Research ArticleOptimal Heat Transfer Coefficient Distributions duringthe Controlled Cooling Process of an H-Shape Steel Beam

Yu-Feng Gan and Jiin-Yuh Jang

Department of Mechanical Engineering National Cheng Kung University Tainan 70101 Taiwan

Correspondence should be addressed to Jiin-Yuh Jang jangjimmailnckuedutw

Received 29 June 2017 Accepted 17 August 2017 Published 27 September 2017

Academic Editor Frederic Dumur

Copyright copy 2017 Yu-Feng Gan and Jiin-Yuh Jang This is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

Three-dimensional thermal-mechanical models for the prediction of heat transfer coefficient distributions with different size beamsare investigated H300 times 300 H250 times 250 and H200 times 200 H-shape steel beams are investigated in a controlled cooling processto obtain the design requirements for maximum uniform temperature distributions and minimal residual stress after controlledcooling An algorithm developed with the conjugated-gradient method is used to optimize the heat transfer coefficient distributionIn a comparison with the three group results the numerical results indicate that with the same model and under the same initialtemperature (119879 = 850∘C) and final temperature (119879 = 550 plusmn 10∘C) the heat transfer coefficients obtained with the conjugated-gradient method can produce more uniform temperature distribution and smaller residual web stress with objective functionsof the final average temperature 119879ave plusmn Δ119879 and maximum temperature difference to minimum minΔ119879max(119909 119910) The maximumtemperature difference is decreased by 57∘C 74∘C and 75∘C for Case 1 Case 2 and Case 3 respectively the surface maximumtemperature difference is decreased by 60sim80∘C for three cases and the residual stress at the web can be reduced by 20sim40MPa forthree cases

1 Introduction

H-shape steel beams are widely utilized in industry and onconstruction sites due to their safety characteristics and thefact that they are economic cross-section steel with a wideflange thin web and superior cross-section properties [1 2]H-shape steel beams have uneven temperature distributionin the cooling process due to their complex geometry whichprobably produces warping and breaking so controlled cool-ing plays a very important role after rolling to obtain high-performance high-strength H-beams

As early as the late 1950s and 1960s studies on theimprovement of the mechanical properties of steel wereperformedmostly in Europewhere in particular such studieswere being performed actively in the BISRA (British Ironand Steel Research Association) Then at the end of the1970s accelerated cooling equipment was first installed in aJapanese steel mill to provide high-strength high-toughenedsteel [3] JFE Steel developed TMCP (thermomechanical con-trol process) technology in 1980 which is a microstructural

control technique combining controlled rolling and cooling[4] In 1998 JFE Steel developed Super-OLAC an advancedaccelerated cooling system capable of cooling plates homo-geneously at high cooling rates close to the theoretical limits[5]

Recently there have beenmany researchers committed tocontrolling the temperature distribution and the temperaturedifferences occurring in the cooling process through simula-tions and experiments intended to improve steel mechanicsIn 2003 Liu et al [6] studied the heat transfer boundaryconditions and analyzed the distribution and changes in thetemperature field ofH-beams during the cooling process andtheir results were compared with the available experimentaldata In 2008 Liu et al [7] analyzed the temperature fieldof H-beams during cooling using a numerical simulationand discussed the temperature distribution and character indifferent parts of the beams during both air cooling and spraycooling It was found that the appropriate cooling mode canobtain a fairly uniform temperature field and thus improvethe steel mechanical properties Zhao et al [8] focused on

HindawiAdvances in Materials Science and EngineeringVolume 2017 Article ID 9873283 15 pageshttpsdoiorg10115520179873283

2 Advances in Materials Science and Engineering

how to improve H-beam cooling distortion and discussedthe uneven temperature distribution occurring in the coolingprocess causing uneven distortion phenomenon in an H-beam cross-section

In 2013 Guo et al [9] analyzed temperature fields duringboth controlled cooling and air cooling and the microstruc-tural mechanical properties for different parts of an H-beamafter it was control cooled for 45 s and the lowest and thehighest temperature were measured at the edge of the flangeand at the R corner respectively The microstructure for thedifferent parts of the H-beam consisted of ferrite and pearliteand the grain size at the R corner was coarser than that atthe flange and web The changes in the microstructure werein good agreement with those of the temperature field andthe results indicated that the uniformity of themicrostructureandmechanical properties can be improved by increasing thewater flow rate at the R corner In 2014Ma et al [10] designedfour kinds of cooling schemes with spray cooling to probe thetransient temperature field and the stress field which havedifferent water flow rate at different parts of an H-beam andit was found that the main reason for residual thermal stressis section temperature differences in the H-shape steel beamduring cooling process after rolling

There have been many studies discussing simulationsof a thermal stress field In 2008 Zhu et al [11] simulatedthe thermal residual stress field of H-beams with differentlengths using air cooling where the results indicated thatthe residual stress at the web exhibited compressive stresswhile that at the web center and the fillet exhibited tensilestress The residual stress condition in the direction of thewidth influenced the distribution of the residual stress in thedirection of the length Liu et al (2008) [12] considered heattransfer and the thermoelastic plastic theory together withthe finite element method to simulate the temperature andthe thermal stress of an H-beam during the cooling processand found that the appropriate cooling mode and parameterscan effectively enable the acquisition of a fairly uniformtemperature field and lower residual stress In 2011 Zhaoet al [13] analyzed the distribution of the stress and strainfield under different thermal boundary and cooling timeconditions and their results showed that the temperature ofan H-beam will decrease with increases in the density of thewater flow and themaximum stress will increase In the sameyear Zhu et al [14] simulated the residual stress distributionof hot rolled H-beams after cooling and the transformationsin residual stress during flange cutting Jang and Gan [15]recently proposed an algorism to calculate the optimum heattransfer coefficient combination for an H-beam

Based on previous studies it can be inferred that thetemperature field significantly affect the stress of H-beamsTherefore the heat transfer coefficients of an H-beam as thethermal boundary condition are very important operatingparameters to control the temperature field In order tooptimize the temperature field and temperature differencesoptimization of the heat transfer coefficients is investigatedand solved numerically using a commercial CFD codeANSYS FLUENT solver along with the conjugate-gradientmethod (CGM) that has been successfully applied to otherapplications [16ndash19]The design variables in the present study

are the heat transfer coefficients which strongly affect thetemperature field and the residual stress field of an H-beamThe optimal temperature field and stress field with the heattransfer coefficients are thus determined using the developedoptimization algorithm

2 Mathematical Analysis

In this work the controlled cooling process of H-beams fromthe China Steel Corporation (CSC) Taiwan is simulated viatransient heat transfer using a three-dimensional model tocontrol cooling after the rolling process The H-beams arecomposed of three parts the web flange and fillet for whichthe physicalmodel and the gridmeshing are shown in Figures1 and 2Three sizes of H-beammodels have been establishedand the sizes of the H-beams (H300 times 300 H250 times 250and H200 times 200) are listed in Figure 1 which correspondto Case 1 Case 2 and Case 3 in this paper respectivelyThe H-beams used in this study were a high strength lowalloy steel ASTM A572 beam Based on the cooling andthe geometrical characteristics of the H-beams symmetryphysical models were built which symmetry with 119909-axis 119910-axis and 119911-axis respectively and the controlled cooling ofthe H-beams was divided into five parts corresponding to theheat transfer coefficients ℎ1 ℎ2 ℎ3 ℎ4 and ℎ5 as shown inFigure 3 In this research two approaches were used to obtainthe objective temperature fields of the H-beams where oneadopted the constant heat transfer coefficients and the otherone adopted the CGM search for the optimal heat transfercoefficients as the boundary condition by which to comparethe temperature fields and stress fields with the same averagecooling temperature

21 Governing Equations The following transient heat con-duction equation governs the temperature field of anH-beam[20]

120597120597119909119895 119896(120597119879120597119909119895) = 120588 sdot 119862119901120597119879120597119905 (1)

where 120588 119862119901 and 119896 are the density heat capacity andthermal conductivity of the H-beam respectively The phasetransformation occurs at 119879 = 550∘C which brings latentheat and internal stress These effects are considered inthe variations of thermal-mechanical physical properties(specific heat thermal conductivity thermal expansion coef-ficient Youngrsquos modulus and yield stress) with temperatureas shown in Figure 4 [21] The diagrams of the variation inmicrostructures are also shown in Figure 4(d) The density 120588is a constant equal to 7750 kgm3 and the initial temperatureof the H-beam is assumed to be uniform at 850∘C

The incremental theory of plasticity governs the residualstress field of an H-beam The plastic deformation of thematerials is assumed to obey the vonMises yield criterion andthe associated flow rule The relationship of the rate com-ponents between thermal stresses 120590119894119895 and strains 120576119894119895 isdescribed by [20]

Advances in Materials Science and Engineering 3

Size Web heightH (mm)

Flange widthB (mm)

Web thickness Flange thickness RadiusR (mm)

LengthL (mm)

Case 1 300 300 10 15 13 1500Case 2 250 250 9 14 13 1200Case 3 200 200 8 12 13 1000

R

H

B

Flan

ge w

idth

Fillet radius

Web thickness

Web height

Flange thickness

L

t1

t2

t1 (mm) t2 (mm)

Figure 1 The physical model

120576119894119895 = 1 + 120592119864 119894119895 minus 120592119864119896119896120575119894119895 + 120582119904119894119895+ [120572 + 120597120572120597119879 (119879119904 minus 119879infin)]

119904119894119895 = 120590119894119895 minus 13120590119896119896120575119894119895120590119890 = ( 32119904119894119895119904119894119895)

12 119889120582 = 31198891205761198942120590119894

(2)

where 119864 is Youngrsquos modulus 120592 is the Poissonrsquos ratio 120572 isthe thermal expansion coefficient 119904119894119895 are the component ofdeviatoric stresses and 120582 is the plastic flow factor 120582 = 0for elastic deformation or 120590119890 lt 120590119904 and 120582 gt 0 for plasticdeformation or 120590119890 ge 120590119904 where 120590119904 is the yield stress and 120590119890is the von Mises effective stress Poissonrsquos ratio 120592 is a constantequal to 03 whereYoungrsquosmodulus119864 the thermal expansioncoefficient 120572 and the yield stress 120590119904 vary with the temperatureand have similar significantly change in about 550∘C withthe heat capacity and the thermal conductivity as shown inFigures 4(b) and 4(c) [20] The initial residual stress of theH-beam is assumed to be uniform and equal to zero

4 Advances in Materials Science and Engineering

Case Size Grid numberCase 1 443034Case 2 359214Case 3 245916

300 times 300250 times 250200 times 200

Figure 2 The meshing grid of the physical model

Tinfin

ℎ1 ℎ2ℎ2

ℎ2 ℎ2

ℎ4 ℎ4

ℎ4ℎ4ℎ4ℎ4

ℎ4

ℎ3

ℎ4

ℎ1

ℎ5ℎ5

ℎ5 ℎ5

ℎ3

Figure 3 The heat transfer coefficients distribution

22 Initial and Boundary Conditions The convective heattransfer between the H-beams surface and the surroundingsis evaluated by using the equation

11990210158401015840 = minus119896 12059711987912059711989910038161003816100381610038161003816100381610038161003816surface = ℎ (119879119904 minus 119879infin) (3)

where ℎ is the convective heat transfer coefficient for thesurface of the H-beam and varies with the different parts ofthe H-beam In addition 119879119904 is the surface temperature of theH-beam and 119879infin is the environmental temperature of 25∘Cin the water cooling and air cooling process The coolingconditions require water cooling for 15 s and then air coolingto the ambient temperature (about 100 minutes) where thewater cooling rate is about 20∘Cs the air cooling heat transfercoefficient is 30Wm2K and the water cooling heat transfercoefficients are determined by the constant and optimizationvalues of the CGM

3 Numerical Method

31 Numerical Algorithm In this study the commercialsoftware ANSYS is adopted to solve the governing equationsand the finite element method is used to simulate thetemperature field and the thermomechanical analysis Thecomputation process is transient so the first-order implicitmethod was adopted to maintain the numerical stability ofevery time step The time step was 01 s for the CGM and theprogram controlled for thermomechanical analysis to satisfythe requirements of the computational accuracyThe compu-tational grids for the three-dimensional models which arecomposed of 3544272 107848 and 94248 cells for Case 1Case 2 and Case 3 respectively were typically adopted inthe computational domain as shown in Figure 2 and Table 2However a careful check of the grid independence of thenumerical solutions was made to ensure the accuracy andvalidity of the numerical results For this purpose thesymmetry model grid systems were tested which comprised541323 443034 and 295556 cells for Case 1 394611 359214and 262200 cells for Case 2 and 385120 245916 and 129276cells for Case 3 respectively It was found that the relativeerrors under the same conditions among the temperaturesolutions obtained with three types of grids were less than05 for the three cases The discretized system was solvediteratively until it satisfied the following residual convergencecriterion

max(10038161003816100381610038161003816119879119899119894119895 minus 119879119899minus1119894119895 10038161003816100381610038161003816119879119899119894119895 ) le 10minus6 (4)

where 119879119899minus1119894119895 is the previous temperature value of 119879119899119894119895 at thesame time level

The simulations were performed as a parallel calculationusing sixteen core central processing units for the three-dimensional models The computer computation times wereapproximately 10 8 and 6 minutes for each search step forCase 1 Case 2 and Case 3 respectively

Advances in Materials Science and Engineering 5

Cp

(KJK

g-K)

k(W

m-K

)

Temperature (∘C)

Thermal conductivity

Specific heat

10

8

6

4

2

00 200 400 600 800

60

50

40

30

20

10

0

minus10

(a) Specific heat and thermal conductivity

Temperature (∘C)0 200 400 600 800

30

25

20

15

10

5

0

(10minus

6∘

C)

(b) Thermal expansion

Temperature (∘C)0 200 400 600 800

0

200

400

600

800

Y

(MPa

)

E(G

Pa)

Youngrsquos modulus

Yield stress

220

200

180

160

140

120

100

80

(c) Youngrsquos modulus and yield stress

Temperature (∘C)0 200 400 600 800

100

80

60

40

20

0

Phas

e vol

()

AusteniteFerrite

BainitePearlite

(d) Phases volume

Figure 4 Variations in the thermal-mechanical properties with temperature

32 Optimization In this study the conjugate-gradientmethod (CGM) was combined with a finite volume method(FVM) code as an optimizer to search for the optimum heattransfer coefficients (ℎ1 ℎ2 ℎ3 ℎ4 ℎ5) The objective func-tions 1198691obj(ℎ1 ℎ2 ℎ3 ℎ4 ℎ5) and 1198692obj(ℎ1 ℎ2 ℎ3 ℎ4 ℎ5) weredefined as the range of the final average temperature (119879ave plusmn120575 = 550 plusmn 10∘C) calculated by vertex average temperatureof the H-beam and the maximum temperature differenceas a minimum (minΔ119879max(119909 119910)) calculated by the vertexmaximum temperature subtracting the vertex minimumtemperature of the H-beam

Above all the CGMmethod evaluates the gradient of theobjective functions and then it sets up a new conjugate direc-tion for the updated design variables with the help of a direct

numerical sensitivity analysis The initial guess for the valueof each search variable was made and in the successive stepsthe conjugate-gradient coefficients and the search directionswere evaluated to estimate the new search variables Thesolutions obtained from the finite difference method werethen used to calculate the value of the objective functionswhich were further transmitted back to the optimizer forthe purpose of calculating the consecutive directions Theprocedure for applying this method is described in thefollowing

(1) Generate an initial guess for five design vari-ables (1199091 1199092 1199093 1199094 1199095) the heat transfer coefficients(ℎ1 ℎ2 ℎ3 ℎ4 ℎ5)

6 Advances in Materials Science and Engineering

(2) Adopt the finite difference method to predict thetemperature fields (119879(119909 119910)) associated with the last(ℎ1 ℎ2 ℎ3 ℎ4 ℎ5) and then calculate the objectivefunctions 1198691obj(ℎ1 ℎ2 ℎ3 ℎ4 ℎ5) and 1198692obj(ℎ1 ℎ2 ℎ3ℎ4 ℎ5)

(3) When the value of 1198691obj(ℎ1 ℎ2 ℎ3 ℎ4 ℎ5) reaches theobject range and 1198692obj(ℎ1 ℎ2 ℎ3 ℎ4 ℎ5) reaches theminimum the optimization process is terminatedOtherwise proceed to step (4)

(4) Determine the gradient functions (1205971198691obj120597119909119894)(119898) and(1205971198692obj120597119909119894)(119898) (119894 = 1 2 3 4 5) by applying a smallperturbation (Δ1199091 Δ1199092 Δ1199093 Δ1199094 Δ1199095) to each valueof (1199091 1199092 1199093 1199094 1199095) and calculate the correspondingchange in objective functions 1198691obj(ℎ1 ℎ2 ℎ3 ℎ4 ℎ5)and 1198692obj(ℎ1 ℎ2 ℎ3 ℎ4 ℎ5) Then the gradient func-tions with respect to each value of the design variables(1199091 1199092 1199093 1199094 1199095) can be calculated by the directnumerical differentiation as

1205971198691obj1205971199091 = Δ1198691objΔ1199091 1205971198691obj1205971199092 = Δ1198691objΔ1199092 1205971198691obj1205971199093 = Δ1198691objΔ1199093 1205971198691obj1205971199094 = Δ1198691objΔ1199094 1205971198691obj1205971199095 = Δ1198691objΔ11990951205971198692obj1205971199091 = Δ1198692objΔ1199091 1205971198692obj1205971199092 = Δ1198692objΔ1199092 1205971198692obj1205971199093 = Δ1198692objΔ1199093 1205971198692obj1205971199094 = Δ1198692objΔ1199094 1205971198692obj1205971199095 = Δ1198692objΔ1199095

(5)

(5) Calculate the conjugate-gradient coefficients 120574(119898)and the search directions 120585(119898+1)1 120585(119898+1)2 120585(119898+1)3 120585(119898+1)4 120585(119898+1)5 for the search variables For the first stepwith 119896 = 1 120574(1) = 0

120574(119898) = [[sum2119899 (120597119869(119898)obj 120597119909119899)sum2119899 (120597119869(119898minus1)obj 120597119909119899)]]

2

119899 = 1 2 3 4 5

120585(119898+1)1 = 120597119869(119898)obj1205971199091 + 120574(119898)120585(119898)1 120585(119898+1)2 = 120597119869(119898)obj1205971199092 + 120574(119898)120585(119898)2 120585(119898+1)3 = 120597119869(119898)obj1205971199093 + 120574(119898)120585(119898)3 120585(119898+1)4 = 120597119869(119898)obj1205971199094 + 120574(119898)120585(119898)4 120585(119898+1)5 = 120597119869(119898)obj1205971199095 + 120574(119898)120585(119898)5

(6)

(6) Assign values to the coefficients in a descendingdirection (1205731 1205732 1205733 1205734 1205735) for all values of the designvariables (1199091 1199092 1199093 1199094 1199095) Specifically these val-ues are chosen with a trial and error process Ingeneral the coefficients in the descending direction(1205731 1205732 1205733 1205734 1205735) range from 1 to 50

(7) Update the design variables with

119909(119898+1)1 = 119909(119898)1 + 1205731120585(119898)1 119909(119898+1)2 = 119909(119898)2 + 1205732120585(119898)2 119909(119898+1)3 = 119909(119898)3 + 1205733120585(119898)3 119909(119898+1)4 = 119909(119898)4 + 1205734120585(119898)4 119909(119898+1)5 = 119909(119898)5 + 1205735120585(119898)5

(7)

A flowchart of the CGM optimization process is plotted asFigure 5

4 Result and Discussion

In order to examine the validity of the results of the CGMmethod two initial guess values were used to search forthe optimal heat transfer coefficient distributions of the H-beams for three cases and they all obtained approximatelythe same result Based on the optimal heat transfer coefficientdistributions using CGM method the temperature field andthe residual stress of the H-beam were calculated to comparewith the constant heat transfer coefficient results Taking intoaccount the operability in the cooling process the area ofthe up surface of the flange is small and not in the watercooling range therefore the heat transfer coefficient ℎ5 is asignificantly smaller value than ℎ1 to ℎ4

Figure 6 shows the final average temperature and themaximum temperature difference search path where theinitial guess value 1 was ℎ1 = ℎ2 = ℎ3 = ℎ4 = 1600 (Wm2K)

Advances in Materials Science and Engineering 7

X design variables (heat transfer coefficient h)

No

YesStop

Start

Conjugate-gradient method

k = k + 1 Initial guess design variablesX1 X2 X3 X4 X5

Direct problem solver(heat transfer conductionthermal-stress equation)F(X1 X2 X3 X4 X5)

Evaluate objective functionJ(k)

Convergence

J(k) minus J(kminus1)

J(k)lt ＬＦＮＣＰ error

N = 1

Sensitivity analysisF(X1 + ΔX1 X2 X3 X4 X5) = J(k)1

J(k)1

X1

=J(k)1 minus J(k)

X1

Conjugate-gradient coefficient

Searching direction

(k)1 =J(k)1

X1

+ (k)1 (kminus1)1

New design variable

X(k+1)1 = X(k)

1 + 1J(k)1

N = 2

Sensitivity analysisF(X1 X2 + ΔX2 X3 X4 X5) = J(k)2

J(k)2

X2

=J(k)2 minus J(k)

X2

Conjugate-gradient coefficient

Searching direction

(k)2 =J(k)2

X2

+ (k)2 (kminus1)2

New design variable

X(k+1)2 = X(k)

2 + 2J(k)2

N = 5

N = 3 4

Sensitivity analysisF(X1 X2 X3 X4 X5 + ΔX5) = J(k)5

J(k)5

X5

=J(k)5 minus J(k)

X5

Searching direction

(k)5 =J(k)n

X5

+ (k)5 (kminus1)5

New design variable

X(k+1)5 = X(k)

5 + 5J(k)5

F(X1 X2 X3 X4 X5) functionsJ(k) objective functions (the average temperature and the temperature difference TＰ plusmn ΔT ＧＣＨ ΔTＧＲ)

Conjugate-gradient coefficient

(k)1 = [ J(k)1 X1

J(kminus1)1 X1

]2

(k)2 = [ J(k)2 X2

J(kminus1)2 X2

]2

(k)5 = [ J(k)5 X5

J(kminus1)5 X5

]2

Figure 5 A flowchart of the CGM optimization process

8 Advances in Materials Science and Engineering

ℎ5 = 160 (Wm2K) and the initial guess value 2 was ℎ1 =ℎ2 = ℎ3 = ℎ4 = 1200 (Wm2K) ℎ5 = 120 (Wm2K) forCase 1 The initial guess values for heat transfer coefficientsℎ1 = ℎ2 = ℎ3 = ℎ4 = 1600Wm2K in water are estimatedfrom the water jet heat transfer coefficient correlations byWendelstorf et al [22] while the ℎ5 = 160Wm2K in airis estimated from the combined natural and radiation heattransfer theory [23] The search trend of the final averagetemperature for these two initial guess values was close to theobjective value step by step and were within the error rangeThe search trend for the temperature difference for the twoinitial guess values was obtained by searching for the lowestvalue step by step with the final average temperature in therange of the objective value until the temperature differencevariable was stable The different search steps were obtainedfor different initial guess values but they all obtained almostthe same result where the resultant deviation for the twoinitial values was below 1∘C It is apparent that Case 2 andCase 3 both had the same search path as that of Case 1 Theinitial guess value 1 was ℎ1 = ℎ2 = ℎ3 = ℎ4 = 1500 (Wm2K)ℎ5 = 150 (Wm2K) and the initial guess value 2was ℎ1 = ℎ2 =ℎ3 = ℎ4 = 1000 (Wm2K) ℎ5 = 100 (Wm2K) for Case 2 theinitial guess value 1 was ℎ1 = ℎ2 = ℎ3 = ℎ4 = 1200 (Wm2K)ℎ5 = 120 (Wm2K) and the initial guess value 2 was ℎ1 =ℎ2 = ℎ3 = ℎ4 = 800 (Wm2K) ℎ5 = 80 (Wm2K) for Case 3respectively For the three cases there were different searchsteps for the difference cases and initial guess values wherethe search steps were simply related to the initial guess valueand the search step size

Tables 1ndash3 list the different distributions of the heattransfer coefficients for an example with ten points in theoptimization search process where the final average tem-peratures were all in a range of 540∘C to 560∘C but thetemperature difference decreased from 201∘C to 144∘C from204∘C to 130∘C and from 179∘C to 104∘C for Case 1 toCase 3 respectively The optimization distribution of theheat transfer coefficients is the tenth point in Tables 1ndash3For Case 1 the optimization distribution of heat transfercoefficients ℎ1 to ℎ5 is 1227 1745 1732 1479 and 101 (Wm2K)where the average temperature is 560∘C and the temperaturedifference is 144∘C For Case 2 the optimization distributionof the heat transfer coefficients ℎ1 to h5 is 1091 1705 16661445 and 37 (Wm2K) where the average temperature is560∘C and the temperature difference is 130∘C For Case 3the optimization distribution of heat transfer coefficients ℎ1to ℎ5 is 913 1579 1556 1166 and 225 (Wm2K)where the finalaverage temperature is 560∘C and the temperature differenceis 104∘C

Figure 7 shows that the variation of the average tempera-ture in the different parts of the beam and that the maximumtemperature difference with time for the three cases undertwo heat boundary conditions satisfied the average tempera-ture of H-beams in the range of 540sim560∘C For Case 1 whenthe heat transfer coefficients were constant (ℎ1 = ℎ2 = ℎ3 =ℎ4 = 1500 (Wm2K) ℎ5 = 150 (Wm2K)) the different partshad different trends in variation For the web the coolingrate was very fast at first and then gradually slowed with thedecrease of the web surface temperature and when the time

was over 12 seconds the cooling rate became as fast as it wasoriginally In the case of the flange the cooling rate was slowerthan that of the web where the cooling rate went from fast toslow In the case of the fillet the cooling rate was the slowestand exhibited the same variable trends as that of the flangeThe maximum temperature difference increased initially inthe first 6 seconds and then decreased slightly until themaximum temperature difference was 201∘C at 15 secondsWhen the heat transfer coefficients were optimized (ℎ1 =1227 (Wm2K) ℎ2 = 1745 (Wm2K) ℎ3 = 1732 (Wm2K)ℎ4 = 1479 (Wm2K) and ℎ5 = 101 (Wm2K)) the variationof the average temperature with time in the different partsexhibited the same trends compared with the results of theconstant heat transfer coefficients except the fillet coolingrate was slightly larger than that of the constant heat transfercoefficients the flange cooling rate was slightly less than thatof the constant heat transfer coefficients the web cooling ratewas obviously slower than that of the constant heat transfercoefficients and the cooling rate of the web went from fastto slow The temperature difference for the optimal heattransfer coefficients increased initially and then decreaseduntil the maximum temperature difference was 144∘C at 15seconds and the peak of the temperature difference wascontrolled at 6 seconds For Case 2 and Case 3 the constantheat transfer coefficients were ℎ1 = ℎ2 = ℎ3 = ℎ4 =1400 (Wm2K) ℎ5 = 140 (Wm2K) and ℎ1 = ℎ2 = ℎ3 =ℎ4 = 1200 (Wm2K) ℎ5 = 120 (Wm2K) and the optimalheat transfer coefficients were ℎ1 = 1091 (Wm2K) ℎ2 =1705 (Wm2K) ℎ3 = 1666 (Wm2K) ℎ4 = 1444 (Wm2K)ℎ5 = 37 (Wm2K) and ℎ1 = 913 (Wm2K) ℎ2 =1579 (Wm2K) ℎ3 = 1556 (Wm2K) ℎ4 = 1166 (Wm2K)and ℎ5 = 225 (Wm2K) respectively The variation of theaverage temperature with time exhibited the same trends asthose found for Case 1 regardless of whether it was deter-mined with constant heat transfer coefficients or optimalheat transfer coefficients The cooling rates of Case 2 amongthe web flange and fillet were closer than in Case 1 andCase 3 was the closest because of the physical dimensionsThe temperature difference for the constant heat transfercoefficients reached a peak at 65 seconds and 7 secondsfor Case 2 and Case 3 and then slowly decreased until thetemperature differences were 204∘C and 180∘C at 15 secondsrespectively The maximum temperature difference of theoptimal heat transfer coefficients reached a peak at 4 secondsand at 3 seconds for Case 2 and Case 3 and then slowlydecreased to 130∘C and 104∘C at 15 seconds respectively

Figure 8 displays the temperature field for three caseswith constant heat transfer coefficients and optimal heattransfer coefficients at 15 seconds ForCase 1 the temperatureranged from 667∘C to 466∘C with the constant heat transfercoefficients and the temperature ranged from 654∘C to 510∘Cwith the optimal heat transfer coefficients For Case 2 thetemperature ranged from 658∘C to 454∘C with the constantheat transfer coefficients and the temperature ranged from639∘C to 509∘Cwith the optimal heat transfer coefficients ForCase 3 the temperature ranged from 646∘C to 466∘C withthe constant heat transfer coefficients and the temperatureranged from 618∘C to 514∘C with the optimal heat transfercoefficients the surface maximum temperature difference

Advances in Materials Science and Engineering 9

TＰ

(∘

C)

Search steps

Initial value 1

Initial value 2

Initial value 1 ℎ1 = ℎ2 = ℎ3 = ℎ4 = 1600 ℎ5 = 160 (WＧ2K)Initial value 2 ℎ1 = ℎ2 = ℎ3 = ℎ4 = 1200 ℎ5 = 120 (WＧ2K)

590

580

570

560

550

540

5300 20 40 60 80 100 120 140 160

Search steps0 50 100 150 200

Initial value 1Initial value 2

Search steps

Search steps

0 20 40 60 80 100 120 140 160

220

200

180

160

140

120

0 50 100 150 200

Initial value 1

Initial value 2

Initial value 1Initial value 2

ΔT

(∘C)

(a) Case 1

220

200

180

160

140

120

Search steps0 50 100 150 200

Search steps0 50 100 150 200 250 300

Initial value 1

Initial value 2

Initial value 1Initial value 2

TＰ

(∘

C)

Search steps

600

580

560

540

0 50 100 150 200

Initial value 1 ℎ1 = ℎ2 = ℎ3 = ℎ4 = 1500 ℎ5 = 150 (WＧ2K)Initial value 2 ℎ1 = ℎ2 = ℎ3 = ℎ4 = 1000 ℎ5 = 100 (WＧ2K)

Search steps0 50 100 150 200 250 300

Initial value 1

Initial value 2

Initial value 1Initial value 2

ΔT

(∘C)

(b) Case 2

TＰ

(∘

C)

Search steps0 20 40 60 80 100 120 140 160

620

600

580

560

540

520

0 100 200 300 400

Initial value 1 ℎ1 = ℎ2 = ℎ3 = ℎ4 = 1200 ℎ5 = 120 (WＧ2 K)Initial value 2 ℎ1 = ℎ2 = ℎ3 = ℎ4 = 800 ℎ5 = 80 (WＧ2K)

Search steps

Initial value 1

Initial value 2

Initial value 1Initial value 2

Search steps0 20 40 60 80 100 120 140 160

220

200

180

160

140

120

100

80

0 100 200 300 400

Search steps

Initial value 1

Initial value 2

Initial value 1Initial value 2

ΔT

(∘C)

(c) Case 3

Figure 6 The search path with the different initial values for the three cases

10 Advances in Materials Science and Engineering

Constant

ℎ1 = ℎ2 = ℎ3 = ℎ4 = 1500 ℎ5 = 150 (WＧ2K)Optimal

ℎ1 = 1227 ℎ2 = 1745 ℎ3 = 1732 ℎ4 = 1479 ℎ5 = 101 (WＧ2K)

Time (s)0 2 4 6 8 10 12 14

250

200

150

100

50

0

ℎ are constantOptimal ℎ

ℎ are constant

Optimal ℎ

Time (s)

Tem

p(∘ C)

900

800

700

600

500

4000 2 4 6 8 10 12 14

Optimal ℎℎ are constant

ΔT

(∘C)

TＰ with filletTＰ with flange

TＰ with web

(a) Case 1

Tem

p(∘ C)

900

800

700

600

500

400

Time (s)0 2 4 6 8 10 12 14

Optimal ℎℎ are constant

TＰ with filletTＰ with flange

TＰ with web

Constantℎ1 = ℎ2 = ℎ3 = ℎ4 = 1400 ℎ5 = 140 (WＧ2K)

Optimalℎ1 = 1091 ℎ2 = 1705 ℎ3 = 1666 ℎ4 = 1444 ℎ5 = 37 (WＧ2K)

Time (s)0 2 4 6 8 10 12 14

ℎ is constantOptimal ℎ

ℎ are constant

Optimal ℎ

200

150

100

50

0

ΔT

(∘C)

(b) Case 2

Constant

ℎ1 = ℎ2 = ℎ3 = ℎ4 = 1200 ℎ5 = 120 (WＧ2K)Optimal

ℎ1 = 913 ℎ2 = 1579 ℎ3 = 1556 ℎ4 = 1166 ℎ5 = 225 (WＧ2K)

Time (s)0 2 4 6 8 10 12 14

200

150

100

50

0

ℎ is constantOptimal ℎ

ℎ are constant

Optimal ℎ

Tem

p(∘ C)

900

800

700

600

500

400

Time (s)0 2 4 6 8 10 12 14

TＰ with filletTＰ with flange

TＰ with web

Optimal ℎℎ are constant

ΔT

(∘C)

(c) Case 3

Figure 7 The temperature in different parts and the temperature difference variable with time

Advances in Materials Science and Engineering 11

Section

L2

Constant heattransfer coefficients

Optimal heattransfer coefficients

667652637621606591576560545530515499484469454

(∘C)

(a) Case 1

Constant heattransfer coefficients

Optimal heattransfer coefficients

667652637621606591576560545530515499484469454

(∘C)

(b) Case 2

Constant heattransfer coefficients

Optimal heattransfer coefficients

667652637621606591576560545530515499484469454

(∘C)

(c) Case 3

Figure 8 The distribution of temperature with the different heat transfer coefficients at 15 seconds in a section of the H-beams

is decreased by 60sim80∘C for three cases For the threecases the temperature field for the optimal heat transfercoefficients was more uniform than for the temperaturefield of the constant heat transfer coefficients Also from

Case 1 to Case 3 the sizes became gradually smaller so thetemperature differences were also smaller

Figure 9 and Table 4 display the residual stress distribu-tion at the web for three cases with constant heat transfer

12 Advances in Materials Science and Engineering

Table 1 The optimization search of ℎ1ndashℎ5 distribution for Case 1 (eg with ten points)

Number ℎ1 ℎ2 ℎ3 ℎ4 ℎ5 119879ave Δ119879Wm2K Wm2K Wm2K Wm2K Wm2K ∘C ∘C

(1) (constant) 1500 1500 1500 1500 150 550 201(2) 1546 1595 1596 1557 157 540 205(3) 1485 1547 1548 1507 107 546 195(4) 1419 1556 1555 1508 107 550 180(5) 1335 1576 1427 1415 167 555 176(6) 1309 1565 1497 1482 163 556 170(7) 1316 1572 1568 1507 106 557 160(8) 1264 1642 1635 1504 111 558 150(9) 1240 1702 1695 1490 105 559 145(10) (optimization) 1227 1745 1732 1479 101 560 144

Table 2 The optimization search of ℎ1ndashℎ5 distribution for Case 2 (eg with ten points)

Number ℎ1 ℎ2 ℎ3 ℎ4 ℎ5 119879ave Δ119879Wm2K Wm2K Wm2K Wm2K Wm2K ∘C ∘C

(1) (constant) 1400 1400 1400 1400 140 548 204(2) 1340 1449 1450 1419 100 548 198(3) 1365 1453 1452 1420 99 549 192(4) 1315 1460 1458 1421 98 551 178(5) 1289 1464 1461 1421 96 553 172(6) 1213 1482 1475 1465 92 555 151(7) 1181 1545 1485 1486 67 556 142(8) 1133 1610 1550 1477 64 558 136(9) 1109 1610 1555 1456 44 559 132(10) (optimization) 1091 1705 1666 1445 37 560 130

Table 3 The optimization search of ℎ1ndashℎ5 distribution for Case 3 (eg with ten points)

Number ℎ1 ℎ2 ℎ3 ℎ4 ℎ5 119879ave Δ119879Wm2K Wm2K Wm2K Wm2K Wm2K ∘C ∘C

(1) (constant) 1200 1200 1200 1200 120 543 179(2) 1213 945 934 1087 101 552 210(3) 1168 1013 1000 1140 151 553 189(4) 1121 1081 1065 1192 201 553 168(5) 1074 1148 1130 1243 201 555 147(6) 1038 1213 1191 1267 182 556 133(7) 1017 1278 1254 1280 204 556 131(8) 991 1353 1323 1265 203 557 123(9) 959 1435 1398 1223 181 559 111(10) (optimization) 913 1579 1556 1166 225 560 104

coefficients and optimal heat transfer coefficients based onthe analysis of the temperature field FromFigure 9 the flangeand the fillet of the residual stress are tensor in nature theweb of the residual stress is compressive in nature with themaximum compressive stress occurring at the junction of theweb and the fillet For Case 1 the residual stress in the webis in the range of minus125MPa to minus107MPa with the constantheat transfer coefficients and is about minus90MPa with optimal

heat transfer coefficients For Case 2 the residual stress inthe web is in the range of minus100MPa to minus80MPa with theconstant heat transfer coefficients and is about minus60MPa withoptimal heat transfer coefficients For Case 3 the residualstress in the web is in the range of minus80MPa to minus70MPa withthe constant heat transfer coefficients and is about minus50MPawith optimal heat transfer coefficients From Table 4 it isobserved that optimum heat transfer coefficients result in

Advances in Materials Science and Engineering 13

Optimal ℎℎ are constant

Resid

ual s

tress

(MPa

)

x (mm)

60

40

20

0

minus20

minus40

minus60

minus80

minus100

minus120

minus140

minus160

minus150 minus100 minus50 0 50 100 150

(a) Case 1Re

sidua

l stre

ss (M

Pa)

x (mm)

60

40

20

0

minus20

minus40

minus60

minus80

minus100

minus120

minus140

minus100 minus50 0 50 100

Optimal ℎℎ are constant

(b) Case 2

Resid

ual s

tress

(MPa

)

60

40

20

0

minus20

minus40

minus60

minus80

minus100

minus120

x (mm)minus100 minus50 0 50 100

Optimal ℎℎ are constant

(c) Case 3

Figure 9 The residual stresses distribution at the web

Table 4 A comparison of the results for the three cases

Case Case 1 Case 2 Case 3Constant Optimal Constant Optimal Constant Optimalℎ1 (Wm2K) 1500 1227 1400 1091 1200 913ℎ2 (Wm2K) 1500 1745 1400 1705 1200 1579ℎ3 (Wm2K) 1500 1732 1400 1666 1200 1556ℎ4 (Wm2K) 1500 1479 1400 1445 1200 1166ℎ5 (Wm2K) 150 101 140 37 120 225Δ119879max (

∘C) 201 144 204 130 180 104Residual stress at web (MPa) minus114 minus91 minus89 minus62 minus73 minus51

14 Advances in Materials Science and Engineering

smaller differences in maximum temperature and therebygenerated a lower residual stress distribution in the webacross all sizes of the beams researched in this paper

5 Conclusions

The conjugate-gradient method (CGM) was used here topredict the optimization heat transfer coefficient distributionof H-beams with different sizes To this end three 3D numer-ical models were analyzed to estimate the temperature fieldand the residual stresses field developed during the coolingprocess The results from the optimal heat transfer coeffi-cients were compared with those constant heat transfer coef-ficients Based on the numerical results the conclusion canbe summarized as follows

(1)The conjugate-gradientmethod (CGM) has accuratelypredicted the heat transfer coefficients and the differentinitial guess values have resulted in very similar result Inthe search process we can get many heat transfer coefficientsto satisfy the average temperature with different temperaturedifferences and there is a group of heat transfer coefficientsthat has the lowest temperature difference using the CGMoptimization search method

(2) A comparison of the constant heat transfer coefficientsand the optimal heat transfer coefficients reveals obviousimprovement in the predicted uniform temperature fieldand temperature difference with the optimal heat transfercoefficients where the maximum temperature difference andthe surface maximum temperature difference are decreasedfor all the cases analyzed

(3)The cooling rate was fastest in the web followed by theflange and the fillet

(4) After the water cooling and the air cooling to theambient temperature the residual stress at the web was calcu-lated and a comparison of the three results for the constantheat transfer coefficients and optimal heat transfer coeffi-cients indicates that the residual stress can be reduced by20sim40MPa for the three cases

