Experimental Thermal and Fluid Science 34 (2010) 1068–1076

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Experimental Thermal and Fluid Science

journal homepage: www.elsevier .com/locate /et fs

Application of an optimization method and experiment in inverse determinationof interfacial heat transfer coefficients in the blade casting process

Weihong Zhang *, Gongnan Xie, Dan ZhangEngineering Simulation and Aerospace Computing (ESAC), The Key Laboratory of Contemporary Design and Integrated Manufacturing Technology,Northwestern Polytechnical University, P.O. Box 552, 710072 Xi’an, Shaanxi, China

a r t i c l e i n f o

Article history:Received 21 December 2009Received in revised form 18 March 2010Accepted 20 March 2010

Keywords:Interfacial heat transfer coefficientNumerical predictionExperimental measurementOptimization method

0894-1777/$ - see front matter � 2010 Elsevier Inc. Adoi:10.1016/j.expthermflusci.2010.03.009

* Corresponding author. Tel./fax: +86 29 88495774E-mail address: zhangwh@nwpu.edu.cn (W. Zhang

a b s t r a c t

In order to effectively improve the numerical prediction accuracy in a blade investment casting process, anew method is proposed to determine the interfacial heat transfer coefficient (IHTC) in a complicatedblade casting by combining the numerical prediction, optimization and limited experimental data. Aninvestment experiment of the blade is conducted to acquire the surface temperature of the casting andthe shell mould. Regarding the complicated mechanism of the interfacial heat transfer in the progressivesolidification, a new continuous model with three-step evolution is established for the casting–mouldIHTC, and a power function is proposed to correlate the mould–environment IHTC with solidificationtime as well. A globally convergent method is employed to search the optimal coefficients involved inthe IHTCs correlations. Results show that the predicted temperature based on proposed models agreeswell with the experimental data with the maximum deviation being less than 5.5%, and a significant var-iation of the casting–mould IHTC is observed. It is concluded that the prediction accuracy and efficiencyassociated with the optimization method can be greatly improved with the present IHTC models.

� 2010 Elsevier Inc. All rights reserved.

1. Introduction

It is known that the shape of the casting depends on the cavitygeometry of the metal die significantly in the investment castingprocess. An exact die profile, which generally takes into accountthe various shrinkages involved in the casting process, is therefore,important to improve the quality of net-shaped products. In thissense, an accurate numerical simulation of the entire casting pro-cess is very helpful to realize optimal designs of the die-cavity pro-file [1]. Many commercial solidification simulation softwares canbe used to obtain reliable simulation results if the appropriate dataof thermal properties and boundary conditions are provided [2].For the heat transfer in solidification, how the heat transfersthrough the casting–mould interface is one of the most importantboundary conditions to be characterized because this problem di-rectly dominates the evolution of solidification and controls thefreezing conditions within the casting. Therefore, the determina-tion of interfacial heat transfer coefficient (IHTC) is vital ahead ofthe simulation of the solidification process. In fact, the IHTC de-pends upon multiple factors such as die coating thickness, insulat-ing pads, chill and casting geometries, pouring temperature,surface roughness, alloy composition, metallostatic head, mould

ll rights reserved.

.).

temperature and other mechanical boundary conditions [3–6]. Itsdetermination is often carried out by manual adjustments to re-duce the difference between the experimental observation andthe numerical prediction.

Generally, two kinds of methods exist. The first one is to mea-sure the size variation of the interfacial gap that usually appearsat the metal/mould interface during the solidification process. Forexample, Prates and Biloni [7] and Nishida et al. [8] measured theIHTCs based on the immersion method, fluidity test, unidirectionalmethod and one-dimensional solidification in a mould. The forma-tion process of the air gap and the involved heat transfer mecha-nism were investigated by measuring the displacements andtemperatures for both cylindrical and flat castings of aluminum al-loys. The second one is to evaluate the IHTC inversely based on thetemperature data measured at selected locations in both the castingand the mould or chill. Note that the surface temperature or heatflux is determined based on the measured temperatures at internalpoints near the surface. Since the solidification of a casting involvesboth the material phase change and the variation of thermal prop-erties with respect to the temperature, the inverse heat conductionis a nonlinear problem and can be solved by means of the nonlinearestimation methods [9,10]. For instance, Lau et al. [11] studied theIHTC between an iron casting and a metallic mould. Souza et al. [12]analyzed the heat transfer along the circumference of cylindersmade up of Sn–Pb alloys in the mould.

