martensitic transformation and its effect on thermalstress can be incorporated.

ACKNOWLEDGMENTS

The authors gratefully acknowledge the support of theNASA/UTA Center for Hypersonic Research, The MAE


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Table IV. Measured and Predicted Cooling Rates at 973 K and Hardness at Room Temperature

ExperimentalCooling

Rate at 973 K(K/s)

PredictedCooling Rate

at 973 K(K/s)

Predicted Hardnessfrom Literature, Rc

(Based on ExperimentallyMeasured Cooling Rate)

Predicted Hardnessfrom Literature, Rc

(Based on FiniteDifference Analysis)

MeasuredHardness

(Rc)

11.5 8 to 20 7 to 14 47 to 53 46 to 50 53 to 562.24 2.1 to 2.3 1.52 to 1.91 38 38 410 2.1 to 2.3 1.07 to 1.13 38 38 35

Department, and Trinity Forge Inc., who supplied the ma-terials for this work.

NOMENCLATUREa, b constants in Eq. (5)Bi Biot number, hR/ks

D0 diameter of the cylinder, 2Rh surface heat-transfer coefficient, W/(m2K)kair thermal conductivity of air, W/(mK)ks thermal conductivity of steel, W/(mK)Cp specific heat capacity, J/(kgK)mair Air, mass-flow rate, kg/s

mwater Water, mass-flow rate, kg/sNu Nusselt number, hD0/kaRe Reynolds number, [au0(1 )D0/m](Tsat/Ta)Pr Prandtl number, Cp/kqs surface heat flux, W/m

2

Q volume flow rate, L/sr radial coordinate

R radius of cylinder, mt time, sT temperature, KTi initial temperature, KTsat saturation temperature, Ku0 maximum velocity 0.41 m from nozzle exit, m/sV volume, m3

L length, mLi initial length, m

fL0 final length without martensitic transformation, mfLT final length incorporating the effects of martensitic

transformation, m

Greek0 Thermal-expansion coefficients of steel without

martensitic transformationT thermal-expansion coefficients of steel with

martensitic transformation emissivity of cylinder angle in cylindrical coordinates viscosity coefficient, Ns/m2

w density of water, kg/m

3

a density of mixture at adiabatic spray temperature,kg/m3

StefanBoltzmann constant mass-flow-rate ratio, mwater/mair

Subscriptsa gas-stream temperaturec convection-dominated regionm mixturemax at maximum heat flux0 stagnation point

R radiation-dominated region

w surface condition ambient temperature

MicrostructureA austeniteB bainiteBs austenite-to-bainite start temperatureF ferriteFs austenite-to-ferrite start temperatureM martensiteMs austenite-to-martensite start temperatureP pearlitePs austenite-to-pearlite start temperature

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