Nomenclature

119861 Flange width mm119862119901 Specific heat Jsdotkgminus1sdotKminus1119864 Youngrsquos modulus GPaℎ Heat transfer coefficient Wsdotmminus2 Kminus1119867 Web height mm119869obj Objective functions119896 Thermal conductivity Wsdotmminus1sdotKminus1119877 Fillet radius mm119904119894119895 The component of deviatoric stresses MPa119879 Temperature ∘C1199051 Web thickness mm1199052 Flange thickness mm119905 Time s

Greek Symbols

120572 Thermal expansion coefficient ∘Cminus1120573 Search step size

120574 Conjugate-gradient coefficient120576119894119895 Strains120585 Search directions120588 Density kgmminus3120592 Poissonrsquos ratio120590119894119895 Thermal stresses MPa120590119890 The von Mises effective stress MPa120590119904 The yield stress MPa120575119894119895 Kronecker delta120575 Error of the average temperature ∘CΔ119879 Temperature difference ∘C

Subscripts

ave Average valueinfin Environment condition119898 Search steps in the CGM119899 The variables number119904 Surface of the H-beam

Disclosure

Part of the abstract of this paper was presented in 19th Inter-national Conference on Aerospace Mechanical Automotiveand Materials Engineering May 11-12 Montreal Canada2017

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

This research was financially supported by the Ministryof Science and Technology and China Steel CorporationTaiwan under Contract no MOST103-2622-E006-037 Theauthors are also grateful to Dr Chao-Hua Wang of ChinaSteel Corporation Taiwan for his valuable suggestions

References

[1] C J Bettles and M A Gibson ldquoCurrent wrought magnesiumalloys strengths and weaknessesrdquo Journal of Metals vol 57 pp46ndash49 2005

[2] Z Li D Wu and W Lu ldquoEffects of rolling and coolingconditions onmicrostructure andmechanical properties of lowcarbon cold heading steelrdquo Journal of Iron and Steel ResearchInternational vol 19 no 11 pp 64ndash70 2012

[3] C Ouchi ldquoDevelopment of steel plates by intensive use ofTMCP and direct quenching processesrdquo ISIJ International vol41 no 6 pp 542ndash553 2001

[4] S Endo and N Nakata ldquoDevelopment of Thermo-MechanicalControl Process (TMCP) and high performance steel in JFESteelrdquo JFE Technical Report no 20 pp 1ndash7 2015

[5] N Shikanai S Mitao and S Endo ldquoRecent developmentin microstructural control technologies through the thermo-mechanical control process (TMCP) with JFE Steelrsquos high-performance platesrdquo JFE Technical Report no 11 pp 1ndash6 2008

Advances in Materials Science and Engineering 15

[6] H W Liu J Yan D H Chen and J Q Qian ldquoFinite elementanalysis of H-beam temperature field during coolingrdquo Journalof Anhui University of Technology vol 20 pp 4ndash7 2003

[7] Q Liu Y ZangQQin and J Q Zhao ldquoAnalysis on temperatureand thermal stress of h-beam during coolingrdquo MetallurgicalEquipment vol 167 pp 17ndash20 2008

[8] X-M Zhao F Guo L-N Wang and D Wu ldquoEffect of ultra-fast cooling on cooling distortion for H-beam steelrdquo Journal ofNortheastern University vol 33 no 7 pp 949ndash952 2012

[9] F Guo X-M Zhao L-N Wang D Wu G-D Wang and Z-L Ning ldquoEffect of controlled cooling on microstructure andmechanical properties of H-beamrdquo Journal of Iron and SteelResearch International vol 20 no 8 pp 60ndash65 2013

[10] J H Ma X H Yao B Tao and S Li ldquoEffects of coolingafter rolling on thermal stress of H-beam steelrdquo Hot WorkingTechnology vol 43 pp 121ndash123 2014

[11] G M Zhu Y L Kang W Chen J L Gui and G T MaldquoThermalmechanical coupled finite element analysis of thermalresidual stress in h-beamrsquos air coolingrdquoMaterials for MechanicalEngineering vol 32 pp 77ndash80 2008

[12] Q Liu Y Zang Q Qin and Z Q Zhao ldquoAnalysis ontemperature and thermal stress of H-beam during coolingrdquoMetallurgical Equipment vol 167 pp 17ndash20 2008

[13] L Zhao X G Yu G B Liao and Y L Hu ldquoSimulation ofStress-strain of H-beam in control cooling processrdquo Journal ofUniversity of Science and Technology Liaoning vol 34 pp 466ndash469 2011

[14] G Zhu Y Kang andGMa ldquoSimulation of residual stress in hotrolled large size H-beamsrdquo Materials Science Forum vol 704-705 pp 1370ndash1378 2012

[15] J Y Jang and Y G Gan ldquoOptimization analysis of controlledcooling process for h-shape steel beamsrdquo in Proceedings ofthe International Conference on Mechatronic Automobile andEnvironment Engineering 2017

[16] J Y Jang L FHsu and J S Leu ldquoOptimization of the span angleand location of vortex generators in a plate-fin and tube heatexchangerrdquo International Journal of Heat andMass Transfer vol67 pp 432ndash444 2013

[17] J-Y Jang and Y-C Tsai ldquoOptimization of thermoelectricgenerator module spacing and spreader thickness used in awaste heat recovery systemrdquo Applied Thermal Engineering vol51 no 1-2 pp 677ndash689 2013

[18] J-Y Jang and C-C Chen ldquoOptimization of louvered-fin heatexchanger with variable louver anglesrdquo Applied Thermal Engi-neering vol 91 pp 138ndash150 2015

[19] J-Y Jang and J-B Huang ldquoOptimization of a slab heatingpattern for minimum energy consumption in a walking-beamtype reheating furnacerdquo Applied Thermal Engineering vol 85article no 6547 pp 313ndash321 2015

[20] X K Zhu and Y J Chao ldquoNumerical simulation of transienttemperature and residual stresses in friction stir welding of 304Lstainless steelrdquo Journal of Materials Processing Technology vol146 no 2 pp 263ndash272 2004

[21] JmatPro 90 Practical software for material properties SenteSoftware Ltd 2017

[22] J Wendelstorf K-H Spitzer and R Wendelstorf ldquoSpray watercooling heat transfer at high temperatures and liquid massfluxesrdquo International Journal of Heat and Mass Transfer vol 51no 19-20 pp 4902ndash4910 2008

[23] F P Incropera D P Dewitt T L Bergman and A S LavineldquoPrinciples of Heat and Mass Transferrdquo in International StudentVersion 7th edition

Submit your manuscripts athttpswwwhindawicom

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CeramicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CompositesJournal of

NanoparticlesJournal of

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International Journal of

Biomaterials

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NanoscienceJournal of

TextilesHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Journal of

NanotechnologyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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CrystallographyJournal of

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Advances in

Materials Science and EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Smart Materials Research

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MetallurgyJournal of

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BioMed Research International

MaterialsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Materials Science and Engineering 3

Size Web heightH (mm)

Flange widthB (mm)

Web thickness Flange thickness RadiusR (mm)

LengthL (mm)

Case 1 300 300 10 15 13 1500Case 2 250 250 9 14 13 1200Case 3 200 200 8 12 13 1000

R

H

B

Flan

ge w

idth

Fillet radius

Web thickness

Web height

Flange thickness

L

t1

t2

t1 (mm) t2 (mm)

Figure 1 The physical model

120576119894119895 = 1 + 120592119864 119894119895 minus 120592119864119896119896120575119894119895 + 120582119904119894119895+ [120572 + 120597120572120597119879 (119879119904 minus 119879infin)]

119904119894119895 = 120590119894119895 minus 13120590119896119896120575119894119895120590119890 = ( 32119904119894119895119904119894119895)

12 119889120582 = 31198891205761198942120590119894

(2)

where 119864 is Youngrsquos modulus 120592 is the Poissonrsquos ratio 120572 isthe thermal expansion coefficient 119904119894119895 are the component ofdeviatoric stresses and 120582 is the plastic flow factor 120582 = 0for elastic deformation or 120590119890 lt 120590119904 and 120582 gt 0 for plasticdeformation or 120590119890 ge 120590119904 where 120590119904 is the yield stress and 120590119890is the von Mises effective stress Poissonrsquos ratio 120592 is a constantequal to 03 whereYoungrsquosmodulus119864 the thermal expansioncoefficient 120572 and the yield stress 120590119904 vary with the temperatureand have similar significantly change in about 550∘C withthe heat capacity and the thermal conductivity as shown inFigures 4(b) and 4(c) [20] The initial residual stress of theH-beam is assumed to be uniform and equal to zero

4 Advances in Materials Science and Engineering

Case Size Grid numberCase 1 443034Case 2 359214Case 3 245916

300 times 300250 times 250200 times 200

Figure 2 The meshing grid of the physical model

Tinfin

ℎ1 ℎ2ℎ2

ℎ2 ℎ2

ℎ4 ℎ4

ℎ4ℎ4ℎ4ℎ4

ℎ4

ℎ3

ℎ4

ℎ1

ℎ5ℎ5

ℎ5 ℎ5

ℎ3

Figure 3 The heat transfer coefficients distribution

22 Initial and Boundary Conditions The convective heattransfer between the H-beams surface and the surroundingsis evaluated by using the equation

11990210158401015840 = minus119896 12059711987912059711989910038161003816100381610038161003816100381610038161003816surface = ℎ (119879119904 minus 119879infin) (3)

where ℎ is the convective heat transfer coefficient for thesurface of the H-beam and varies with the different parts ofthe H-beam In addition 119879119904 is the surface temperature of theH-beam and 119879infin is the environmental temperature of 25∘Cin the water cooling and air cooling process The coolingconditions require water cooling for 15 s and then air coolingto the ambient temperature (about 100 minutes) where thewater cooling rate is about 20∘Cs the air cooling heat transfercoefficient is 30Wm2K and the water cooling heat transfercoefficients are determined by the constant and optimizationvalues of the CGM

3 Numerical Method

31 Numerical Algorithm In this study the commercialsoftware ANSYS is adopted to solve the governing equationsand the finite element method is used to simulate thetemperature field and the thermomechanical analysis Thecomputation process is transient so the first-order implicitmethod was adopted to maintain the numerical stability ofevery time step The time step was 01 s for the CGM and theprogram controlled for thermomechanical analysis to satisfythe requirements of the computational accuracyThe compu-tational grids for the three-dimensional models which arecomposed of 3544272 107848 and 94248 cells for Case 1Case 2 and Case 3 respectively were typically adopted inthe computational domain as shown in Figure 2 and Table 2However a careful check of the grid independence of thenumerical solutions was made to ensure the accuracy andvalidity of the numerical results For this purpose thesymmetry model grid systems were tested which comprised541323 443034 and 295556 cells for Case 1 394611 359214and 262200 cells for Case 2 and 385120 245916 and 129276cells for Case 3 respectively It was found that the relativeerrors under the same conditions among the temperaturesolutions obtained with three types of grids were less than05 for the three cases The discretized system was solvediteratively until it satisfied the following residual convergencecriterion

max(10038161003816100381610038161003816119879119899119894119895 minus 119879119899minus1119894119895 10038161003816100381610038161003816119879119899119894119895 ) le 10minus6 (4)

where 119879119899minus1119894119895 is the previous temperature value of 119879119899119894119895 at thesame time level

The simulations were performed as a parallel calculationusing sixteen core central processing units for the three-dimensional models The computer computation times wereapproximately 10 8 and 6 minutes for each search step forCase 1 Case 2 and Case 3 respectively

Advances in Materials Science and Engineering 5

Cp

(KJK

g-K)

k(W

m-K

)

Temperature (∘C)

Thermal conductivity

Specific heat

10

8

6

4

2

00 200 400 600 800

60

50

40

30

20

10

0

minus10

(a) Specific heat and thermal conductivity

Temperature (∘C)0 200 400 600 800

30

25

20

15

10

5

0

(10minus

6∘

C)

(b) Thermal expansion

Temperature (∘C)0 200 400 600 800

0

200

400

600

800

Y

(MPa

)

E(G

Pa)

Youngrsquos modulus

Yield stress

220

200

180

160

140

120

100

80

(c) Youngrsquos modulus and yield stress

Temperature (∘C)0 200 400 600 800

100

80

60

40

20

0

Phas

e vol

()

AusteniteFerrite

BainitePearlite

(d) Phases volume

Figure 4 Variations in the thermal-mechanical properties with temperature

32 Optimization In this study the conjugate-gradientmethod (CGM) was combined with a finite volume method(FVM) code as an optimizer to search for the optimum heattransfer coefficients (ℎ1 ℎ2 ℎ3 ℎ4 ℎ5) The objective func-tions 1198691obj(ℎ1 ℎ2 ℎ3 ℎ4 ℎ5) and 1198692obj(ℎ1 ℎ2 ℎ3 ℎ4 ℎ5) weredefined as the range of the final average temperature (119879ave plusmn120575 = 550 plusmn 10∘C) calculated by vertex average temperatureof the H-beam and the maximum temperature differenceas a minimum (minΔ119879max(119909 119910)) calculated by the vertexmaximum temperature subtracting the vertex minimumtemperature of the H-beam

Above all the CGMmethod evaluates the gradient of theobjective functions and then it sets up a new conjugate direc-tion for the updated design variables with the help of a direct

numerical sensitivity analysis The initial guess for the valueof each search variable was made and in the successive stepsthe conjugate-gradient coefficients and the search directionswere evaluated to estimate the new search variables Thesolutions obtained from the finite difference method werethen used to calculate the value of the objective functionswhich were further transmitted back to the optimizer forthe purpose of calculating the consecutive directions Theprocedure for applying this method is described in thefollowing

(1) Generate an initial guess for five design vari-ables (1199091 1199092 1199093 1199094 1199095) the heat transfer coefficients(ℎ1 ℎ2 ℎ3 ℎ4 ℎ5)

6 Advances in Materials Science and Engineering

(2) Adopt the finite difference method to predict thetemperature fields (119879(119909 119910)) associated with the last(ℎ1 ℎ2 ℎ3 ℎ4 ℎ5) and then calculate the objectivefunctions 1198691obj(ℎ1 ℎ2 ℎ3 ℎ4 ℎ5) and 1198692obj(ℎ1 ℎ2 ℎ3ℎ4 ℎ5)

(3) When the value of 1198691obj(ℎ1 ℎ2 ℎ3 ℎ4 ℎ5) reaches theobject range and 1198692obj(ℎ1 ℎ2 ℎ3 ℎ4 ℎ5) reaches theminimum the optimization process is terminatedOtherwise proceed to step (4)

(4) Determine the gradient functions (1205971198691obj120597119909119894)(119898) and(1205971198692obj120597119909119894)(119898) (119894 = 1 2 3 4 5) by applying a smallperturbation (Δ1199091 Δ1199092 Δ1199093 Δ1199094 Δ1199095) to each valueof (1199091 1199092 1199093 1199094 1199095) and calculate the correspondingchange in objective functions 1198691obj(ℎ1 ℎ2 ℎ3 ℎ4 ℎ5)and 1198692obj(ℎ1 ℎ2 ℎ3 ℎ4 ℎ5) Then the gradient func-tions with respect to each value of the design variables(1199091 1199092 1199093 1199094 1199095) can be calculated by the directnumerical differentiation as

1205971198691obj1205971199091 = Δ1198691objΔ1199091 1205971198691obj1205971199092 = Δ1198691objΔ1199092 1205971198691obj1205971199093 = Δ1198691objΔ1199093 1205971198691obj1205971199094 = Δ1198691objΔ1199094 1205971198691obj1205971199095 = Δ1198691objΔ11990951205971198692obj1205971199091 = Δ1198692objΔ1199091 1205971198692obj1205971199092 = Δ1198692objΔ1199092 1205971198692obj1205971199093 = Δ1198692objΔ1199093 1205971198692obj1205971199094 = Δ1198692objΔ1199094 1205971198692obj1205971199095 = Δ1198692objΔ1199095

(5)

(5) Calculate the conjugate-gradient coefficients 120574(119898)and the search directions 120585(119898+1)1 120585(119898+1)2 120585(119898+1)3 120585(119898+1)4 120585(119898+1)5 for the search variables For the first stepwith 119896 = 1 120574(1) = 0

120574(119898) = [[sum2119899 (120597119869(119898)obj 120597119909119899)sum2119899 (120597119869(119898minus1)obj 120597119909119899)]]

2

119899 = 1 2 3 4 5

120585(119898+1)1 = 120597119869(119898)obj1205971199091 + 120574(119898)120585(119898)1 120585(119898+1)2 = 120597119869(119898)obj1205971199092 + 120574(119898)120585(119898)2 120585(119898+1)3 = 120597119869(119898)obj1205971199093 + 120574(119898)120585(119898)3 120585(119898+1)4 = 120597119869(119898)obj1205971199094 + 120574(119898)120585(119898)4 120585(119898+1)5 = 120597119869(119898)obj1205971199095 + 120574(119898)120585(119898)5

(6)

(6) Assign values to the coefficients in a descendingdirection (1205731 1205732 1205733 1205734 1205735) for all values of the designvariables (1199091 1199092 1199093 1199094 1199095) Specifically these val-ues are chosen with a trial and error process Ingeneral the coefficients in the descending direction(1205731 1205732 1205733 1205734 1205735) range from 1 to 50

(7) Update the design variables with

119909(119898+1)1 = 119909(119898)1 + 1205731120585(119898)1 119909(119898+1)2 = 119909(119898)2 + 1205732120585(119898)2 119909(119898+1)3 = 119909(119898)3 + 1205733120585(119898)3 119909(119898+1)4 = 119909(119898)4 + 1205734120585(119898)4 119909(119898+1)5 = 119909(119898)5 + 1205735120585(119898)5

(7)

A flowchart of the CGM optimization process is plotted asFigure 5

4 Result and Discussion

In order to examine the validity of the results of the CGMmethod two initial guess values were used to search forthe optimal heat transfer coefficient distributions of the H-beams for three cases and they all obtained approximatelythe same result Based on the optimal heat transfer coefficientdistributions using CGM method the temperature field andthe residual stress of the H-beam were calculated to comparewith the constant heat transfer coefficient results Taking intoaccount the operability in the cooling process the area ofthe up surface of the flange is small and not in the watercooling range therefore the heat transfer coefficient ℎ5 is asignificantly smaller value than ℎ1 to ℎ4

Figure 6 shows the final average temperature and themaximum temperature difference search path where theinitial guess value 1 was ℎ1 = ℎ2 = ℎ3 = ℎ4 = 1600 (Wm2K)

Advances in Materials Science and Engineering 7

X design variables (heat transfer coefficient h)

No

YesStop

Start

Conjugate-gradient method

k = k + 1 Initial guess design variablesX1 X2 X3 X4 X5

Direct problem solver(heat transfer conductionthermal-stress equation)F(X1 X2 X3 X4 X5)

Evaluate objective functionJ(k)

Convergence

J(k) minus J(kminus1)

J(k)lt ＬＦＮＣＰ error

N = 1

Sensitivity analysisF(X1 + ΔX1 X2 X3 X4 X5) = J(k)1

J(k)1

X1

=J(k)1 minus J(k)

X1

Conjugate-gradient coefficient

Searching direction

(k)1 =J(k)1

X1

+ (k)1 (kminus1)1

New design variable

X(k+1)1 = X(k)

1 + 1J(k)1

N = 2

Sensitivity analysisF(X1 X2 + ΔX2 X3 X4 X5) = J(k)2

J(k)2

X2

=J(k)2 minus J(k)

X2

Conjugate-gradient coefficient

Searching direction

(k)2 =J(k)2

X2

+ (k)2 (kminus1)2

New design variable

X(k+1)2 = X(k)

2 + 2J(k)2

N = 5

N = 3 4

Sensitivity analysisF(X1 X2 X3 X4 X5 + ΔX5) = J(k)5

J(k)5

X5

=J(k)5 minus J(k)

X5

Searching direction

(k)5 =J(k)n

X5

+ (k)5 (kminus1)5

New design variable

X(k+1)5 = X(k)

5 + 5J(k)5

F(X1 X2 X3 X4 X5) functionsJ(k) objective functions (the average temperature and the temperature difference TＰ plusmn ΔT ＧＣＨ ΔTＧＲ)

Conjugate-gradient coefficient

(k)1 = [ J(k)1 X1

J(kminus1)1 X1

]2

(k)2 = [ J(k)2 X2

J(kminus1)2 X2

]2

(k)5 = [ J(k)5 X5

J(kminus1)5 X5

]2

Figure 5 A flowchart of the CGM optimization process

8 Advances in Materials Science and Engineering

ℎ5 = 160 (Wm2K) and the initial guess value 2 was ℎ1 =ℎ2 = ℎ3 = ℎ4 = 1200 (Wm2K) ℎ5 = 120 (Wm2K) forCase 1 The initial guess values for heat transfer coefficientsℎ1 = ℎ2 = ℎ3 = ℎ4 = 1600Wm2K in water are estimatedfrom the water jet heat transfer coefficient correlations byWendelstorf et al [22] while the ℎ5 = 160Wm2K in airis estimated from the combined natural and radiation heattransfer theory [23] The search trend of the final averagetemperature for these two initial guess values was close to theobjective value step by step and were within the error rangeThe search trend for the temperature difference for the twoinitial guess values was obtained by searching for the lowestvalue step by step with the final average temperature in therange of the objective value until the temperature differencevariable was stable The different search steps were obtainedfor different initial guess values but they all obtained almostthe same result where the resultant deviation for the twoinitial values was below 1∘C It is apparent that Case 2 andCase 3 both had the same search path as that of Case 1 Theinitial guess value 1 was ℎ1 = ℎ2 = ℎ3 = ℎ4 = 1500 (Wm2K)ℎ5 = 150 (Wm2K) and the initial guess value 2was ℎ1 = ℎ2 =ℎ3 = ℎ4 = 1000 (Wm2K) ℎ5 = 100 (Wm2K) for Case 2 theinitial guess value 1 was ℎ1 = ℎ2 = ℎ3 = ℎ4 = 1200 (Wm2K)ℎ5 = 120 (Wm2K) and the initial guess value 2 was ℎ1 =ℎ2 = ℎ3 = ℎ4 = 800 (Wm2K) ℎ5 = 80 (Wm2K) for Case 3respectively For the three cases there were different searchsteps for the difference cases and initial guess values wherethe search steps were simply related to the initial guess valueand the search step size

Tables 1ndash3 list the different distributions of the heattransfer coefficients for an example with ten points in theoptimization search process where the final average tem-peratures were all in a range of 540∘C to 560∘C but thetemperature difference decreased from 201∘C to 144∘C from204∘C to 130∘C and from 179∘C to 104∘C for Case 1 toCase 3 respectively The optimization distribution of theheat transfer coefficients is the tenth point in Tables 1ndash3For Case 1 the optimization distribution of heat transfercoefficients ℎ1 to ℎ5 is 1227 1745 1732 1479 and 101 (Wm2K)where the average temperature is 560∘C and the temperaturedifference is 144∘C For Case 2 the optimization distributionof the heat transfer coefficients ℎ1 to h5 is 1091 1705 16661445 and 37 (Wm2K) where the average temperature is560∘C and the temperature difference is 130∘C For Case 3the optimization distribution of heat transfer coefficients ℎ1to ℎ5 is 913 1579 1556 1166 and 225 (Wm2K)where the finalaverage temperature is 560∘C and the temperature differenceis 104∘C

Figure 7 shows that the variation of the average tempera-ture in the different parts of the beam and that the maximumtemperature difference with time for the three cases undertwo heat boundary conditions satisfied the average tempera-ture of H-beams in the range of 540sim560∘C For Case 1 whenthe heat transfer coefficients were constant (ℎ1 = ℎ2 = ℎ3 =ℎ4 = 1500 (Wm2K) ℎ5 = 150 (Wm2K)) the different partshad different trends in variation For the web the coolingrate was very fast at first and then gradually slowed with thedecrease of the web surface temperature and when the time

was over 12 seconds the cooling rate became as fast as it wasoriginally In the case of the flange the cooling rate was slowerthan that of the web where the cooling rate went from fast toslow In the case of the fillet the cooling rate was the slowestand exhibited the same variable trends as that of the flangeThe maximum temperature difference increased initially inthe first 6 seconds and then decreased slightly until themaximum temperature difference was 201∘C at 15 secondsWhen the heat transfer coefficients were optimized (ℎ1 =1227 (Wm2K) ℎ2 = 1745 (Wm2K) ℎ3 = 1732 (Wm2K)ℎ4 = 1479 (Wm2K) and ℎ5 = 101 (Wm2K)) the variationof the average temperature with time in the different partsexhibited the same trends compared with the results of theconstant heat transfer coefficients except the fillet coolingrate was slightly larger than that of the constant heat transfercoefficients the flange cooling rate was slightly less than thatof the constant heat transfer coefficients the web cooling ratewas obviously slower than that of the constant heat transfercoefficients and the cooling rate of the web went from fastto slow The temperature difference for the optimal heattransfer coefficients increased initially and then decreaseduntil the maximum temperature difference was 144∘C at 15seconds and the peak of the temperature difference wascontrolled at 6 seconds For Case 2 and Case 3 the constantheat transfer coefficients were ℎ1 = ℎ2 = ℎ3 = ℎ4 =1400 (Wm2K) ℎ5 = 140 (Wm2K) and ℎ1 = ℎ2 = ℎ3 =ℎ4 = 1200 (Wm2K) ℎ5 = 120 (Wm2K) and the optimalheat transfer coefficients were ℎ1 = 1091 (Wm2K) ℎ2 =1705 (Wm2K) ℎ3 = 1666 (Wm2K) ℎ4 = 1444 (Wm2K)ℎ5 = 37 (Wm2K) and ℎ1 = 913 (Wm2K) ℎ2 =1579 (Wm2K) ℎ3 = 1556 (Wm2K) ℎ4 = 1166 (Wm2K)and ℎ5 = 225 (Wm2K) respectively The variation of theaverage temperature with time exhibited the same trends asthose found for Case 1 regardless of whether it was deter-mined with constant heat transfer coefficients or optimalheat transfer coefficients The cooling rates of Case 2 amongthe web flange and fillet were closer than in Case 1 andCase 3 was the closest because of the physical dimensionsThe temperature difference for the constant heat transfercoefficients reached a peak at 65 seconds and 7 secondsfor Case 2 and Case 3 and then slowly decreased until thetemperature differences were 204∘C and 180∘C at 15 secondsrespectively The maximum temperature difference of theoptimal heat transfer coefficients reached a peak at 4 secondsand at 3 seconds for Case 2 and Case 3 and then slowlydecreased to 130∘C and 104∘C at 15 seconds respectively

Figure 8 displays the temperature field for three caseswith constant heat transfer coefficients and optimal heattransfer coefficients at 15 seconds ForCase 1 the temperatureranged from 667∘C to 466∘C with the constant heat transfercoefficients and the temperature ranged from 654∘C to 510∘Cwith the optimal heat transfer coefficients For Case 2 thetemperature ranged from 658∘C to 454∘C with the constantheat transfer coefficients and the temperature ranged from639∘C to 509∘Cwith the optimal heat transfer coefficients ForCase 3 the temperature ranged from 646∘C to 466∘C withthe constant heat transfer coefficients and the temperatureranged from 618∘C to 514∘C with the optimal heat transfercoefficients the surface maximum temperature difference

Advances in Materials Science and Engineering 9

TＰ

(∘

C)

Search steps

Initial value 1

Initial value 2

Initial value 1 ℎ1 = ℎ2 = ℎ3 = ℎ4 = 1600 ℎ5 = 160 (WＧ2K)Initial value 2 ℎ1 = ℎ2 = ℎ3 = ℎ4 = 1200 ℎ5 = 120 (WＧ2K)

590

580

570

560

550

540

5300 20 40 60 80 100 120 140 160

Search steps0 50 100 150 200

Initial value 1Initial value 2

Search steps

Search steps

0 20 40 60 80 100 120 140 160

220

200

180

160

140

120

0 50 100 150 200

Initial value 1

Initial value 2

Initial value 1Initial value 2

ΔT

(∘C)

(a) Case 1

220

200

180

160

140

120

Search steps0 50 100 150 200

Search steps0 50 100 150 200 250 300

Initial value 1

Initial value 2

Initial value 1Initial value 2

TＰ

(∘

C)

Search steps

600

580

560

540

0 50 100 150 200

Initial value 1 ℎ1 = ℎ2 = ℎ3 = ℎ4 = 1500 ℎ5 = 150 (WＧ2K)Initial value 2 ℎ1 = ℎ2 = ℎ3 = ℎ4 = 1000 ℎ5 = 100 (WＧ2K)

Search steps0 50 100 150 200 250 300

Initial value 1

Initial value 2

Initial value 1Initial value 2

ΔT

(∘C)

(b) Case 2

TＰ

(∘

C)

Search steps0 20 40 60 80 100 120 140 160

620

600

580

560

540

520

0 100 200 300 400

Initial value 1 ℎ1 = ℎ2 = ℎ3 = ℎ4 = 1200 ℎ5 = 120 (WＧ2 K)Initial value 2 ℎ1 = ℎ2 = ℎ3 = ℎ4 = 800 ℎ5 = 80 (WＧ2K)

Search steps

Initial value 1

Initial value 2

Initial value 1Initial value 2

Search steps0 20 40 60 80 100 120 140 160

220

200

180

160

140

120

100

80

0 100 200 300 400

Search steps

Initial value 1

Initial value 2

Initial value 1Initial value 2

ΔT

(∘C)

(c) Case 3

Figure 6 The search path with the different initial values for the three cases

10 Advances in Materials Science and Engineering

Constant

ℎ1 = ℎ2 = ℎ3 = ℎ4 = 1500 ℎ5 = 150 (WＧ2K)Optimal

ℎ1 = 1227 ℎ2 = 1745 ℎ3 = 1732 ℎ4 = 1479 ℎ5 = 101 (WＧ2K)

Time (s)0 2 4 6 8 10 12 14

250

200

150

100

50

0

ℎ are constantOptimal ℎ

ℎ are constant

Optimal ℎ

Time (s)

Tem

p(∘ C)

900

800

700

600

500

4000 2 4 6 8 10 12 14

Optimal ℎℎ are constant

ΔT

(∘C)

TＰ with filletTＰ with flange

TＰ with web

(a) Case 1

Tem

p(∘ C)

900

800

700

600

500

400

Time (s)0 2 4 6 8 10 12 14

Optimal ℎℎ are constant

TＰ with filletTＰ with flange

TＰ with web

Constantℎ1 = ℎ2 = ℎ3 = ℎ4 = 1400 ℎ5 = 140 (WＧ2K)

Optimalℎ1 = 1091 ℎ2 = 1705 ℎ3 = 1666 ℎ4 = 1444 ℎ5 = 37 (WＧ2K)

Time (s)0 2 4 6 8 10 12 14

ℎ is constantOptimal ℎ

ℎ are constant

Optimal ℎ

200

150

100

50

0

ΔT

(∘C)

(b) Case 2

Constant

ℎ1 = ℎ2 = ℎ3 = ℎ4 = 1200 ℎ5 = 120 (WＧ2K)Optimal

ℎ1 = 913 ℎ2 = 1579 ℎ3 = 1556 ℎ4 = 1166 ℎ5 = 225 (WＧ2K)

Time (s)0 2 4 6 8 10 12 14

200

150

100

50

0

ℎ is constantOptimal ℎ

ℎ are constant

Optimal ℎ

Tem

p(∘ C)

900

800

700

600

500

400

Time (s)0 2 4 6 8 10 12 14

TＰ with filletTＰ with flange

TＰ with web

Optimal ℎℎ are constant

ΔT

(∘C)

(c) Case 3

Figure 7 The temperature in different parts and the temperature difference variable with time

Advances in Materials Science and Engineering 11

Section

L2

Constant heattransfer coefficients

Optimal heattransfer coefficients

667652637621606591576560545530515499484469454

(∘C)

(a) Case 1

Constant heattransfer coefficients

Optimal heattransfer coefficients

667652637621606591576560545530515499484469454

(∘C)

(b) Case 2

Constant heattransfer coefficients

Optimal heattransfer coefficients

667652637621606591576560545530515499484469454

(∘C)

(c) Case 3

Figure 8 The distribution of temperature with the different heat transfer coefficients at 15 seconds in a section of the H-beams

is decreased by 60sim80∘C for three cases For the threecases the temperature field for the optimal heat transfercoefficients was more uniform than for the temperaturefield of the constant heat transfer coefficients Also from

Case 1 to Case 3 the sizes became gradually smaller so thetemperature differences were also smaller

Figure 9 and Table 4 display the residual stress distribu-tion at the web for three cases with constant heat transfer

12 Advances in Materials Science and Engineering

Table 1 The optimization search of ℎ1ndashℎ5 distribution for Case 1 (eg with ten points)

Number ℎ1 ℎ2 ℎ3 ℎ4 ℎ5 119879ave Δ119879Wm2K Wm2K Wm2K Wm2K Wm2K ∘C ∘C

(1) (constant) 1500 1500 1500 1500 150 550 201(2) 1546 1595 1596 1557 157 540 205(3) 1485 1547 1548 1507 107 546 195(4) 1419 1556 1555 1508 107 550 180(5) 1335 1576 1427 1415 167 555 176(6) 1309 1565 1497 1482 163 556 170(7) 1316 1572 1568 1507 106 557 160(8) 1264 1642 1635 1504 111 558 150(9) 1240 1702 1695 1490 105 559 145(10) (optimization) 1227 1745 1732 1479 101 560 144

Table 2 The optimization search of ℎ1ndashℎ5 distribution for Case 2 (eg with ten points)

Number ℎ1 ℎ2 ℎ3 ℎ4 ℎ5 119879ave Δ119879Wm2K Wm2K Wm2K Wm2K Wm2K ∘C ∘C

(1) (constant) 1400 1400 1400 1400 140 548 204(2) 1340 1449 1450 1419 100 548 198(3) 1365 1453 1452 1420 99 549 192(4) 1315 1460 1458 1421 98 551 178(5) 1289 1464 1461 1421 96 553 172(6) 1213 1482 1475 1465 92 555 151(7) 1181 1545 1485 1486 67 556 142(8) 1133 1610 1550 1477 64 558 136(9) 1109 1610 1555 1456 44 559 132(10) (optimization) 1091 1705 1666 1445 37 560 130

Table 3 The optimization search of ℎ1ndashℎ5 distribution for Case 3 (eg with ten points)

Number ℎ1 ℎ2 ℎ3 ℎ4 ℎ5 119879ave Δ119879Wm2K Wm2K Wm2K Wm2K Wm2K ∘C ∘C

(1) (constant) 1200 1200 1200 1200 120 543 179(2) 1213 945 934 1087 101 552 210(3) 1168 1013 1000 1140 151 553 189(4) 1121 1081 1065 1192 201 553 168(5) 1074 1148 1130 1243 201 555 147(6) 1038 1213 1191 1267 182 556 133(7) 1017 1278 1254 1280 204 556 131(8) 991 1353 1323 1265 203 557 123(9) 959 1435 1398 1223 181 559 111(10) (optimization) 913 1579 1556 1166 225 560 104

coefficients and optimal heat transfer coefficients based onthe analysis of the temperature field FromFigure 9 the flangeand the fillet of the residual stress are tensor in nature theweb of the residual stress is compressive in nature with themaximum compressive stress occurring at the junction of theweb and the fillet For Case 1 the residual stress in the webis in the range of minus125MPa to minus107MPa with the constantheat transfer coefficients and is about minus90MPa with optimal

heat transfer coefficients For Case 2 the residual stress inthe web is in the range of minus100MPa to minus80MPa with theconstant heat transfer coefficients and is about minus60MPa withoptimal heat transfer coefficients For Case 3 the residualstress in the web is in the range of minus80MPa to minus70MPa withthe constant heat transfer coefficients and is about minus50MPawith optimal heat transfer coefficients From Table 4 it isobserved that optimum heat transfer coefficients result in

Advances in Materials Science and Engineering 13

Optimal ℎℎ are constant

Resid

ual s

tress

(MPa

)

x (mm)

60

40

20

0

minus20

minus40

minus60

minus80

minus100

minus120

minus140

minus160

minus150 minus100 minus50 0 50 100 150

(a) Case 1Re

sidua

l stre

ss (M

Pa)

x (mm)

60

40

20

0

minus20

minus40

minus60

minus80

minus100

minus120

minus140

minus100 minus50 0 50 100

Optimal ℎℎ are constant

(b) Case 2

Resid

ual s

tress

(MPa

)

60

40

20

0

minus20

minus40

minus60

minus80

minus100

minus120

x (mm)minus100 minus50 0 50 100

Optimal ℎℎ are constant

(c) Case 3

Figure 9 The residual stresses distribution at the web

Table 4 A comparison of the results for the three cases

Case Case 1 Case 2 Case 3Constant Optimal Constant Optimal Constant Optimalℎ1 (Wm2K) 1500 1227 1400 1091 1200 913ℎ2 (Wm2K) 1500 1745 1400 1705 1200 1579ℎ3 (Wm2K) 1500 1732 1400 1666 1200 1556ℎ4 (Wm2K) 1500 1479 1400 1445 1200 1166ℎ5 (Wm2K) 150 101 140 37 120 225Δ119879max (

∘C) 201 144 204 130 180 104Residual stress at web (MPa) minus114 minus91 minus89 minus62 minus73 minus51

14 Advances in Materials Science and Engineering

smaller differences in maximum temperature and therebygenerated a lower residual stress distribution in the webacross all sizes of the beams researched in this paper

5 Conclusions

The conjugate-gradient method (CGM) was used here topredict the optimization heat transfer coefficient distributionof H-beams with different sizes To this end three 3D numer-ical models were analyzed to estimate the temperature fieldand the residual stresses field developed during the coolingprocess The results from the optimal heat transfer coeffi-cients were compared with those constant heat transfer coef-ficients Based on the numerical results the conclusion canbe summarized as follows

(1)The conjugate-gradientmethod (CGM) has accuratelypredicted the heat transfer coefficients and the differentinitial guess values have resulted in very similar result Inthe search process we can get many heat transfer coefficientsto satisfy the average temperature with different temperaturedifferences and there is a group of heat transfer coefficientsthat has the lowest temperature difference using the CGMoptimization search method

(2) A comparison of the constant heat transfer coefficientsand the optimal heat transfer coefficients reveals obviousimprovement in the predicted uniform temperature fieldand temperature difference with the optimal heat transfercoefficients where the maximum temperature difference andthe surface maximum temperature difference are decreasedfor all the cases analyzed

(3)The cooling rate was fastest in the web followed by theflange and the fillet

(4) After the water cooling and the air cooling to theambient temperature the residual stress at the web was calcu-lated and a comparison of the three results for the constantheat transfer coefficients and optimal heat transfer coeffi-cients indicates that the residual stress can be reduced by20sim40MPa for the three cases

Nomenclature

119861 Flange width mm119862119901 Specific heat Jsdotkgminus1sdotKminus1119864 Youngrsquos modulus GPaℎ Heat transfer coefficient Wsdotmminus2 Kminus1119867 Web height mm119869obj Objective functions119896 Thermal conductivity Wsdotmminus1sdotKminus1119877 Fillet radius mm119904119894119895 The component of deviatoric stresses MPa119879 Temperature ∘C1199051 Web thickness mm1199052 Flange thickness mm119905 Time s

Greek Symbols

120572 Thermal expansion coefficient ∘Cminus1120573 Search step size

120574 Conjugate-gradient coefficient120576119894119895 Strains120585 Search directions120588 Density kgmminus3120592 Poissonrsquos ratio120590119894119895 Thermal stresses MPa120590119890 The von Mises effective stress MPa120590119904 The yield stress MPa120575119894119895 Kronecker delta120575 Error of the average temperature ∘CΔ119879 Temperature difference ∘C

Subscripts

ave Average valueinfin Environment condition119898 Search steps in the CGM119899 The variables number119904 Surface of the H-beam

Disclosure

Part of the abstract of this paper was presented in 19th Inter-national Conference on Aerospace Mechanical Automotiveand Materials Engineering May 11-12 Montreal Canada2017

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

This research was financially supported by the Ministryof Science and Technology and China Steel CorporationTaiwan under Contract no MOST103-2622-E006-037 Theauthors are also grateful to Dr Chao-Hua Wang of ChinaSteel Corporation Taiwan for his valuable suggestions

References

[1] C J Bettles and M A Gibson ldquoCurrent wrought magnesiumalloys strengths and weaknessesrdquo Journal of Metals vol 57 pp46ndash49 2005