Nomenclature

a1, a2 coefficient of casting–mould IHTC functionb1, b2 coefficient of mould–environment IHTC functionfs solid fractionh heat transfer coefficientk number of thermocouples in the mouldL latent heat of fusionm number of time stepsn number of thermocouples in the castingq interface heat fluxt solidification timetc critical solidification timeT temperature

Greek symbolk thermal conductivity

Subscripts0 initial statec castingcr criticalh heat transfer coefficientl liquidusm moulds solidusT temperature

Superscriptsest predicted valuesexp experimental datamax maximummin minimum

Casting

Insulating heat material

MouldmT

cT

q

Casting Mould

c m( )q h T T= −

Fig. 1. Schematic of a casting–mould interface.

W. Zhang et al. / Experimental Thermal and Fluid Science 34 (2010) 1068–1076 1069

However, few reliable data of the IHTCs are available for theinvestment casting process in practice. Sturm and Kallien [13]identified the IHTC involved in the model of an aluminum alloyinvestment casting where the resultant data of IHTC (1000 W/m2 K) was assumed to be unchanged throughout the solidification.Anderson et al. [14] combined the simulation and experiment tostudy thermal behaviors of a two-dimensional symmetrical alumi-num casting where the IHTC was buried in an overall heat transfercoefficient. Based on the nonlinear estimation technique men-tioned above, Sahai and Overfelt [15] completed a study of theIHTC for both cylindrical and plate investment castings of anickel-based alloy. For the cylindrical casting (mould preheatedto 745 �C), it was found that the IHTC varied linearly from200 W/m2 K at 1300 �C to 100 W/m2 K at 850 �C. For the plate cast-ing, the IHTC was found to vary between 5000 W/m2 K at 1400 �Cand 100 W/m2 K at 1100 �C. The results showed that the castingshape had a great impact upon the IHTC in the investment casting.O’Mahoney and Browne [16] suggested that cares should be takenof the solidification process, the alloy type and the metallostatichead effect. The aluminum casting alloys, 413, A356, 319, wereused in their study.

For these reasons, this work is to develop a simple and universalinverse methodology, which makes use of the existing simulationsoftwares such as ProCAST to resolve the IHTCs in the investmentcasting process of a complicated blade. Based on a switch functionof solidification time, a novel model of IHTC is proposed to replacethe original power function. With the obtained IHTCs, the pre-dicted temperature is compared with the experimental data. Be-sides, thermocouples are placed in a very thin mould cavitywithout manufacturing a special mould. This methodology is help-ful for a foundry engineer to look for a reference effectively on howto apply boundary conditions for simulation of a specific castingprocess.

2. Mathematical model of casting process

Fig. 1 depicts the heat transfer through between the two con-tacting surfaces. When the mould is suddenly filled with the liquidmetal, the effects of fluid flow in the liquid phase, the convectiveheat transfer and the radiative heat transfer are negligible. There-fore, the direct problem for the casting region is formulated onlyin terms of unsteady-state heat conduction.

qc@T@t¼ @

@xk@T@x

� �þ @

@yk@T@y

� �þ @

@zk@T@z

� �þ qL

@fs

@tð1Þ

where q is the cast density, c and k are specific heat and thermalconductivity, respectively. L is the latent heat of fusion and fs isthe solid fraction. Note that the thermal properties are known dur-ing the investment process. The initial and boundary conditions forthe casting region are

initial condition Tjt¼t0¼ T0ðx; y; zÞ ð2aÞ

at cast—mould interface � k@T@n¼ q ¼ hcðT � TmÞ ð2bÞ

The casting temperature field is governed by the above heat con-duction equation and boundary conditions. Numerical solutionscan be obtained by means of the finite element method.