[2] Z Li D Wu and W Lu ldquoEffects of rolling and coolingconditions onmicrostructure andmechanical properties of lowcarbon cold heading steelrdquo Journal of Iron and Steel ResearchInternational vol 19 no 11 pp 64ndash70 2012

[3] C Ouchi ldquoDevelopment of steel plates by intensive use ofTMCP and direct quenching processesrdquo ISIJ International vol41 no 6 pp 542ndash553 2001

[4] S Endo and N Nakata ldquoDevelopment of Thermo-MechanicalControl Process (TMCP) and high performance steel in JFESteelrdquo JFE Technical Report no 20 pp 1ndash7 2015

[5] N Shikanai S Mitao and S Endo ldquoRecent developmentin microstructural control technologies through the thermo-mechanical control process (TMCP) with JFE Steelrsquos high-performance platesrdquo JFE Technical Report no 11 pp 1ndash6 2008

Advances in Materials Science and Engineering 15

[6] H W Liu J Yan D H Chen and J Q Qian ldquoFinite elementanalysis of H-beam temperature field during coolingrdquo Journalof Anhui University of Technology vol 20 pp 4ndash7 2003

[7] Q Liu Y ZangQQin and J Q Zhao ldquoAnalysis on temperatureand thermal stress of h-beam during coolingrdquo MetallurgicalEquipment vol 167 pp 17ndash20 2008

[8] X-M Zhao F Guo L-N Wang and D Wu ldquoEffect of ultra-fast cooling on cooling distortion for H-beam steelrdquo Journal ofNortheastern University vol 33 no 7 pp 949ndash952 2012

[9] F Guo X-M Zhao L-N Wang D Wu G-D Wang and Z-L Ning ldquoEffect of controlled cooling on microstructure andmechanical properties of H-beamrdquo Journal of Iron and SteelResearch International vol 20 no 8 pp 60ndash65 2013

[10] J H Ma X H Yao B Tao and S Li ldquoEffects of coolingafter rolling on thermal stress of H-beam steelrdquo Hot WorkingTechnology vol 43 pp 121ndash123 2014

[11] G M Zhu Y L Kang W Chen J L Gui and G T MaldquoThermalmechanical coupled finite element analysis of thermalresidual stress in h-beamrsquos air coolingrdquoMaterials for MechanicalEngineering vol 32 pp 77ndash80 2008

[12] Q Liu Y Zang Q Qin and Z Q Zhao ldquoAnalysis ontemperature and thermal stress of H-beam during coolingrdquoMetallurgical Equipment vol 167 pp 17ndash20 2008

[13] L Zhao X G Yu G B Liao and Y L Hu ldquoSimulation ofStress-strain of H-beam in control cooling processrdquo Journal ofUniversity of Science and Technology Liaoning vol 34 pp 466ndash469 2011

[14] G Zhu Y Kang andGMa ldquoSimulation of residual stress in hotrolled large size H-beamsrdquo Materials Science Forum vol 704-705 pp 1370ndash1378 2012

[15] J Y Jang and Y G Gan ldquoOptimization analysis of controlledcooling process for h-shape steel beamsrdquo in Proceedings ofthe International Conference on Mechatronic Automobile andEnvironment Engineering 2017

[16] J Y Jang L FHsu and J S Leu ldquoOptimization of the span angleand location of vortex generators in a plate-fin and tube heatexchangerrdquo International Journal of Heat andMass Transfer vol67 pp 432ndash444 2013

[17] J-Y Jang and Y-C Tsai ldquoOptimization of thermoelectricgenerator module spacing and spreader thickness used in awaste heat recovery systemrdquo Applied Thermal Engineering vol51 no 1-2 pp 677ndash689 2013

[18] J-Y Jang and C-C Chen ldquoOptimization of louvered-fin heatexchanger with variable louver anglesrdquo Applied Thermal Engi-neering vol 91 pp 138ndash150 2015

[19] J-Y Jang and J-B Huang ldquoOptimization of a slab heatingpattern for minimum energy consumption in a walking-beamtype reheating furnacerdquo Applied Thermal Engineering vol 85article no 6547 pp 313ndash321 2015

[20] X K Zhu and Y J Chao ldquoNumerical simulation of transienttemperature and residual stresses in friction stir welding of 304Lstainless steelrdquo Journal of Materials Processing Technology vol146 no 2 pp 263ndash272 2004

[21] JmatPro 90 Practical software for material properties SenteSoftware Ltd 2017

[22] J Wendelstorf K-H Spitzer and R Wendelstorf ldquoSpray watercooling heat transfer at high temperatures and liquid massfluxesrdquo International Journal of Heat and Mass Transfer vol 51no 19-20 pp 4902ndash4910 2008

[23] F P Incropera D P Dewitt T L Bergman and A S LavineldquoPrinciples of Heat and Mass Transferrdquo in International StudentVersion 7th edition

Submit your manuscripts athttpswwwhindawicom

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4 Advances in Materials Science and Engineering

Case Size Grid numberCase 1 443034Case 2 359214Case 3 245916

300 times 300250 times 250200 times 200

Figure 2 The meshing grid of the physical model

Tinfin

ℎ1 ℎ2ℎ2

ℎ2 ℎ2

ℎ4 ℎ4

ℎ4ℎ4ℎ4ℎ4

ℎ4

ℎ3

ℎ4

ℎ1

ℎ5ℎ5

ℎ5 ℎ5

ℎ3

Figure 3 The heat transfer coefficients distribution

22 Initial and Boundary Conditions The convective heattransfer between the H-beams surface and the surroundingsis evaluated by using the equation

11990210158401015840 = minus119896 12059711987912059711989910038161003816100381610038161003816100381610038161003816surface = ℎ (119879119904 minus 119879infin) (3)

where ℎ is the convective heat transfer coefficient for thesurface of the H-beam and varies with the different parts ofthe H-beam In addition 119879119904 is the surface temperature of theH-beam and 119879infin is the environmental temperature of 25∘Cin the water cooling and air cooling process The coolingconditions require water cooling for 15 s and then air coolingto the ambient temperature (about 100 minutes) where thewater cooling rate is about 20∘Cs the air cooling heat transfercoefficient is 30Wm2K and the water cooling heat transfercoefficients are determined by the constant and optimizationvalues of the CGM

3 Numerical Method

31 Numerical Algorithm In this study the commercialsoftware ANSYS is adopted to solve the governing equationsand the finite element method is used to simulate thetemperature field and the thermomechanical analysis Thecomputation process is transient so the first-order implicitmethod was adopted to maintain the numerical stability ofevery time step The time step was 01 s for the CGM and theprogram controlled for thermomechanical analysis to satisfythe requirements of the computational accuracyThe compu-tational grids for the three-dimensional models which arecomposed of 3544272 107848 and 94248 cells for Case 1Case 2 and Case 3 respectively were typically adopted inthe computational domain as shown in Figure 2 and Table 2However a careful check of the grid independence of thenumerical solutions was made to ensure the accuracy andvalidity of the numerical results For this purpose thesymmetry model grid systems were tested which comprised541323 443034 and 295556 cells for Case 1 394611 359214and 262200 cells for Case 2 and 385120 245916 and 129276cells for Case 3 respectively It was found that the relativeerrors under the same conditions among the temperaturesolutions obtained with three types of grids were less than05 for the three cases The discretized system was solvediteratively until it satisfied the following residual convergencecriterion

max(10038161003816100381610038161003816119879119899119894119895 minus 119879119899minus1119894119895 10038161003816100381610038161003816119879119899119894119895 ) le 10minus6 (4)

where 119879119899minus1119894119895 is the previous temperature value of 119879119899119894119895 at thesame time level

The simulations were performed as a parallel calculationusing sixteen core central processing units for the three-dimensional models The computer computation times wereapproximately 10 8 and 6 minutes for each search step forCase 1 Case 2 and Case 3 respectively

Advances in Materials Science and Engineering 5

Cp

(KJK

g-K)

k(W

m-K

)

Temperature (∘C)

Thermal conductivity

Specific heat

10

8

6

4

2

00 200 400 600 800

60

50

40

30

20

10

0

minus10

(a) Specific heat and thermal conductivity

Temperature (∘C)0 200 400 600 800

30

25

20

15

10

5

0

(10minus

6∘

C)

(b) Thermal expansion

Temperature (∘C)0 200 400 600 800

0

200

400

600

800

Y

(MPa

)

E(G

Pa)

Youngrsquos modulus

Yield stress

220

200

180

160

140

120

100

80

(c) Youngrsquos modulus and yield stress

Temperature (∘C)0 200 400 600 800

100

80

60

40

20

0

Phas

e vol

()

AusteniteFerrite

BainitePearlite

(d) Phases volume

Figure 4 Variations in the thermal-mechanical properties with temperature

32 Optimization In this study the conjugate-gradientmethod (CGM) was combined with a finite volume method(FVM) code as an optimizer to search for the optimum heattransfer coefficients (ℎ1 ℎ2 ℎ3 ℎ4 ℎ5) The objective func-tions 1198691obj(ℎ1 ℎ2 ℎ3 ℎ4 ℎ5) and 1198692obj(ℎ1 ℎ2 ℎ3 ℎ4 ℎ5) weredefined as the range of the final average temperature (119879ave plusmn120575 = 550 plusmn 10∘C) calculated by vertex average temperatureof the H-beam and the maximum temperature differenceas a minimum (minΔ119879max(119909 119910)) calculated by the vertexmaximum temperature subtracting the vertex minimumtemperature of the H-beam

Above all the CGMmethod evaluates the gradient of theobjective functions and then it sets up a new conjugate direc-tion for the updated design variables with the help of a direct

numerical sensitivity analysis The initial guess for the valueof each search variable was made and in the successive stepsthe conjugate-gradient coefficients and the search directionswere evaluated to estimate the new search variables Thesolutions obtained from the finite difference method werethen used to calculate the value of the objective functionswhich were further transmitted back to the optimizer forthe purpose of calculating the consecutive directions Theprocedure for applying this method is described in thefollowing

(1) Generate an initial guess for five design vari-ables (1199091 1199092 1199093 1199094 1199095) the heat transfer coefficients(ℎ1 ℎ2 ℎ3 ℎ4 ℎ5)

6 Advances in Materials Science and Engineering

(2) Adopt the finite difference method to predict thetemperature fields (119879(119909 119910)) associated with the last(ℎ1 ℎ2 ℎ3 ℎ4 ℎ5) and then calculate the objectivefunctions 1198691obj(ℎ1 ℎ2 ℎ3 ℎ4 ℎ5) and 1198692obj(ℎ1 ℎ2 ℎ3ℎ4 ℎ5)

(3) When the value of 1198691obj(ℎ1 ℎ2 ℎ3 ℎ4 ℎ5) reaches theobject range and 1198692obj(ℎ1 ℎ2 ℎ3 ℎ4 ℎ5) reaches theminimum the optimization process is terminatedOtherwise proceed to step (4)

(4) Determine the gradient functions (1205971198691obj120597119909119894)(119898) and(1205971198692obj120597119909119894)(119898) (119894 = 1 2 3 4 5) by applying a smallperturbation (Δ1199091 Δ1199092 Δ1199093 Δ1199094 Δ1199095) to each valueof (1199091 1199092 1199093 1199094 1199095) and calculate the correspondingchange in objective functions 1198691obj(ℎ1 ℎ2 ℎ3 ℎ4 ℎ5)and 1198692obj(ℎ1 ℎ2 ℎ3 ℎ4 ℎ5) Then the gradient func-tions with respect to each value of the design variables(1199091 1199092 1199093 1199094 1199095) can be calculated by the directnumerical differentiation as

1205971198691obj1205971199091 = Δ1198691objΔ1199091 1205971198691obj1205971199092 = Δ1198691objΔ1199092 1205971198691obj1205971199093 = Δ1198691objΔ1199093 1205971198691obj1205971199094 = Δ1198691objΔ1199094 1205971198691obj1205971199095 = Δ1198691objΔ11990951205971198692obj1205971199091 = Δ1198692objΔ1199091 1205971198692obj1205971199092 = Δ1198692objΔ1199092 1205971198692obj1205971199093 = Δ1198692objΔ1199093 1205971198692obj1205971199094 = Δ1198692objΔ1199094 1205971198692obj1205971199095 = Δ1198692objΔ1199095

(5)

(5) Calculate the conjugate-gradient coefficients 120574(119898)and the search directions 120585(119898+1)1 120585(119898+1)2 120585(119898+1)3 120585(119898+1)4 120585(119898+1)5 for the search variables For the first stepwith 119896 = 1 120574(1) = 0

120574(119898) = [[sum2119899 (120597119869(119898)obj 120597119909119899)sum2119899 (120597119869(119898minus1)obj 120597119909119899)]]

2

119899 = 1 2 3 4 5

120585(119898+1)1 = 120597119869(119898)obj1205971199091 + 120574(119898)120585(119898)1 120585(119898+1)2 = 120597119869(119898)obj1205971199092 + 120574(119898)120585(119898)2 120585(119898+1)3 = 120597119869(119898)obj1205971199093 + 120574(119898)120585(119898)3 120585(119898+1)4 = 120597119869(119898)obj1205971199094 + 120574(119898)120585(119898)4 120585(119898+1)5 = 120597119869(119898)obj1205971199095 + 120574(119898)120585(119898)5

(6)

(6) Assign values to the coefficients in a descendingdirection (1205731 1205732 1205733 1205734 1205735) for all values of the designvariables (1199091 1199092 1199093 1199094 1199095) Specifically these val-ues are chosen with a trial and error process Ingeneral the coefficients in the descending direction(1205731 1205732 1205733 1205734 1205735) range from 1 to 50

(7) Update the design variables with

119909(119898+1)1 = 119909(119898)1 + 1205731120585(119898)1 119909(119898+1)2 = 119909(119898)2 + 1205732120585(119898)2 119909(119898+1)3 = 119909(119898)3 + 1205733120585(119898)3 119909(119898+1)4 = 119909(119898)4 + 1205734120585(119898)4 119909(119898+1)5 = 119909(119898)5 + 1205735120585(119898)5

(7)

A flowchart of the CGM optimization process is plotted asFigure 5

4 Result and Discussion

In order to examine the validity of the results of the CGMmethod two initial guess values were used to search forthe optimal heat transfer coefficient distributions of the H-beams for three cases and they all obtained approximatelythe same result Based on the optimal heat transfer coefficientdistributions using CGM method the temperature field andthe residual stress of the H-beam were calculated to comparewith the constant heat transfer coefficient results Taking intoaccount the operability in the cooling process the area ofthe up surface of the flange is small and not in the watercooling range therefore the heat transfer coefficient ℎ5 is asignificantly smaller value than ℎ1 to ℎ4

Figure 6 shows the final average temperature and themaximum temperature difference search path where theinitial guess value 1 was ℎ1 = ℎ2 = ℎ3 = ℎ4 = 1600 (Wm2K)

Advances in Materials Science and Engineering 7

X design variables (heat transfer coefficient h)

No

YesStop

Start

Conjugate-gradient method

k = k + 1 Initial guess design variablesX1 X2 X3 X4 X5

Direct problem solver(heat transfer conductionthermal-stress equation)F(X1 X2 X3 X4 X5)

Evaluate objective functionJ(k)

Convergence

J(k) minus J(kminus1)

J(k)lt ＬＦＮＣＰ error

N = 1

Sensitivity analysisF(X1 + ΔX1 X2 X3 X4 X5) = J(k)1

J(k)1

X1

=J(k)1 minus J(k)

X1

Conjugate-gradient coefficient

Searching direction

(k)1 =J(k)1

X1

+ (k)1 (kminus1)1

New design variable

X(k+1)1 = X(k)

1 + 1J(k)1

N = 2

Sensitivity analysisF(X1 X2 + ΔX2 X3 X4 X5) = J(k)2

J(k)2

X2

=J(k)2 minus J(k)

X2

Conjugate-gradient coefficient

Searching direction

(k)2 =J(k)2

X2

+ (k)2 (kminus1)2

New design variable

X(k+1)2 = X(k)

2 + 2J(k)2

N = 5

N = 3 4

Sensitivity analysisF(X1 X2 X3 X4 X5 + ΔX5) = J(k)5

J(k)5

X5

=J(k)5 minus J(k)

X5

Searching direction

(k)5 =J(k)n

X5

+ (k)5 (kminus1)5

New design variable

X(k+1)5 = X(k)

5 + 5J(k)5

F(X1 X2 X3 X4 X5) functionsJ(k) objective functions (the average temperature and the temperature difference TＰ plusmn ΔT ＧＣＨ ΔTＧＲ)

Conjugate-gradient coefficient

(k)1 = [ J(k)1 X1

J(kminus1)1 X1

]2

(k)2 = [ J(k)2 X2

J(kminus1)2 X2

]2

(k)5 = [ J(k)5 X5

J(kminus1)5 X5

]2

Figure 5 A flowchart of the CGM optimization process

8 Advances in Materials Science and Engineering

ℎ5 = 160 (Wm2K) and the initial guess value 2 was ℎ1 =ℎ2 = ℎ3 = ℎ4 = 1200 (Wm2K) ℎ5 = 120 (Wm2K) forCase 1 The initial guess values for heat transfer coefficientsℎ1 = ℎ2 = ℎ3 = ℎ4 = 1600Wm2K in water are estimatedfrom the water jet heat transfer coefficient correlations byWendelstorf et al [22] while the ℎ5 = 160Wm2K in airis estimated from the combined natural and radiation heattransfer theory [23] The search trend of the final averagetemperature for these two initial guess values was close to theobjective value step by step and were within the error rangeThe search trend for the temperature difference for the twoinitial guess values was obtained by searching for the lowestvalue step by step with the final average temperature in therange of the objective value until the temperature differencevariable was stable The different search steps were obtainedfor different initial guess values but they all obtained almostthe same result where the resultant deviation for the twoinitial values was below 1∘C It is apparent that Case 2 andCase 3 both had the same search path as that of Case 1 Theinitial guess value 1 was ℎ1 = ℎ2 = ℎ3 = ℎ4 = 1500 (Wm2K)ℎ5 = 150 (Wm2K) and the initial guess value 2was ℎ1 = ℎ2 =ℎ3 = ℎ4 = 1000 (Wm2K) ℎ5 = 100 (Wm2K) for Case 2 theinitial guess value 1 was ℎ1 = ℎ2 = ℎ3 = ℎ4 = 1200 (Wm2K)ℎ5 = 120 (Wm2K) and the initial guess value 2 was ℎ1 =ℎ2 = ℎ3 = ℎ4 = 800 (Wm2K) ℎ5 = 80 (Wm2K) for Case 3respectively For the three cases there were different searchsteps for the difference cases and initial guess values wherethe search steps were simply related to the initial guess valueand the search step size

Tables 1ndash3 list the different distributions of the heattransfer coefficients for an example with ten points in theoptimization search process where the final average tem-peratures were all in a range of 540∘C to 560∘C but thetemperature difference decreased from 201∘C to 144∘C from204∘C to 130∘C and from 179∘C to 104∘C for Case 1 toCase 3 respectively The optimization distribution of theheat transfer coefficients is the tenth point in Tables 1ndash3For Case 1 the optimization distribution of heat transfercoefficients ℎ1 to ℎ5 is 1227 1745 1732 1479 and 101 (Wm2K)where the average temperature is 560∘C and the temperaturedifference is 144∘C For Case 2 the optimization distributionof the heat transfer coefficients ℎ1 to h5 is 1091 1705 16661445 and 37 (Wm2K) where the average temperature is560∘C and the temperature difference is 130∘C For Case 3the optimization distribution of heat transfer coefficients ℎ1to ℎ5 is 913 1579 1556 1166 and 225 (Wm2K)where the finalaverage temperature is 560∘C and the temperature differenceis 104∘C

Figure 7 shows that the variation of the average tempera-ture in the different parts of the beam and that the maximumtemperature difference with time for the three cases undertwo heat boundary conditions satisfied the average tempera-ture of H-beams in the range of 540sim560∘C For Case 1 whenthe heat transfer coefficients were constant (ℎ1 = ℎ2 = ℎ3 =ℎ4 = 1500 (Wm2K) ℎ5 = 150 (Wm2K)) the different partshad different trends in variation For the web the coolingrate was very fast at first and then gradually slowed with thedecrease of the web surface temperature and when the time

was over 12 seconds the cooling rate became as fast as it wasoriginally In the case of the flange the cooling rate was slowerthan that of the web where the cooling rate went from fast toslow In the case of the fillet the cooling rate was the slowestand exhibited the same variable trends as that of the flangeThe maximum temperature difference increased initially inthe first 6 seconds and then decreased slightly until themaximum temperature difference was 201∘C at 15 secondsWhen the heat transfer coefficients were optimized (ℎ1 =1227 (Wm2K) ℎ2 = 1745 (Wm2K) ℎ3 = 1732 (Wm2K)ℎ4 = 1479 (Wm2K) and ℎ5 = 101 (Wm2K)) the variationof the average temperature with time in the different partsexhibited the same trends compared with the results of theconstant heat transfer coefficients except the fillet coolingrate was slightly larger than that of the constant heat transfercoefficients the flange cooling rate was slightly less than thatof the constant heat transfer coefficients the web cooling ratewas obviously slower than that of the constant heat transfercoefficients and the cooling rate of the web went from fastto slow The temperature difference for the optimal heattransfer coefficients increased initially and then decreaseduntil the maximum temperature difference was 144∘C at 15seconds and the peak of the temperature difference wascontrolled at 6 seconds For Case 2 and Case 3 the constantheat transfer coefficients were ℎ1 = ℎ2 = ℎ3 = ℎ4 =1400 (Wm2K) ℎ5 = 140 (Wm2K) and ℎ1 = ℎ2 = ℎ3 =ℎ4 = 1200 (Wm2K) ℎ5 = 120 (Wm2K) and the optimalheat transfer coefficients were ℎ1 = 1091 (Wm2K) ℎ2 =1705 (Wm2K) ℎ3 = 1666 (Wm2K) ℎ4 = 1444 (Wm2K)ℎ5 = 37 (Wm2K) and ℎ1 = 913 (Wm2K) ℎ2 =1579 (Wm2K) ℎ3 = 1556 (Wm2K) ℎ4 = 1166 (Wm2K)and ℎ5 = 225 (Wm2K) respectively The variation of theaverage temperature with time exhibited the same trends asthose found for Case 1 regardless of whether it was deter-mined with constant heat transfer coefficients or optimalheat transfer coefficients The cooling rates of Case 2 amongthe web flange and fillet were closer than in Case 1 andCase 3 was the closest because of the physical dimensionsThe temperature difference for the constant heat transfercoefficients reached a peak at 65 seconds and 7 secondsfor Case 2 and Case 3 and then slowly decreased until thetemperature differences were 204∘C and 180∘C at 15 secondsrespectively The maximum temperature difference of theoptimal heat transfer coefficients reached a peak at 4 secondsand at 3 seconds for Case 2 and Case 3 and then slowlydecreased to 130∘C and 104∘C at 15 seconds respectively

Figure 8 displays the temperature field for three caseswith constant heat transfer coefficients and optimal heattransfer coefficients at 15 seconds ForCase 1 the temperatureranged from 667∘C to 466∘C with the constant heat transfercoefficients and the temperature ranged from 654∘C to 510∘Cwith the optimal heat transfer coefficients For Case 2 thetemperature ranged from 658∘C to 454∘C with the constantheat transfer coefficients and the temperature ranged from639∘C to 509∘Cwith the optimal heat transfer coefficients ForCase 3 the temperature ranged from 646∘C to 466∘C withthe constant heat transfer coefficients and the temperatureranged from 618∘C to 514∘C with the optimal heat transfercoefficients the surface maximum temperature difference

Advances in Materials Science and Engineering 9

TＰ

(∘

C)

Search steps

Initial value 1

Initial value 2

Initial value 1 ℎ1 = ℎ2 = ℎ3 = ℎ4 = 1600 ℎ5 = 160 (WＧ2K)Initial value 2 ℎ1 = ℎ2 = ℎ3 = ℎ4 = 1200 ℎ5 = 120 (WＧ2K)

590

580

570

560

550

540

5300 20 40 60 80 100 120 140 160

Search steps0 50 100 150 200

Initial value 1Initial value 2

Search steps

Search steps

0 20 40 60 80 100 120 140 160

220

200

180

160

140

120

0 50 100 150 200

Initial value 1

Initial value 2

Initial value 1Initial value 2

ΔT

(∘C)

(a) Case 1

220

200

180

160

140

120

Search steps0 50 100 150 200

Search steps0 50 100 150 200 250 300

Initial value 1

Initial value 2

Initial value 1Initial value 2

TＰ

(∘

C)

Search steps

600

580

560

540

0 50 100 150 200

Initial value 1 ℎ1 = ℎ2 = ℎ3 = ℎ4 = 1500 ℎ5 = 150 (WＧ2K)Initial value 2 ℎ1 = ℎ2 = ℎ3 = ℎ4 = 1000 ℎ5 = 100 (WＧ2K)

Search steps0 50 100 150 200 250 300

Initial value 1

Initial value 2

Initial value 1Initial value 2

ΔT

(∘C)

(b) Case 2

TＰ

(∘

C)

Search steps0 20 40 60 80 100 120 140 160

620

600

580

560

540

520

0 100 200 300 400

Initial value 1 ℎ1 = ℎ2 = ℎ3 = ℎ4 = 1200 ℎ5 = 120 (WＧ2 K)Initial value 2 ℎ1 = ℎ2 = ℎ3 = ℎ4 = 800 ℎ5 = 80 (WＧ2K)

Search steps

Initial value 1

Initial value 2

Initial value 1Initial value 2

Search steps0 20 40 60 80 100 120 140 160

220

200

180

160

140

120

100

80

0 100 200 300 400

Search steps

Initial value 1

Initial value 2

Initial value 1Initial value 2

ΔT

(∘C)

(c) Case 3

Figure 6 The search path with the different initial values for the three cases

10 Advances in Materials Science and Engineering

Constant

ℎ1 = ℎ2 = ℎ3 = ℎ4 = 1500 ℎ5 = 150 (WＧ2K)Optimal

ℎ1 = 1227 ℎ2 = 1745 ℎ3 = 1732 ℎ4 = 1479 ℎ5 = 101 (WＧ2K)

Time (s)0 2 4 6 8 10 12 14

250

200

150

100

50

0

ℎ are constantOptimal ℎ

ℎ are constant

Optimal ℎ

Time (s)

Tem

p(∘ C)

900

800

700

600

500

4000 2 4 6 8 10 12 14

Optimal ℎℎ are constant

ΔT

(∘C)

TＰ with filletTＰ with flange

TＰ with web

(a) Case 1

Tem

p(∘ C)

900

800

700

600

500

400

Time (s)0 2 4 6 8 10 12 14

Optimal ℎℎ are constant

TＰ with filletTＰ with flange

TＰ with web

Constantℎ1 = ℎ2 = ℎ3 = ℎ4 = 1400 ℎ5 = 140 (WＧ2K)

Optimalℎ1 = 1091 ℎ2 = 1705 ℎ3 = 1666 ℎ4 = 1444 ℎ5 = 37 (WＧ2K)

Time (s)0 2 4 6 8 10 12 14

ℎ is constantOptimal ℎ

ℎ are constant

Optimal ℎ

200

150

100

50

0

ΔT

(∘C)

(b) Case 2

Constant

ℎ1 = ℎ2 = ℎ3 = ℎ4 = 1200 ℎ5 = 120 (WＧ2K)Optimal

ℎ1 = 913 ℎ2 = 1579 ℎ3 = 1556 ℎ4 = 1166 ℎ5 = 225 (WＧ2K)

Time (s)0 2 4 6 8 10 12 14

200

150

100

50

0

ℎ is constantOptimal ℎ

ℎ are constant

Optimal ℎ

Tem

p(∘ C)

900

800

700

600

500

400

Time (s)0 2 4 6 8 10 12 14

TＰ with filletTＰ with flange

TＰ with web

Optimal ℎℎ are constant

ΔT

(∘C)

(c) Case 3

Figure 7 The temperature in different parts and the temperature difference variable with time

Advances in Materials Science and Engineering 11

Section

L2

Constant heattransfer coefficients

Optimal heattransfer coefficients

667652637621606591576560545530515499484469454

(∘C)

(a) Case 1

Constant heattransfer coefficients

Optimal heattransfer coefficients

667652637621606591576560545530515499484469454

(∘C)

(b) Case 2

Constant heattransfer coefficients

Optimal heattransfer coefficients

667652637621606591576560545530515499484469454

(∘C)

(c) Case 3

Figure 8 The distribution of temperature with the different heat transfer coefficients at 15 seconds in a section of the H-beams

is decreased by 60sim80∘C for three cases For the threecases the temperature field for the optimal heat transfercoefficients was more uniform than for the temperaturefield of the constant heat transfer coefficients Also from

Case 1 to Case 3 the sizes became gradually smaller so thetemperature differences were also smaller

Figure 9 and Table 4 display the residual stress distribu-tion at the web for three cases with constant heat transfer

12 Advances in Materials Science and Engineering

Table 1 The optimization search of ℎ1ndashℎ5 distribution for Case 1 (eg with ten points)

Number ℎ1 ℎ2 ℎ3 ℎ4 ℎ5 119879ave Δ119879Wm2K Wm2K Wm2K Wm2K Wm2K ∘C ∘C

(1) (constant) 1500 1500 1500 1500 150 550 201(2) 1546 1595 1596 1557 157 540 205(3) 1485 1547 1548 1507 107 546 195(4) 1419 1556 1555 1508 107 550 180(5) 1335 1576 1427 1415 167 555 176(6) 1309 1565 1497 1482 163 556 170(7) 1316 1572 1568 1507 106 557 160(8) 1264 1642 1635 1504 111 558 150(9) 1240 1702 1695 1490 105 559 145(10) (optimization) 1227 1745 1732 1479 101 560 144

Table 2 The optimization search of ℎ1ndashℎ5 distribution for Case 2 (eg with ten points)

Number ℎ1 ℎ2 ℎ3 ℎ4 ℎ5 119879ave Δ119879Wm2K Wm2K Wm2K Wm2K Wm2K ∘C ∘C

(1) (constant) 1400 1400 1400 1400 140 548 204(2) 1340 1449 1450 1419 100 548 198(3) 1365 1453 1452 1420 99 549 192(4) 1315 1460 1458 1421 98 551 178(5) 1289 1464 1461 1421 96 553 172(6) 1213 1482 1475 1465 92 555 151(7) 1181 1545 1485 1486 67 556 142(8) 1133 1610 1550 1477 64 558 136(9) 1109 1610 1555 1456 44 559 132(10) (optimization) 1091 1705 1666 1445 37 560 130

Table 3 The optimization search of ℎ1ndashℎ5 distribution for Case 3 (eg with ten points)

Number ℎ1 ℎ2 ℎ3 ℎ4 ℎ5 119879ave Δ119879Wm2K Wm2K Wm2K Wm2K Wm2K ∘C ∘C

(1) (constant) 1200 1200 1200 1200 120 543 179(2) 1213 945 934 1087 101 552 210(3) 1168 1013 1000 1140 151 553 189(4) 1121 1081 1065 1192 201 553 168(5) 1074 1148 1130 1243 201 555 147(6) 1038 1213 1191 1267 182 556 133(7) 1017 1278 1254 1280 204 556 131(8) 991 1353 1323 1265 203 557 123(9) 959 1435 1398 1223 181 559 111(10) (optimization) 913 1579 1556 1166 225 560 104

coefficients and optimal heat transfer coefficients based onthe analysis of the temperature field FromFigure 9 the flangeand the fillet of the residual stress are tensor in nature theweb of the residual stress is compressive in nature with themaximum compressive stress occurring at the junction of theweb and the fillet For Case 1 the residual stress in the webis in the range of minus125MPa to minus107MPa with the constantheat transfer coefficients and is about minus90MPa with optimal

heat transfer coefficients For Case 2 the residual stress inthe web is in the range of minus100MPa to minus80MPa with theconstant heat transfer coefficients and is about minus60MPa withoptimal heat transfer coefficients For Case 3 the residualstress in the web is in the range of minus80MPa to minus70MPa withthe constant heat transfer coefficients and is about minus50MPawith optimal heat transfer coefficients From Table 4 it isobserved that optimum heat transfer coefficients result in

Advances in Materials Science and Engineering 13

Optimal ℎℎ are constant

Resid

ual s

tress

(MPa

)

x (mm)

60

40

20

0

minus20

minus40

minus60

minus80

minus100

minus120

minus140

minus160

minus150 minus100 minus50 0 50 100 150

(a) Case 1Re

sidua

l stre

ss (M

Pa)

x (mm)

60

40

20

0

minus20

minus40

minus60

minus80

minus100

minus120

minus140

minus100 minus50 0 50 100

Optimal ℎℎ are constant

(b) Case 2

Resid

ual s

tress

(MPa

)

60

40

20

0

minus20

minus40

minus60

minus80

minus100

minus120

x (mm)minus100 minus50 0 50 100

Optimal ℎℎ are constant

(c) Case 3

Figure 9 The residual stresses distribution at the web

Table 4 A comparison of the results for the three cases

Case Case 1 Case 2 Case 3Constant Optimal Constant Optimal Constant Optimalℎ1 (Wm2K) 1500 1227 1400 1091 1200 913ℎ2 (Wm2K) 1500 1745 1400 1705 1200 1579ℎ3 (Wm2K) 1500 1732 1400 1666 1200 1556ℎ4 (Wm2K) 1500 1479 1400 1445 1200 1166ℎ5 (Wm2K) 150 101 140 37 120 225Δ119879max (

∘C) 201 144 204 130 180 104Residual stress at web (MPa) minus114 minus91 minus89 minus62 minus73 minus51

14 Advances in Materials Science and Engineering

smaller differences in maximum temperature and therebygenerated a lower residual stress distribution in the webacross all sizes of the beams researched in this paper

5 Conclusions

The conjugate-gradient method (CGM) was used here topredict the optimization heat transfer coefficient distributionof H-beams with different sizes To this end three 3D numer-ical models were analyzed to estimate the temperature fieldand the residual stresses field developed during the coolingprocess The results from the optimal heat transfer coeffi-cients were compared with those constant heat transfer coef-ficients Based on the numerical results the conclusion canbe summarized as follows

(1)The conjugate-gradientmethod (CGM) has accuratelypredicted the heat transfer coefficients and the differentinitial guess values have resulted in very similar result Inthe search process we can get many heat transfer coefficientsto satisfy the average temperature with different temperaturedifferences and there is a group of heat transfer coefficientsthat has the lowest temperature difference using the CGMoptimization search method

(2) A comparison of the constant heat transfer coefficientsand the optimal heat transfer coefficients reveals obviousimprovement in the predicted uniform temperature fieldand temperature difference with the optimal heat transfercoefficients where the maximum temperature difference andthe surface maximum temperature difference are decreasedfor all the cases analyzed

(3)The cooling rate was fastest in the web followed by theflange and the fillet

(4) After the water cooling and the air cooling to theambient temperature the residual stress at the web was calcu-lated and a comparison of the three results for the constantheat transfer coefficients and optimal heat transfer coeffi-cients indicates that the residual stress can be reduced by20sim40MPa for the three cases

Nomenclature

119861 Flange width mm119862119901 Specific heat Jsdotkgminus1sdotKminus1119864 Youngrsquos modulus GPaℎ Heat transfer coefficient Wsdotmminus2 Kminus1119867 Web height mm119869obj Objective functions119896 Thermal conductivity Wsdotmminus1sdotKminus1119877 Fillet radius mm119904119894119895 The component of deviatoric stresses MPa119879 Temperature ∘C1199051 Web thickness mm1199052 Flange thickness mm119905 Time s

Greek Symbols

120572 Thermal expansion coefficient ∘Cminus1120573 Search step size

120574 Conjugate-gradient coefficient120576119894119895 Strains120585 Search directions120588 Density kgmminus3120592 Poissonrsquos ratio120590119894119895 Thermal stresses MPa120590119890 The von Mises effective stress MPa120590119904 The yield stress MPa120575119894119895 Kronecker delta120575 Error of the average temperature ∘CΔ119879 Temperature difference ∘C

Subscripts

ave Average valueinfin Environment condition119898 Search steps in the CGM119899 The variables number119904 Surface of the H-beam

Disclosure

Part of the abstract of this paper was presented in 19th Inter-national Conference on Aerospace Mechanical Automotiveand Materials Engineering May 11-12 Montreal Canada2017

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

This research was financially supported by the Ministryof Science and Technology and China Steel CorporationTaiwan under Contract no MOST103-2622-E006-037 Theauthors are also grateful to Dr Chao-Hua Wang of ChinaSteel Corporation Taiwan for his valuable suggestions

References

[1] C J Bettles and M A Gibson ldquoCurrent wrought magnesiumalloys strengths and weaknessesrdquo Journal of Metals vol 57 pp46ndash49 2005

[2] Z Li D Wu and W Lu ldquoEffects of rolling and coolingconditions onmicrostructure andmechanical properties of lowcarbon cold heading steelrdquo Journal of Iron and Steel ResearchInternational vol 19 no 11 pp 64ndash70 2012

[3] C Ouchi ldquoDevelopment of steel plates by intensive use ofTMCP and direct quenching processesrdquo ISIJ International vol41 no 6 pp 542ndash553 2001

[4] S Endo and N Nakata ldquoDevelopment of Thermo-MechanicalControl Process (TMCP) and high performance steel in JFESteelrdquo JFE Technical Report no 20 pp 1ndash7 2015

[5] N Shikanai S Mitao and S Endo ldquoRecent developmentin microstructural control technologies through the thermo-mechanical control process (TMCP) with JFE Steelrsquos high-performance platesrdquo JFE Technical Report no 11 pp 1ndash6 2008

Advances in Materials Science and Engineering 15

[6] H W Liu J Yan D H Chen and J Q Qian ldquoFinite elementanalysis of H-beam temperature field during coolingrdquo Journalof Anhui University of Technology vol 20 pp 4ndash7 2003

[7] Q Liu Y ZangQQin and J Q Zhao ldquoAnalysis on temperatureand thermal stress of h-beam during coolingrdquo MetallurgicalEquipment vol 167 pp 17ndash20 2008

[8] X-M Zhao F Guo L-N Wang and D Wu ldquoEffect of ultra-fast cooling on cooling distortion for H-beam steelrdquo Journal ofNortheastern University vol 33 no 7 pp 949ndash952 2012

[9] F Guo X-M Zhao L-N Wang D Wu G-D Wang and Z-L Ning ldquoEffect of controlled cooling on microstructure andmechanical properties of H-beamrdquo Journal of Iron and SteelResearch International vol 20 no 8 pp 60ndash65 2013

[10] J H Ma X H Yao B Tao and S Li ldquoEffects of coolingafter rolling on thermal stress of H-beam steelrdquo Hot WorkingTechnology vol 43 pp 121ndash123 2014

[11] G M Zhu Y L Kang W Chen J L Gui and G T MaldquoThermalmechanical coupled finite element analysis of thermalresidual stress in h-beamrsquos air coolingrdquoMaterials for MechanicalEngineering vol 32 pp 77ndash80 2008

[12] Q Liu Y Zang Q Qin and Z Q Zhao ldquoAnalysis ontemperature and thermal stress of H-beam during coolingrdquoMetallurgical Equipment vol 167 pp 17ndash20 2008

[13] L Zhao X G Yu G B Liao and Y L Hu ldquoSimulation ofStress-strain of H-beam in control cooling processrdquo Journal ofUniversity of Science and Technology Liaoning vol 34 pp 466ndash469 2011

[14] G Zhu Y Kang andGMa ldquoSimulation of residual stress in hotrolled large size H-beamsrdquo Materials Science Forum vol 704-705 pp 1370ndash1378 2012

[15] J Y Jang and Y G Gan ldquoOptimization analysis of controlledcooling process for h-shape steel beamsrdquo in Proceedings ofthe International Conference on Mechatronic Automobile andEnvironment Engineering 2017

[16] J Y Jang L FHsu and J S Leu ldquoOptimization of the span angleand location of vortex generators in a plate-fin and tube heatexchangerrdquo International Journal of Heat andMass Transfer vol67 pp 432ndash444 2013

[17] J-Y Jang and Y-C Tsai ldquoOptimization of thermoelectricgenerator module spacing and spreader thickness used in awaste heat recovery systemrdquo Applied Thermal Engineering vol51 no 1-2 pp 677ndash689 2013

[18] J-Y Jang and C-C Chen ldquoOptimization of louvered-fin heatexchanger with variable louver anglesrdquo Applied Thermal Engi-neering vol 91 pp 138ndash150 2015

[19] J-Y Jang and J-B Huang ldquoOptimization of a slab heatingpattern for minimum energy consumption in a walking-beamtype reheating furnacerdquo Applied Thermal Engineering vol 85article no 6547 pp 313ndash321 2015

[20] X K Zhu and Y J Chao ldquoNumerical simulation of transienttemperature and residual stresses in friction stir welding of 304Lstainless steelrdquo Journal of Materials Processing Technology vol146 no 2 pp 263ndash272 2004

[21] JmatPro 90 Practical software for material properties SenteSoftware Ltd 2017

[22] J Wendelstorf K-H Spitzer and R Wendelstorf ldquoSpray watercooling heat transfer at high temperatures and liquid massfluxesrdquo International Journal of Heat and Mass Transfer vol 51no 19-20 pp 4902ndash4910 2008

[23] F P Incropera D P Dewitt T L Bergman and A S LavineldquoPrinciples of Heat and Mass Transferrdquo in International StudentVersion 7th edition

Submit your manuscripts athttpswwwhindawicom

ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CorrosionInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Polymer ScienceInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CeramicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CompositesJournal of

NanoparticlesJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Biomaterials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

NanoscienceJournal of

TextilesHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Journal of

NanotechnologyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

CrystallographyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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CoatingsJournal of

Advances in

Materials Science and EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Smart Materials Research

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MetallurgyJournal of

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BioMed Research International

MaterialsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Materials Science and Engineering 5

Cp

(KJK

g-K)

k(W

m-K

)

Temperature (∘C)

Thermal conductivity

Specific heat

10

8

6

4

2

00 200 400 600 800

60

50

40

30

20

10

0

minus10

(a) Specific heat and thermal conductivity

Temperature (∘C)0 200 400 600 800

30

25

20

15

10

5

0

(10minus

6∘

C)

(b) Thermal expansion

Temperature (∘C)0 200 400 600 800

0

200

400

600

800

Y

(MPa

)

E(G

Pa)

Youngrsquos modulus

Yield stress

220

200

180

160

140

120

100

80

(c) Youngrsquos modulus and yield stress

Temperature (∘C)0 200 400 600 800

100

80

60

40

20

0

Phas

e vol

()

AusteniteFerrite

BainitePearlite

(d) Phases volume

Figure 4 Variations in the thermal-mechanical properties with temperature

32 Optimization In this study the conjugate-gradientmethod (CGM) was combined with a finite volume method(FVM) code as an optimizer to search for the optimum heattransfer coefficients (ℎ1 ℎ2 ℎ3 ℎ4 ℎ5) The objective func-tions 1198691obj(ℎ1 ℎ2 ℎ3 ℎ4 ℎ5) and 1198692obj(ℎ1 ℎ2 ℎ3 ℎ4 ℎ5) weredefined as the range of the final average temperature (119879ave plusmn120575 = 550 plusmn 10∘C) calculated by vertex average temperatureof the H-beam and the maximum temperature differenceas a minimum (minΔ119879max(119909 119910)) calculated by the vertexmaximum temperature subtracting the vertex minimumtemperature of the H-beam

Above all the CGMmethod evaluates the gradient of theobjective functions and then it sets up a new conjugate direc-tion for the updated design variables with the help of a direct

numerical sensitivity analysis The initial guess for the valueof each search variable was made and in the successive stepsthe conjugate-gradient coefficients and the search directionswere evaluated to estimate the new search variables Thesolutions obtained from the finite difference method werethen used to calculate the value of the objective functionswhich were further transmitted back to the optimizer forthe purpose of calculating the consecutive directions Theprocedure for applying this method is described in thefollowing

(1) Generate an initial guess for five design vari-ables (1199091 1199092 1199093 1199094 1199095) the heat transfer coefficients(ℎ1 ℎ2 ℎ3 ℎ4 ℎ5)

6 Advances in Materials Science and Engineering

(2) Adopt the finite difference method to predict thetemperature fields (119879(119909 119910)) associated with the last(ℎ1 ℎ2 ℎ3 ℎ4 ℎ5) and then calculate the objectivefunctions 1198691obj(ℎ1 ℎ2 ℎ3 ℎ4 ℎ5) and 1198692obj(ℎ1 ℎ2 ℎ3ℎ4 ℎ5)

(3) When the value of 1198691obj(ℎ1 ℎ2 ℎ3 ℎ4 ℎ5) reaches theobject range and 1198692obj(ℎ1 ℎ2 ℎ3 ℎ4 ℎ5) reaches theminimum the optimization process is terminatedOtherwise proceed to step (4)

(4) Determine the gradient functions (1205971198691obj120597119909119894)(119898) and(1205971198692obj120597119909119894)(119898) (119894 = 1 2 3 4 5) by applying a smallperturbation (Δ1199091 Δ1199092 Δ1199093 Δ1199094 Δ1199095) to each valueof (1199091 1199092 1199093 1199094 1199095) and calculate the correspondingchange in objective functions 1198691obj(ℎ1 ℎ2 ℎ3 ℎ4 ℎ5)and 1198692obj(ℎ1 ℎ2 ℎ3 ℎ4 ℎ5) Then the gradient func-tions with respect to each value of the design variables(1199091 1199092 1199093 1199094 1199095) can be calculated by the directnumerical differentiation as

1205971198691obj1205971199091 = Δ1198691objΔ1199091 1205971198691obj1205971199092 = Δ1198691objΔ1199092 1205971198691obj1205971199093 = Δ1198691objΔ1199093 1205971198691obj1205971199094 = Δ1198691objΔ1199094 1205971198691obj1205971199095 = Δ1198691objΔ11990951205971198692obj1205971199091 = Δ1198692objΔ1199091 1205971198692obj1205971199092 = Δ1198692objΔ1199092 1205971198692obj1205971199093 = Δ1198692objΔ1199093 1205971198692obj1205971199094 = Δ1198692objΔ1199094 1205971198692obj1205971199095 = Δ1198692objΔ1199095

(5)

(5) Calculate the conjugate-gradient coefficients 120574(119898)and the search directions 120585(119898+1)1 120585(119898+1)2 120585(119898+1)3 120585(119898+1)4 120585(119898+1)5 for the search variables For the first stepwith 119896 = 1 120574(1) = 0

120574(119898) = [[sum2119899 (120597119869(119898)obj 120597119909119899)sum2119899 (120597119869(119898minus1)obj 120597119909119899)]]

2

119899 = 1 2 3 4 5

120585(119898+1)1 = 120597119869(119898)obj1205971199091 + 120574(119898)120585(119898)1 120585(119898+1)2 = 120597119869(119898)obj1205971199092 + 120574(119898)120585(119898)2 120585(119898+1)3 = 120597119869(119898)obj1205971199093 + 120574(119898)120585(119898)3 120585(119898+1)4 = 120597119869(119898)obj1205971199094 + 120574(119898)120585(119898)4 120585(119898+1)5 = 120597119869(119898)obj1205971199095 + 120574(119898)120585(119898)5

(6)

(6) Assign values to the coefficients in a descendingdirection (1205731 1205732 1205733 1205734 1205735) for all values of the designvariables (1199091 1199092 1199093 1199094 1199095) Specifically these val-ues are chosen with a trial and error process Ingeneral the coefficients in the descending direction(1205731 1205732 1205733 1205734 1205735) range from 1 to 50

(7) Update the design variables with

119909(119898+1)1 = 119909(119898)1 + 1205731120585(119898)1 119909(119898+1)2 = 119909(119898)2 + 1205732120585(119898)2 119909(119898+1)3 = 119909(119898)3 + 1205733120585(119898)3 119909(119898+1)4 = 119909(119898)4 + 1205734120585(119898)4 119909(119898+1)5 = 119909(119898)5 + 1205735120585(119898)5

(7)

A flowchart of the CGM optimization process is plotted asFigure 5

4 Result and Discussion

In order to examine the validity of the results of the CGMmethod two initial guess values were used to search forthe optimal heat transfer coefficient distributions of the H-beams for three cases and they all obtained approximatelythe same result Based on the optimal heat transfer coefficientdistributions using CGM method the temperature field andthe residual stress of the H-beam were calculated to comparewith the constant heat transfer coefficient results Taking intoaccount the operability in the cooling process the area ofthe up surface of the flange is small and not in the watercooling range therefore the heat transfer coefficient ℎ5 is asignificantly smaller value than ℎ1 to ℎ4

Figure 6 shows the final average temperature and themaximum temperature difference search path where theinitial guess value 1 was ℎ1 = ℎ2 = ℎ3 = ℎ4 = 1600 (Wm2K)

Advances in Materials Science and Engineering 7

X design variables (heat transfer coefficient h)

No

YesStop

Start

Conjugate-gradient method

k = k + 1 Initial guess design variablesX1 X2 X3 X4 X5

Direct problem solver(heat transfer conductionthermal-stress equation)F(X1 X2 X3 X4 X5)

Evaluate objective functionJ(k)

Convergence

J(k) minus J(kminus1)

J(k)lt ＬＦＮＣＰ error

N = 1

Sensitivity analysisF(X1 + ΔX1 X2 X3 X4 X5) = J(k)1

J(k)1

X1

=J(k)1 minus J(k)

X1

Conjugate-gradient coefficient

Searching direction

(k)1 =J(k)1

X1

+ (k)1 (kminus1)1

New design variable

X(k+1)1 = X(k)

1 + 1J(k)1

N = 2

Sensitivity analysisF(X1 X2 + ΔX2 X3 X4 X5) = J(k)2

J(k)2

X2

=J(k)2 minus J(k)

X2

Conjugate-gradient coefficient

Searching direction

(k)2 =J(k)2

X2

+ (k)2 (kminus1)2

New design variable

X(k+1)2 = X(k)

2 + 2J(k)2

N = 5

N = 3 4

Sensitivity analysisF(X1 X2 X3 X4 X5 + ΔX5) = J(k)5

J(k)5

X5

=J(k)5 minus J(k)

X5

Searching direction

(k)5 =J(k)n

X5

+ (k)5 (kminus1)5

New design variable

X(k+1)5 = X(k)

5 + 5J(k)5

F(X1 X2 X3 X4 X5) functionsJ(k) objective functions (the average temperature and the temperature difference TＰ plusmn ΔT ＧＣＨ ΔTＧＲ)

Conjugate-gradient coefficient

(k)1 = [ J(k)1 X1

J(kminus1)1 X1

]2

(k)2 = [ J(k)2 X2

J(kminus1)2 X2

]2

(k)5 = [ J(k)5 X5

J(kminus1)5 X5

]2

Figure 5 A flowchart of the CGM optimization process

8 Advances in Materials Science and Engineering

ℎ5 = 160 (Wm2K) and the initial guess value 2 was ℎ1 =ℎ2 = ℎ3 = ℎ4 = 1200 (Wm2K) ℎ5 = 120 (Wm2K) forCase 1 The initial guess values for heat transfer coefficientsℎ1 = ℎ2 = ℎ3 = ℎ4 = 1600Wm2K in water are estimatedfrom the water jet heat transfer coefficient correlations byWendelstorf et al [22] while the ℎ5 = 160Wm2K in airis estimated from the combined natural and radiation heattransfer theory [23] The search trend of the final averagetemperature for these two initial guess values was close to theobjective value step by step and were within the error rangeThe search trend for the temperature difference for the twoinitial guess values was obtained by searching for the lowestvalue step by step with the final average temperature in therange of the objective value until the temperature differencevariable was stable The different search steps were obtainedfor different initial guess values but they all obtained almostthe same result where the resultant deviation for the twoinitial values was below 1∘C It is apparent that Case 2 andCase 3 both had the same search path as that of Case 1 Theinitial guess value 1 was ℎ1 = ℎ2 = ℎ3 = ℎ4 = 1500 (Wm2K)ℎ5 = 150 (Wm2K) and the initial guess value 2was ℎ1 = ℎ2 =ℎ3 = ℎ4 = 1000 (Wm2K) ℎ5 = 100 (Wm2K) for Case 2 theinitial guess value 1 was ℎ1 = ℎ2 = ℎ3 = ℎ4 = 1200 (Wm2K)ℎ5 = 120 (Wm2K) and the initial guess value 2 was ℎ1 =ℎ2 = ℎ3 = ℎ4 = 800 (Wm2K) ℎ5 = 80 (Wm2K) for Case 3respectively For the three cases there were different searchsteps for the difference cases and initial guess values wherethe search steps were simply related to the initial guess valueand the search step size

Tables 1ndash3 list the different distributions of the heattransfer coefficients for an example with ten points in theoptimization search process where the final average tem-peratures were all in a range of 540∘C to 560∘C but thetemperature difference decreased from 201∘C to 144∘C from204∘C to 130∘C and from 179∘C to 104∘C for Case 1 toCase 3 respectively The optimization distribution of theheat transfer coefficients is the tenth point in Tables 1ndash3For Case 1 the optimization distribution of heat transfercoefficients ℎ1 to ℎ5 is 1227 1745 1732 1479 and 101 (Wm2K)where the average temperature is 560∘C and the temperaturedifference is 144∘C For Case 2 the optimization distributionof the heat transfer coefficients ℎ1 to h5 is 1091 1705 16661445 and 37 (Wm2K) where the average temperature is560∘C and the temperature difference is 130∘C For Case 3the optimization distribution of heat transfer coefficients ℎ1to ℎ5 is 913 1579 1556 1166 and 225 (Wm2K)where the finalaverage temperature is 560∘C and the temperature differenceis 104∘C

Figure 7 shows that the variation of the average tempera-ture in the different parts of the beam and that the maximumtemperature difference with time for the three cases undertwo heat boundary conditions satisfied the average tempera-ture of H-beams in the range of 540sim560∘C For Case 1 whenthe heat transfer coefficients were constant (ℎ1 = ℎ2 = ℎ3 =ℎ4 = 1500 (Wm2K) ℎ5 = 150 (Wm2K)) the different partshad different trends in variation For the web the coolingrate was very fast at first and then gradually slowed with thedecrease of the web surface temperature and when the time

was over 12 seconds the cooling rate became as fast as it wasoriginally In the case of the flange the cooling rate was slowerthan that of the web where the cooling rate went from fast toslow In the case of the fillet the cooling rate was the slowestand exhibited the same variable trends as that of the flangeThe maximum temperature difference increased initially inthe first 6 seconds and then decreased slightly until themaximum temperature difference was 201∘C at 15 secondsWhen the heat transfer coefficients were optimized (ℎ1 =1227 (Wm2K) ℎ2 = 1745 (Wm2K) ℎ3 = 1732 (Wm2K)ℎ4 = 1479 (Wm2K) and ℎ5 = 101 (Wm2K)) the variationof the average temperature with time in the different partsexhibited the same trends compared with the results of theconstant heat transfer coefficients except the fillet coolingrate was slightly larger than that of the constant heat transfercoefficients the flange cooling rate was slightly less than thatof the constant heat transfer coefficients the web cooling ratewas obviously slower than that of the constant heat transfercoefficients and the cooling rate of the web went from fastto slow The temperature difference for the optimal heattransfer coefficients increased initially and then decreaseduntil the maximum temperature difference was 144∘C at 15seconds and the peak of the temperature difference wascontrolled at 6 seconds For Case 2 and Case 3 the constantheat transfer coefficients were ℎ1 = ℎ2 = ℎ3 = ℎ4 =1400 (Wm2K) ℎ5 = 140 (Wm2K) and ℎ1 = ℎ2 = ℎ3 =ℎ4 = 1200 (Wm2K) ℎ5 = 120 (Wm2K) and the optimalheat transfer coefficients were ℎ1 = 1091 (Wm2K) ℎ2 =1705 (Wm2K) ℎ3 = 1666 (Wm2K) ℎ4 = 1444 (Wm2K)ℎ5 = 37 (Wm2K) and ℎ1 = 913 (Wm2K) ℎ2 =1579 (Wm2K) ℎ3 = 1556 (Wm2K) ℎ4 = 1166 (Wm2K)and ℎ5 = 225 (Wm2K) respectively The variation of theaverage temperature with time exhibited the same trends asthose found for Case 1 regardless of whether it was deter-mined with constant heat transfer coefficients or optimalheat transfer coefficients The cooling rates of Case 2 amongthe web flange and fillet were closer than in Case 1 andCase 3 was the closest because of the physical dimensionsThe temperature difference for the constant heat transfercoefficients reached a peak at 65 seconds and 7 secondsfor Case 2 and Case 3 and then slowly decreased until thetemperature differences were 204∘C and 180∘C at 15 secondsrespectively The maximum temperature difference of theoptimal heat transfer coefficients reached a peak at 4 secondsand at 3 seconds for Case 2 and Case 3 and then slowlydecreased to 130∘C and 104∘C at 15 seconds respectively

Figure 8 displays the temperature field for three caseswith constant heat transfer coefficients and optimal heattransfer coefficients at 15 seconds ForCase 1 the temperatureranged from 667∘C to 466∘C with the constant heat transfercoefficients and the temperature ranged from 654∘C to 510∘Cwith the optimal heat transfer coefficients For Case 2 thetemperature ranged from 658∘C to 454∘C with the constantheat transfer coefficients and the temperature ranged from639∘C to 509∘Cwith the optimal heat transfer coefficients ForCase 3 the temperature ranged from 646∘C to 466∘C withthe constant heat transfer coefficients and the temperatureranged from 618∘C to 514∘C with the optimal heat transfercoefficients the surface maximum temperature difference

Advances in Materials Science and Engineering 9

TＰ

(∘

C)

Search steps

Initial value 1

Initial value 2

Initial value 1 ℎ1 = ℎ2 = ℎ3 = ℎ4 = 1600 ℎ5 = 160 (WＧ2K)Initial value 2 ℎ1 = ℎ2 = ℎ3 = ℎ4 = 1200 ℎ5 = 120 (WＧ2K)

590

580

570

560

550

540

5300 20 40 60 80 100 120 140 160

Search steps0 50 100 150 200

Initial value 1Initial value 2

Search steps

Search steps

0 20 40 60 80 100 120 140 160

220

200

180

160

140

120

0 50 100 150 200

Initial value 1

Initial value 2

Initial value 1Initial value 2

ΔT

(∘C)

(a) Case 1

220

200

180

160

140

120

Search steps0 50 100 150 200

Search steps0 50 100 150 200 250 300

Initial value 1

Initial value 2

Initial value 1Initial value 2

TＰ

(∘

C)

Search steps

600

580

560

540

0 50 100 150 200

Initial value 1 ℎ1 = ℎ2 = ℎ3 = ℎ4 = 1500 ℎ5 = 150 (WＧ2K)Initial value 2 ℎ1 = ℎ2 = ℎ3 = ℎ4 = 1000 ℎ5 = 100 (WＧ2K)

Search steps0 50 100 150 200 250 300

Initial value 1

Initial value 2

Initial value 1Initial value 2

ΔT

(∘C)

(b) Case 2

TＰ

(∘

C)

Search steps0 20 40 60 80 100 120 140 160

620

600

580

560

540

520

0 100 200 300 400

Initial value 1 ℎ1 = ℎ2 = ℎ3 = ℎ4 = 1200 ℎ5 = 120 (WＧ2 K)Initial value 2 ℎ1 = ℎ2 = ℎ3 = ℎ4 = 800 ℎ5 = 80 (WＧ2K)

Search steps

Initial value 1

Initial value 2

Initial value 1Initial value 2

Search steps0 20 40 60 80 100 120 140 160

220

200

180

160

140

120

100

80

0 100 200 300 400

Search steps

Initial value 1

Initial value 2

Initial value 1Initial value 2

ΔT

(∘C)

(c) Case 3

Figure 6 The search path with the different initial values for the three cases

10 Advances in Materials Science and Engineering

Constant

ℎ1 = ℎ2 = ℎ3 = ℎ4 = 1500 ℎ5 = 150 (WＧ2K)Optimal

ℎ1 = 1227 ℎ2 = 1745 ℎ3 = 1732 ℎ4 = 1479 ℎ5 = 101 (WＧ2K)

Time (s)0 2 4 6 8 10 12 14

250

200

150

100

50

0

ℎ are constantOptimal ℎ

ℎ are constant

Optimal ℎ

Time (s)

Tem

p(∘ C)

900

800

700

600

500

4000 2 4 6 8 10 12 14

Optimal ℎℎ are constant

ΔT

(∘C)

TＰ with filletTＰ with flange

TＰ with web

(a) Case 1

Tem

p(∘ C)

900

800

700

600

500

400

Time (s)0 2 4 6 8 10 12 14

Optimal ℎℎ are constant

TＰ with filletTＰ with flange

TＰ with web

Constantℎ1 = ℎ2 = ℎ3 = ℎ4 = 1400 ℎ5 = 140 (WＧ2K)

Optimalℎ1 = 1091 ℎ2 = 1705 ℎ3 = 1666 ℎ4 = 1444 ℎ5 = 37 (WＧ2K)

Time (s)0 2 4 6 8 10 12 14

ℎ is constantOptimal ℎ

ℎ are constant

Optimal ℎ

200

150

100

50

0

ΔT

(∘C)

(b) Case 2

Constant

ℎ1 = ℎ2 = ℎ3 = ℎ4 = 1200 ℎ5 = 120 (WＧ2K)Optimal

ℎ1 = 913 ℎ2 = 1579 ℎ3 = 1556 ℎ4 = 1166 ℎ5 = 225 (WＧ2K)

Time (s)0 2 4 6 8 10 12 14

200

150

100

50

0

ℎ is constantOptimal ℎ

ℎ are constant

Optimal ℎ

Tem

p(∘ C)

900

800

700

600

500

400

Time (s)0 2 4 6 8 10 12 14

TＰ with filletTＰ with flange

TＰ with web

Optimal ℎℎ are constant

ΔT

(∘C)

(c) Case 3

Figure 7 The temperature in different parts and the temperature difference variable with time

Advances in Materials Science and Engineering 11

Section

L2

Constant heattransfer coefficients

Optimal heattransfer coefficients

667652637621606591576560545530515499484469454

(∘C)

(a) Case 1

Constant heattransfer coefficients

Optimal heattransfer coefficients

667652637621606591576560545530515499484469454

(∘C)

(b) Case 2

Constant heattransfer coefficients

Optimal heattransfer coefficients

667652637621606591576560545530515499484469454

(∘C)

(c) Case 3

Figure 8 The distribution of temperature with the different heat transfer coefficients at 15 seconds in a section of the H-beams

is decreased by 60sim80∘C for three cases For the threecases the temperature field for the optimal heat transfercoefficients was more uniform than for the temperaturefield of the constant heat transfer coefficients Also from

Case 1 to Case 3 the sizes became gradually smaller so thetemperature differences were also smaller

Figure 9 and Table 4 display the residual stress distribu-tion at the web for three cases with constant heat transfer

12 Advances in Materials Science and Engineering

Table 1 The optimization search of ℎ1ndashℎ5 distribution for Case 1 (eg with ten points)

Number ℎ1 ℎ2 ℎ3 ℎ4 ℎ5 119879ave Δ119879Wm2K Wm2K Wm2K Wm2K Wm2K ∘C ∘C

(1) (constant) 1500 1500 1500 1500 150 550 201(2) 1546 1595 1596 1557 157 540 205(3) 1485 1547 1548 1507 107 546 195(4) 1419 1556 1555 1508 107 550 180(5) 1335 1576 1427 1415 167 555 176(6) 1309 1565 1497 1482 163 556 170(7) 1316 1572 1568 1507 106 557 160(8) 1264 1642 1635 1504 111 558 150(9) 1240 1702 1695 1490 105 559 145(10) (optimization) 1227 1745 1732 1479 101 560 144

Table 2 The optimization search of ℎ1ndashℎ5 distribution for Case 2 (eg with ten points)

Number ℎ1 ℎ2 ℎ3 ℎ4 ℎ5 119879ave Δ119879Wm2K Wm2K Wm2K Wm2K Wm2K ∘C ∘C

(1) (constant) 1400 1400 1400 1400 140 548 204(2) 1340 1449 1450 1419 100 548 198(3) 1365 1453 1452 1420 99 549 192(4) 1315 1460 1458 1421 98 551 178(5) 1289 1464 1461 1421 96 553 172(6) 1213 1482 1475 1465 92 555 151(7) 1181 1545 1485 1486 67 556 142(8) 1133 1610 1550 1477 64 558 136(9) 1109 1610 1555 1456 44 559 132(10) (optimization) 1091 1705 1666 1445 37 560 130

Table 3 The optimization search of ℎ1ndashℎ5 distribution for Case 3 (eg with ten points)

Number ℎ1 ℎ2 ℎ3 ℎ4 ℎ5 119879ave Δ119879Wm2K Wm2K Wm2K Wm2K Wm2K ∘C ∘C

(1) (constant) 1200 1200 1200 1200 120 543 179(2) 1213 945 934 1087 101 552 210(3) 1168 1013 1000 1140 151 553 189(4) 1121 1081 1065 1192 201 553 168(5) 1074 1148 1130 1243 201 555 147(6) 1038 1213 1191 1267 182 556 133(7) 1017 1278 1254 1280 204 556 131(8) 991 1353 1323 1265 203 557 123(9) 959 1435 1398 1223 181 559 111(10) (optimization) 913 1579 1556 1166 225 560 104

coefficients and optimal heat transfer coefficients based onthe analysis of the temperature field FromFigure 9 the flangeand the fillet of the residual stress are tensor in nature theweb of the residual stress is compressive in nature with themaximum compressive stress occurring at the junction of theweb and the fillet For Case 1 the residual stress in the webis in the range of minus125MPa to minus107MPa with the constantheat transfer coefficients and is about minus90MPa with optimal

heat transfer coefficients For Case 2 the residual stress inthe web is in the range of minus100MPa to minus80MPa with theconstant heat transfer coefficients and is about minus60MPa withoptimal heat transfer coefficients For Case 3 the residualstress in the web is in the range of minus80MPa to minus70MPa withthe constant heat transfer coefficients and is about minus50MPawith optimal heat transfer coefficients From Table 4 it isobserved that optimum heat transfer coefficients result in

Advances in Materials Science and Engineering 13

Optimal ℎℎ are constant

Resid

ual s

tress

(MPa

)

x (mm)

60

40

20

0

minus20

minus40

minus60

minus80

minus100

minus120

minus140

minus160

minus150 minus100 minus50 0 50 100 150

(a) Case 1Re

sidua

l stre

ss (M

Pa)

x (mm)

60

40

20

0

minus20

minus40

minus60

minus80

minus100

minus120

minus140

minus100 minus50 0 50 100

Optimal ℎℎ are constant

(b) Case 2

Resid

ual s

tress

(MPa

)

60

40

20

0

minus20

minus40

minus60

minus80

minus100

minus120

x (mm)minus100 minus50 0 50 100

Optimal ℎℎ are constant

(c) Case 3

Figure 9 The residual stresses distribution at the web

Table 4 A comparison of the results for the three cases

Case Case 1 Case 2 Case 3Constant Optimal Constant Optimal Constant Optimalℎ1 (Wm2K) 1500 1227 1400 1091 1200 913ℎ2 (Wm2K) 1500 1745 1400 1705 1200 1579ℎ3 (Wm2K) 1500 1732 1400 1666 1200 1556ℎ4 (Wm2K) 1500 1479 1400 1445 1200 1166ℎ5 (Wm2K) 150 101 140 37 120 225Δ119879max (

∘C) 201 144 204 130 180 104Residual stress at web (MPa) minus114 minus91 minus89 minus62 minus73 minus51

14 Advances in Materials Science and Engineering

smaller differences in maximum temperature and therebygenerated a lower residual stress distribution in the webacross all sizes of the beams researched in this paper

5 Conclusions

The conjugate-gradient method (CGM) was used here topredict the optimization heat transfer coefficient distributionof H-beams with different sizes To this end three 3D numer-ical models were analyzed to estimate the temperature fieldand the residual stresses field developed during the coolingprocess The results from the optimal heat transfer coeffi-cients were compared with those constant heat transfer coef-ficients Based on the numerical results the conclusion canbe summarized as follows

(1)The conjugate-gradientmethod (CGM) has accuratelypredicted the heat transfer coefficients and the differentinitial guess values have resulted in very similar result Inthe search process we can get many heat transfer coefficientsto satisfy the average temperature with different temperaturedifferences and there is a group of heat transfer coefficientsthat has the lowest temperature difference using the CGMoptimization search method

(2) A comparison of the constant heat transfer coefficientsand the optimal heat transfer coefficients reveals obviousimprovement in the predicted uniform temperature fieldand temperature difference with the optimal heat transfercoefficients where the maximum temperature difference andthe surface maximum temperature difference are decreasedfor all the cases analyzed

(3)The cooling rate was fastest in the web followed by theflange and the fillet

(4) After the water cooling and the air cooling to theambient temperature the residual stress at the web was calcu-lated and a comparison of the three results for the constantheat transfer coefficients and optimal heat transfer coeffi-cients indicates that the residual stress can be reduced by20sim40MPa for the three cases

Nomenclature

119861 Flange width mm119862119901 Specific heat Jsdotkgminus1sdotKminus1119864 Youngrsquos modulus GPaℎ Heat transfer coefficient Wsdotmminus2 Kminus1119867 Web height mm119869obj Objective functions119896 Thermal conductivity Wsdotmminus1sdotKminus1119877 Fillet radius mm119904119894119895 The component of deviatoric stresses MPa119879 Temperature ∘C1199051 Web thickness mm1199052 Flange thickness mm119905 Time s

Greek Symbols

120572 Thermal expansion coefficient ∘Cminus1120573 Search step size

120574 Conjugate-gradient coefficient120576119894119895 Strains120585 Search directions120588 Density kgmminus3120592 Poissonrsquos ratio120590119894119895 Thermal stresses MPa120590119890 The von Mises effective stress MPa120590119904 The yield stress MPa120575119894119895 Kronecker delta120575 Error of the average temperature ∘CΔ119879 Temperature difference ∘C

Subscripts

ave Average valueinfin Environment condition119898 Search steps in the CGM119899 The variables number119904 Surface of the H-beam

Disclosure

Part of the abstract of this paper was presented in 19th Inter-national Conference on Aerospace Mechanical Automotiveand Materials Engineering May 11-12 Montreal Canada2017

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

This research was financially supported by the Ministryof Science and Technology and China Steel CorporationTaiwan under Contract no MOST103-2622-E006-037 Theauthors are also grateful to Dr Chao-Hua Wang of ChinaSteel Corporation Taiwan for his valuable suggestions

References

[1] C J Bettles and M A Gibson ldquoCurrent wrought magnesiumalloys strengths and weaknessesrdquo Journal of Metals vol 57 pp46ndash49 2005

[2] Z Li D Wu and W Lu ldquoEffects of rolling and coolingconditions onmicrostructure andmechanical properties of lowcarbon cold heading steelrdquo Journal of Iron and Steel ResearchInternational vol 19 no 11 pp 64ndash70 2012

[3] C Ouchi ldquoDevelopment of steel plates by intensive use ofTMCP and direct quenching processesrdquo ISIJ International vol41 no 6 pp 542ndash553 2001

[4] S Endo and N Nakata ldquoDevelopment of Thermo-MechanicalControl Process (TMCP) and high performance steel in JFESteelrdquo JFE Technical Report no 20 pp 1ndash7 2015

[5] N Shikanai S Mitao and S Endo ldquoRecent developmentin microstructural control technologies through the thermo-mechanical control process (TMCP) with JFE Steelrsquos high-performance platesrdquo JFE Technical Report no 11 pp 1ndash6 2008

Advances in Materials Science and Engineering 15

[6] H W Liu J Yan D H Chen and J Q Qian ldquoFinite elementanalysis of H-beam temperature field during coolingrdquo Journalof Anhui University of Technology vol 20 pp 4ndash7 2003

[7] Q Liu Y ZangQQin and J Q Zhao ldquoAnalysis on temperatureand thermal stress of h-beam during coolingrdquo MetallurgicalEquipment vol 167 pp 17ndash20 2008

[8] X-M Zhao F Guo L-N Wang and D Wu ldquoEffect of ultra-fast cooling on cooling distortion for H-beam steelrdquo Journal ofNortheastern University vol 33 no 7 pp 949ndash952 2012

[9] F Guo X-M Zhao L-N Wang D Wu G-D Wang and Z-L Ning ldquoEffect of controlled cooling on microstructure andmechanical properties of H-beamrdquo Journal of Iron and SteelResearch International vol 20 no 8 pp 60ndash65 2013

[10] J H Ma X H Yao B Tao and S Li ldquoEffects of coolingafter rolling on thermal stress of H-beam steelrdquo Hot WorkingTechnology vol 43 pp 121ndash123 2014

[11] G M Zhu Y L Kang W Chen J L Gui and G T MaldquoThermalmechanical coupled finite element analysis of thermalresidual stress in h-beamrsquos air coolingrdquoMaterials for MechanicalEngineering vol 32 pp 77ndash80 2008

[12] Q Liu Y Zang Q Qin and Z Q Zhao ldquoAnalysis ontemperature and thermal stress of H-beam during coolingrdquoMetallurgical Equipment vol 167 pp 17ndash20 2008

[13] L Zhao X G Yu G B Liao and Y L Hu ldquoSimulation ofStress-strain of H-beam in control cooling processrdquo Journal ofUniversity of Science and Technology Liaoning vol 34 pp 466ndash469 2011

[14] G Zhu Y Kang andGMa ldquoSimulation of residual stress in hotrolled large size H-beamsrdquo Materials Science Forum vol 704-705 pp 1370ndash1378 2012

[15] J Y Jang and Y G Gan ldquoOptimization analysis of controlledcooling process for h-shape steel beamsrdquo in Proceedings ofthe International Conference on Mechatronic Automobile andEnvironment Engineering 2017

[16] J Y Jang L FHsu and J S Leu ldquoOptimization of the span angleand location of vortex generators in a plate-fin and tube heatexchangerrdquo International Journal of Heat andMass Transfer vol67 pp 432ndash444 2013

[17] J-Y Jang and Y-C Tsai ldquoOptimization of thermoelectricgenerator module spacing and spreader thickness used in awaste heat recovery systemrdquo Applied Thermal Engineering vol51 no 1-2 pp 677ndash689 2013

[18] J-Y Jang and C-C Chen ldquoOptimization of louvered-fin heatexchanger with variable louver anglesrdquo Applied Thermal Engi-neering vol 91 pp 138ndash150 2015

[19] J-Y Jang and J-B Huang ldquoOptimization of a slab heatingpattern for minimum energy consumption in a walking-beamtype reheating furnacerdquo Applied Thermal Engineering vol 85article no 6547 pp 313ndash321 2015

[20] X K Zhu and Y J Chao ldquoNumerical simulation of transienttemperature and residual stresses in friction stir welding of 304Lstainless steelrdquo Journal of Materials Processing Technology vol146 no 2 pp 263ndash272 2004

[21] JmatPro 90 Practical software for material properties SenteSoftware Ltd 2017

[22] J Wendelstorf K-H Spitzer and R Wendelstorf ldquoSpray watercooling heat transfer at high temperatures and liquid massfluxesrdquo International Journal of Heat and Mass Transfer vol 51no 19-20 pp 4902ndash4910 2008

[23] F P Incropera D P Dewitt T L Bergman and A S LavineldquoPrinciples of Heat and Mass Transferrdquo in International StudentVersion 7th edition

Submit your manuscripts athttpswwwhindawicom

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CeramicsJournal of

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Biomaterials

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TextilesHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Journal of

NanotechnologyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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CrystallographyJournal of

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Advances in

Materials Science and EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Smart Materials Research

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MetallurgyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

BioMed Research International

MaterialsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

6 Advances in Materials Science and Engineering

(2) Adopt the finite difference method to predict thetemperature fields (119879(119909 119910)) associated with the last(ℎ1 ℎ2 ℎ3 ℎ4 ℎ5) and then calculate the objectivefunctions 1198691obj(ℎ1 ℎ2 ℎ3 ℎ4 ℎ5) and 1198692obj(ℎ1 ℎ2 ℎ3ℎ4 ℎ5)

(3) When the value of 1198691obj(ℎ1 ℎ2 ℎ3 ℎ4 ℎ5) reaches theobject range and 1198692obj(ℎ1 ℎ2 ℎ3 ℎ4 ℎ5) reaches theminimum the optimization process is terminatedOtherwise proceed to step (4)

(4) Determine the gradient functions (1205971198691obj120597119909119894)(119898) and(1205971198692obj120597119909119894)(119898) (119894 = 1 2 3 4 5) by applying a smallperturbation (Δ1199091 Δ1199092 Δ1199093 Δ1199094 Δ1199095) to each valueof (1199091 1199092 1199093 1199094 1199095) and calculate the correspondingchange in objective functions 1198691obj(ℎ1 ℎ2 ℎ3 ℎ4 ℎ5)and 1198692obj(ℎ1 ℎ2 ℎ3 ℎ4 ℎ5) Then the gradient func-tions with respect to each value of the design variables(1199091 1199092 1199093 1199094 1199095) can be calculated by the directnumerical differentiation as

1205971198691obj1205971199091 = Δ1198691objΔ1199091 1205971198691obj1205971199092 = Δ1198691objΔ1199092 1205971198691obj1205971199093 = Δ1198691objΔ1199093 1205971198691obj1205971199094 = Δ1198691objΔ1199094 1205971198691obj1205971199095 = Δ1198691objΔ11990951205971198692obj1205971199091 = Δ1198692objΔ1199091 1205971198692obj1205971199092 = Δ1198692objΔ1199092 1205971198692obj1205971199093 = Δ1198692objΔ1199093 1205971198692obj1205971199094 = Δ1198692objΔ1199094 1205971198692obj1205971199095 = Δ1198692objΔ1199095

(5)

(5) Calculate the conjugate-gradient coefficients 120574(119898)and the search directions 120585(119898+1)1 120585(119898+1)2 120585(119898+1)3 120585(119898+1)4 120585(119898+1)5 for the search variables For the first stepwith 119896 = 1 120574(1) = 0

120574(119898) = [[sum2119899 (120597119869(119898)obj 120597119909119899)sum2119899 (120597119869(119898minus1)obj 120597119909119899)]]

2

119899 = 1 2 3 4 5

120585(119898+1)1 = 120597119869(119898)obj1205971199091 + 120574(119898)120585(119898)1 120585(119898+1)2 = 120597119869(119898)obj1205971199092 + 120574(119898)120585(119898)2 120585(119898+1)3 = 120597119869(119898)obj1205971199093 + 120574(119898)120585(119898)3 120585(119898+1)4 = 120597119869(119898)obj1205971199094 + 120574(119898)120585(119898)4 120585(119898+1)5 = 120597119869(119898)obj1205971199095 + 120574(119898)120585(119898)5

(6)

(6) Assign values to the coefficients in a descendingdirection (1205731 1205732 1205733 1205734 1205735) for all values of the designvariables (1199091 1199092 1199093 1199094 1199095) Specifically these val-ues are chosen with a trial and error process Ingeneral the coefficients in the descending direction(1205731 1205732 1205733 1205734 1205735) range from 1 to 50

(7) Update the design variables with

119909(119898+1)1 = 119909(119898)1 + 1205731120585(119898)1 119909(119898+1)2 = 119909(119898)2 + 1205732120585(119898)2 119909(119898+1)3 = 119909(119898)3 + 1205733120585(119898)3 119909(119898+1)4 = 119909(119898)4 + 1205734120585(119898)4 119909(119898+1)5 = 119909(119898)5 + 1205735120585(119898)5

(7)

A flowchart of the CGM optimization process is plotted asFigure 5

4 Result and Discussion

In order to examine the validity of the results of the CGMmethod two initial guess values were used to search forthe optimal heat transfer coefficient distributions of the H-beams for three cases and they all obtained approximatelythe same result Based on the optimal heat transfer coefficientdistributions using CGM method the temperature field andthe residual stress of the H-beam were calculated to comparewith the constant heat transfer coefficient results Taking intoaccount the operability in the cooling process the area ofthe up surface of the flange is small and not in the watercooling range therefore the heat transfer coefficient ℎ5 is asignificantly smaller value than ℎ1 to ℎ4

Figure 6 shows the final average temperature and themaximum temperature difference search path where theinitial guess value 1 was ℎ1 = ℎ2 = ℎ3 = ℎ4 = 1600 (Wm2K)

Advances in Materials Science and Engineering 7

X design variables (heat transfer coefficient h)

No

YesStop

Start

Conjugate-gradient method

k = k + 1 Initial guess design variablesX1 X2 X3 X4 X5

Direct problem solver(heat transfer conductionthermal-stress equation)F(X1 X2 X3 X4 X5)

Evaluate objective functionJ(k)

Convergence

J(k) minus J(kminus1)

J(k)lt ＬＦＮＣＰ error

N = 1

Sensitivity analysisF(X1 + ΔX1 X2 X3 X4 X5) = J(k)1

J(k)1

X1

=J(k)1 minus J(k)