Obviously, hc, the IHTC at the casting–mould interface, affectsthe calculated temperature field and is thus of importance forthe numerical solution of the casting temperature. Likewise, thegoverning equation related to the mould region is similar to theabove one except that the source term, qL

@fs@t , is not included. More-

over, hm at the mould–environment interface has to be determinedin advance. For an inverse heat transfer problem, the aim is to pre-dict the unknown IHTCs from the knowledge of measured or/andcalculated temperatures at specific positions on the interface. Thispaper is to determine hc and hm in the blade investment castingprocess.

3. Determination of interfacial heat transfer coefficient

3.1. Inverse parameter estimation

Inverse estimation methods are based on the minimization ofan objective function containing both estimated and measured

0 5 10 15 200.0

0.2

0.4

0.6

0.8

1.0

α=100

α=10

α=3

1+eα

t-t c

r-1

t

α=1

tcr=10

Switch function

Fig. 2. Typical switch functions for heat transfer coefficient.

1070 W. Zhang et al. / Experimental Thermal and Fluid Science 34 (2010) 1068–1076

temperatures. Solving one such optimization problem withoutconstraints can identify unknown parameters involved in themodel. In this study, a globally convergent method (GCM), origi-nally proposed by Svanberg [17], is used to find the optimal valuesof unknown parameters by minimizing an objective function de-fined by

SðhÞ ¼ w1

Pmi¼1

Pnj¼1 Texp

ij � Testij

� �2

nþw2

Pmi¼1

Pkj¼1 Texp

ij � Testij

� �2

kð3Þ

where Testij and Texp

ij denote the estimated and the experimental dataof the temperature field at various thermocouple locations and timeincrements, respectively. m is the number of time steps, n is thenumber of thermocouples in the casting, and k is the number ofthermocouples in the shell mould. w1 and w2 are the weightings.Because the minimization must ensure the accuracy of the temper-ature field over the casting as much as possible, the value of w1 isoften larger than that of w2. Here, w1 and w2 are assumed to be0.7 and 0.3, respectively.

In order to minimize S(h), the first-order sensitivity coefficientsare usually calculated by finite difference scheme with

@Testij ðhÞ@hr

ffiTest

ij ðh1; . . . ;hr þ dhr ; . . . hnÞ � Testij ðh1; . . . ; hr ; . . . hnÞ

dhrð4Þ

Then an iterative procedure is designed to find the minimiza-tion solution of S(h). It must be pointed out that finite differencemethod used for the sensitivity analysis suffers from two majordrawbacks. Firstly, the approximation accuracy depends on themagnitude of the perturbation dhr. If dhr is too small, the round-off errors will be significant. Oppositely, if dhr is too large, thetruncation errors will degrade the accuracy. In this work, themagnitude of perturbations is automatically chosen by an optimi-zation method. Secondly, the use of finite difference method isexpensive because the finite element reanalysis must be runn + 1 times for each iteration. At this point, an efficient way ofdecreasing the computing cost is to parameterize the IHTC onlyas a function of time because of the interdependence betweenthe IHTC and the temperature.

3.2. Continuous IHTC model with three-step evolution

To achieve a reasonable model of the casting–mould IHTC, thecomplicated mechanism of the interfacial heat transfer in the pro-gressive solidification should be discussed firstly. In general, thevariation of the casting–mould IHTC with time can be divided intofour stages: (i) At the first stage, the IHTC increases rapidly whenthe molten alloy is poured into the mould. Although the flow inthe alloy has a great influence on the IHTC, it is not considered inthis study due to the limitation of high frequency acquisition dis-posals. (ii) At the second stage, the IHTC is higher in longer mushyzones with the temperature variations between liquidus tempera-ture and solidus temperature, as pointed out by Santos et al. [18].The magnitude of the IHTC almost remains unchanged because themacro air gap does not appear during such a short period of time.(iii) At the third stage, the IHTC starts to decrease rapidly as longas the casting thickness becomes larger and larger with a decreaseof the velocity of heat transfer from the casting to the mould. (iv)At the fourth stage, a gradual decrease of the IHTC is observeddue to the further increase of the air gap.