X1

Conjugate-gradient coefficient

Searching direction

(k)1 =J(k)1

X1

+ (k)1 (kminus1)1

New design variable

X(k+1)1 = X(k)

1 + 1J(k)1

N = 2

Sensitivity analysisF(X1 X2 + ΔX2 X3 X4 X5) = J(k)2

J(k)2

X2

=J(k)2 minus J(k)

X2

Conjugate-gradient coefficient

Searching direction

(k)2 =J(k)2

X2

+ (k)2 (kminus1)2

New design variable

X(k+1)2 = X(k)

2 + 2J(k)2

N = 5

N = 3 4

Sensitivity analysisF(X1 X2 X3 X4 X5 + ΔX5) = J(k)5

J(k)5

X5

=J(k)5 minus J(k)

X5

Searching direction

(k)5 =J(k)n

X5

+ (k)5 (kminus1)5

New design variable

X(k+1)5 = X(k)

5 + 5J(k)5

F(X1 X2 X3 X4 X5) functionsJ(k) objective functions (the average temperature and the temperature difference TＰ plusmn ΔT ＧＣＨ ΔTＧＲ)

Conjugate-gradient coefficient

(k)1 = [ J(k)1 X1

J(kminus1)1 X1

]2

(k)2 = [ J(k)2 X2

J(kminus1)2 X2

]2

(k)5 = [ J(k)5 X5

J(kminus1)5 X5

]2

Figure 5 A flowchart of the CGM optimization process

8 Advances in Materials Science and Engineering

ℎ5 = 160 (Wm2K) and the initial guess value 2 was ℎ1 =ℎ2 = ℎ3 = ℎ4 = 1200 (Wm2K) ℎ5 = 120 (Wm2K) forCase 1 The initial guess values for heat transfer coefficientsℎ1 = ℎ2 = ℎ3 = ℎ4 = 1600Wm2K in water are estimatedfrom the water jet heat transfer coefficient correlations byWendelstorf et al [22] while the ℎ5 = 160Wm2K in airis estimated from the combined natural and radiation heattransfer theory [23] The search trend of the final averagetemperature for these two initial guess values was close to theobjective value step by step and were within the error rangeThe search trend for the temperature difference for the twoinitial guess values was obtained by searching for the lowestvalue step by step with the final average temperature in therange of the objective value until the temperature differencevariable was stable The different search steps were obtainedfor different initial guess values but they all obtained almostthe same result where the resultant deviation for the twoinitial values was below 1∘C It is apparent that Case 2 andCase 3 both had the same search path as that of Case 1 Theinitial guess value 1 was ℎ1 = ℎ2 = ℎ3 = ℎ4 = 1500 (Wm2K)ℎ5 = 150 (Wm2K) and the initial guess value 2was ℎ1 = ℎ2 =ℎ3 = ℎ4 = 1000 (Wm2K) ℎ5 = 100 (Wm2K) for Case 2 theinitial guess value 1 was ℎ1 = ℎ2 = ℎ3 = ℎ4 = 1200 (Wm2K)ℎ5 = 120 (Wm2K) and the initial guess value 2 was ℎ1 =ℎ2 = ℎ3 = ℎ4 = 800 (Wm2K) ℎ5 = 80 (Wm2K) for Case 3respectively For the three cases there were different searchsteps for the difference cases and initial guess values wherethe search steps were simply related to the initial guess valueand the search step size

Tables 1ndash3 list the different distributions of the heattransfer coefficients for an example with ten points in theoptimization search process where the final average tem-peratures were all in a range of 540∘C to 560∘C but thetemperature difference decreased from 201∘C to 144∘C from204∘C to 130∘C and from 179∘C to 104∘C for Case 1 toCase 3 respectively The optimization distribution of theheat transfer coefficients is the tenth point in Tables 1ndash3For Case 1 the optimization distribution of heat transfercoefficients ℎ1 to ℎ5 is 1227 1745 1732 1479 and 101 (Wm2K)where the average temperature is 560∘C and the temperaturedifference is 144∘C For Case 2 the optimization distributionof the heat transfer coefficients ℎ1 to h5 is 1091 1705 16661445 and 37 (Wm2K) where the average temperature is560∘C and the temperature difference is 130∘C For Case 3the optimization distribution of heat transfer coefficients ℎ1to ℎ5 is 913 1579 1556 1166 and 225 (Wm2K)where the finalaverage temperature is 560∘C and the temperature differenceis 104∘C

Figure 7 shows that the variation of the average tempera-ture in the different parts of the beam and that the maximumtemperature difference with time for the three cases undertwo heat boundary conditions satisfied the average tempera-ture of H-beams in the range of 540sim560∘C For Case 1 whenthe heat transfer coefficients were constant (ℎ1 = ℎ2 = ℎ3 =ℎ4 = 1500 (Wm2K) ℎ5 = 150 (Wm2K)) the different partshad different trends in variation For the web the coolingrate was very fast at first and then gradually slowed with thedecrease of the web surface temperature and when the time

was over 12 seconds the cooling rate became as fast as it wasoriginally In the case of the flange the cooling rate was slowerthan that of the web where the cooling rate went from fast toslow In the case of the fillet the cooling rate was the slowestand exhibited the same variable trends as that of the flangeThe maximum temperature difference increased initially inthe first 6 seconds and then decreased slightly until themaximum temperature difference was 201∘C at 15 secondsWhen the heat transfer coefficients were optimized (ℎ1 =1227 (Wm2K) ℎ2 = 1745 (Wm2K) ℎ3 = 1732 (Wm2K)ℎ4 = 1479 (Wm2K) and ℎ5 = 101 (Wm2K)) the variationof the average temperature with time in the different partsexhibited the same trends compared with the results of theconstant heat transfer coefficients except the fillet coolingrate was slightly larger than that of the constant heat transfercoefficients the flange cooling rate was slightly less than thatof the constant heat transfer coefficients the web cooling ratewas obviously slower than that of the constant heat transfercoefficients and the cooling rate of the web went from fastto slow The temperature difference for the optimal heattransfer coefficients increased initially and then decreaseduntil the maximum temperature difference was 144∘C at 15seconds and the peak of the temperature difference wascontrolled at 6 seconds For Case 2 and Case 3 the constantheat transfer coefficients were ℎ1 = ℎ2 = ℎ3 = ℎ4 =1400 (Wm2K) ℎ5 = 140 (Wm2K) and ℎ1 = ℎ2 = ℎ3 =ℎ4 = 1200 (Wm2K) ℎ5 = 120 (Wm2K) and the optimalheat transfer coefficients were ℎ1 = 1091 (Wm2K) ℎ2 =1705 (Wm2K) ℎ3 = 1666 (Wm2K) ℎ4 = 1444 (Wm2K)ℎ5 = 37 (Wm2K) and ℎ1 = 913 (Wm2K) ℎ2 =1579 (Wm2K) ℎ3 = 1556 (Wm2K) ℎ4 = 1166 (Wm2K)and ℎ5 = 225 (Wm2K) respectively The variation of theaverage temperature with time exhibited the same trends asthose found for Case 1 regardless of whether it was deter-mined with constant heat transfer coefficients or optimalheat transfer coefficients The cooling rates of Case 2 amongthe web flange and fillet were closer than in Case 1 andCase 3 was the closest because of the physical dimensionsThe temperature difference for the constant heat transfercoefficients reached a peak at 65 seconds and 7 secondsfor Case 2 and Case 3 and then slowly decreased until thetemperature differences were 204∘C and 180∘C at 15 secondsrespectively The maximum temperature difference of theoptimal heat transfer coefficients reached a peak at 4 secondsand at 3 seconds for Case 2 and Case 3 and then slowlydecreased to 130∘C and 104∘C at 15 seconds respectively

Figure 8 displays the temperature field for three caseswith constant heat transfer coefficients and optimal heattransfer coefficients at 15 seconds ForCase 1 the temperatureranged from 667∘C to 466∘C with the constant heat transfercoefficients and the temperature ranged from 654∘C to 510∘Cwith the optimal heat transfer coefficients For Case 2 thetemperature ranged from 658∘C to 454∘C with the constantheat transfer coefficients and the temperature ranged from639∘C to 509∘Cwith the optimal heat transfer coefficients ForCase 3 the temperature ranged from 646∘C to 466∘C withthe constant heat transfer coefficients and the temperatureranged from 618∘C to 514∘C with the optimal heat transfercoefficients the surface maximum temperature difference

Advances in Materials Science and Engineering 9

TＰ

(∘

C)

Search steps

Initial value 1

Initial value 2

Initial value 1 ℎ1 = ℎ2 = ℎ3 = ℎ4 = 1600 ℎ5 = 160 (WＧ2K)Initial value 2 ℎ1 = ℎ2 = ℎ3 = ℎ4 = 1200 ℎ5 = 120 (WＧ2K)

590

580

570

560

550

540

5300 20 40 60 80 100 120 140 160

Search steps0 50 100 150 200

Initial value 1Initial value 2

Search steps

Search steps

0 20 40 60 80 100 120 140 160

220

200

180

160

140

120

0 50 100 150 200

Initial value 1

Initial value 2

Initial value 1Initial value 2

ΔT

(∘C)

(a) Case 1

220

200

180

160

140

120

Search steps0 50 100 150 200

Search steps0 50 100 150 200 250 300

Initial value 1

Initial value 2

Initial value 1Initial value 2

TＰ

(∘

C)

Search steps

600

580

560

540

0 50 100 150 200

Initial value 1 ℎ1 = ℎ2 = ℎ3 = ℎ4 = 1500 ℎ5 = 150 (WＧ2K)Initial value 2 ℎ1 = ℎ2 = ℎ3 = ℎ4 = 1000 ℎ5 = 100 (WＧ2K)

Search steps0 50 100 150 200 250 300

Initial value 1

Initial value 2

Initial value 1Initial value 2

ΔT

(∘C)

(b) Case 2

TＰ

(∘

C)

Search steps0 20 40 60 80 100 120 140 160

620

600

580

560

540

520

0 100 200 300 400

Initial value 1 ℎ1 = ℎ2 = ℎ3 = ℎ4 = 1200 ℎ5 = 120 (WＧ2 K)Initial value 2 ℎ1 = ℎ2 = ℎ3 = ℎ4 = 800 ℎ5 = 80 (WＧ2K)

Search steps

Initial value 1

Initial value 2

Initial value 1Initial value 2

Search steps0 20 40 60 80 100 120 140 160

220

200

180

160

140

120

100

80

0 100 200 300 400

Search steps

Initial value 1

Initial value 2

Initial value 1Initial value 2

ΔT

(∘C)

(c) Case 3

Figure 6 The search path with the different initial values for the three cases

10 Advances in Materials Science and Engineering

Constant

ℎ1 = ℎ2 = ℎ3 = ℎ4 = 1500 ℎ5 = 150 (WＧ2K)Optimal

ℎ1 = 1227 ℎ2 = 1745 ℎ3 = 1732 ℎ4 = 1479 ℎ5 = 101 (WＧ2K)

Time (s)0 2 4 6 8 10 12 14

250

200

150

100

50

0

ℎ are constantOptimal ℎ

ℎ are constant

Optimal ℎ

Time (s)

Tem

p(∘ C)

900

800

700

600

500

4000 2 4 6 8 10 12 14

Optimal ℎℎ are constant

ΔT

(∘C)

TＰ with filletTＰ with flange

TＰ with web

(a) Case 1

Tem

p(∘ C)

900

800

700

600

500

400

Time (s)0 2 4 6 8 10 12 14

Optimal ℎℎ are constant

TＰ with filletTＰ with flange

TＰ with web

Constantℎ1 = ℎ2 = ℎ3 = ℎ4 = 1400 ℎ5 = 140 (WＧ2K)

Optimalℎ1 = 1091 ℎ2 = 1705 ℎ3 = 1666 ℎ4 = 1444 ℎ5 = 37 (WＧ2K)

Time (s)0 2 4 6 8 10 12 14

ℎ is constantOptimal ℎ

ℎ are constant

Optimal ℎ

200

150

100

50

0

ΔT

(∘C)

(b) Case 2

Constant

ℎ1 = ℎ2 = ℎ3 = ℎ4 = 1200 ℎ5 = 120 (WＧ2K)Optimal

ℎ1 = 913 ℎ2 = 1579 ℎ3 = 1556 ℎ4 = 1166 ℎ5 = 225 (WＧ2K)

Time (s)0 2 4 6 8 10 12 14

200

150

100

50

0

ℎ is constantOptimal ℎ

ℎ are constant

Optimal ℎ

Tem

p(∘ C)

900

800

700

600

500

400

Time (s)0 2 4 6 8 10 12 14

TＰ with filletTＰ with flange

TＰ with web

Optimal ℎℎ are constant

ΔT

(∘C)

(c) Case 3

Figure 7 The temperature in different parts and the temperature difference variable with time

Advances in Materials Science and Engineering 11

Section

L2

Constant heattransfer coefficients

Optimal heattransfer coefficients

667652637621606591576560545530515499484469454

(∘C)

(a) Case 1

Constant heattransfer coefficients

Optimal heattransfer coefficients

667652637621606591576560545530515499484469454

(∘C)

(b) Case 2

Constant heattransfer coefficients

Optimal heattransfer coefficients

667652637621606591576560545530515499484469454

(∘C)

(c) Case 3

Figure 8 The distribution of temperature with the different heat transfer coefficients at 15 seconds in a section of the H-beams

is decreased by 60sim80∘C for three cases For the threecases the temperature field for the optimal heat transfercoefficients was more uniform than for the temperaturefield of the constant heat transfer coefficients Also from

Case 1 to Case 3 the sizes became gradually smaller so thetemperature differences were also smaller

Figure 9 and Table 4 display the residual stress distribu-tion at the web for three cases with constant heat transfer

12 Advances in Materials Science and Engineering

Table 1 The optimization search of ℎ1ndashℎ5 distribution for Case 1 (eg with ten points)

Number ℎ1 ℎ2 ℎ3 ℎ4 ℎ5 119879ave Δ119879Wm2K Wm2K Wm2K Wm2K Wm2K ∘C ∘C

(1) (constant) 1500 1500 1500 1500 150 550 201(2) 1546 1595 1596 1557 157 540 205(3) 1485 1547 1548 1507 107 546 195(4) 1419 1556 1555 1508 107 550 180(5) 1335 1576 1427 1415 167 555 176(6) 1309 1565 1497 1482 163 556 170(7) 1316 1572 1568 1507 106 557 160(8) 1264 1642 1635 1504 111 558 150(9) 1240 1702 1695 1490 105 559 145(10) (optimization) 1227 1745 1732 1479 101 560 144

Table 2 The optimization search of ℎ1ndashℎ5 distribution for Case 2 (eg with ten points)

Number ℎ1 ℎ2 ℎ3 ℎ4 ℎ5 119879ave Δ119879Wm2K Wm2K Wm2K Wm2K Wm2K ∘C ∘C

(1) (constant) 1400 1400 1400 1400 140 548 204(2) 1340 1449 1450 1419 100 548 198(3) 1365 1453 1452 1420 99 549 192(4) 1315 1460 1458 1421 98 551 178(5) 1289 1464 1461 1421 96 553 172(6) 1213 1482 1475 1465 92 555 151(7) 1181 1545 1485 1486 67 556 142(8) 1133 1610 1550 1477 64 558 136(9) 1109 1610 1555 1456 44 559 132(10) (optimization) 1091 1705 1666 1445 37 560 130

Table 3 The optimization search of ℎ1ndashℎ5 distribution for Case 3 (eg with ten points)

Number ℎ1 ℎ2 ℎ3 ℎ4 ℎ5 119879ave Δ119879Wm2K Wm2K Wm2K Wm2K Wm2K ∘C ∘C

(1) (constant) 1200 1200 1200 1200 120 543 179(2) 1213 945 934 1087 101 552 210(3) 1168 1013 1000 1140 151 553 189(4) 1121 1081 1065 1192 201 553 168(5) 1074 1148 1130 1243 201 555 147(6) 1038 1213 1191 1267 182 556 133(7) 1017 1278 1254 1280 204 556 131(8) 991 1353 1323 1265 203 557 123(9) 959 1435 1398 1223 181 559 111(10) (optimization) 913 1579 1556 1166 225 560 104

coefficients and optimal heat transfer coefficients based onthe analysis of the temperature field FromFigure 9 the flangeand the fillet of the residual stress are tensor in nature theweb of the residual stress is compressive in nature with themaximum compressive stress occurring at the junction of theweb and the fillet For Case 1 the residual stress in the webis in the range of minus125MPa to minus107MPa with the constantheat transfer coefficients and is about minus90MPa with optimal

heat transfer coefficients For Case 2 the residual stress inthe web is in the range of minus100MPa to minus80MPa with theconstant heat transfer coefficients and is about minus60MPa withoptimal heat transfer coefficients For Case 3 the residualstress in the web is in the range of minus80MPa to minus70MPa withthe constant heat transfer coefficients and is about minus50MPawith optimal heat transfer coefficients From Table 4 it isobserved that optimum heat transfer coefficients result in

Advances in Materials Science and Engineering 13

Optimal ℎℎ are constant

Resid

ual s

tress

(MPa

)

x (mm)

60

40

20

0

minus20

minus40

minus60

minus80

minus100

minus120

minus140

minus160

minus150 minus100 minus50 0 50 100 150

(a) Case 1Re

sidua

l stre

ss (M

Pa)

x (mm)

60

40

20

0

minus20

minus40

minus60

minus80

minus100

minus120

minus140

minus100 minus50 0 50 100

Optimal ℎℎ are constant

(b) Case 2

Resid

ual s

tress

(MPa

)

60

40

20

0

minus20

minus40

minus60

minus80

minus100

minus120

x (mm)minus100 minus50 0 50 100

Optimal ℎℎ are constant

(c) Case 3

Figure 9 The residual stresses distribution at the web

Table 4 A comparison of the results for the three cases

Case Case 1 Case 2 Case 3Constant Optimal Constant Optimal Constant Optimalℎ1 (Wm2K) 1500 1227 1400 1091 1200 913ℎ2 (Wm2K) 1500 1745 1400 1705 1200 1579ℎ3 (Wm2K) 1500 1732 1400 1666 1200 1556ℎ4 (Wm2K) 1500 1479 1400 1445 1200 1166ℎ5 (Wm2K) 150 101 140 37 120 225Δ119879max (

∘C) 201 144 204 130 180 104Residual stress at web (MPa) minus114 minus91 minus89 minus62 minus73 minus51

14 Advances in Materials Science and Engineering

smaller differences in maximum temperature and therebygenerated a lower residual stress distribution in the webacross all sizes of the beams researched in this paper

5 Conclusions

The conjugate-gradient method (CGM) was used here topredict the optimization heat transfer coefficient distributionof H-beams with different sizes To this end three 3D numer-ical models were analyzed to estimate the temperature fieldand the residual stresses field developed during the coolingprocess The results from the optimal heat transfer coeffi-cients were compared with those constant heat transfer coef-ficients Based on the numerical results the conclusion canbe summarized as follows

(1)The conjugate-gradientmethod (CGM) has accuratelypredicted the heat transfer coefficients and the differentinitial guess values have resulted in very similar result Inthe search process we can get many heat transfer coefficientsto satisfy the average temperature with different temperaturedifferences and there is a group of heat transfer coefficientsthat has the lowest temperature difference using the CGMoptimization search method

(2) A comparison of the constant heat transfer coefficientsand the optimal heat transfer coefficients reveals obviousimprovement in the predicted uniform temperature fieldand temperature difference with the optimal heat transfercoefficients where the maximum temperature difference andthe surface maximum temperature difference are decreasedfor all the cases analyzed

(3)The cooling rate was fastest in the web followed by theflange and the fillet

(4) After the water cooling and the air cooling to theambient temperature the residual stress at the web was calcu-lated and a comparison of the three results for the constantheat transfer coefficients and optimal heat transfer coeffi-cients indicates that the residual stress can be reduced by20sim40MPa for the three cases

Nomenclature

119861 Flange width mm119862119901 Specific heat Jsdotkgminus1sdotKminus1119864 Youngrsquos modulus GPaℎ Heat transfer coefficient Wsdotmminus2 Kminus1119867 Web height mm119869obj Objective functions119896 Thermal conductivity Wsdotmminus1sdotKminus1119877 Fillet radius mm119904119894119895 The component of deviatoric stresses MPa119879 Temperature ∘C1199051 Web thickness mm1199052 Flange thickness mm119905 Time s

Greek Symbols

120572 Thermal expansion coefficient ∘Cminus1120573 Search step size

120574 Conjugate-gradient coefficient120576119894119895 Strains120585 Search directions120588 Density kgmminus3120592 Poissonrsquos ratio120590119894119895 Thermal stresses MPa120590119890 The von Mises effective stress MPa120590119904 The yield stress MPa120575119894119895 Kronecker delta120575 Error of the average temperature ∘CΔ119879 Temperature difference ∘C

Subscripts

ave Average valueinfin Environment condition119898 Search steps in the CGM119899 The variables number119904 Surface of the H-beam

Disclosure

Part of the abstract of this paper was presented in 19th Inter-national Conference on Aerospace Mechanical Automotiveand Materials Engineering May 11-12 Montreal Canada2017

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

This research was financially supported by the Ministryof Science and Technology and China Steel CorporationTaiwan under Contract no MOST103-2622-E006-037 Theauthors are also grateful to Dr Chao-Hua Wang of ChinaSteel Corporation Taiwan for his valuable suggestions

References

[1] C J Bettles and M A Gibson ldquoCurrent wrought magnesiumalloys strengths and weaknessesrdquo Journal of Metals vol 57 pp46ndash49 2005

[2] Z Li D Wu and W Lu ldquoEffects of rolling and coolingconditions onmicrostructure andmechanical properties of lowcarbon cold heading steelrdquo Journal of Iron and Steel ResearchInternational vol 19 no 11 pp 64ndash70 2012

[3] C Ouchi ldquoDevelopment of steel plates by intensive use ofTMCP and direct quenching processesrdquo ISIJ International vol41 no 6 pp 542ndash553 2001

[4] S Endo and N Nakata ldquoDevelopment of Thermo-MechanicalControl Process (TMCP) and high performance steel in JFESteelrdquo JFE Technical Report no 20 pp 1ndash7 2015

[5] N Shikanai S Mitao and S Endo ldquoRecent developmentin microstructural control technologies through the thermo-mechanical control process (TMCP) with JFE Steelrsquos high-performance platesrdquo JFE Technical Report no 11 pp 1ndash6 2008

Advances in Materials Science and Engineering 15

[6] H W Liu J Yan D H Chen and J Q Qian ldquoFinite elementanalysis of H-beam temperature field during coolingrdquo Journalof Anhui University of Technology vol 20 pp 4ndash7 2003

[7] Q Liu Y ZangQQin and J Q Zhao ldquoAnalysis on temperatureand thermal stress of h-beam during coolingrdquo MetallurgicalEquipment vol 167 pp 17ndash20 2008

[8] X-M Zhao F Guo L-N Wang and D Wu ldquoEffect of ultra-fast cooling on cooling distortion for H-beam steelrdquo Journal ofNortheastern University vol 33 no 7 pp 949ndash952 2012

[9] F Guo X-M Zhao L-N Wang D Wu G-D Wang and Z-L Ning ldquoEffect of controlled cooling on microstructure andmechanical properties of H-beamrdquo Journal of Iron and SteelResearch International vol 20 no 8 pp 60ndash65 2013

[10] J H Ma X H Yao B Tao and S Li ldquoEffects of coolingafter rolling on thermal stress of H-beam steelrdquo Hot WorkingTechnology vol 43 pp 121ndash123 2014

[11] G M Zhu Y L Kang W Chen J L Gui and G T MaldquoThermalmechanical coupled finite element analysis of thermalresidual stress in h-beamrsquos air coolingrdquoMaterials for MechanicalEngineering vol 32 pp 77ndash80 2008

[12] Q Liu Y Zang Q Qin and Z Q Zhao ldquoAnalysis ontemperature and thermal stress of H-beam during coolingrdquoMetallurgical Equipment vol 167 pp 17ndash20 2008

[13] L Zhao X G Yu G B Liao and Y L Hu ldquoSimulation ofStress-strain of H-beam in control cooling processrdquo Journal ofUniversity of Science and Technology Liaoning vol 34 pp 466ndash469 2011

[14] G Zhu Y Kang andGMa ldquoSimulation of residual stress in hotrolled large size H-beamsrdquo Materials Science Forum vol 704-705 pp 1370ndash1378 2012

[15] J Y Jang and Y G Gan ldquoOptimization analysis of controlledcooling process for h-shape steel beamsrdquo in Proceedings ofthe International Conference on Mechatronic Automobile andEnvironment Engineering 2017

[16] J Y Jang L FHsu and J S Leu ldquoOptimization of the span angleand location of vortex generators in a plate-fin and tube heatexchangerrdquo International Journal of Heat andMass Transfer vol67 pp 432ndash444 2013

[17] J-Y Jang and Y-C Tsai ldquoOptimization of thermoelectricgenerator module spacing and spreader thickness used in awaste heat recovery systemrdquo Applied Thermal Engineering vol51 no 1-2 pp 677ndash689 2013

[18] J-Y Jang and C-C Chen ldquoOptimization of louvered-fin heatexchanger with variable louver anglesrdquo Applied Thermal Engi-neering vol 91 pp 138ndash150 2015

[19] J-Y Jang and J-B Huang ldquoOptimization of a slab heatingpattern for minimum energy consumption in a walking-beamtype reheating furnacerdquo Applied Thermal Engineering vol 85article no 6547 pp 313ndash321 2015

[20] X K Zhu and Y J Chao ldquoNumerical simulation of transienttemperature and residual stresses in friction stir welding of 304Lstainless steelrdquo Journal of Materials Processing Technology vol146 no 2 pp 263ndash272 2004

[21] JmatPro 90 Practical software for material properties SenteSoftware Ltd 2017

[22] J Wendelstorf K-H Spitzer and R Wendelstorf ldquoSpray watercooling heat transfer at high temperatures and liquid massfluxesrdquo International Journal of Heat and Mass Transfer vol 51no 19-20 pp 4902ndash4910 2008

[23] F P Incropera D P Dewitt T L Bergman and A S LavineldquoPrinciples of Heat and Mass Transferrdquo in International StudentVersion 7th edition

Submit your manuscripts athttpswwwhindawicom

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CeramicsJournal of

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Biomaterials

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TextilesHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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BioMed Research International

MaterialsJournal of

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Advances in Materials Science and Engineering 7

X design variables (heat transfer coefficient h)

No

YesStop

Start

Conjugate-gradient method

k = k + 1 Initial guess design variablesX1 X2 X3 X4 X5

Direct problem solver(heat transfer conductionthermal-stress equation)F(X1 X2 X3 X4 X5)

Evaluate objective functionJ(k)

Convergence

J(k) minus J(kminus1)

J(k)lt ＬＦＮＣＰ error

N = 1

Sensitivity analysisF(X1 + ΔX1 X2 X3 X4 X5) = J(k)1

J(k)1

X1

=J(k)1 minus J(k)

X1

Conjugate-gradient coefficient

Searching direction

(k)1 =J(k)1

X1

+ (k)1 (kminus1)1

New design variable

X(k+1)1 = X(k)

1 + 1J(k)1

N = 2

Sensitivity analysisF(X1 X2 + ΔX2 X3 X4 X5) = J(k)2

J(k)2

X2

=J(k)2 minus J(k)

X2

Conjugate-gradient coefficient

Searching direction

(k)2 =J(k)2

X2

+ (k)2 (kminus1)2

New design variable

X(k+1)2 = X(k)

2 + 2J(k)2

N = 5

N = 3 4

Sensitivity analysisF(X1 X2 X3 X4 X5 + ΔX5) = J(k)5

J(k)5

X5

=J(k)5 minus J(k)

X5

Searching direction

(k)5 =J(k)n

X5

+ (k)5 (kminus1)5

New design variable

X(k+1)5 = X(k)

5 + 5J(k)5

F(X1 X2 X3 X4 X5) functionsJ(k) objective functions (the average temperature and the temperature difference TＰ plusmn ΔT ＧＣＨ ΔTＧＲ)

Conjugate-gradient coefficient

(k)1 = [ J(k)1 X1

J(kminus1)1 X1

]2

(k)2 = [ J(k)2 X2

J(kminus1)2 X2

]2

(k)5 = [ J(k)5 X5

J(kminus1)5 X5

]2

Figure 5 A flowchart of the CGM optimization process

8 Advances in Materials Science and Engineering

ℎ5 = 160 (Wm2K) and the initial guess value 2 was ℎ1 =ℎ2 = ℎ3 = ℎ4 = 1200 (Wm2K) ℎ5 = 120 (Wm2K) forCase 1 The initial guess values for heat transfer coefficientsℎ1 = ℎ2 = ℎ3 = ℎ4 = 1600Wm2K in water are estimatedfrom the water jet heat transfer coefficient correlations byWendelstorf et al [22] while the ℎ5 = 160Wm2K in airis estimated from the combined natural and radiation heattransfer theory [23] The search trend of the final averagetemperature for these two initial guess values was close to theobjective value step by step and were within the error rangeThe search trend for the temperature difference for the twoinitial guess values was obtained by searching for the lowestvalue step by step with the final average temperature in therange of the objective value until the temperature differencevariable was stable The different search steps were obtainedfor different initial guess values but they all obtained almostthe same result where the resultant deviation for the twoinitial values was below 1∘C It is apparent that Case 2 andCase 3 both had the same search path as that of Case 1 Theinitial guess value 1 was ℎ1 = ℎ2 = ℎ3 = ℎ4 = 1500 (Wm2K)ℎ5 = 150 (Wm2K) and the initial guess value 2was ℎ1 = ℎ2 =ℎ3 = ℎ4 = 1000 (Wm2K) ℎ5 = 100 (Wm2K) for Case 2 theinitial guess value 1 was ℎ1 = ℎ2 = ℎ3 = ℎ4 = 1200 (Wm2K)ℎ5 = 120 (Wm2K) and the initial guess value 2 was ℎ1 =ℎ2 = ℎ3 = ℎ4 = 800 (Wm2K) ℎ5 = 80 (Wm2K) for Case 3respectively For the three cases there were different searchsteps for the difference cases and initial guess values wherethe search steps were simply related to the initial guess valueand the search step size

Tables 1ndash3 list the different distributions of the heattransfer coefficients for an example with ten points in theoptimization search process where the final average tem-peratures were all in a range of 540∘C to 560∘C but thetemperature difference decreased from 201∘C to 144∘C from204∘C to 130∘C and from 179∘C to 104∘C for Case 1 toCase 3 respectively The optimization distribution of theheat transfer coefficients is the tenth point in Tables 1ndash3For Case 1 the optimization distribution of heat transfercoefficients ℎ1 to ℎ5 is 1227 1745 1732 1479 and 101 (Wm2K)where the average temperature is 560∘C and the temperaturedifference is 144∘C For Case 2 the optimization distributionof the heat transfer coefficients ℎ1 to h5 is 1091 1705 16661445 and 37 (Wm2K) where the average temperature is560∘C and the temperature difference is 130∘C For Case 3the optimization distribution of heat transfer coefficients ℎ1to ℎ5 is 913 1579 1556 1166 and 225 (Wm2K)where the finalaverage temperature is 560∘C and the temperature differenceis 104∘C

Figure 7 shows that the variation of the average tempera-ture in the different parts of the beam and that the maximumtemperature difference with time for the three cases undertwo heat boundary conditions satisfied the average tempera-ture of H-beams in the range of 540sim560∘C For Case 1 whenthe heat transfer coefficients were constant (ℎ1 = ℎ2 = ℎ3 =ℎ4 = 1500 (Wm2K) ℎ5 = 150 (Wm2K)) the different partshad different trends in variation For the web the coolingrate was very fast at first and then gradually slowed with thedecrease of the web surface temperature and when the time

was over 12 seconds the cooling rate became as fast as it wasoriginally In the case of the flange the cooling rate was slowerthan that of the web where the cooling rate went from fast toslow In the case of the fillet the cooling rate was the slowestand exhibited the same variable trends as that of the flangeThe maximum temperature difference increased initially inthe first 6 seconds and then decreased slightly until themaximum temperature difference was 201∘C at 15 secondsWhen the heat transfer coefficients were optimized (ℎ1 =1227 (Wm2K) ℎ2 = 1745 (Wm2K) ℎ3 = 1732 (Wm2K)ℎ4 = 1479 (Wm2K) and ℎ5 = 101 (Wm2K)) the variationof the average temperature with time in the different partsexhibited the same trends compared with the results of theconstant heat transfer coefficients except the fillet coolingrate was slightly larger than that of the constant heat transfercoefficients the flange cooling rate was slightly less than thatof the constant heat transfer coefficients the web cooling ratewas obviously slower than that of the constant heat transfercoefficients and the cooling rate of the web went from fastto slow The temperature difference for the optimal heattransfer coefficients increased initially and then decreaseduntil the maximum temperature difference was 144∘C at 15seconds and the peak of the temperature difference wascontrolled at 6 seconds For Case 2 and Case 3 the constantheat transfer coefficients were ℎ1 = ℎ2 = ℎ3 = ℎ4 =1400 (Wm2K) ℎ5 = 140 (Wm2K) and ℎ1 = ℎ2 = ℎ3 =ℎ4 = 1200 (Wm2K) ℎ5 = 120 (Wm2K) and the optimalheat transfer coefficients were ℎ1 = 1091 (Wm2K) ℎ2 =1705 (Wm2K) ℎ3 = 1666 (Wm2K) ℎ4 = 1444 (Wm2K)ℎ5 = 37 (Wm2K) and ℎ1 = 913 (Wm2K) ℎ2 =1579 (Wm2K) ℎ3 = 1556 (Wm2K) ℎ4 = 1166 (Wm2K)and ℎ5 = 225 (Wm2K) respectively The variation of theaverage temperature with time exhibited the same trends asthose found for Case 1 regardless of whether it was deter-mined with constant heat transfer coefficients or optimalheat transfer coefficients The cooling rates of Case 2 amongthe web flange and fillet were closer than in Case 1 andCase 3 was the closest because of the physical dimensionsThe temperature difference for the constant heat transfercoefficients reached a peak at 65 seconds and 7 secondsfor Case 2 and Case 3 and then slowly decreased until thetemperature differences were 204∘C and 180∘C at 15 secondsrespectively The maximum temperature difference of theoptimal heat transfer coefficients reached a peak at 4 secondsand at 3 seconds for Case 2 and Case 3 and then slowlydecreased to 130∘C and 104∘C at 15 seconds respectively

Figure 8 displays the temperature field for three caseswith constant heat transfer coefficients and optimal heattransfer coefficients at 15 seconds ForCase 1 the temperatureranged from 667∘C to 466∘C with the constant heat transfercoefficients and the temperature ranged from 654∘C to 510∘Cwith the optimal heat transfer coefficients For Case 2 thetemperature ranged from 658∘C to 454∘C with the constantheat transfer coefficients and the temperature ranged from639∘C to 509∘Cwith the optimal heat transfer coefficients ForCase 3 the temperature ranged from 646∘C to 466∘C withthe constant heat transfer coefficients and the temperatureranged from 618∘C to 514∘C with the optimal heat transfercoefficients the surface maximum temperature difference

Advances in Materials Science and Engineering 9

TＰ

(∘

C)

Search steps

Initial value 1

Initial value 2

Initial value 1 ℎ1 = ℎ2 = ℎ3 = ℎ4 = 1600 ℎ5 = 160 (WＧ2K)Initial value 2 ℎ1 = ℎ2 = ℎ3 = ℎ4 = 1200 ℎ5 = 120 (WＧ2K)

590

580

570

560

550

540

5300 20 40 60 80 100 120 140 160

Search steps0 50 100 150 200

Initial value 1Initial value 2

Search steps

Search steps

0 20 40 60 80 100 120 140 160

220

200

180

160

140

120

0 50 100 150 200

Initial value 1

Initial value 2

Initial value 1Initial value 2

ΔT

(∘C)

(a) Case 1

220

200

180

160

140

120

Search steps0 50 100 150 200

Search steps0 50 100 150 200 250 300

Initial value 1

Initial value 2

Initial value 1Initial value 2

TＰ

(∘

C)

Search steps

600

580

560

540

0 50 100 150 200

Initial value 1 ℎ1 = ℎ2 = ℎ3 = ℎ4 = 1500 ℎ5 = 150 (WＧ2K)Initial value 2 ℎ1 = ℎ2 = ℎ3 = ℎ4 = 1000 ℎ5 = 100 (WＧ2K)

Search steps0 50 100 150 200 250 300

Initial value 1

Initial value 2

Initial value 1Initial value 2

ΔT

(∘C)

(b) Case 2

TＰ

(∘

C)

Search steps0 20 40 60 80 100 120 140 160

620

600

580

560

540

520

0 100 200 300 400

Initial value 1 ℎ1 = ℎ2 = ℎ3 = ℎ4 = 1200 ℎ5 = 120 (WＧ2 K)Initial value 2 ℎ1 = ℎ2 = ℎ3 = ℎ4 = 800 ℎ5 = 80 (WＧ2K)

Search steps

Initial value 1

Initial value 2

Initial value 1Initial value 2

Search steps0 20 40 60 80 100 120 140 160

220

200

180

160

140

120

100

80

0 100 200 300 400

Search steps

Initial value 1

Initial value 2

Initial value 1Initial value 2

ΔT

(∘C)

(c) Case 3

Figure 6 The search path with the different initial values for the three cases

10 Advances in Materials Science and Engineering

Constant

ℎ1 = ℎ2 = ℎ3 = ℎ4 = 1500 ℎ5 = 150 (WＧ2K)Optimal

ℎ1 = 1227 ℎ2 = 1745 ℎ3 = 1732 ℎ4 = 1479 ℎ5 = 101 (WＧ2K)

Time (s)0 2 4 6 8 10 12 14

250

200

150

100

50

0

ℎ are constantOptimal ℎ

ℎ are constant

Optimal ℎ

Time (s)

Tem

p(∘ C)

900

800

700

600

500

4000 2 4 6 8 10 12 14

Optimal ℎℎ are constant

ΔT

(∘C)

TＰ with filletTＰ with flange

TＰ with web

(a) Case 1

Tem

p(∘ C)

900

800

700

600

500

400

Time (s)0 2 4 6 8 10 12 14

Optimal ℎℎ are constant

TＰ with filletTＰ with flange

TＰ with web

Constantℎ1 = ℎ2 = ℎ3 = ℎ4 = 1400 ℎ5 = 140 (WＧ2K)

Optimalℎ1 = 1091 ℎ2 = 1705 ℎ3 = 1666 ℎ4 = 1444 ℎ5 = 37 (WＧ2K)

Time (s)0 2 4 6 8 10 12 14

ℎ is constantOptimal ℎ

ℎ are constant

Optimal ℎ

200

150

100

50

0

ΔT

(∘C)

(b) Case 2

Constant

ℎ1 = ℎ2 = ℎ3 = ℎ4 = 1200 ℎ5 = 120 (WＧ2K)Optimal

ℎ1 = 913 ℎ2 = 1579 ℎ3 = 1556 ℎ4 = 1166 ℎ5 = 225 (WＧ2K)

Time (s)0 2 4 6 8 10 12 14

200

150

100

50

0

ℎ is constantOptimal ℎ

ℎ are constant

Optimal ℎ

Tem

p(∘ C)

900

800

700

600

500

400

Time (s)0 2 4 6 8 10 12 14

TＰ with filletTＰ with flange

TＰ with web

Optimal ℎℎ are constant

ΔT

(∘C)

(c) Case 3

Figure 7 The temperature in different parts and the temperature difference variable with time

Advances in Materials Science and Engineering 11

Section

L2

Constant heattransfer coefficients

Optimal heattransfer coefficients

667652637621606591576560545530515499484469454

(∘C)

(a) Case 1

Constant heattransfer coefficients

Optimal heattransfer coefficients

667652637621606591576560545530515499484469454

(∘C)

(b) Case 2

Constant heattransfer coefficients

Optimal heattransfer coefficients

667652637621606591576560545530515499484469454

(∘C)

(c) Case 3

Figure 8 The distribution of temperature with the different heat transfer coefficients at 15 seconds in a section of the H-beams

is decreased by 60sim80∘C for three cases For the threecases the temperature field for the optimal heat transfercoefficients was more uniform than for the temperaturefield of the constant heat transfer coefficients Also from

Case 1 to Case 3 the sizes became gradually smaller so thetemperature differences were also smaller

Figure 9 and Table 4 display the residual stress distribu-tion at the web for three cases with constant heat transfer

12 Advances in Materials Science and Engineering

Table 1 The optimization search of ℎ1ndashℎ5 distribution for Case 1 (eg with ten points)

Number ℎ1 ℎ2 ℎ3 ℎ4 ℎ5 119879ave Δ119879Wm2K Wm2K Wm2K Wm2K Wm2K ∘C ∘C

(1) (constant) 1500 1500 1500 1500 150 550 201(2) 1546 1595 1596 1557 157 540 205(3) 1485 1547 1548 1507 107 546 195(4) 1419 1556 1555 1508 107 550 180(5) 1335 1576 1427 1415 167 555 176(6) 1309 1565 1497 1482 163 556 170(7) 1316 1572 1568 1507 106 557 160(8) 1264 1642 1635 1504 111 558 150(9) 1240 1702 1695 1490 105 559 145(10) (optimization) 1227 1745 1732 1479 101 560 144

Table 2 The optimization search of ℎ1ndashℎ5 distribution for Case 2 (eg with ten points)

Number ℎ1 ℎ2 ℎ3 ℎ4 ℎ5 119879ave Δ119879Wm2K Wm2K Wm2K Wm2K Wm2K ∘C ∘C

(1) (constant) 1400 1400 1400 1400 140 548 204(2) 1340 1449 1450 1419 100 548 198(3) 1365 1453 1452 1420 99 549 192(4) 1315 1460 1458 1421 98 551 178(5) 1289 1464 1461 1421 96 553 172(6) 1213 1482 1475 1465 92 555 151(7) 1181 1545 1485 1486 67 556 142(8) 1133 1610 1550 1477 64 558 136(9) 1109 1610 1555 1456 44 559 132(10) (optimization) 1091 1705 1666 1445 37 560 130

Table 3 The optimization search of ℎ1ndashℎ5 distribution for Case 3 (eg with ten points)

Number ℎ1 ℎ2 ℎ3 ℎ4 ℎ5 119879ave Δ119879Wm2K Wm2K Wm2K Wm2K Wm2K ∘C ∘C

(1) (constant) 1200 1200 1200 1200 120 543 179(2) 1213 945 934 1087 101 552 210(3) 1168 1013 1000 1140 151 553 189(4) 1121 1081 1065 1192 201 553 168(5) 1074 1148 1130 1243 201 555 147(6) 1038 1213 1191 1267 182 556 133(7) 1017 1278 1254 1280 204 556 131(8) 991 1353 1323 1265 203 557 123(9) 959 1435 1398 1223 181 559 111(10) (optimization) 913 1579 1556 1166 225 560 104

coefficients and optimal heat transfer coefficients based onthe analysis of the temperature field FromFigure 9 the flangeand the fillet of the residual stress are tensor in nature theweb of the residual stress is compressive in nature with themaximum compressive stress occurring at the junction of theweb and the fillet For Case 1 the residual stress in the webis in the range of minus125MPa to minus107MPa with the constantheat transfer coefficients and is about minus90MPa with optimal

heat transfer coefficients For Case 2 the residual stress inthe web is in the range of minus100MPa to minus80MPa with theconstant heat transfer coefficients and is about minus60MPa withoptimal heat transfer coefficients For Case 3 the residualstress in the web is in the range of minus80MPa to minus70MPa withthe constant heat transfer coefficients and is about minus50MPawith optimal heat transfer coefficients From Table 4 it isobserved that optimum heat transfer coefficients result in

Advances in Materials Science and Engineering 13

Optimal ℎℎ are constant

Resid

ual s

tress

(MPa

)

x (mm)

60

40

20

0

minus20

minus40

minus60

minus80

minus100

minus120

minus140

minus160

minus150 minus100 minus50 0 50 100 150

(a) Case 1Re

sidua

l stre

ss (M

Pa)

x (mm)

60

40

20

0

minus20

minus40

minus60

minus80

minus100

minus120

minus140

minus100 minus50 0 50 100

Optimal ℎℎ are constant

(b) Case 2

Resid

ual s

tress

(MPa

)

60

40

20

0

minus20

minus40

minus60

minus80

minus100

minus120

x (mm)minus100 minus50 0 50 100

Optimal ℎℎ are constant

(c) Case 3

Figure 9 The residual stresses distribution at the web

Table 4 A comparison of the results for the three cases

Case Case 1 Case 2 Case 3Constant Optimal Constant Optimal Constant Optimalℎ1 (Wm2K) 1500 1227 1400 1091 1200 913ℎ2 (Wm2K) 1500 1745 1400 1705 1200 1579ℎ3 (Wm2K) 1500 1732 1400 1666 1200 1556ℎ4 (Wm2K) 1500 1479 1400 1445 1200 1166ℎ5 (Wm2K) 150 101 140 37 120 225Δ119879max (

∘C) 201 144 204 130 180 104Residual stress at web (MPa) minus114 minus91 minus89 minus62 minus73 minus51

14 Advances in Materials Science and Engineering

smaller differences in maximum temperature and therebygenerated a lower residual stress distribution in the webacross all sizes of the beams researched in this paper

5 Conclusions

The conjugate-gradient method (CGM) was used here topredict the optimization heat transfer coefficient distributionof H-beams with different sizes To this end three 3D numer-ical models were analyzed to estimate the temperature fieldand the residual stresses field developed during the coolingprocess The results from the optimal heat transfer coeffi-cients were compared with those constant heat transfer coef-ficients Based on the numerical results the conclusion canbe summarized as follows

(1)The conjugate-gradientmethod (CGM) has accuratelypredicted the heat transfer coefficients and the differentinitial guess values have resulted in very similar result Inthe search process we can get many heat transfer coefficientsto satisfy the average temperature with different temperaturedifferences and there is a group of heat transfer coefficientsthat has the lowest temperature difference using the CGMoptimization search method

(2) A comparison of the constant heat transfer coefficientsand the optimal heat transfer coefficients reveals obviousimprovement in the predicted uniform temperature fieldand temperature difference with the optimal heat transfercoefficients where the maximum temperature difference andthe surface maximum temperature difference are decreasedfor all the cases analyzed

(3)The cooling rate was fastest in the web followed by theflange and the fillet

(4) After the water cooling and the air cooling to theambient temperature the residual stress at the web was calcu-lated and a comparison of the three results for the constantheat transfer coefficients and optimal heat transfer coeffi-cients indicates that the residual stress can be reduced by20sim40MPa for the three cases

Nomenclature

119861 Flange width mm119862119901 Specific heat Jsdotkgminus1sdotKminus1119864 Youngrsquos modulus GPaℎ Heat transfer coefficient Wsdotmminus2 Kminus1119867 Web height mm119869obj Objective functions119896 Thermal conductivity Wsdotmminus1sdotKminus1119877 Fillet radius mm119904119894119895 The component of deviatoric stresses MPa119879 Temperature ∘C1199051 Web thickness mm1199052 Flange thickness mm119905 Time s

Greek Symbols

120572 Thermal expansion coefficient ∘Cminus1120573 Search step size

120574 Conjugate-gradient coefficient120576119894119895 Strains120585 Search directions120588 Density kgmminus3120592 Poissonrsquos ratio120590119894119895 Thermal stresses MPa120590119890 The von Mises effective stress MPa120590119904 The yield stress MPa120575119894119895 Kronecker delta120575 Error of the average temperature ∘CΔ119879 Temperature difference ∘C

Subscripts

ave Average valueinfin Environment condition119898 Search steps in the CGM119899 The variables number119904 Surface of the H-beam

Disclosure

Part of the abstract of this paper was presented in 19th Inter-national Conference on Aerospace Mechanical Automotiveand Materials Engineering May 11-12 Montreal Canada2017

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

This research was financially supported by the Ministryof Science and Technology and China Steel CorporationTaiwan under Contract no MOST103-2622-E006-037 Theauthors are also grateful to Dr Chao-Hua Wang of ChinaSteel Corporation Taiwan for his valuable suggestions

References

[1] C J Bettles and M A Gibson ldquoCurrent wrought magnesiumalloys strengths and weaknessesrdquo Journal of Metals vol 57 pp46ndash49 2005

[2] Z Li D Wu and W Lu ldquoEffects of rolling and coolingconditions onmicrostructure andmechanical properties of lowcarbon cold heading steelrdquo Journal of Iron and Steel ResearchInternational vol 19 no 11 pp 64ndash70 2012

[3] C Ouchi ldquoDevelopment of steel plates by intensive use ofTMCP and direct quenching processesrdquo ISIJ International vol41 no 6 pp 542ndash553 2001

[4] S Endo and N Nakata ldquoDevelopment of Thermo-MechanicalControl Process (TMCP) and high performance steel in JFESteelrdquo JFE Technical Report no 20 pp 1ndash7 2015

[5] N Shikanai S Mitao and S Endo ldquoRecent developmentin microstructural control technologies through the thermo-mechanical control process (TMCP) with JFE Steelrsquos high-performance platesrdquo JFE Technical Report no 11 pp 1ndash6 2008

Advances in Materials Science and Engineering 15

[6] H W Liu J Yan D H Chen and J Q Qian ldquoFinite elementanalysis of H-beam temperature field during coolingrdquo Journalof Anhui University of Technology vol 20 pp 4ndash7 2003

[7] Q Liu Y ZangQQin and J Q Zhao ldquoAnalysis on temperatureand thermal stress of h-beam during coolingrdquo MetallurgicalEquipment vol 167 pp 17ndash20 2008

[8] X-M Zhao F Guo L-N Wang and D Wu ldquoEffect of ultra-fast cooling on cooling distortion for H-beam steelrdquo Journal ofNortheastern University vol 33 no 7 pp 949ndash952 2012

[9] F Guo X-M Zhao L-N Wang D Wu G-D Wang and Z-L Ning ldquoEffect of controlled cooling on microstructure andmechanical properties of H-beamrdquo Journal of Iron and SteelResearch International vol 20 no 8 pp 60ndash65 2013

[10] J H Ma X H Yao B Tao and S Li ldquoEffects of coolingafter rolling on thermal stress of H-beam steelrdquo Hot WorkingTechnology vol 43 pp 121ndash123 2014

[11] G M Zhu Y L Kang W Chen J L Gui and G T MaldquoThermalmechanical coupled finite element analysis of thermalresidual stress in h-beamrsquos air coolingrdquoMaterials for MechanicalEngineering vol 32 pp 77ndash80 2008

[12] Q Liu Y Zang Q Qin and Z Q Zhao ldquoAnalysis ontemperature and thermal stress of H-beam during coolingrdquoMetallurgical Equipment vol 167 pp 17ndash20 2008

[13] L Zhao X G Yu G B Liao and Y L Hu ldquoSimulation ofStress-strain of H-beam in control cooling processrdquo Journal ofUniversity of Science and Technology Liaoning vol 34 pp 466ndash469 2011

[14] G Zhu Y Kang andGMa ldquoSimulation of residual stress in hotrolled large size H-beamsrdquo Materials Science Forum vol 704-705 pp 1370ndash1378 2012

[15] J Y Jang and Y G Gan ldquoOptimization analysis of controlledcooling process for h-shape steel beamsrdquo in Proceedings ofthe International Conference on Mechatronic Automobile andEnvironment Engineering 2017

[16] J Y Jang L FHsu and J S Leu ldquoOptimization of the span angleand location of vortex generators in a plate-fin and tube heatexchangerrdquo International Journal of Heat andMass Transfer vol67 pp 432ndash444 2013

[17] J-Y Jang and Y-C Tsai ldquoOptimization of thermoelectricgenerator module spacing and spreader thickness used in awaste heat recovery systemrdquo Applied Thermal Engineering vol51 no 1-2 pp 677ndash689 2013

[18] J-Y Jang and C-C Chen ldquoOptimization of louvered-fin heatexchanger with variable louver anglesrdquo Applied Thermal Engi-neering vol 91 pp 138ndash150 2015

[19] J-Y Jang and J-B Huang ldquoOptimization of a slab heatingpattern for minimum energy consumption in a walking-beamtype reheating furnacerdquo Applied Thermal Engineering vol 85article no 6547 pp 313ndash321 2015

[20] X K Zhu and Y J Chao ldquoNumerical simulation of transienttemperature and residual stresses in friction stir welding of 304Lstainless steelrdquo Journal of Materials Processing Technology vol146 no 2 pp 263ndash272 2004

[21] JmatPro 90 Practical software for material properties SenteSoftware Ltd 2017

[22] J Wendelstorf K-H Spitzer and R Wendelstorf ldquoSpray watercooling heat transfer at high temperatures and liquid massfluxesrdquo International Journal of Heat and Mass Transfer vol 51no 19-20 pp 4902ndash4910 2008

[23] F P Incropera D P Dewitt T L Bergman and A S LavineldquoPrinciples of Heat and Mass Transferrdquo in International StudentVersion 7th edition

Submit your manuscripts athttpswwwhindawicom

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8 Advances in Materials Science and Engineering

ℎ5 = 160 (Wm2K) and the initial guess value 2 was ℎ1 =ℎ2 = ℎ3 = ℎ4 = 1200 (Wm2K) ℎ5 = 120 (Wm2K) forCase 1 The initial guess values for heat transfer coefficientsℎ1 = ℎ2 = ℎ3 = ℎ4 = 1600Wm2K in water are estimatedfrom the water jet heat transfer coefficient correlations byWendelstorf et al [22] while the ℎ5 = 160Wm2K in airis estimated from the combined natural and radiation heattransfer theory [23] The search trend of the final averagetemperature for these two initial guess values was close to theobjective value step by step and were within the error rangeThe search trend for the temperature difference for the twoinitial guess values was obtained by searching for the lowestvalue step by step with the final average temperature in therange of the objective value until the temperature differencevariable was stable The different search steps were obtainedfor different initial guess values but they all obtained almostthe same result where the resultant deviation for the twoinitial values was below 1∘C It is apparent that Case 2 andCase 3 both had the same search path as that of Case 1 Theinitial guess value 1 was ℎ1 = ℎ2 = ℎ3 = ℎ4 = 1500 (Wm2K)ℎ5 = 150 (Wm2K) and the initial guess value 2was ℎ1 = ℎ2 =ℎ3 = ℎ4 = 1000 (Wm2K) ℎ5 = 100 (Wm2K) for Case 2 theinitial guess value 1 was ℎ1 = ℎ2 = ℎ3 = ℎ4 = 1200 (Wm2K)ℎ5 = 120 (Wm2K) and the initial guess value 2 was ℎ1 =ℎ2 = ℎ3 = ℎ4 = 800 (Wm2K) ℎ5 = 80 (Wm2K) for Case 3respectively For the three cases there were different searchsteps for the difference cases and initial guess values wherethe search steps were simply related to the initial guess valueand the search step size

Tables 1ndash3 list the different distributions of the heattransfer coefficients for an example with ten points in theoptimization search process where the final average tem-peratures were all in a range of 540∘C to 560∘C but thetemperature difference decreased from 201∘C to 144∘C from204∘C to 130∘C and from 179∘C to 104∘C for Case 1 toCase 3 respectively The optimization distribution of theheat transfer coefficients is the tenth point in Tables 1ndash3For Case 1 the optimization distribution of heat transfercoefficients ℎ1 to ℎ5 is 1227 1745 1732 1479 and 101 (Wm2K)where the average temperature is 560∘C and the temperaturedifference is 144∘C For Case 2 the optimization distributionof the heat transfer coefficients ℎ1 to h5 is 1091 1705 16661445 and 37 (Wm2K) where the average temperature is560∘C and the temperature difference is 130∘C For Case 3the optimization distribution of heat transfer coefficients ℎ1to ℎ5 is 913 1579 1556 1166 and 225 (Wm2K)where the finalaverage temperature is 560∘C and the temperature differenceis 104∘C

Figure 7 shows that the variation of the average tempera-ture in the different parts of the beam and that the maximumtemperature difference with time for the three cases undertwo heat boundary conditions satisfied the average tempera-ture of H-beams in the range of 540sim560∘C For Case 1 whenthe heat transfer coefficients were constant (ℎ1 = ℎ2 = ℎ3 =ℎ4 = 1500 (Wm2K) ℎ5 = 150 (Wm2K)) the different partshad different trends in variation For the web the coolingrate was very fast at first and then gradually slowed with thedecrease of the web surface temperature and when the time

was over 12 seconds the cooling rate became as fast as it wasoriginally In the case of the flange the cooling rate was slowerthan that of the web where the cooling rate went from fast toslow In the case of the fillet the cooling rate was the slowestand exhibited the same variable trends as that of the flangeThe maximum temperature difference increased initially inthe first 6 seconds and then decreased slightly until themaximum temperature difference was 201∘C at 15 secondsWhen the heat transfer coefficients were optimized (ℎ1 =1227 (Wm2K) ℎ2 = 1745 (Wm2K) ℎ3 = 1732 (Wm2K)ℎ4 = 1479 (Wm2K) and ℎ5 = 101 (Wm2K)) the variationof the average temperature with time in the different partsexhibited the same trends compared with the results of theconstant heat transfer coefficients except the fillet coolingrate was slightly larger than that of the constant heat transfercoefficients the flange cooling rate was slightly less than thatof the constant heat transfer coefficients the web cooling ratewas obviously slower than that of the constant heat transfercoefficients and the cooling rate of the web went from fastto slow The temperature difference for the optimal heattransfer coefficients increased initially and then decreaseduntil the maximum temperature difference was 144∘C at 15seconds and the peak of the temperature difference wascontrolled at 6 seconds For Case 2 and Case 3 the constantheat transfer coefficients were ℎ1 = ℎ2 = ℎ3 = ℎ4 =1400 (Wm2K) ℎ5 = 140 (Wm2K) and ℎ1 = ℎ2 = ℎ3 =ℎ4 = 1200 (Wm2K) ℎ5 = 120 (Wm2K) and the optimalheat transfer coefficients were ℎ1 = 1091 (Wm2K) ℎ2 =1705 (Wm2K) ℎ3 = 1666 (Wm2K) ℎ4 = 1444 (Wm2K)ℎ5 = 37 (Wm2K) and ℎ1 = 913 (Wm2K) ℎ2 =1579 (Wm2K) ℎ3 = 1556 (Wm2K) ℎ4 = 1166 (Wm2K)and ℎ5 = 225 (Wm2K) respectively The variation of theaverage temperature with time exhibited the same trends asthose found for Case 1 regardless of whether it was deter-mined with constant heat transfer coefficients or optimalheat transfer coefficients The cooling rates of Case 2 amongthe web flange and fillet were closer than in Case 1 andCase 3 was the closest because of the physical dimensionsThe temperature difference for the constant heat transfercoefficients reached a peak at 65 seconds and 7 secondsfor Case 2 and Case 3 and then slowly decreased until thetemperature differences were 204∘C and 180∘C at 15 secondsrespectively The maximum temperature difference of theoptimal heat transfer coefficients reached a peak at 4 secondsand at 3 seconds for Case 2 and Case 3 and then slowlydecreased to 130∘C and 104∘C at 15 seconds respectively

Figure 8 displays the temperature field for three caseswith constant heat transfer coefficients and optimal heattransfer coefficients at 15 seconds ForCase 1 the temperatureranged from 667∘C to 466∘C with the constant heat transfercoefficients and the temperature ranged from 654∘C to 510∘Cwith the optimal heat transfer coefficients For Case 2 thetemperature ranged from 658∘C to 454∘C with the constantheat transfer coefficients and the temperature ranged from639∘C to 509∘Cwith the optimal heat transfer coefficients ForCase 3 the temperature ranged from 646∘C to 466∘C withthe constant heat transfer coefficients and the temperatureranged from 618∘C to 514∘C with the optimal heat transfercoefficients the surface maximum temperature difference

Advances in Materials Science and Engineering 9

TＰ

(∘

C)

Search steps

Initial value 1

Initial value 2

Initial value 1 ℎ1 = ℎ2 = ℎ3 = ℎ4 = 1600 ℎ5 = 160 (WＧ2K)Initial value 2 ℎ1 = ℎ2 = ℎ3 = ℎ4 = 1200 ℎ5 = 120 (WＧ2K)

590

580

570

560

550

540

5300 20 40 60 80 100 120 140 160

Search steps0 50 100 150 200

Initial value 1Initial value 2

Search steps

Search steps

0 20 40 60 80 100 120 140 160

220

200

180

160

140

120

0 50 100 150 200

Initial value 1

Initial value 2

Initial value 1Initial value 2

ΔT

(∘C)

(a) Case 1

220

200

180

160

140

120

Search steps0 50 100 150 200

Search steps0 50 100 150 200 250 300

Initial value 1

Initial value 2

Initial value 1Initial value 2

TＰ

(∘

C)

Search steps

600

580

560

540

0 50 100 150 200

Initial value 1 ℎ1 = ℎ2 = ℎ3 = ℎ4 = 1500 ℎ5 = 150 (WＧ2K)Initial value 2 ℎ1 = ℎ2 = ℎ3 = ℎ4 = 1000 ℎ5 = 100 (WＧ2K)

Search steps0 50 100 150 200 250 300

Initial value 1

Initial value 2

Initial value 1Initial value 2

ΔT

(∘C)

(b) Case 2

TＰ

(∘

C)

Search steps0 20 40 60 80 100 120 140 160

620

600

580

560

540

520

0 100 200 300 400

Initial value 1 ℎ1 = ℎ2 = ℎ3 = ℎ4 = 1200 ℎ5 = 120 (WＧ2 K)Initial value 2 ℎ1 = ℎ2 = ℎ3 = ℎ4 = 800 ℎ5 = 80 (WＧ2K)

Search steps

Initial value 1

Initial value 2

Initial value 1Initial value 2

Search steps0 20 40 60 80 100 120 140 160

220

200

180

160

140

120

100

80

0 100 200 300 400

Search steps

Initial value 1

Initial value 2

Initial value 1Initial value 2

ΔT

(∘C)

(c) Case 3

Figure 6 The search path with the different initial values for the three cases

10 Advances in Materials Science and Engineering

Constant

ℎ1 = ℎ2 = ℎ3 = ℎ4 = 1500 ℎ5 = 150 (WＧ2K)Optimal

ℎ1 = 1227 ℎ2 = 1745 ℎ3 = 1732 ℎ4 = 1479 ℎ5 = 101 (WＧ2K)

Time (s)0 2 4 6 8 10 12 14

250

200

150

100

50

0

ℎ are constantOptimal ℎ

ℎ are constant

Optimal ℎ

Time (s)

Tem

p(∘ C)

900

800

700

600

500

4000 2 4 6 8 10 12 14

Optimal ℎℎ are constant

ΔT

(∘C)

TＰ with filletTＰ with flange

TＰ with web

(a) Case 1

Tem

p(∘ C)

900

800

700

600

500

400

Time (s)0 2 4 6 8 10 12 14

Optimal ℎℎ are constant

TＰ with filletTＰ with flange

TＰ with web

Constantℎ1 = ℎ2 = ℎ3 = ℎ4 = 1400 ℎ5 = 140 (WＧ2K)

Optimalℎ1 = 1091 ℎ2 = 1705 ℎ3 = 1666 ℎ4 = 1444 ℎ5 = 37 (WＧ2K)

Time (s)0 2 4 6 8 10 12 14

ℎ is constantOptimal ℎ

ℎ are constant

Optimal ℎ

200

150

100

50

0

ΔT

(∘C)

(b) Case 2

Constant

ℎ1 = ℎ2 = ℎ3 = ℎ4 = 1200 ℎ5 = 120 (WＧ2K)Optimal

ℎ1 = 913 ℎ2 = 1579 ℎ3 = 1556 ℎ4 = 1166 ℎ5 = 225 (WＧ2K)

Time (s)0 2 4 6 8 10 12 14

200

150

100

50

0

ℎ is constantOptimal ℎ

ℎ are constant

Optimal ℎ

Tem

p(∘ C)

900

800

700

600

500

400

Time (s)0 2 4 6 8 10 12 14

TＰ with filletTＰ with flange

TＰ with web

Optimal ℎℎ are constant

ΔT

(∘C)

(c) Case 3

Figure 7 The temperature in different parts and the temperature difference variable with time

Advances in Materials Science and Engineering 11

Section

L2

Constant heattransfer coefficients

Optimal heattransfer coefficients

667652637621606591576560545530515499484469454

(∘C)

(a) Case 1

Constant heattransfer coefficients

Optimal heattransfer coefficients

667652637621606591576560545530515499484469454

(∘C)

(b) Case 2

Constant heattransfer coefficients

Optimal heattransfer coefficients

667652637621606591576560545530515499484469454

(∘C)

(c) Case 3

Figure 8 The distribution of temperature with the different heat transfer coefficients at 15 seconds in a section of the H-beams

is decreased by 60sim80∘C for three cases For the threecases the temperature field for the optimal heat transfercoefficients was more uniform than for the temperaturefield of the constant heat transfer coefficients Also from

Case 1 to Case 3 the sizes became gradually smaller so thetemperature differences were also smaller

Figure 9 and Table 4 display the residual stress distribu-tion at the web for three cases with constant heat transfer

12 Advances in Materials Science and Engineering

Table 1 The optimization search of ℎ1ndashℎ5 distribution for Case 1 (eg with ten points)

Number ℎ1 ℎ2 ℎ3 ℎ4 ℎ5 119879ave Δ119879Wm2K Wm2K Wm2K Wm2K Wm2K ∘C ∘C

(1) (constant) 1500 1500 1500 1500 150 550 201(2) 1546 1595 1596 1557 157 540 205(3) 1485 1547 1548 1507 107 546 195(4) 1419 1556 1555 1508 107 550 180(5) 1335 1576 1427 1415 167 555 176(6) 1309 1565 1497 1482 163 556 170(7) 1316 1572 1568 1507 106 557 160(8) 1264 1642 1635 1504 111 558 150(9) 1240 1702 1695 1490 105 559 145(10) (optimization) 1227 1745 1732 1479 101 560 144

Table 2 The optimization search of ℎ1ndashℎ5 distribution for Case 2 (eg with ten points)

Number ℎ1 ℎ2 ℎ3 ℎ4 ℎ5 119879ave Δ119879Wm2K Wm2K Wm2K Wm2K Wm2K ∘C ∘C

(1) (constant) 1400 1400 1400 1400 140 548 204(2) 1340 1449 1450 1419 100 548 198(3) 1365 1453 1452 1420 99 549 192(4) 1315 1460 1458 1421 98 551 178(5) 1289 1464 1461 1421 96 553 172(6) 1213 1482 1475 1465 92 555 151(7) 1181 1545 1485 1486 67 556 142(8) 1133 1610 1550 1477 64 558 136(9) 1109 1610 1555 1456 44 559 132(10) (optimization) 1091 1705 1666 1445 37 560 130

Table 3 The optimization search of ℎ1ndashℎ5 distribution for Case 3 (eg with ten points)

Number ℎ1 ℎ2 ℎ3 ℎ4 ℎ5 119879ave Δ119879Wm2K Wm2K Wm2K Wm2K Wm2K ∘C ∘C

(1) (constant) 1200 1200 1200 1200 120 543 179(2) 1213 945 934 1087 101 552 210(3) 1168 1013 1000 1140 151 553 189(4) 1121 1081 1065 1192 201 553 168(5) 1074 1148 1130 1243 201 555 147(6) 1038 1213 1191 1267 182 556 133(7) 1017 1278 1254 1280 204 556 131(8) 991 1353 1323 1265 203 557 123(9) 959 1435 1398 1223 181 559 111(10) (optimization) 913 1579 1556 1166 225 560 104

coefficients and optimal heat transfer coefficients based onthe analysis of the temperature field FromFigure 9 the flangeand the fillet of the residual stress are tensor in nature theweb of the residual stress is compressive in nature with themaximum compressive stress occurring at the junction of theweb and the fillet For Case 1 the residual stress in the webis in the range of minus125MPa to minus107MPa with the constantheat transfer coefficients and is about minus90MPa with optimal

heat transfer coefficients For Case 2 the residual stress inthe web is in the range of minus100MPa to minus80MPa with theconstant heat transfer coefficients and is about minus60MPa withoptimal heat transfer coefficients For Case 3 the residualstress in the web is in the range of minus80MPa to minus70MPa withthe constant heat transfer coefficients and is about minus50MPawith optimal heat transfer coefficients From Table 4 it isobserved that optimum heat transfer coefficients result in

Advances in Materials Science and Engineering 13

Optimal ℎℎ are constant

Resid

ual s

tress

(MPa

)

x (mm)

60

40

20

0

minus20

minus40

minus60

minus80

minus100

minus120

minus140

minus160

minus150 minus100 minus50 0 50 100 150

(a) Case 1Re

sidua

l stre

ss (M

Pa)

x (mm)

60

40

20

0

minus20

minus40

minus60

minus80

minus100

minus120

minus140

minus100 minus50 0 50 100

Optimal ℎℎ are constant

(b) Case 2

Resid

ual s

tress

(MPa

)

60

40

20

0

minus20

minus40

minus60

minus80

minus100

minus120

x (mm)minus100 minus50 0 50 100

Optimal ℎℎ are constant

(c) Case 3

Figure 9 The residual stresses distribution at the web

Table 4 A comparison of the results for the three cases

Case Case 1 Case 2 Case 3Constant Optimal Constant Optimal Constant Optimalℎ1 (Wm2K) 1500 1227 1400 1091 1200 913ℎ2 (Wm2K) 1500 1745 1400 1705 1200 1579ℎ3 (Wm2K) 1500 1732 1400 1666 1200 1556ℎ4 (Wm2K) 1500 1479 1400 1445 1200 1166ℎ5 (Wm2K) 150 101 140 37 120 225Δ119879max (

∘C) 201 144 204 130 180 104Residual stress at web (MPa) minus114 minus91 minus89 minus62 minus73 minus51

14 Advances in Materials Science and Engineering

smaller differences in maximum temperature and therebygenerated a lower residual stress distribution in the webacross all sizes of the beams researched in this paper

5 Conclusions

The conjugate-gradient method (CGM) was used here topredict the optimization heat transfer coefficient distributionof H-beams with different sizes To this end three 3D numer-ical models were analyzed to estimate the temperature fieldand the residual stresses field developed during the coolingprocess The results from the optimal heat transfer coeffi-cients were compared with those constant heat transfer coef-ficients Based on the numerical results the conclusion canbe summarized as follows

(1)The conjugate-gradientmethod (CGM) has accuratelypredicted the heat transfer coefficients and the differentinitial guess values have resulted in very similar result Inthe search process we can get many heat transfer coefficientsto satisfy the average temperature with different temperaturedifferences and there is a group of heat transfer coefficientsthat has the lowest temperature difference using the CGMoptimization search method

(2) A comparison of the constant heat transfer coefficientsand the optimal heat transfer coefficients reveals obviousimprovement in the predicted uniform temperature fieldand temperature difference with the optimal heat transfercoefficients where the maximum temperature difference andthe surface maximum temperature difference are decreasedfor all the cases analyzed

(3)The cooling rate was fastest in the web followed by theflange and the fillet

(4) After the water cooling and the air cooling to theambient temperature the residual stress at the web was calcu-lated and a comparison of the three results for the constantheat transfer coefficients and optimal heat transfer coeffi-cients indicates that the residual stress can be reduced by20sim40MPa for the three cases

Nomenclature

119861 Flange width mm119862119901 Specific heat Jsdotkgminus1sdotKminus1119864 Youngrsquos modulus GPaℎ Heat transfer coefficient Wsdotmminus2 Kminus1119867 Web height mm119869obj Objective functions119896 Thermal conductivity Wsdotmminus1sdotKminus1119877 Fillet radius mm119904119894119895 The component of deviatoric stresses MPa119879 Temperature ∘C1199051 Web thickness mm1199052 Flange thickness mm119905 Time s

Greek Symbols

120572 Thermal expansion coefficient ∘Cminus1120573 Search step size

120574 Conjugate-gradient coefficient120576119894119895 Strains120585 Search directions120588 Density kgmminus3120592 Poissonrsquos ratio120590119894119895 Thermal stresses MPa120590119890 The von Mises effective stress MPa120590119904 The yield stress MPa120575119894119895 Kronecker delta120575 Error of the average temperature ∘CΔ119879 Temperature difference ∘C

Subscripts

ave Average valueinfin Environment condition119898 Search steps in the CGM119899 The variables number119904 Surface of the H-beam

Disclosure

Part of the abstract of this paper was presented in 19th Inter-national Conference on Aerospace Mechanical Automotiveand Materials Engineering May 11-12 Montreal Canada2017

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

This research was financially supported by the Ministryof Science and Technology and China Steel CorporationTaiwan under Contract no MOST103-2622-E006-037 Theauthors are also grateful to Dr Chao-Hua Wang of ChinaSteel Corporation Taiwan for his valuable suggestions

References

[1] C J Bettles and M A Gibson ldquoCurrent wrought magnesiumalloys strengths and weaknessesrdquo Journal of Metals vol 57 pp46ndash49 2005

[2] Z Li D Wu and W Lu ldquoEffects of rolling and coolingconditions onmicrostructure andmechanical properties of lowcarbon cold heading steelrdquo Journal of Iron and Steel ResearchInternational vol 19 no 11 pp 64ndash70 2012

[3] C Ouchi ldquoDevelopment of steel plates by intensive use ofTMCP and direct quenching processesrdquo ISIJ International vol41 no 6 pp 542ndash553 2001

[4] S Endo and N Nakata ldquoDevelopment of Thermo-MechanicalControl Process (TMCP) and high performance steel in JFESteelrdquo JFE Technical Report no 20 pp 1ndash7 2015

[5] N Shikanai S Mitao and S Endo ldquoRecent developmentin microstructural control technologies through the thermo-mechanical control process (TMCP) with JFE Steelrsquos high-performance platesrdquo JFE Technical Report no 11 pp 1ndash6 2008

Advances in Materials Science and Engineering 15

[6] H W Liu J Yan D H Chen and J Q Qian ldquoFinite elementanalysis of H-beam temperature field during coolingrdquo Journalof Anhui University of Technology vol 20 pp 4ndash7 2003

[7] Q Liu Y ZangQQin and J Q Zhao ldquoAnalysis on temperatureand thermal stress of h-beam during coolingrdquo MetallurgicalEquipment vol 167 pp 17ndash20 2008

[8] X-M Zhao F Guo L-N Wang and D Wu ldquoEffect of ultra-fast cooling on cooling distortion for H-beam steelrdquo Journal ofNortheastern University vol 33 no 7 pp 949ndash952 2012

[9] F Guo X-M Zhao L-N Wang D Wu G-D Wang and Z-L Ning ldquoEffect of controlled cooling on microstructure andmechanical properties of H-beamrdquo Journal of Iron and SteelResearch International vol 20 no 8 pp 60ndash65 2013

[10] J H Ma X H Yao B Tao and S Li ldquoEffects of coolingafter rolling on thermal stress of H-beam steelrdquo Hot WorkingTechnology vol 43 pp 121ndash123 2014

[11] G M Zhu Y L Kang W Chen J L Gui and G T MaldquoThermalmechanical coupled finite element analysis of thermalresidual stress in h-beamrsquos air coolingrdquoMaterials for MechanicalEngineering vol 32 pp 77ndash80 2008

[12] Q Liu Y Zang Q Qin and Z Q Zhao ldquoAnalysis ontemperature and thermal stress of H-beam during coolingrdquoMetallurgical Equipment vol 167 pp 17ndash20 2008

[13] L Zhao X G Yu G B Liao and Y L Hu ldquoSimulation ofStress-strain of H-beam in control cooling processrdquo Journal ofUniversity of Science and Technology Liaoning vol 34 pp 466ndash469 2011

[14] G Zhu Y Kang andGMa ldquoSimulation of residual stress in hotrolled large size H-beamsrdquo Materials Science Forum vol 704-705 pp 1370ndash1378 2012

[15] J Y Jang and Y G Gan ldquoOptimization analysis of controlledcooling process for h-shape steel beamsrdquo in Proceedings ofthe International Conference on Mechatronic Automobile andEnvironment Engineering 2017

[16] J Y Jang L FHsu and J S Leu ldquoOptimization of the span angleand location of vortex generators in a plate-fin and tube heatexchangerrdquo International Journal of Heat andMass Transfer vol67 pp 432ndash444 2013

[17] J-Y Jang and Y-C Tsai ldquoOptimization of thermoelectricgenerator module spacing and spreader thickness used in awaste heat recovery systemrdquo Applied Thermal Engineering vol51 no 1-2 pp 677ndash689 2013

[18] J-Y Jang and C-C Chen ldquoOptimization of louvered-fin heatexchanger with variable louver anglesrdquo Applied Thermal Engi-neering vol 91 pp 138ndash150 2015

[19] J-Y Jang and J-B Huang ldquoOptimization of a slab heatingpattern for minimum energy consumption in a walking-beamtype reheating furnacerdquo Applied Thermal Engineering vol 85article no 6547 pp 313ndash321 2015

[20] X K Zhu and Y J Chao ldquoNumerical simulation of transienttemperature and residual stresses in friction stir welding of 304Lstainless steelrdquo Journal of Materials Processing Technology vol146 no 2 pp 263ndash272 2004

[21] JmatPro 90 Practical software for material properties SenteSoftware Ltd 2017

[22] J Wendelstorf K-H Spitzer and R Wendelstorf ldquoSpray watercooling heat transfer at high temperatures and liquid massfluxesrdquo International Journal of Heat and Mass Transfer vol 51no 19-20 pp 4902ndash4910 2008

[23] F P Incropera D P Dewitt T L Bergman and A S LavineldquoPrinciples of Heat and Mass Transferrdquo in International StudentVersion 7th edition

Submit your manuscripts athttpswwwhindawicom

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CorrosionInternational Journal of

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Polymer ScienceInternational Journal of

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CeramicsJournal of

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CompositesJournal of

NanoparticlesJournal of

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Biomaterials

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NanoscienceJournal of

TextilesHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Journal of

NanotechnologyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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CrystallographyJournal of

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Advances in

Materials Science and EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Smart Materials Research

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MetallurgyJournal of

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BioMed Research International

MaterialsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Materials Science and Engineering 9

TＰ

(∘

C)

Search steps

Initial value 1

Initial value 2

Initial value 1 ℎ1 = ℎ2 = ℎ3 = ℎ4 = 1600 ℎ5 = 160 (WＧ2K)Initial value 2 ℎ1 = ℎ2 = ℎ3 = ℎ4 = 1200 ℎ5 = 120 (WＧ2K)

590

580

570

560

550

540

5300 20 40 60 80 100 120 140 160

Search steps0 50 100 150 200

Initial value 1Initial value 2

Search steps

Search steps

0 20 40 60 80 100 120 140 160

220

200

180

160

140

120

0 50 100 150 200

Initial value 1

Initial value 2

Initial value 1Initial value 2

ΔT

(∘C)

(a) Case 1

220

200

180

160

140

120

Search steps0 50 100 150 200

Search steps0 50 100 150 200 250 300

Initial value 1

Initial value 2

Initial value 1Initial value 2

TＰ

(∘

C)

Search steps

600

580

560

540

0 50 100 150 200

Initial value 1 ℎ1 = ℎ2 = ℎ3 = ℎ4 = 1500 ℎ5 = 150 (WＧ2K)Initial value 2 ℎ1 = ℎ2 = ℎ3 = ℎ4 = 1000 ℎ5 = 100 (WＧ2K)

Search steps0 50 100 150 200 250 300

Initial value 1

Initial value 2

Initial value 1Initial value 2

ΔT

(∘C)

(b) Case 2

TＰ

(∘

C)

Search steps0 20 40 60 80 100 120 140 160

620

600

580

560

540

520

0 100 200 300 400

Initial value 1 ℎ1 = ℎ2 = ℎ3 = ℎ4 = 1200 ℎ5 = 120 (WＧ2 K)Initial value 2 ℎ1 = ℎ2 = ℎ3 = ℎ4 = 800 ℎ5 = 80 (WＧ2K)

Search steps

Initial value 1

Initial value 2

Initial value 1Initial value 2

Search steps0 20 40 60 80 100 120 140 160

220

200

180

160

140

120

100

80

0 100 200 300 400

Search steps

Initial value 1

Initial value 2

Initial value 1Initial value 2

ΔT

(∘C)

(c) Case 3

Figure 6 The search path with the different initial values for the three cases

10 Advances in Materials Science and Engineering

Constant

ℎ1 = ℎ2 = ℎ3 = ℎ4 = 1500 ℎ5 = 150 (WＧ2K)Optimal

ℎ1 = 1227 ℎ2 = 1745 ℎ3 = 1732 ℎ4 = 1479 ℎ5 = 101 (WＧ2K)

Time (s)0 2 4 6 8 10 12 14

250

200

150

100

50

0

ℎ are constantOptimal ℎ

ℎ are constant

Optimal ℎ

Time (s)

Tem

p(∘ C)