Based on the above interface heat transfer mechanism, a newmodel of the IHTC is proposed with the negligence of the firststage. The IHTC could be assumed to be a constant at the secondstage, whose initial value may change from case to case. A powerfunction of time is used to characterize the significant drop of

the IHTC caused by the appearance of the macroscopic air gap dur-ing the third stage and the fourth stage. Therefore, a piecewisefunction is thus proposed to formulate the casting–mould IHTC.

h ¼h0 t � tcr

a1t�a2 t > tcr

�ð5Þ

where h0 is an initial value of the IHTC in the initial stage; a1 and a2

are the parameters to be determined. tcr is the critical time corre-sponding to the intersection between the second stage and the thirdstage.

In order to improve the prediction and optimization efficiency, acontinuous and differentiable switch function is devised to expressthe heat transfer coefficient. The casting–mould IHTC is now re-formed as

hc ¼ h01

1þ eaðt�tcrÞþ a1t�a2

11þ eaðtcr�tÞ ð6Þ

where a refers to a large positive number. The term, 11þeaðt�tcr Þ, de-

notes a typical switch function, as illustrated in Fig. 2 for differentvalues of a when tcr = 10. Clearly, a large a results in a closedapproximation of the unity once t is less than tcr, or of the zero whent is larger than tcr. Thus, a moderate value of a = 10 is chosen in themodel of the casting–mould IHTC and the three unknown parame-ters, a1, a2 and tcr, are to be determined.

As to hm, the external surface temperature of the shell mould isinitially low. It rises rapidly to a peak value at the beginning ofsolidification and then declines. According to the experimentaldata [18], the values of the IHTC are 22 W/(m2 K) and 34 W/(m2 K) when the temperature of the mould surface is 300 �C and600 �C, respectively. Similarly, a power function is given to corre-late the IHTC with the process time

hm ¼ b1t�b2 ð7Þ

where b1 and b2 are the unknown parameters to be determined.

3.3. Description of the optimization procedure

The optimization procedure is shown in Fig. 3, where the GCMis used as the optimizer. The process starts by initializing the basicdata for the direct heat transfer analysis and the optimization pro-grams. Based on initially estimated parameters and sensitivity val-ues, a proper search direction and a step size will be evaluated bythe optimizer to update design parameters. In this study, fiveparameters tcr, a1, a2, b1 and b2 are optimized. Because the objec-tive function is highly nonlinear, the finite difference method isapplied in sensitivity analysis.

DATA INPUTInitialize design parameter & their upper and lower bounds

FEM ANALYSISCall ProCAST to simulate solidification

and calculate objective function

Converged ?

SENSITIVITY ANALYSISCalculate the sensitivity of the design parameter

MODIFY INPUTRefresh the design parameter by GCM

Stop

Yes

No

Fig. 3. The flowchart of optimization procedure.

Support

Fig. 4. The core and the exterior of the wax pattern.

W. Zhang et al. / Experimental Thermal and Fluid Science 34 (2010) 1068–1076 1071

Optimization problems are usually written in the followingform:

Min f0ðxÞs:t: fiðxÞ � 0

x2Xi ¼ 1; . . . ;m ð8Þ

f0, f1, . . . , fm are real valued functions which are assumed to be sec-ond-order continuously differentiable on the set X ¼ x 2 Rjxmin

j

n� x � xmax

j ; j ¼ 1; . . . ; ng.To avoid the situation that the feasible design space is empty, a

modified optimization problem is considered:

Min f0ðxÞ þPmi¼1

Miyi þ y2i =2

� �

s:t: fiðxÞ � yi � 0 i ¼ 1; . . . ;mð9Þ

where Mi are often assigned by ‘‘very large” real numbers and yi areso-called artificial variables. All yi are usually zeros at the optimumunless some of them are relaxed to take positive values.

The GCM works iteratively according to the following generalscheme: Assume x represents the set of design parameters. Duringiteration k, an explicit approximate sub-problem is generated at acurrent iteration point (x(k), y(k)). In the sub-problem, the functionsfi(x) are replaced by approximate convex functions based on thegradient information and the information from the previous itera-tion points. Once this sub-problem is solved, the optimal solutionbecomes the starting point of the next iteration for the new sub-problem. A description of the GCM can be found in [17].

4. Experimental setup of the investment casting process of ablade

To validate the proposed IHTCs models, aluminum alloy A355 isused instead of super-alloy in the present work. Moreover, to re-duce experimental cost, the gravity casting process is adopted.The procedure starts with a blade fabricated by an investmentcasting wax (the pattern). The wax is heated above its melting tem-perature and then pressed into a steel die. The wax pattern is madeup of two parts: the core and the exterior, as shown in Fig. 4.

Six thermocouples are positioned in the pattern’s middle cross-section as shown in Fig. 5. Six wood sticks are selected to drill someholes of 1.5 mm in diameter. Then these sticks are inserted into sixholes of the wax blade that is fixed to the feed system, as shown inFig. 6. Finally the pattern is cleaned to allow the adherence of themould material. The investment shell moulds are composed of twolayers. Firstly the pattern is dipped into the ceramic slurry anddrained, and then rained by fine ceramic and finally dried in avent-pipe. This procedure is repeated until a desired thickness of2.0 mm attains for the primary shell. The other six sticks are usedto measure the temperature of the primary shell. A secondary layerwith a thickness of 4.0 mm is formed in the same way. When thewood sticks are burned out, 12 K-type thermocouples are thenplaced into the small holes with a depth of about 2.0 mm fromthe interface to metal region and from the interface to the mouldregion, as shown in Fig. 5. Moreover, two thermocouples are placedon the external surface of the shell mould so as to acquire the

thermocouple

1

2

3

4

5

6

7

8

9

10

11

12

casting

sand_zircon

sand_silica

axial

chor

d

support

A

A

The middle cross-section

2. 02. 0

The interface

60

A-A

Fig. 5. The cross-section of the blade and thermocouple positions.

Feed System

CastingBladeThe middle

cross-section

thermocouple

110

200

Fig. 6. The 3D model of the blade and the feeder.

1072 W. Zhang et al. / Experimental Thermal and Fluid Science 34 (2010) 1068–1076

temperature data for the prediction of the IHTC between the mouldand the environment.

The mould shell is not preheated in gas furnace. Molten alumi-num alloy is poured into the mould shell at a temperature of about624 �C by the gravity method, and the mould shell is cooled by theair with the insulating heat materials on the top and the bottom, sothat the heat flux from the casting to the mould shell can only takeplace along the periphery of the turbine blade cross-section asshown in Fig. 7. The temperature is recorded by sampling frequencyof 1 Hz using a temperature instrument HR3200 (YOKOGAWA,

He

Hea

Fig. 7. Schematic representation

Japan). Before the thermal profiles are measured, they must besmoothed out using a digital filter.

Although the experimental setup should be designed to be asrepresentative as possible of the real process, one should realizein mind that it is impossible to consider performing an inverse cal-culation on a real 3D casting geometry due to the prohibitive com-puting time. Thus, a 2D geometry is considered in the parameteroptimization process. In addition, a certain number of thermocou-ples have to be properly located rather than ‘‘flooding” the exper-iments with many thermocouples. With the experimental setup,the maximum absolute error is about 12 �C.

The relevant properties and chemical composition of aluminumalloys are widely available in the literature [19], but relatively littleaccurate information of the investment of the casting and shell

at insulating material

The Shell

mould

The Casting

t insulating material

Atmosphere

of the experimental setup.

Table 1Thermal data for aluminum alloy A355.

Alloy Liquidustemperature (�C)

Eutectictemperature (�C)

Solidustemperature (�C)

Freezingrange (�C)

A355 624 582 540 84

Table 2Thermal data for the shell mould.

Shellmould

Conductivity(W/(m K))

Density(kg/m3)

Special heat(kJ/kg/K)

Sand silica 0.59 1520 1.20Sand zircon 0.83 2780 0.77

0 200 400 600 800

80

100

120

140

160

180

200

0

200

400

600

800

1000

1200

1400

Ethalphy

Eth

alph

y (k

J/kg

)

Con

duct

ivit

y (W

/(m

.K))

Temperature (oC)

Conductivity

Fig. 8. Conductivity and enthalpy of aluminum alloy A355.

Time (s)

Experiment thermocouple 1,7

Simulated

The shell mold

The casting

200

250

300

350

400

450

500

550

600

650

0 100 200 300 400 500

Tem

pera

ture

(o C

)

Fig. 9. Experimental and predicted temperature at the external surface of the shellmould.