900

800

700

600

500

4000 2 4 6 8 10 12 14

Optimal ℎℎ are constant

ΔT

(∘C)

TＰ with filletTＰ with flange

TＰ with web

(a) Case 1

Tem

p(∘ C)

900

800

700

600

500

400

Time (s)0 2 4 6 8 10 12 14

Optimal ℎℎ are constant

TＰ with filletTＰ with flange

TＰ with web

Constantℎ1 = ℎ2 = ℎ3 = ℎ4 = 1400 ℎ5 = 140 (WＧ2K)

Optimalℎ1 = 1091 ℎ2 = 1705 ℎ3 = 1666 ℎ4 = 1444 ℎ5 = 37 (WＧ2K)

Time (s)0 2 4 6 8 10 12 14

ℎ is constantOptimal ℎ

ℎ are constant

Optimal ℎ

200

150

100

50

0

ΔT

(∘C)

(b) Case 2

Constant

ℎ1 = ℎ2 = ℎ3 = ℎ4 = 1200 ℎ5 = 120 (WＧ2K)Optimal

ℎ1 = 913 ℎ2 = 1579 ℎ3 = 1556 ℎ4 = 1166 ℎ5 = 225 (WＧ2K)

Time (s)0 2 4 6 8 10 12 14

200

150

100

50

0

ℎ is constantOptimal ℎ

ℎ are constant

Optimal ℎ

Tem

p(∘ C)

900

800

700

600

500

400

Time (s)0 2 4 6 8 10 12 14

TＰ with filletTＰ with flange

TＰ with web

Optimal ℎℎ are constant

ΔT

(∘C)

(c) Case 3

Figure 7 The temperature in different parts and the temperature difference variable with time

Advances in Materials Science and Engineering 11

Section

L2

Constant heattransfer coefficients

Optimal heattransfer coefficients

667652637621606591576560545530515499484469454

(∘C)

(a) Case 1

Constant heattransfer coefficients

Optimal heattransfer coefficients

667652637621606591576560545530515499484469454

(∘C)

(b) Case 2

Constant heattransfer coefficients

Optimal heattransfer coefficients

667652637621606591576560545530515499484469454

(∘C)

(c) Case 3

Figure 8 The distribution of temperature with the different heat transfer coefficients at 15 seconds in a section of the H-beams

is decreased by 60sim80∘C for three cases For the threecases the temperature field for the optimal heat transfercoefficients was more uniform than for the temperaturefield of the constant heat transfer coefficients Also from

Case 1 to Case 3 the sizes became gradually smaller so thetemperature differences were also smaller

Figure 9 and Table 4 display the residual stress distribu-tion at the web for three cases with constant heat transfer

12 Advances in Materials Science and Engineering

Table 1 The optimization search of ℎ1ndashℎ5 distribution for Case 1 (eg with ten points)

Number ℎ1 ℎ2 ℎ3 ℎ4 ℎ5 119879ave Δ119879Wm2K Wm2K Wm2K Wm2K Wm2K ∘C ∘C

(1) (constant) 1500 1500 1500 1500 150 550 201(2) 1546 1595 1596 1557 157 540 205(3) 1485 1547 1548 1507 107 546 195(4) 1419 1556 1555 1508 107 550 180(5) 1335 1576 1427 1415 167 555 176(6) 1309 1565 1497 1482 163 556 170(7) 1316 1572 1568 1507 106 557 160(8) 1264 1642 1635 1504 111 558 150(9) 1240 1702 1695 1490 105 559 145(10) (optimization) 1227 1745 1732 1479 101 560 144

Table 2 The optimization search of ℎ1ndashℎ5 distribution for Case 2 (eg with ten points)

Number ℎ1 ℎ2 ℎ3 ℎ4 ℎ5 119879ave Δ119879Wm2K Wm2K Wm2K Wm2K Wm2K ∘C ∘C

(1) (constant) 1400 1400 1400 1400 140 548 204(2) 1340 1449 1450 1419 100 548 198(3) 1365 1453 1452 1420 99 549 192(4) 1315 1460 1458 1421 98 551 178(5) 1289 1464 1461 1421 96 553 172(6) 1213 1482 1475 1465 92 555 151(7) 1181 1545 1485 1486 67 556 142(8) 1133 1610 1550 1477 64 558 136(9) 1109 1610 1555 1456 44 559 132(10) (optimization) 1091 1705 1666 1445 37 560 130

Table 3 The optimization search of ℎ1ndashℎ5 distribution for Case 3 (eg with ten points)

Number ℎ1 ℎ2 ℎ3 ℎ4 ℎ5 119879ave Δ119879Wm2K Wm2K Wm2K Wm2K Wm2K ∘C ∘C

(1) (constant) 1200 1200 1200 1200 120 543 179(2) 1213 945 934 1087 101 552 210(3) 1168 1013 1000 1140 151 553 189(4) 1121 1081 1065 1192 201 553 168(5) 1074 1148 1130 1243 201 555 147(6) 1038 1213 1191 1267 182 556 133(7) 1017 1278 1254 1280 204 556 131(8) 991 1353 1323 1265 203 557 123(9) 959 1435 1398 1223 181 559 111(10) (optimization) 913 1579 1556 1166 225 560 104

coefficients and optimal heat transfer coefficients based onthe analysis of the temperature field FromFigure 9 the flangeand the fillet of the residual stress are tensor in nature theweb of the residual stress is compressive in nature with themaximum compressive stress occurring at the junction of theweb and the fillet For Case 1 the residual stress in the webis in the range of minus125MPa to minus107MPa with the constantheat transfer coefficients and is about minus90MPa with optimal

heat transfer coefficients For Case 2 the residual stress inthe web is in the range of minus100MPa to minus80MPa with theconstant heat transfer coefficients and is about minus60MPa withoptimal heat transfer coefficients For Case 3 the residualstress in the web is in the range of minus80MPa to minus70MPa withthe constant heat transfer coefficients and is about minus50MPawith optimal heat transfer coefficients From Table 4 it isobserved that optimum heat transfer coefficients result in

Advances in Materials Science and Engineering 13

Optimal ℎℎ are constant

Resid

ual s

tress

(MPa

)

x (mm)

60

40

20

0

minus20

minus40

minus60

minus80

minus100

minus120

minus140

minus160

minus150 minus100 minus50 0 50 100 150

(a) Case 1Re

sidua

l stre

ss (M

Pa)

x (mm)

60

40

20

0

minus20

minus40

minus60

minus80

minus100

minus120

minus140

minus100 minus50 0 50 100

Optimal ℎℎ are constant

(b) Case 2

Resid

ual s

tress

(MPa

)

60

40

20

0

minus20

minus40

minus60

minus80

minus100

minus120

x (mm)minus100 minus50 0 50 100

Optimal ℎℎ are constant

(c) Case 3

Figure 9 The residual stresses distribution at the web

Table 4 A comparison of the results for the three cases

Case Case 1 Case 2 Case 3Constant Optimal Constant Optimal Constant Optimalℎ1 (Wm2K) 1500 1227 1400 1091 1200 913ℎ2 (Wm2K) 1500 1745 1400 1705 1200 1579ℎ3 (Wm2K) 1500 1732 1400 1666 1200 1556ℎ4 (Wm2K) 1500 1479 1400 1445 1200 1166ℎ5 (Wm2K) 150 101 140 37 120 225Δ119879max (

∘C) 201 144 204 130 180 104Residual stress at web (MPa) minus114 minus91 minus89 minus62 minus73 minus51

14 Advances in Materials Science and Engineering

smaller differences in maximum temperature and therebygenerated a lower residual stress distribution in the webacross all sizes of the beams researched in this paper

5 Conclusions

The conjugate-gradient method (CGM) was used here topredict the optimization heat transfer coefficient distributionof H-beams with different sizes To this end three 3D numer-ical models were analyzed to estimate the temperature fieldand the residual stresses field developed during the coolingprocess The results from the optimal heat transfer coeffi-cients were compared with those constant heat transfer coef-ficients Based on the numerical results the conclusion canbe summarized as follows

(1)The conjugate-gradientmethod (CGM) has accuratelypredicted the heat transfer coefficients and the differentinitial guess values have resulted in very similar result Inthe search process we can get many heat transfer coefficientsto satisfy the average temperature with different temperaturedifferences and there is a group of heat transfer coefficientsthat has the lowest temperature difference using the CGMoptimization search method

(2) A comparison of the constant heat transfer coefficientsand the optimal heat transfer coefficients reveals obviousimprovement in the predicted uniform temperature fieldand temperature difference with the optimal heat transfercoefficients where the maximum temperature difference andthe surface maximum temperature difference are decreasedfor all the cases analyzed

(3)The cooling rate was fastest in the web followed by theflange and the fillet

(4) After the water cooling and the air cooling to theambient temperature the residual stress at the web was calcu-lated and a comparison of the three results for the constantheat transfer coefficients and optimal heat transfer coeffi-cients indicates that the residual stress can be reduced by20sim40MPa for the three cases

Nomenclature

119861 Flange width mm119862119901 Specific heat Jsdotkgminus1sdotKminus1119864 Youngrsquos modulus GPaℎ Heat transfer coefficient Wsdotmminus2 Kminus1119867 Web height mm119869obj Objective functions119896 Thermal conductivity Wsdotmminus1sdotKminus1119877 Fillet radius mm119904119894119895 The component of deviatoric stresses MPa119879 Temperature ∘C1199051 Web thickness mm1199052 Flange thickness mm119905 Time s

Greek Symbols

120572 Thermal expansion coefficient ∘Cminus1120573 Search step size

120574 Conjugate-gradient coefficient120576119894119895 Strains120585 Search directions120588 Density kgmminus3120592 Poissonrsquos ratio120590119894119895 Thermal stresses MPa120590119890 The von Mises effective stress MPa120590119904 The yield stress MPa120575119894119895 Kronecker delta120575 Error of the average temperature ∘CΔ119879 Temperature difference ∘C

Subscripts

ave Average valueinfin Environment condition119898 Search steps in the CGM119899 The variables number119904 Surface of the H-beam

Disclosure

Part of the abstract of this paper was presented in 19th Inter-national Conference on Aerospace Mechanical Automotiveand Materials Engineering May 11-12 Montreal Canada2017

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

This research was financially supported by the Ministryof Science and Technology and China Steel CorporationTaiwan under Contract no MOST103-2622-E006-037 Theauthors are also grateful to Dr Chao-Hua Wang of ChinaSteel Corporation Taiwan for his valuable suggestions

References

[1] C J Bettles and M A Gibson ldquoCurrent wrought magnesiumalloys strengths and weaknessesrdquo Journal of Metals vol 57 pp46ndash49 2005

[2] Z Li D Wu and W Lu ldquoEffects of rolling and coolingconditions onmicrostructure andmechanical properties of lowcarbon cold heading steelrdquo Journal of Iron and Steel ResearchInternational vol 19 no 11 pp 64ndash70 2012

[3] C Ouchi ldquoDevelopment of steel plates by intensive use ofTMCP and direct quenching processesrdquo ISIJ International vol41 no 6 pp 542ndash553 2001

[4] S Endo and N Nakata ldquoDevelopment of Thermo-MechanicalControl Process (TMCP) and high performance steel in JFESteelrdquo JFE Technical Report no 20 pp 1ndash7 2015

[5] N Shikanai S Mitao and S Endo ldquoRecent developmentin microstructural control technologies through the thermo-mechanical control process (TMCP) with JFE Steelrsquos high-performance platesrdquo JFE Technical Report no 11 pp 1ndash6 2008

Advances in Materials Science and Engineering 15

[6] H W Liu J Yan D H Chen and J Q Qian ldquoFinite elementanalysis of H-beam temperature field during coolingrdquo Journalof Anhui University of Technology vol 20 pp 4ndash7 2003

[7] Q Liu Y ZangQQin and J Q Zhao ldquoAnalysis on temperatureand thermal stress of h-beam during coolingrdquo MetallurgicalEquipment vol 167 pp 17ndash20 2008

[8] X-M Zhao F Guo L-N Wang and D Wu ldquoEffect of ultra-fast cooling on cooling distortion for H-beam steelrdquo Journal ofNortheastern University vol 33 no 7 pp 949ndash952 2012

[9] F Guo X-M Zhao L-N Wang D Wu G-D Wang and Z-L Ning ldquoEffect of controlled cooling on microstructure andmechanical properties of H-beamrdquo Journal of Iron and SteelResearch International vol 20 no 8 pp 60ndash65 2013

[10] J H Ma X H Yao B Tao and S Li ldquoEffects of coolingafter rolling on thermal stress of H-beam steelrdquo Hot WorkingTechnology vol 43 pp 121ndash123 2014

[11] G M Zhu Y L Kang W Chen J L Gui and G T MaldquoThermalmechanical coupled finite element analysis of thermalresidual stress in h-beamrsquos air coolingrdquoMaterials for MechanicalEngineering vol 32 pp 77ndash80 2008

[12] Q Liu Y Zang Q Qin and Z Q Zhao ldquoAnalysis ontemperature and thermal stress of H-beam during coolingrdquoMetallurgical Equipment vol 167 pp 17ndash20 2008

[13] L Zhao X G Yu G B Liao and Y L Hu ldquoSimulation ofStress-strain of H-beam in control cooling processrdquo Journal ofUniversity of Science and Technology Liaoning vol 34 pp 466ndash469 2011

[14] G Zhu Y Kang andGMa ldquoSimulation of residual stress in hotrolled large size H-beamsrdquo Materials Science Forum vol 704-705 pp 1370ndash1378 2012

[15] J Y Jang and Y G Gan ldquoOptimization analysis of controlledcooling process for h-shape steel beamsrdquo in Proceedings ofthe International Conference on Mechatronic Automobile andEnvironment Engineering 2017

[16] J Y Jang L FHsu and J S Leu ldquoOptimization of the span angleand location of vortex generators in a plate-fin and tube heatexchangerrdquo International Journal of Heat andMass Transfer vol67 pp 432ndash444 2013

[17] J-Y Jang and Y-C Tsai ldquoOptimization of thermoelectricgenerator module spacing and spreader thickness used in awaste heat recovery systemrdquo Applied Thermal Engineering vol51 no 1-2 pp 677ndash689 2013

[18] J-Y Jang and C-C Chen ldquoOptimization of louvered-fin heatexchanger with variable louver anglesrdquo Applied Thermal Engi-neering vol 91 pp 138ndash150 2015

[19] J-Y Jang and J-B Huang ldquoOptimization of a slab heatingpattern for minimum energy consumption in a walking-beamtype reheating furnacerdquo Applied Thermal Engineering vol 85article no 6547 pp 313ndash321 2015

[20] X K Zhu and Y J Chao ldquoNumerical simulation of transienttemperature and residual stresses in friction stir welding of 304Lstainless steelrdquo Journal of Materials Processing Technology vol146 no 2 pp 263ndash272 2004

[21] JmatPro 90 Practical software for material properties SenteSoftware Ltd 2017

[22] J Wendelstorf K-H Spitzer and R Wendelstorf ldquoSpray watercooling heat transfer at high temperatures and liquid massfluxesrdquo International Journal of Heat and Mass Transfer vol 51no 19-20 pp 4902ndash4910 2008

[23] F P Incropera D P Dewitt T L Bergman and A S LavineldquoPrinciples of Heat and Mass Transferrdquo in International StudentVersion 7th edition

Submit your manuscripts athttpswwwhindawicom

ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CorrosionInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Polymer ScienceInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CeramicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CompositesJournal of

NanoparticlesJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Biomaterials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

NanoscienceJournal of

TextilesHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Journal of

NanotechnologyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

CrystallographyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CoatingsJournal of

Advances in

Materials Science and EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Smart Materials Research

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MetallurgyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

BioMed Research International

MaterialsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

10 Advances in Materials Science and Engineering

Constant

ℎ1 = ℎ2 = ℎ3 = ℎ4 = 1500 ℎ5 = 150 (WＧ2K)Optimal

ℎ1 = 1227 ℎ2 = 1745 ℎ3 = 1732 ℎ4 = 1479 ℎ5 = 101 (WＧ2K)

Time (s)0 2 4 6 8 10 12 14

250

200

150

100

50

0

ℎ are constantOptimal ℎ

ℎ are constant

Optimal ℎ

Time (s)

Tem

p(∘ C)

900

800

700

600

500

4000 2 4 6 8 10 12 14

Optimal ℎℎ are constant

ΔT

(∘C)

TＰ with filletTＰ with flange

TＰ with web

(a) Case 1

Tem

p(∘ C)

900

800

700

600

500

400

Time (s)0 2 4 6 8 10 12 14

Optimal ℎℎ are constant

TＰ with filletTＰ with flange

TＰ with web

Constantℎ1 = ℎ2 = ℎ3 = ℎ4 = 1400 ℎ5 = 140 (WＧ2K)

Optimalℎ1 = 1091 ℎ2 = 1705 ℎ3 = 1666 ℎ4 = 1444 ℎ5 = 37 (WＧ2K)

Time (s)0 2 4 6 8 10 12 14

ℎ is constantOptimal ℎ

ℎ are constant

Optimal ℎ

200

150

100

50

0

ΔT

(∘C)

(b) Case 2

Constant

ℎ1 = ℎ2 = ℎ3 = ℎ4 = 1200 ℎ5 = 120 (WＧ2K)Optimal

ℎ1 = 913 ℎ2 = 1579 ℎ3 = 1556 ℎ4 = 1166 ℎ5 = 225 (WＧ2K)

Time (s)0 2 4 6 8 10 12 14

200

150

100

50

0

ℎ is constantOptimal ℎ

ℎ are constant

Optimal ℎ

Tem

p(∘ C)

900

800

700

600

500

400

Time (s)0 2 4 6 8 10 12 14

TＰ with filletTＰ with flange

TＰ with web

Optimal ℎℎ are constant

ΔT

(∘C)

(c) Case 3

Figure 7 The temperature in different parts and the temperature difference variable with time

Advances in Materials Science and Engineering 11

Section

L2

Constant heattransfer coefficients

Optimal heattransfer coefficients

667652637621606591576560545530515499484469454

(∘C)

(a) Case 1

Constant heattransfer coefficients

Optimal heattransfer coefficients

667652637621606591576560545530515499484469454

(∘C)

(b) Case 2

Constant heattransfer coefficients

Optimal heattransfer coefficients

667652637621606591576560545530515499484469454

(∘C)

(c) Case 3

Figure 8 The distribution of temperature with the different heat transfer coefficients at 15 seconds in a section of the H-beams

is decreased by 60sim80∘C for three cases For the threecases the temperature field for the optimal heat transfercoefficients was more uniform than for the temperaturefield of the constant heat transfer coefficients Also from

Case 1 to Case 3 the sizes became gradually smaller so thetemperature differences were also smaller

Figure 9 and Table 4 display the residual stress distribu-tion at the web for three cases with constant heat transfer

12 Advances in Materials Science and Engineering

Table 1 The optimization search of ℎ1ndashℎ5 distribution for Case 1 (eg with ten points)

Number ℎ1 ℎ2 ℎ3 ℎ4 ℎ5 119879ave Δ119879Wm2K Wm2K Wm2K Wm2K Wm2K ∘C ∘C

(1) (constant) 1500 1500 1500 1500 150 550 201(2) 1546 1595 1596 1557 157 540 205(3) 1485 1547 1548 1507 107 546 195(4) 1419 1556 1555 1508 107 550 180(5) 1335 1576 1427 1415 167 555 176(6) 1309 1565 1497 1482 163 556 170(7) 1316 1572 1568 1507 106 557 160(8) 1264 1642 1635 1504 111 558 150(9) 1240 1702 1695 1490 105 559 145(10) (optimization) 1227 1745 1732 1479 101 560 144

Table 2 The optimization search of ℎ1ndashℎ5 distribution for Case 2 (eg with ten points)

Number ℎ1 ℎ2 ℎ3 ℎ4 ℎ5 119879ave Δ119879Wm2K Wm2K Wm2K Wm2K Wm2K ∘C ∘C

(1) (constant) 1400 1400 1400 1400 140 548 204(2) 1340 1449 1450 1419 100 548 198(3) 1365 1453 1452 1420 99 549 192(4) 1315 1460 1458 1421 98 551 178(5) 1289 1464 1461 1421 96 553 172(6) 1213 1482 1475 1465 92 555 151(7) 1181 1545 1485 1486 67 556 142(8) 1133 1610 1550 1477 64 558 136(9) 1109 1610 1555 1456 44 559 132(10) (optimization) 1091 1705 1666 1445 37 560 130

Table 3 The optimization search of ℎ1ndashℎ5 distribution for Case 3 (eg with ten points)

Number ℎ1 ℎ2 ℎ3 ℎ4 ℎ5 119879ave Δ119879Wm2K Wm2K Wm2K Wm2K Wm2K ∘C ∘C

(1) (constant) 1200 1200 1200 1200 120 543 179(2) 1213 945 934 1087 101 552 210(3) 1168 1013 1000 1140 151 553 189(4) 1121 1081 1065 1192 201 553 168(5) 1074 1148 1130 1243 201 555 147(6) 1038 1213 1191 1267 182 556 133(7) 1017 1278 1254 1280 204 556 131(8) 991 1353 1323 1265 203 557 123(9) 959 1435 1398 1223 181 559 111(10) (optimization) 913 1579 1556 1166 225 560 104

coefficients and optimal heat transfer coefficients based onthe analysis of the temperature field FromFigure 9 the flangeand the fillet of the residual stress are tensor in nature theweb of the residual stress is compressive in nature with themaximum compressive stress occurring at the junction of theweb and the fillet For Case 1 the residual stress in the webis in the range of minus125MPa to minus107MPa with the constantheat transfer coefficients and is about minus90MPa with optimal

heat transfer coefficients For Case 2 the residual stress inthe web is in the range of minus100MPa to minus80MPa with theconstant heat transfer coefficients and is about minus60MPa withoptimal heat transfer coefficients For Case 3 the residualstress in the web is in the range of minus80MPa to minus70MPa withthe constant heat transfer coefficients and is about minus50MPawith optimal heat transfer coefficients From Table 4 it isobserved that optimum heat transfer coefficients result in

Advances in Materials Science and Engineering 13

Optimal ℎℎ are constant

Resid

ual s

tress

(MPa

)

x (mm)

60

40

20

0

minus20

minus40

minus60

minus80

minus100

minus120

minus140

minus160

minus150 minus100 minus50 0 50 100 150

(a) Case 1Re

sidua

l stre

ss (M

Pa)

x (mm)

60

40

20

0

minus20

minus40

minus60

minus80

minus100

minus120

minus140

minus100 minus50 0 50 100

Optimal ℎℎ are constant

(b) Case 2

Resid

ual s

tress

(MPa

)

60

40

20

0

minus20

minus40

minus60

minus80

minus100

minus120

x (mm)minus100 minus50 0 50 100

Optimal ℎℎ are constant

(c) Case 3

Figure 9 The residual stresses distribution at the web

Table 4 A comparison of the results for the three cases

Case Case 1 Case 2 Case 3Constant Optimal Constant Optimal Constant Optimalℎ1 (Wm2K) 1500 1227 1400 1091 1200 913ℎ2 (Wm2K) 1500 1745 1400 1705 1200 1579ℎ3 (Wm2K) 1500 1732 1400 1666 1200 1556ℎ4 (Wm2K) 1500 1479 1400 1445 1200 1166ℎ5 (Wm2K) 150 101 140 37 120 225Δ119879max (

∘C) 201 144 204 130 180 104Residual stress at web (MPa) minus114 minus91 minus89 minus62 minus73 minus51

14 Advances in Materials Science and Engineering

smaller differences in maximum temperature and therebygenerated a lower residual stress distribution in the webacross all sizes of the beams researched in this paper

5 Conclusions

The conjugate-gradient method (CGM) was used here topredict the optimization heat transfer coefficient distributionof H-beams with different sizes To this end three 3D numer-ical models were analyzed to estimate the temperature fieldand the residual stresses field developed during the coolingprocess The results from the optimal heat transfer coeffi-cients were compared with those constant heat transfer coef-ficients Based on the numerical results the conclusion canbe summarized as follows

(1)The conjugate-gradientmethod (CGM) has accuratelypredicted the heat transfer coefficients and the differentinitial guess values have resulted in very similar result Inthe search process we can get many heat transfer coefficientsto satisfy the average temperature with different temperaturedifferences and there is a group of heat transfer coefficientsthat has the lowest temperature difference using the CGMoptimization search method

(2) A comparison of the constant heat transfer coefficientsand the optimal heat transfer coefficients reveals obviousimprovement in the predicted uniform temperature fieldand temperature difference with the optimal heat transfercoefficients where the maximum temperature difference andthe surface maximum temperature difference are decreasedfor all the cases analyzed

(3)The cooling rate was fastest in the web followed by theflange and the fillet

(4) After the water cooling and the air cooling to theambient temperature the residual stress at the web was calcu-lated and a comparison of the three results for the constantheat transfer coefficients and optimal heat transfer coeffi-cients indicates that the residual stress can be reduced by20sim40MPa for the three cases

Nomenclature

119861 Flange width mm119862119901 Specific heat Jsdotkgminus1sdotKminus1119864 Youngrsquos modulus GPaℎ Heat transfer coefficient Wsdotmminus2 Kminus1119867 Web height mm119869obj Objective functions119896 Thermal conductivity Wsdotmminus1sdotKminus1119877 Fillet radius mm119904119894119895 The component of deviatoric stresses MPa119879 Temperature ∘C1199051 Web thickness mm1199052 Flange thickness mm119905 Time s

Greek Symbols

120572 Thermal expansion coefficient ∘Cminus1120573 Search step size

120574 Conjugate-gradient coefficient120576119894119895 Strains120585 Search directions120588 Density kgmminus3120592 Poissonrsquos ratio120590119894119895 Thermal stresses MPa120590119890 The von Mises effective stress MPa120590119904 The yield stress MPa120575119894119895 Kronecker delta120575 Error of the average temperature ∘CΔ119879 Temperature difference ∘C

Subscripts

ave Average valueinfin Environment condition119898 Search steps in the CGM119899 The variables number119904 Surface of the H-beam

Disclosure

Part of the abstract of this paper was presented in 19th Inter-national Conference on Aerospace Mechanical Automotiveand Materials Engineering May 11-12 Montreal Canada2017

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

This research was financially supported by the Ministryof Science and Technology and China Steel CorporationTaiwan under Contract no MOST103-2622-E006-037 Theauthors are also grateful to Dr Chao-Hua Wang of ChinaSteel Corporation Taiwan for his valuable suggestions

References

[1] C J Bettles and M A Gibson ldquoCurrent wrought magnesiumalloys strengths and weaknessesrdquo Journal of Metals vol 57 pp46ndash49 2005

[2] Z Li D Wu and W Lu ldquoEffects of rolling and coolingconditions onmicrostructure andmechanical properties of lowcarbon cold heading steelrdquo Journal of Iron and Steel ResearchInternational vol 19 no 11 pp 64ndash70 2012

[3] C Ouchi ldquoDevelopment of steel plates by intensive use ofTMCP and direct quenching processesrdquo ISIJ International vol41 no 6 pp 542ndash553 2001

[4] S Endo and N Nakata ldquoDevelopment of Thermo-MechanicalControl Process (TMCP) and high performance steel in JFESteelrdquo JFE Technical Report no 20 pp 1ndash7 2015

[5] N Shikanai S Mitao and S Endo ldquoRecent developmentin microstructural control technologies through the thermo-mechanical control process (TMCP) with JFE Steelrsquos high-performance platesrdquo JFE Technical Report no 11 pp 1ndash6 2008

Advances in Materials Science and Engineering 15

[6] H W Liu J Yan D H Chen and J Q Qian ldquoFinite elementanalysis of H-beam temperature field during coolingrdquo Journalof Anhui University of Technology vol 20 pp 4ndash7 2003

[7] Q Liu Y ZangQQin and J Q Zhao ldquoAnalysis on temperatureand thermal stress of h-beam during coolingrdquo MetallurgicalEquipment vol 167 pp 17ndash20 2008

[8] X-M Zhao F Guo L-N Wang and D Wu ldquoEffect of ultra-fast cooling on cooling distortion for H-beam steelrdquo Journal ofNortheastern University vol 33 no 7 pp 949ndash952 2012

[9] F Guo X-M Zhao L-N Wang D Wu G-D Wang and Z-L Ning ldquoEffect of controlled cooling on microstructure andmechanical properties of H-beamrdquo Journal of Iron and SteelResearch International vol 20 no 8 pp 60ndash65 2013

[10] J H Ma X H Yao B Tao and S Li ldquoEffects of coolingafter rolling on thermal stress of H-beam steelrdquo Hot WorkingTechnology vol 43 pp 121ndash123 2014

[11] G M Zhu Y L Kang W Chen J L Gui and G T MaldquoThermalmechanical coupled finite element analysis of thermalresidual stress in h-beamrsquos air coolingrdquoMaterials for MechanicalEngineering vol 32 pp 77ndash80 2008

[12] Q Liu Y Zang Q Qin and Z Q Zhao ldquoAnalysis ontemperature and thermal stress of H-beam during coolingrdquoMetallurgical Equipment vol 167 pp 17ndash20 2008

[13] L Zhao X G Yu G B Liao and Y L Hu ldquoSimulation ofStress-strain of H-beam in control cooling processrdquo Journal ofUniversity of Science and Technology Liaoning vol 34 pp 466ndash469 2011

[14] G Zhu Y Kang andGMa ldquoSimulation of residual stress in hotrolled large size H-beamsrdquo Materials Science Forum vol 704-705 pp 1370ndash1378 2012

[15] J Y Jang and Y G Gan ldquoOptimization analysis of controlledcooling process for h-shape steel beamsrdquo in Proceedings ofthe International Conference on Mechatronic Automobile andEnvironment Engineering 2017

[16] J Y Jang L FHsu and J S Leu ldquoOptimization of the span angleand location of vortex generators in a plate-fin and tube heatexchangerrdquo International Journal of Heat andMass Transfer vol67 pp 432ndash444 2013

[17] J-Y Jang and Y-C Tsai ldquoOptimization of thermoelectricgenerator module spacing and spreader thickness used in awaste heat recovery systemrdquo Applied Thermal Engineering vol51 no 1-2 pp 677ndash689 2013

[18] J-Y Jang and C-C Chen ldquoOptimization of louvered-fin heatexchanger with variable louver anglesrdquo Applied Thermal Engi-neering vol 91 pp 138ndash150 2015

[19] J-Y Jang and J-B Huang ldquoOptimization of a slab heatingpattern for minimum energy consumption in a walking-beamtype reheating furnacerdquo Applied Thermal Engineering vol 85article no 6547 pp 313ndash321 2015

[20] X K Zhu and Y J Chao ldquoNumerical simulation of transienttemperature and residual stresses in friction stir welding of 304Lstainless steelrdquo Journal of Materials Processing Technology vol146 no 2 pp 263ndash272 2004

[21] JmatPro 90 Practical software for material properties SenteSoftware Ltd 2017

[22] J Wendelstorf K-H Spitzer and R Wendelstorf ldquoSpray watercooling heat transfer at high temperatures and liquid massfluxesrdquo International Journal of Heat and Mass Transfer vol 51no 19-20 pp 4902ndash4910 2008

[23] F P Incropera D P Dewitt T L Bergman and A S LavineldquoPrinciples of Heat and Mass Transferrdquo in International StudentVersion 7th edition

Submit your manuscripts athttpswwwhindawicom

ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CorrosionInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Polymer ScienceInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CeramicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CompositesJournal of

NanoparticlesJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Biomaterials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

NanoscienceJournal of

TextilesHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Journal of

NanotechnologyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

CrystallographyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CoatingsJournal of

Advances in

Materials Science and EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Smart Materials Research

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MetallurgyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

BioMed Research International

MaterialsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Materials Science and Engineering 11

Section

L2

Constant heattransfer coefficients

Optimal heattransfer coefficients

667652637621606591576560545530515499484469454

(∘C)

(a) Case 1

Constant heattransfer coefficients

Optimal heattransfer coefficients

667652637621606591576560545530515499484469454

(∘C)

(b) Case 2

Constant heattransfer coefficients

Optimal heattransfer coefficients

667652637621606591576560545530515499484469454

(∘C)

(c) Case 3

Figure 8 The distribution of temperature with the different heat transfer coefficients at 15 seconds in a section of the H-beams

is decreased by 60sim80∘C for three cases For the threecases the temperature field for the optimal heat transfercoefficients was more uniform than for the temperaturefield of the constant heat transfer coefficients Also from

Case 1 to Case 3 the sizes became gradually smaller so thetemperature differences were also smaller

Figure 9 and Table 4 display the residual stress distribu-tion at the web for three cases with constant heat transfer

12 Advances in Materials Science and Engineering

Table 1 The optimization search of ℎ1ndashℎ5 distribution for Case 1 (eg with ten points)

Number ℎ1 ℎ2 ℎ3 ℎ4 ℎ5 119879ave Δ119879Wm2K Wm2K Wm2K Wm2K Wm2K ∘C ∘C

(1) (constant) 1500 1500 1500 1500 150 550 201(2) 1546 1595 1596 1557 157 540 205(3) 1485 1547 1548 1507 107 546 195(4) 1419 1556 1555 1508 107 550 180(5) 1335 1576 1427 1415 167 555 176(6) 1309 1565 1497 1482 163 556 170(7) 1316 1572 1568 1507 106 557 160(8) 1264 1642 1635 1504 111 558 150(9) 1240 1702 1695 1490 105 559 145(10) (optimization) 1227 1745 1732 1479 101 560 144

Table 2 The optimization search of ℎ1ndashℎ5 distribution for Case 2 (eg with ten points)

Number ℎ1 ℎ2 ℎ3 ℎ4 ℎ5 119879ave Δ119879Wm2K Wm2K Wm2K Wm2K Wm2K ∘C ∘C

(1) (constant) 1400 1400 1400 1400 140 548 204(2) 1340 1449 1450 1419 100 548 198(3) 1365 1453 1452 1420 99 549 192(4) 1315 1460 1458 1421 98 551 178(5) 1289 1464 1461 1421 96 553 172(6) 1213 1482 1475 1465 92 555 151(7) 1181 1545 1485 1486 67 556 142(8) 1133 1610 1550 1477 64 558 136(9) 1109 1610 1555 1456 44 559 132(10) (optimization) 1091 1705 1666 1445 37 560 130

Table 3 The optimization search of ℎ1ndashℎ5 distribution for Case 3 (eg with ten points)

Number ℎ1 ℎ2 ℎ3 ℎ4 ℎ5 119879ave Δ119879Wm2K Wm2K Wm2K Wm2K Wm2K ∘C ∘C

(1) (constant) 1200 1200 1200 1200 120 543 179(2) 1213 945 934 1087 101 552 210(3) 1168 1013 1000 1140 151 553 189(4) 1121 1081 1065 1192 201 553 168(5) 1074 1148 1130 1243 201 555 147(6) 1038 1213 1191 1267 182 556 133(7) 1017 1278 1254 1280 204 556 131(8) 991 1353 1323 1265 203 557 123(9) 959 1435 1398 1223 181 559 111(10) (optimization) 913 1579 1556 1166 225 560 104

coefficients and optimal heat transfer coefficients based onthe analysis of the temperature field FromFigure 9 the flangeand the fillet of the residual stress are tensor in nature theweb of the residual stress is compressive in nature with themaximum compressive stress occurring at the junction of theweb and the fillet For Case 1 the residual stress in the webis in the range of minus125MPa to minus107MPa with the constantheat transfer coefficients and is about minus90MPa with optimal

heat transfer coefficients For Case 2 the residual stress inthe web is in the range of minus100MPa to minus80MPa with theconstant heat transfer coefficients and is about minus60MPa withoptimal heat transfer coefficients For Case 3 the residualstress in the web is in the range of minus80MPa to minus70MPa withthe constant heat transfer coefficients and is about minus50MPawith optimal heat transfer coefficients From Table 4 it isobserved that optimum heat transfer coefficients result in

Advances in Materials Science and Engineering 13

Optimal ℎℎ are constant

Resid

ual s

tress

(MPa

)

x (mm)

60

40

20

0

minus20

minus40

minus60

minus80

minus100

minus120

minus140

minus160

minus150 minus100 minus50 0 50 100 150

(a) Case 1Re

sidua

l stre

ss (M

Pa)

x (mm)

60

40

20

0

minus20

minus40

minus60

minus80

minus100

minus120

minus140

minus100 minus50 0 50 100

Optimal ℎℎ are constant

(b) Case 2

Resid

ual s

tress

(MPa

)

60

40

20

0

minus20

minus40

minus60

minus80

minus100

minus120

x (mm)minus100 minus50 0 50 100

Optimal ℎℎ are constant

(c) Case 3

Figure 9 The residual stresses distribution at the web

Table 4 A comparison of the results for the three cases

Case Case 1 Case 2 Case 3Constant Optimal Constant Optimal Constant Optimalℎ1 (Wm2K) 1500 1227 1400 1091 1200 913ℎ2 (Wm2K) 1500 1745 1400 1705 1200 1579ℎ3 (Wm2K) 1500 1732 1400 1666 1200 1556ℎ4 (Wm2K) 1500 1479 1400 1445 1200 1166ℎ5 (Wm2K) 150 101 140 37 120 225Δ119879max (

∘C) 201 144 204 130 180 104Residual stress at web (MPa) minus114 minus91 minus89 minus62 minus73 minus51

14 Advances in Materials Science and Engineering

smaller differences in maximum temperature and therebygenerated a lower residual stress distribution in the webacross all sizes of the beams researched in this paper

5 Conclusions

The conjugate-gradient method (CGM) was used here topredict the optimization heat transfer coefficient distributionof H-beams with different sizes To this end three 3D numer-ical models were analyzed to estimate the temperature fieldand the residual stresses field developed during the coolingprocess The results from the optimal heat transfer coeffi-cients were compared with those constant heat transfer coef-ficients Based on the numerical results the conclusion canbe summarized as follows

(1)The conjugate-gradientmethod (CGM) has accuratelypredicted the heat transfer coefficients and the differentinitial guess values have resulted in very similar result Inthe search process we can get many heat transfer coefficientsto satisfy the average temperature with different temperaturedifferences and there is a group of heat transfer coefficientsthat has the lowest temperature difference using the CGMoptimization search method

(2) A comparison of the constant heat transfer coefficientsand the optimal heat transfer coefficients reveals obviousimprovement in the predicted uniform temperature fieldand temperature difference with the optimal heat transfercoefficients where the maximum temperature difference andthe surface maximum temperature difference are decreasedfor all the cases analyzed

(3)The cooling rate was fastest in the web followed by theflange and the fillet

(4) After the water cooling and the air cooling to theambient temperature the residual stress at the web was calcu-lated and a comparison of the three results for the constantheat transfer coefficients and optimal heat transfer coeffi-cients indicates that the residual stress can be reduced by20sim40MPa for the three cases

Nomenclature

119861 Flange width mm119862119901 Specific heat Jsdotkgminus1sdotKminus1119864 Youngrsquos modulus GPaℎ Heat transfer coefficient Wsdotmminus2 Kminus1119867 Web height mm119869obj Objective functions119896 Thermal conductivity Wsdotmminus1sdotKminus1119877 Fillet radius mm119904119894119895 The component of deviatoric stresses MPa119879 Temperature ∘C1199051 Web thickness mm1199052 Flange thickness mm119905 Time s

Greek Symbols

120572 Thermal expansion coefficient ∘Cminus1120573 Search step size

120574 Conjugate-gradient coefficient120576119894119895 Strains120585 Search directions120588 Density kgmminus3120592 Poissonrsquos ratio120590119894119895 Thermal stresses MPa120590119890 The von Mises effective stress MPa120590119904 The yield stress MPa120575119894119895 Kronecker delta120575 Error of the average temperature ∘CΔ119879 Temperature difference ∘C

Subscripts

ave Average valueinfin Environment condition119898 Search steps in the CGM119899 The variables number119904 Surface of the H-beam

Disclosure

Part of the abstract of this paper was presented in 19th Inter-national Conference on Aerospace Mechanical Automotiveand Materials Engineering May 11-12 Montreal Canada2017

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

This research was financially supported by the Ministryof Science and Technology and China Steel CorporationTaiwan under Contract no MOST103-2622-E006-037 Theauthors are also grateful to Dr Chao-Hua Wang of ChinaSteel Corporation Taiwan for his valuable suggestions

References

[1] C J Bettles and M A Gibson ldquoCurrent wrought magnesiumalloys strengths and weaknessesrdquo Journal of Metals vol 57 pp46ndash49 2005