0 100 200 300 400

0

50

100

150

200

250

300

Mou

ld-E

nvir

omen

t H

TC

(W

/(m

2 .K))

Time (s)

Mould-enviroment heat transfer coeffecient

hm=82.06 t-0.26

Fig. 10. Variation of the mould–environment IHTC of alloy A355.

650

W. Zhang et al. / Experimental Thermal and Fluid Science 34 (2010) 1068–1076 1073

mould is available. In this paper, the thermal data of the shellmould is referred from the thermo-physical properties given inthe database of the ProCAST and the literature [20]. Thermo-phys-ical properties of the casting and the shell mould are given inTables 1 and 2 and Fig. 8.

Time (s)

Thermocouple 6

Thermocouple 2,4

stc 58=

582=sT

624=lT

400

450

500

550

600

0 50 100 150 200 250 300 350 400 450 500

Tem

pera

ture

(o C

)

Thermocouple 1,3,5

Fig. 11. Experimental temperature profiles of the thermocouples.

5. Results and discussion

5.1. The mould–environment IHTC

Fig. 9 shows the temperature variations of thermocouples 1 and7 sampled in the primary layer of the casting and shell mould (asshown in Fig. 5) during the solidification experiment, respectively.Since the heat transferred between the shell mould and the envi-ronment includes convective and radiative heat, it can be observedthat the temperature on the external surface of the shell mouldrises rapidly from the beginning of solidification to a peak valueand then declines. With the help of the power function introducedin Eq. (7), a docent agreement is achieved between the measuredtemperatures and the numerical predictions, as shown in Fig. 9.In this case, the mould–environment IHTC by the optimization pro-cess is presented in Fig. 10.

5.2. The metal casting–mould IHTC

Considering the geometrical feature of the blade as shown inFig. 11, the cross-section of the blade can be divided into fourparts: the trailing edge (thermocouple 6), the leading edge (ther-mocouple 3), the concave side (thermocouples 1 and 2) and the

convex side (thermocouples 4 and 5). The slopes of different tem-perature curves represent the cooling rates at the correspondingmeasured positions. In general, the cooling curve of the castingconsists of three stages: beyond the liquidus, between the liquidusand the solidus, and below the solidus. At the first and third stages

Table 3Optimization results by GCM.

Model Lewis and Ransing [21] Santos et al. [18] This study

Parameters C n a1 a2 a3 h0 (W/m2 K) a1 a2 b1 b2

Values 7100.21 0.21 13.40 1189.94 0.53 12160.36 1245.61 0.52 82.06 0.28

500

520

540

560

580

600

620

640

660

Time (s)

Time (s)

Tem

pera

ture

()

Tem

pera

ture

()

Thermocouple 6 simulated

Thermocouple 3 simulated

Thermocouple 3 experiment

Thermocouple 6 experiment

(a) Thermocouples 3 and 6.

520

540

560

580

600

620

640

0 10 20 30 40 50 60 70 80 90

0 10 20 30 40 50 60 70 80 90

Thermocouple 2 simulatedThermocouple 4 simulatedThermocouple 2 experimentThermocouple 4 experiment

(b) Thermocouples 2 and 4.

Fig. 14. Comparisons between the experimental temperatures and the predictedtemperatures.

0 100 200 300 400 500

400

450

500

550

600

650

By Santos

This work By Lewis et al. [21] By Santos et al. [18] Experimental data

Time (s)

By Lewis

Tem

pera

ture

(o C

)

Fig. 12. Comparisons between the experimental temperature and the predictedtemperature by different models at thermocouple 1.

Time (s)

Err

or (

)

average error=33.27

average error=20.00

average error =7.85

Lewis et al. [21]

Santos et al. [18]

This work

0

10

20

30

40

50

60

70

80

90

0 100 200 300 400 500

Fig. 13. Absolute errors of predicted temperatures with different models.