[2] Z Li D Wu and W Lu ldquoEffects of rolling and coolingconditions onmicrostructure andmechanical properties of lowcarbon cold heading steelrdquo Journal of Iron and Steel ResearchInternational vol 19 no 11 pp 64ndash70 2012

[3] C Ouchi ldquoDevelopment of steel plates by intensive use ofTMCP and direct quenching processesrdquo ISIJ International vol41 no 6 pp 542ndash553 2001

[4] S Endo and N Nakata ldquoDevelopment of Thermo-MechanicalControl Process (TMCP) and high performance steel in JFESteelrdquo JFE Technical Report no 20 pp 1ndash7 2015

[5] N Shikanai S Mitao and S Endo ldquoRecent developmentin microstructural control technologies through the thermo-mechanical control process (TMCP) with JFE Steelrsquos high-performance platesrdquo JFE Technical Report no 11 pp 1ndash6 2008

Advances in Materials Science and Engineering 15

[6] H W Liu J Yan D H Chen and J Q Qian ldquoFinite elementanalysis of H-beam temperature field during coolingrdquo Journalof Anhui University of Technology vol 20 pp 4ndash7 2003

[7] Q Liu Y ZangQQin and J Q Zhao ldquoAnalysis on temperatureand thermal stress of h-beam during coolingrdquo MetallurgicalEquipment vol 167 pp 17ndash20 2008

[8] X-M Zhao F Guo L-N Wang and D Wu ldquoEffect of ultra-fast cooling on cooling distortion for H-beam steelrdquo Journal ofNortheastern University vol 33 no 7 pp 949ndash952 2012

[9] F Guo X-M Zhao L-N Wang D Wu G-D Wang and Z-L Ning ldquoEffect of controlled cooling on microstructure andmechanical properties of H-beamrdquo Journal of Iron and SteelResearch International vol 20 no 8 pp 60ndash65 2013

[10] J H Ma X H Yao B Tao and S Li ldquoEffects of coolingafter rolling on thermal stress of H-beam steelrdquo Hot WorkingTechnology vol 43 pp 121ndash123 2014

[11] G M Zhu Y L Kang W Chen J L Gui and G T MaldquoThermalmechanical coupled finite element analysis of thermalresidual stress in h-beamrsquos air coolingrdquoMaterials for MechanicalEngineering vol 32 pp 77ndash80 2008

[12] Q Liu Y Zang Q Qin and Z Q Zhao ldquoAnalysis ontemperature and thermal stress of H-beam during coolingrdquoMetallurgical Equipment vol 167 pp 17ndash20 2008

[13] L Zhao X G Yu G B Liao and Y L Hu ldquoSimulation ofStress-strain of H-beam in control cooling processrdquo Journal ofUniversity of Science and Technology Liaoning vol 34 pp 466ndash469 2011

[14] G Zhu Y Kang andGMa ldquoSimulation of residual stress in hotrolled large size H-beamsrdquo Materials Science Forum vol 704-705 pp 1370ndash1378 2012

[15] J Y Jang and Y G Gan ldquoOptimization analysis of controlledcooling process for h-shape steel beamsrdquo in Proceedings ofthe International Conference on Mechatronic Automobile andEnvironment Engineering 2017

[16] J Y Jang L FHsu and J S Leu ldquoOptimization of the span angleand location of vortex generators in a plate-fin and tube heatexchangerrdquo International Journal of Heat andMass Transfer vol67 pp 432ndash444 2013

[17] J-Y Jang and Y-C Tsai ldquoOptimization of thermoelectricgenerator module spacing and spreader thickness used in awaste heat recovery systemrdquo Applied Thermal Engineering vol51 no 1-2 pp 677ndash689 2013

[18] J-Y Jang and C-C Chen ldquoOptimization of louvered-fin heatexchanger with variable louver anglesrdquo Applied Thermal Engi-neering vol 91 pp 138ndash150 2015

[19] J-Y Jang and J-B Huang ldquoOptimization of a slab heatingpattern for minimum energy consumption in a walking-beamtype reheating furnacerdquo Applied Thermal Engineering vol 85article no 6547 pp 313ndash321 2015

[20] X K Zhu and Y J Chao ldquoNumerical simulation of transienttemperature and residual stresses in friction stir welding of 304Lstainless steelrdquo Journal of Materials Processing Technology vol146 no 2 pp 263ndash272 2004

[21] JmatPro 90 Practical software for material properties SenteSoftware Ltd 2017

[22] J Wendelstorf K-H Spitzer and R Wendelstorf ldquoSpray watercooling heat transfer at high temperatures and liquid massfluxesrdquo International Journal of Heat and Mass Transfer vol 51no 19-20 pp 4902ndash4910 2008

[23] F P Incropera D P Dewitt T L Bergman and A S LavineldquoPrinciples of Heat and Mass Transferrdquo in International StudentVersion 7th edition

Submit your manuscripts athttpswwwhindawicom

ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CorrosionInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Polymer ScienceInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CeramicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CompositesJournal of

NanoparticlesJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Biomaterials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

NanoscienceJournal of

TextilesHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Journal of

NanotechnologyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

CrystallographyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CoatingsJournal of

Advances in

Materials Science and EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Smart Materials Research

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MetallurgyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

BioMed Research International

MaterialsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

12 Advances in Materials Science and Engineering

Table 1 The optimization search of ℎ1ndashℎ5 distribution for Case 1 (eg with ten points)

Number ℎ1 ℎ2 ℎ3 ℎ4 ℎ5 119879ave Δ119879Wm2K Wm2K Wm2K Wm2K Wm2K ∘C ∘C

(1) (constant) 1500 1500 1500 1500 150 550 201(2) 1546 1595 1596 1557 157 540 205(3) 1485 1547 1548 1507 107 546 195(4) 1419 1556 1555 1508 107 550 180(5) 1335 1576 1427 1415 167 555 176(6) 1309 1565 1497 1482 163 556 170(7) 1316 1572 1568 1507 106 557 160(8) 1264 1642 1635 1504 111 558 150(9) 1240 1702 1695 1490 105 559 145(10) (optimization) 1227 1745 1732 1479 101 560 144

Table 2 The optimization search of ℎ1ndashℎ5 distribution for Case 2 (eg with ten points)

Number ℎ1 ℎ2 ℎ3 ℎ4 ℎ5 119879ave Δ119879Wm2K Wm2K Wm2K Wm2K Wm2K ∘C ∘C

(1) (constant) 1400 1400 1400 1400 140 548 204(2) 1340 1449 1450 1419 100 548 198(3) 1365 1453 1452 1420 99 549 192(4) 1315 1460 1458 1421 98 551 178(5) 1289 1464 1461 1421 96 553 172(6) 1213 1482 1475 1465 92 555 151(7) 1181 1545 1485 1486 67 556 142(8) 1133 1610 1550 1477 64 558 136(9) 1109 1610 1555 1456 44 559 132(10) (optimization) 1091 1705 1666 1445 37 560 130

Table 3 The optimization search of ℎ1ndashℎ5 distribution for Case 3 (eg with ten points)

Number ℎ1 ℎ2 ℎ3 ℎ4 ℎ5 119879ave Δ119879Wm2K Wm2K Wm2K Wm2K Wm2K ∘C ∘C

(1) (constant) 1200 1200 1200 1200 120 543 179(2) 1213 945 934 1087 101 552 210(3) 1168 1013 1000 1140 151 553 189(4) 1121 1081 1065 1192 201 553 168(5) 1074 1148 1130 1243 201 555 147(6) 1038 1213 1191 1267 182 556 133(7) 1017 1278 1254 1280 204 556 131(8) 991 1353 1323 1265 203 557 123(9) 959 1435 1398 1223 181 559 111(10) (optimization) 913 1579 1556 1166 225 560 104

coefficients and optimal heat transfer coefficients based onthe analysis of the temperature field FromFigure 9 the flangeand the fillet of the residual stress are tensor in nature theweb of the residual stress is compressive in nature with themaximum compressive stress occurring at the junction of theweb and the fillet For Case 1 the residual stress in the webis in the range of minus125MPa to minus107MPa with the constantheat transfer coefficients and is about minus90MPa with optimal

heat transfer coefficients For Case 2 the residual stress inthe web is in the range of minus100MPa to minus80MPa with theconstant heat transfer coefficients and is about minus60MPa withoptimal heat transfer coefficients For Case 3 the residualstress in the web is in the range of minus80MPa to minus70MPa withthe constant heat transfer coefficients and is about minus50MPawith optimal heat transfer coefficients From Table 4 it isobserved that optimum heat transfer coefficients result in

Advances in Materials Science and Engineering 13

Optimal ℎℎ are constant

Resid

ual s

tress

(MPa

)

x (mm)

60

40

20

0

minus20

minus40

minus60

minus80

minus100

minus120

minus140

minus160

minus150 minus100 minus50 0 50 100 150

(a) Case 1Re

sidua

l stre

ss (M

Pa)

x (mm)

60

40

20

0

minus20

minus40

minus60

minus80

minus100

minus120

minus140

minus100 minus50 0 50 100

Optimal ℎℎ are constant

(b) Case 2

Resid

ual s

tress

(MPa

)

60

40

20

0

minus20

minus40

minus60

minus80

minus100

minus120

x (mm)minus100 minus50 0 50 100

Optimal ℎℎ are constant

(c) Case 3

Figure 9 The residual stresses distribution at the web

Table 4 A comparison of the results for the three cases

Case Case 1 Case 2 Case 3Constant Optimal Constant Optimal Constant Optimalℎ1 (Wm2K) 1500 1227 1400 1091 1200 913ℎ2 (Wm2K) 1500 1745 1400 1705 1200 1579ℎ3 (Wm2K) 1500 1732 1400 1666 1200 1556ℎ4 (Wm2K) 1500 1479 1400 1445 1200 1166ℎ5 (Wm2K) 150 101 140 37 120 225Δ119879max (

∘C) 201 144 204 130 180 104Residual stress at web (MPa) minus114 minus91 minus89 minus62 minus73 minus51

14 Advances in Materials Science and Engineering

smaller differences in maximum temperature and therebygenerated a lower residual stress distribution in the webacross all sizes of the beams researched in this paper

5 Conclusions

The conjugate-gradient method (CGM) was used here topredict the optimization heat transfer coefficient distributionof H-beams with different sizes To this end three 3D numer-ical models were analyzed to estimate the temperature fieldand the residual stresses field developed during the coolingprocess The results from the optimal heat transfer coeffi-cients were compared with those constant heat transfer coef-ficients Based on the numerical results the conclusion canbe summarized as follows

(1)The conjugate-gradientmethod (CGM) has accuratelypredicted the heat transfer coefficients and the differentinitial guess values have resulted in very similar result Inthe search process we can get many heat transfer coefficientsto satisfy the average temperature with different temperaturedifferences and there is a group of heat transfer coefficientsthat has the lowest temperature difference using the CGMoptimization search method

(2) A comparison of the constant heat transfer coefficientsand the optimal heat transfer coefficients reveals obviousimprovement in the predicted uniform temperature fieldand temperature difference with the optimal heat transfercoefficients where the maximum temperature difference andthe surface maximum temperature difference are decreasedfor all the cases analyzed

(3)The cooling rate was fastest in the web followed by theflange and the fillet

(4) After the water cooling and the air cooling to theambient temperature the residual stress at the web was calcu-lated and a comparison of the three results for the constantheat transfer coefficients and optimal heat transfer coeffi-cients indicates that the residual stress can be reduced by20sim40MPa for the three cases

Nomenclature

119861 Flange width mm119862119901 Specific heat Jsdotkgminus1sdotKminus1119864 Youngrsquos modulus GPaℎ Heat transfer coefficient Wsdotmminus2 Kminus1119867 Web height mm119869obj Objective functions119896 Thermal conductivity Wsdotmminus1sdotKminus1119877 Fillet radius mm119904119894119895 The component of deviatoric stresses MPa119879 Temperature ∘C1199051 Web thickness mm1199052 Flange thickness mm119905 Time s

Greek Symbols

120572 Thermal expansion coefficient ∘Cminus1120573 Search step size

120574 Conjugate-gradient coefficient120576119894119895 Strains120585 Search directions120588 Density kgmminus3120592 Poissonrsquos ratio120590119894119895 Thermal stresses MPa120590119890 The von Mises effective stress MPa120590119904 The yield stress MPa120575119894119895 Kronecker delta120575 Error of the average temperature ∘CΔ119879 Temperature difference ∘C

Subscripts

ave Average valueinfin Environment condition119898 Search steps in the CGM119899 The variables number119904 Surface of the H-beam

Disclosure

Part of the abstract of this paper was presented in 19th Inter-national Conference on Aerospace Mechanical Automotiveand Materials Engineering May 11-12 Montreal Canada2017

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

This research was financially supported by the Ministryof Science and Technology and China Steel CorporationTaiwan under Contract no MOST103-2622-E006-037 Theauthors are also grateful to Dr Chao-Hua Wang of ChinaSteel Corporation Taiwan for his valuable suggestions

References

[1] C J Bettles and M A Gibson ldquoCurrent wrought magnesiumalloys strengths and weaknessesrdquo Journal of Metals vol 57 pp46ndash49 2005

[2] Z Li D Wu and W Lu ldquoEffects of rolling and coolingconditions onmicrostructure andmechanical properties of lowcarbon cold heading steelrdquo Journal of Iron and Steel ResearchInternational vol 19 no 11 pp 64ndash70 2012

[3] C Ouchi ldquoDevelopment of steel plates by intensive use ofTMCP and direct quenching processesrdquo ISIJ International vol41 no 6 pp 542ndash553 2001

[4] S Endo and N Nakata ldquoDevelopment of Thermo-MechanicalControl Process (TMCP) and high performance steel in JFESteelrdquo JFE Technical Report no 20 pp 1ndash7 2015

[5] N Shikanai S Mitao and S Endo ldquoRecent developmentin microstructural control technologies through the thermo-mechanical control process (TMCP) with JFE Steelrsquos high-performance platesrdquo JFE Technical Report no 11 pp 1ndash6 2008

Advances in Materials Science and Engineering 15

[6] H W Liu J Yan D H Chen and J Q Qian ldquoFinite elementanalysis of H-beam temperature field during coolingrdquo Journalof Anhui University of Technology vol 20 pp 4ndash7 2003

[7] Q Liu Y ZangQQin and J Q Zhao ldquoAnalysis on temperatureand thermal stress of h-beam during coolingrdquo MetallurgicalEquipment vol 167 pp 17ndash20 2008

[8] X-M Zhao F Guo L-N Wang and D Wu ldquoEffect of ultra-fast cooling on cooling distortion for H-beam steelrdquo Journal ofNortheastern University vol 33 no 7 pp 949ndash952 2012

[9] F Guo X-M Zhao L-N Wang D Wu G-D Wang and Z-L Ning ldquoEffect of controlled cooling on microstructure andmechanical properties of H-beamrdquo Journal of Iron and SteelResearch International vol 20 no 8 pp 60ndash65 2013

[10] J H Ma X H Yao B Tao and S Li ldquoEffects of coolingafter rolling on thermal stress of H-beam steelrdquo Hot WorkingTechnology vol 43 pp 121ndash123 2014

[11] G M Zhu Y L Kang W Chen J L Gui and G T MaldquoThermalmechanical coupled finite element analysis of thermalresidual stress in h-beamrsquos air coolingrdquoMaterials for MechanicalEngineering vol 32 pp 77ndash80 2008

[12] Q Liu Y Zang Q Qin and Z Q Zhao ldquoAnalysis ontemperature and thermal stress of H-beam during coolingrdquoMetallurgical Equipment vol 167 pp 17ndash20 2008

[13] L Zhao X G Yu G B Liao and Y L Hu ldquoSimulation ofStress-strain of H-beam in control cooling processrdquo Journal ofUniversity of Science and Technology Liaoning vol 34 pp 466ndash469 2011

[14] G Zhu Y Kang andGMa ldquoSimulation of residual stress in hotrolled large size H-beamsrdquo Materials Science Forum vol 704-705 pp 1370ndash1378 2012

[15] J Y Jang and Y G Gan ldquoOptimization analysis of controlledcooling process for h-shape steel beamsrdquo in Proceedings ofthe International Conference on Mechatronic Automobile andEnvironment Engineering 2017

[16] J Y Jang L FHsu and J S Leu ldquoOptimization of the span angleand location of vortex generators in a plate-fin and tube heatexchangerrdquo International Journal of Heat andMass Transfer vol67 pp 432ndash444 2013

[17] J-Y Jang and Y-C Tsai ldquoOptimization of thermoelectricgenerator module spacing and spreader thickness used in awaste heat recovery systemrdquo Applied Thermal Engineering vol51 no 1-2 pp 677ndash689 2013

[18] J-Y Jang and C-C Chen ldquoOptimization of louvered-fin heatexchanger with variable louver anglesrdquo Applied Thermal Engi-neering vol 91 pp 138ndash150 2015

[19] J-Y Jang and J-B Huang ldquoOptimization of a slab heatingpattern for minimum energy consumption in a walking-beamtype reheating furnacerdquo Applied Thermal Engineering vol 85article no 6547 pp 313ndash321 2015

[20] X K Zhu and Y J Chao ldquoNumerical simulation of transienttemperature and residual stresses in friction stir welding of 304Lstainless steelrdquo Journal of Materials Processing Technology vol146 no 2 pp 263ndash272 2004

[21] JmatPro 90 Practical software for material properties SenteSoftware Ltd 2017

[22] J Wendelstorf K-H Spitzer and R Wendelstorf ldquoSpray watercooling heat transfer at high temperatures and liquid massfluxesrdquo International Journal of Heat and Mass Transfer vol 51no 19-20 pp 4902ndash4910 2008

[23] F P Incropera D P Dewitt T L Bergman and A S LavineldquoPrinciples of Heat and Mass Transferrdquo in International StudentVersion 7th edition

Submit your manuscripts athttpswwwhindawicom

ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CorrosionInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Polymer ScienceInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CeramicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CompositesJournal of

NanoparticlesJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Biomaterials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

NanoscienceJournal of

TextilesHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Journal of

NanotechnologyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

CrystallographyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CoatingsJournal of

Advances in

Materials Science and EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Smart Materials Research

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MetallurgyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

BioMed Research International

MaterialsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Materials Science and Engineering 13

Optimal ℎℎ are constant

Resid

ual s

tress

(MPa

)

x (mm)

60

40

20

0

minus20

minus40

minus60

minus80

minus100

minus120

minus140

minus160

minus150 minus100 minus50 0 50 100 150

(a) Case 1Re

sidua

l stre

ss (M

Pa)

x (mm)

60

40

20

0

minus20

minus40

minus60

minus80

minus100

minus120

minus140

minus100 minus50 0 50 100

Optimal ℎℎ are constant

(b) Case 2

Resid

ual s

tress

(MPa

)

60

40

20

0

minus20

minus40

minus60

minus80

minus100

minus120

x (mm)minus100 minus50 0 50 100

Optimal ℎℎ are constant

(c) Case 3

Figure 9 The residual stresses distribution at the web

Table 4 A comparison of the results for the three cases

Case Case 1 Case 2 Case 3Constant Optimal Constant Optimal Constant Optimalℎ1 (Wm2K) 1500 1227 1400 1091 1200 913ℎ2 (Wm2K) 1500 1745 1400 1705 1200 1579ℎ3 (Wm2K) 1500 1732 1400 1666 1200 1556ℎ4 (Wm2K) 1500 1479 1400 1445 1200 1166ℎ5 (Wm2K) 150 101 140 37 120 225Δ119879max (

∘C) 201 144 204 130 180 104Residual stress at web (MPa) minus114 minus91 minus89 minus62 minus73 minus51

14 Advances in Materials Science and Engineering

smaller differences in maximum temperature and therebygenerated a lower residual stress distribution in the webacross all sizes of the beams researched in this paper

5 Conclusions

The conjugate-gradient method (CGM) was used here topredict the optimization heat transfer coefficient distributionof H-beams with different sizes To this end three 3D numer-ical models were analyzed to estimate the temperature fieldand the residual stresses field developed during the coolingprocess The results from the optimal heat transfer coeffi-cients were compared with those constant heat transfer coef-ficients Based on the numerical results the conclusion canbe summarized as follows

(1)The conjugate-gradientmethod (CGM) has accuratelypredicted the heat transfer coefficients and the differentinitial guess values have resulted in very similar result Inthe search process we can get many heat transfer coefficientsto satisfy the average temperature with different temperaturedifferences and there is a group of heat transfer coefficientsthat has the lowest temperature difference using the CGMoptimization search method

(2) A comparison of the constant heat transfer coefficientsand the optimal heat transfer coefficients reveals obviousimprovement in the predicted uniform temperature fieldand temperature difference with the optimal heat transfercoefficients where the maximum temperature difference andthe surface maximum temperature difference are decreasedfor all the cases analyzed

(3)The cooling rate was fastest in the web followed by theflange and the fillet

(4) After the water cooling and the air cooling to theambient temperature the residual stress at the web was calcu-lated and a comparison of the three results for the constantheat transfer coefficients and optimal heat transfer coeffi-cients indicates that the residual stress can be reduced by20sim40MPa for the three cases

Nomenclature

119861 Flange width mm119862119901 Specific heat Jsdotkgminus1sdotKminus1119864 Youngrsquos modulus GPaℎ Heat transfer coefficient Wsdotmminus2 Kminus1119867 Web height mm119869obj Objective functions119896 Thermal conductivity Wsdotmminus1sdotKminus1119877 Fillet radius mm119904119894119895 The component of deviatoric stresses MPa119879 Temperature ∘C1199051 Web thickness mm1199052 Flange thickness mm119905 Time s

Greek Symbols

120572 Thermal expansion coefficient ∘Cminus1120573 Search step size

120574 Conjugate-gradient coefficient120576119894119895 Strains120585 Search directions120588 Density kgmminus3120592 Poissonrsquos ratio120590119894119895 Thermal stresses MPa120590119890 The von Mises effective stress MPa120590119904 The yield stress MPa120575119894119895 Kronecker delta120575 Error of the average temperature ∘CΔ119879 Temperature difference ∘C

Subscripts

ave Average valueinfin Environment condition119898 Search steps in the CGM119899 The variables number119904 Surface of the H-beam

Disclosure

Part of the abstract of this paper was presented in 19th Inter-national Conference on Aerospace Mechanical Automotiveand Materials Engineering May 11-12 Montreal Canada2017

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

This research was financially supported by the Ministryof Science and Technology and China Steel CorporationTaiwan under Contract no MOST103-2622-E006-037 Theauthors are also grateful to Dr Chao-Hua Wang of ChinaSteel Corporation Taiwan for his valuable suggestions

References

[1] C J Bettles and M A Gibson ldquoCurrent wrought magnesiumalloys strengths and weaknessesrdquo Journal of Metals vol 57 pp46ndash49 2005

[2] Z Li D Wu and W Lu ldquoEffects of rolling and coolingconditions onmicrostructure andmechanical properties of lowcarbon cold heading steelrdquo Journal of Iron and Steel ResearchInternational vol 19 no 11 pp 64ndash70 2012

[3] C Ouchi ldquoDevelopment of steel plates by intensive use ofTMCP and direct quenching processesrdquo ISIJ International vol41 no 6 pp 542ndash553 2001

[4] S Endo and N Nakata ldquoDevelopment of Thermo-MechanicalControl Process (TMCP) and high performance steel in JFESteelrdquo JFE Technical Report no 20 pp 1ndash7 2015

[5] N Shikanai S Mitao and S Endo ldquoRecent developmentin microstructural control technologies through the thermo-mechanical control process (TMCP) with JFE Steelrsquos high-performance platesrdquo JFE Technical Report no 11 pp 1ndash6 2008

Advances in Materials Science and Engineering 15

[6] H W Liu J Yan D H Chen and J Q Qian ldquoFinite elementanalysis of H-beam temperature field during coolingrdquo Journalof Anhui University of Technology vol 20 pp 4ndash7 2003

[7] Q Liu Y ZangQQin and J Q Zhao ldquoAnalysis on temperatureand thermal stress of h-beam during coolingrdquo MetallurgicalEquipment vol 167 pp 17ndash20 2008

[8] X-M Zhao F Guo L-N Wang and D Wu ldquoEffect of ultra-fast cooling on cooling distortion for H-beam steelrdquo Journal ofNortheastern University vol 33 no 7 pp 949ndash952 2012

[9] F Guo X-M Zhao L-N Wang D Wu G-D Wang and Z-L Ning ldquoEffect of controlled cooling on microstructure andmechanical properties of H-beamrdquo Journal of Iron and SteelResearch International vol 20 no 8 pp 60ndash65 2013

[10] J H Ma X H Yao B Tao and S Li ldquoEffects of coolingafter rolling on thermal stress of H-beam steelrdquo Hot WorkingTechnology vol 43 pp 121ndash123 2014

[11] G M Zhu Y L Kang W Chen J L Gui and G T MaldquoThermalmechanical coupled finite element analysis of thermalresidual stress in h-beamrsquos air coolingrdquoMaterials for MechanicalEngineering vol 32 pp 77ndash80 2008

[12] Q Liu Y Zang Q Qin and Z Q Zhao ldquoAnalysis ontemperature and thermal stress of H-beam during coolingrdquoMetallurgical Equipment vol 167 pp 17ndash20 2008

[13] L Zhao X G Yu G B Liao and Y L Hu ldquoSimulation ofStress-strain of H-beam in control cooling processrdquo Journal ofUniversity of Science and Technology Liaoning vol 34 pp 466ndash469 2011

[14] G Zhu Y Kang andGMa ldquoSimulation of residual stress in hotrolled large size H-beamsrdquo Materials Science Forum vol 704-705 pp 1370ndash1378 2012

[15] J Y Jang and Y G Gan ldquoOptimization analysis of controlledcooling process for h-shape steel beamsrdquo in Proceedings ofthe International Conference on Mechatronic Automobile andEnvironment Engineering 2017

[16] J Y Jang L FHsu and J S Leu ldquoOptimization of the span angleand location of vortex generators in a plate-fin and tube heatexchangerrdquo International Journal of Heat andMass Transfer vol67 pp 432ndash444 2013

[17] J-Y Jang and Y-C Tsai ldquoOptimization of thermoelectricgenerator module spacing and spreader thickness used in awaste heat recovery systemrdquo Applied Thermal Engineering vol51 no 1-2 pp 677ndash689 2013

[18] J-Y Jang and C-C Chen ldquoOptimization of louvered-fin heatexchanger with variable louver anglesrdquo Applied Thermal Engi-neering vol 91 pp 138ndash150 2015

[19] J-Y Jang and J-B Huang ldquoOptimization of a slab heatingpattern for minimum energy consumption in a walking-beamtype reheating furnacerdquo Applied Thermal Engineering vol 85article no 6547 pp 313ndash321 2015

[20] X K Zhu and Y J Chao ldquoNumerical simulation of transienttemperature and residual stresses in friction stir welding of 304Lstainless steelrdquo Journal of Materials Processing Technology vol146 no 2 pp 263ndash272 2004

[21] JmatPro 90 Practical software for material properties SenteSoftware Ltd 2017

[22] J Wendelstorf K-H Spitzer and R Wendelstorf ldquoSpray watercooling heat transfer at high temperatures and liquid massfluxesrdquo International Journal of Heat and Mass Transfer vol 51no 19-20 pp 4902ndash4910 2008

[23] F P Incropera D P Dewitt T L Bergman and A S LavineldquoPrinciples of Heat and Mass Transferrdquo in International StudentVersion 7th edition

Submit your manuscripts athttpswwwhindawicom

ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CorrosionInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Polymer ScienceInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CeramicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CompositesJournal of

NanoparticlesJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Biomaterials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

NanoscienceJournal of

TextilesHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Journal of

NanotechnologyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

CrystallographyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CoatingsJournal of

Advances in

Materials Science and EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Smart Materials Research

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MetallurgyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

BioMed Research International

MaterialsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

14 Advances in Materials Science and Engineering

smaller differences in maximum temperature and therebygenerated a lower residual stress distribution in the webacross all sizes of the beams researched in this paper

5 Conclusions

The conjugate-gradient method (CGM) was used here topredict the optimization heat transfer coefficient distributionof H-beams with different sizes To this end three 3D numer-ical models were analyzed to estimate the temperature fieldand the residual stresses field developed during the coolingprocess The results from the optimal heat transfer coeffi-cients were compared with those constant heat transfer coef-ficients Based on the numerical results the conclusion canbe summarized as follows

(1)The conjugate-gradientmethod (CGM) has accuratelypredicted the heat transfer coefficients and the differentinitial guess values have resulted in very similar result Inthe search process we can get many heat transfer coefficientsto satisfy the average temperature with different temperaturedifferences and there is a group of heat transfer coefficientsthat has the lowest temperature difference using the CGMoptimization search method

(2) A comparison of the constant heat transfer coefficientsand the optimal heat transfer coefficients reveals obviousimprovement in the predicted uniform temperature fieldand temperature difference with the optimal heat transfercoefficients where the maximum temperature difference andthe surface maximum temperature difference are decreasedfor all the cases analyzed

(3)The cooling rate was fastest in the web followed by theflange and the fillet

(4) After the water cooling and the air cooling to theambient temperature the residual stress at the web was calcu-lated and a comparison of the three results for the constantheat transfer coefficients and optimal heat transfer coeffi-cients indicates that the residual stress can be reduced by20sim40MPa for the three cases

Nomenclature

119861 Flange width mm119862119901 Specific heat Jsdotkgminus1sdotKminus1119864 Youngrsquos modulus GPaℎ Heat transfer coefficient Wsdotmminus2 Kminus1119867 Web height mm119869obj Objective functions119896 Thermal conductivity Wsdotmminus1sdotKminus1119877 Fillet radius mm119904119894119895 The component of deviatoric stresses MPa119879 Temperature ∘C1199051 Web thickness mm1199052 Flange thickness mm119905 Time s

Greek Symbols

120572 Thermal expansion coefficient ∘Cminus1120573 Search step size

120574 Conjugate-gradient coefficient120576119894119895 Strains120585 Search directions120588 Density kgmminus3120592 Poissonrsquos ratio120590119894119895 Thermal stresses MPa120590119890 The von Mises effective stress MPa120590119904 The yield stress MPa120575119894119895 Kronecker delta120575 Error of the average temperature ∘CΔ119879 Temperature difference ∘C

Subscripts

ave Average valueinfin Environment condition119898 Search steps in the CGM119899 The variables number119904 Surface of the H-beam

Disclosure

Part of the abstract of this paper was presented in 19th Inter-national Conference on Aerospace Mechanical Automotiveand Materials Engineering May 11-12 Montreal Canada2017

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

This research was financially supported by the Ministryof Science and Technology and China Steel CorporationTaiwan under Contract no MOST103-2622-E006-037 Theauthors are also grateful to Dr Chao-Hua Wang of ChinaSteel Corporation Taiwan for his valuable suggestions

References

[1] C J Bettles and M A Gibson ldquoCurrent wrought magnesiumalloys strengths and weaknessesrdquo Journal of Metals vol 57 pp46ndash49 2005

[2] Z Li D Wu and W Lu ldquoEffects of rolling and coolingconditions onmicrostructure andmechanical properties of lowcarbon cold heading steelrdquo Journal of Iron and Steel ResearchInternational vol 19 no 11 pp 64ndash70 2012

[3] C Ouchi ldquoDevelopment of steel plates by intensive use ofTMCP and direct quenching processesrdquo ISIJ International vol41 no 6 pp 542ndash553 2001

[4] S Endo and N Nakata ldquoDevelopment of Thermo-MechanicalControl Process (TMCP) and high performance steel in JFESteelrdquo JFE Technical Report no 20 pp 1ndash7 2015

[5] N Shikanai S Mitao and S Endo ldquoRecent developmentin microstructural control technologies through the thermo-mechanical control process (TMCP) with JFE Steelrsquos high-performance platesrdquo JFE Technical Report no 11 pp 1ndash6 2008

Advances in Materials Science and Engineering 15

[6] H W Liu J Yan D H Chen and J Q Qian ldquoFinite elementanalysis of H-beam temperature field during coolingrdquo Journalof Anhui University of Technology vol 20 pp 4ndash7 2003

[7] Q Liu Y ZangQQin and J Q Zhao ldquoAnalysis on temperatureand thermal stress of h-beam during coolingrdquo MetallurgicalEquipment vol 167 pp 17ndash20 2008

[8] X-M Zhao F Guo L-N Wang and D Wu ldquoEffect of ultra-fast cooling on cooling distortion for H-beam steelrdquo Journal ofNortheastern University vol 33 no 7 pp 949ndash952 2012

[9] F Guo X-M Zhao L-N Wang D Wu G-D Wang and Z-L Ning ldquoEffect of controlled cooling on microstructure andmechanical properties of H-beamrdquo Journal of Iron and SteelResearch International vol 20 no 8 pp 60ndash65 2013

[10] J H Ma X H Yao B Tao and S Li ldquoEffects of coolingafter rolling on thermal stress of H-beam steelrdquo Hot WorkingTechnology vol 43 pp 121ndash123 2014

[11] G M Zhu Y L Kang W Chen J L Gui and G T MaldquoThermalmechanical coupled finite element analysis of thermalresidual stress in h-beamrsquos air coolingrdquoMaterials for MechanicalEngineering vol 32 pp 77ndash80 2008

[12] Q Liu Y Zang Q Qin and Z Q Zhao ldquoAnalysis ontemperature and thermal stress of H-beam during coolingrdquoMetallurgical Equipment vol 167 pp 17ndash20 2008

[13] L Zhao X G Yu G B Liao and Y L Hu ldquoSimulation ofStress-strain of H-beam in control cooling processrdquo Journal ofUniversity of Science and Technology Liaoning vol 34 pp 466ndash469 2011

[14] G Zhu Y Kang andGMa ldquoSimulation of residual stress in hotrolled large size H-beamsrdquo Materials Science Forum vol 704-705 pp 1370ndash1378 2012

[15] J Y Jang and Y G Gan ldquoOptimization analysis of controlledcooling process for h-shape steel beamsrdquo in Proceedings ofthe International Conference on Mechatronic Automobile andEnvironment Engineering 2017

[16] J Y Jang L FHsu and J S Leu ldquoOptimization of the span angleand location of vortex generators in a plate-fin and tube heatexchangerrdquo International Journal of Heat andMass Transfer vol67 pp 432ndash444 2013

[17] J-Y Jang and Y-C Tsai ldquoOptimization of thermoelectricgenerator module spacing and spreader thickness used in awaste heat recovery systemrdquo Applied Thermal Engineering vol51 no 1-2 pp 677ndash689 2013

[18] J-Y Jang and C-C Chen ldquoOptimization of louvered-fin heatexchanger with variable louver anglesrdquo Applied Thermal Engi-neering vol 91 pp 138ndash150 2015

[19] J-Y Jang and J-B Huang ldquoOptimization of a slab heatingpattern for minimum energy consumption in a walking-beamtype reheating furnacerdquo Applied Thermal Engineering vol 85article no 6547 pp 313ndash321 2015

[20] X K Zhu and Y J Chao ldquoNumerical simulation of transienttemperature and residual stresses in friction stir welding of 304Lstainless steelrdquo Journal of Materials Processing Technology vol146 no 2 pp 263ndash272 2004

[21] JmatPro 90 Practical software for material properties SenteSoftware Ltd 2017

[22] J Wendelstorf K-H Spitzer and R Wendelstorf ldquoSpray watercooling heat transfer at high temperatures and liquid massfluxesrdquo International Journal of Heat and Mass Transfer vol 51no 19-20 pp 4902ndash4910 2008

[23] F P Incropera D P Dewitt T L Bergman and A S LavineldquoPrinciples of Heat and Mass Transferrdquo in International StudentVersion 7th edition

Submit your manuscripts athttpswwwhindawicom

ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CorrosionInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Polymer ScienceInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CeramicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CompositesJournal of

NanoparticlesJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Biomaterials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

NanoscienceJournal of

TextilesHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Journal of

NanotechnologyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

CrystallographyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CoatingsJournal of

Advances in

Materials Science and EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Smart Materials Research

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MetallurgyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

BioMed Research International

MaterialsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Materials Science and Engineering 15

[6] H W Liu J Yan D H Chen and J Q Qian ldquoFinite elementanalysis of H-beam temperature field during coolingrdquo Journalof Anhui University of Technology vol 20 pp 4ndash7 2003

[7] Q Liu Y ZangQQin and J Q Zhao ldquoAnalysis on temperatureand thermal stress of h-beam during coolingrdquo MetallurgicalEquipment vol 167 pp 17ndash20 2008

[8] X-M Zhao F Guo L-N Wang and D Wu ldquoEffect of ultra-fast cooling on cooling distortion for H-beam steelrdquo Journal ofNortheastern University vol 33 no 7 pp 949ndash952 2012

[9] F Guo X-M Zhao L-N Wang D Wu G-D Wang and Z-L Ning ldquoEffect of controlled cooling on microstructure andmechanical properties of H-beamrdquo Journal of Iron and SteelResearch International vol 20 no 8 pp 60ndash65 2013

[10] J H Ma X H Yao B Tao and S Li ldquoEffects of coolingafter rolling on thermal stress of H-beam steelrdquo Hot WorkingTechnology vol 43 pp 121ndash123 2014

[11] G M Zhu Y L Kang W Chen J L Gui and G T MaldquoThermalmechanical coupled finite element analysis of thermalresidual stress in h-beamrsquos air coolingrdquoMaterials for MechanicalEngineering vol 32 pp 77ndash80 2008

[12] Q Liu Y Zang Q Qin and Z Q Zhao ldquoAnalysis ontemperature and thermal stress of H-beam during coolingrdquoMetallurgical Equipment vol 167 pp 17ndash20 2008

[13] L Zhao X G Yu G B Liao and Y L Hu ldquoSimulation ofStress-strain of H-beam in control cooling processrdquo Journal ofUniversity of Science and Technology Liaoning vol 34 pp 466ndash469 2011

[14] G Zhu Y Kang andGMa ldquoSimulation of residual stress in hotrolled large size H-beamsrdquo Materials Science Forum vol 704-705 pp 1370ndash1378 2012

[15] J Y Jang and Y G Gan ldquoOptimization analysis of controlledcooling process for h-shape steel beamsrdquo in Proceedings ofthe International Conference on Mechatronic Automobile andEnvironment Engineering 2017

[16] J Y Jang L FHsu and J S Leu ldquoOptimization of the span angleand location of vortex generators in a plate-fin and tube heatexchangerrdquo International Journal of Heat andMass Transfer vol67 pp 432ndash444 2013

[17] J-Y Jang and Y-C Tsai ldquoOptimization of thermoelectricgenerator module spacing and spreader thickness used in awaste heat recovery systemrdquo Applied Thermal Engineering vol51 no 1-2 pp 677ndash689 2013

[18] J-Y Jang and C-C Chen ldquoOptimization of louvered-fin heatexchanger with variable louver anglesrdquo Applied Thermal Engi-neering vol 91 pp 138ndash150 2015

[19] J-Y Jang and J-B Huang ldquoOptimization of a slab heatingpattern for minimum energy consumption in a walking-beamtype reheating furnacerdquo Applied Thermal Engineering vol 85article no 6547 pp 313ndash321 2015

[20] X K Zhu and Y J Chao ldquoNumerical simulation of transienttemperature and residual stresses in friction stir welding of 304Lstainless steelrdquo Journal of Materials Processing Technology vol146 no 2 pp 263ndash272 2004

[21] JmatPro 90 Practical software for material properties SenteSoftware Ltd 2017

[22] J Wendelstorf K-H Spitzer and R Wendelstorf ldquoSpray watercooling heat transfer at high temperatures and liquid massfluxesrdquo International Journal of Heat and Mass Transfer vol 51no 19-20 pp 4902ndash4910 2008

[23] F P Incropera D P Dewitt T L Bergman and A S LavineldquoPrinciples of Heat and Mass Transferrdquo in International StudentVersion 7th edition

Submit your manuscripts athttpswwwhindawicom

ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CorrosionInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Polymer ScienceInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CeramicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CompositesJournal of

NanoparticlesJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Biomaterials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

NanoscienceJournal of

TextilesHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Journal of

NanotechnologyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

CrystallographyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CoatingsJournal of

Advances in

Materials Science and EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Smart Materials Research

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MetallurgyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

BioMed Research International

MaterialsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Submit your manuscripts athttpswwwhindawicom

ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CorrosionInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Polymer ScienceInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CeramicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CompositesJournal of

NanoparticlesJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Biomaterials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

NanoscienceJournal of

TextilesHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Journal of

NanotechnologyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

CrystallographyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CoatingsJournal of

Advances in

Materials Science and EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Smart Materials Research

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MetallurgyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

BioMed Research International

MaterialsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014