1074 W. Zhang et al. / Experimental Thermal and Fluid Science 34 (2010) 1068–1076

the temperature of the casting falls rapidly, while at the secondstage the cooling rate decreases slowly because of the release ofthe latent heat. However, both the experiment data and those datapublished in the literature [2,20] showed that due to the high tem-perature difference between the casting and the shell mould, thetemperature of the casting surface decreased so quickly that thesecond stage almost disappeared. Once the temperature falls be-low the solidus, the effect of IHTC on the cooling rate is muchstronger because the heat conduction between the casting andthe shell mould dominates the cooling rate of the casting.

Note that the cooling rate is the fastest in the trailing edgewhere the blade profile is the thinnest. Therefore, the shrinkagehappens earlier in the trailing edge than in other positions andthereby the macro gap forms earlier in this position. Accordingly,the slope of its cooling curve changes more early. The cooling rateof the concave side is slower than that of the convex side due to theheat radiation. The rapid decrease of these cooling profile slopes atabout 58 s indicates that the macro air–gap forms at this criticaltime rather than at the solidification time of the interface. Suchcritical time is later than the solidification time of the interface.Therefore, the macro air–gap will not be formed until the interfa-cial metal skin between the casting and the shell mould becomeseffective to resist the action of the metallostatic head from liquidmetal. As all cooling rates are not so distinct, a unique equivalentinterfacial heat transfer coefficient is used for simulation. Accord-ing to the above analysis, the value of h is constant for t < 58 sand then modeled by a power expression of time for t < 58 s. Basedon this feature, Eq. (6) is used to resolve the IHTC in this study.

The other two models proposed by Santos et al. [18] and Lewisand Ransing [21] are also tested to calculate the IHTC, and the opti-mization results are given in Table 3. The corresponding tempera-ture calculated at thermocouple 1 is shown in Fig. 12. Differencesbetween the predicted results and the measured data are shownin Fig. 13. Among the three IHTC models, the model proposed inthis study is found to provide the best fitting to the experimentaldata with an average error of about 7.85 �C, while the other twomodels produce the average errors of about 33 �C and 20 �C,respectively.

6. 28s

10.28s

14.28s

18.28s

22.28s

22. 28s

24.28s

24. 28s

26.28s

Solidus

The inscribed circle

Rmax

Fig. 15. Variations of solidus isotherms as a function of time.

0 100 200 300 4000

50

100

150

200

250

300

350

400

Cas

ting

-Mou

ld I

HT

C(W

/(m

2 .K))

Time (s)

Casting-Mould heat transfer coeffecient

hc=1245.61 t-0.52

Fig. 16. The casting–mould IHTC of alloy A355 at last two stages.

W. Zhang et al. / Experimental Thermal and Fluid Science 34 (2010) 1068–1076 1075

Comparisons between the experimental temperature and thepredicted temperature in other locations are shown in Fig. 14. Itis observed that an acceptable agreement is achieved betweenthe experimental data and those predicted results. The maximumdeviation is less than 5.5%. Solidus isotherm varying as a functionof time is shown in Fig. 15. It can be seen that the solidificationends around the center of the maximal inscribed circle of the bladeprofile. This indicates the shrinkage center of the blade.

From optimization results in Table 3, it is found that the IHTCtakes a very high value, i.e., up to 12160.36 W/m2 K, at the initialstage of solidification as a result of the tight surface contact be-tween the liquid metal and the shell mould. The shell temperaturerises rapidly from the beginning of solidification, since the shellmould is very thick and is not preheated. As a result, the mouldexpansion favors the thermal contact between the metal and theshell surface. Therefore, the initial value of the IHTC in the invest-ment process is higher. Moreover, it might be deduced that the ini-tial value increases with increasing values of superheat, and thefirst stage in which the value keeps a constant will be prolonged.Once the air gap has been formed, the heat transferring across

the interface decreases rapidly due to the increase of the transientthermal resistance between the metal and the mould. Conse-quently, the IHTC reaches a relatively lower value of about 50 W/m2 K, as shown in Fig. 16.

Furthermore, based on numerical tests, the same convergentsolution of the objective function is achieved even if the GCMmethod starts with different initial values of design parameters.The computational cost is decreased significantly owing to theparameterization of the IHTC, which reduces the number of designparameters effectively.

6. Conclusions

The interfacial heat transfer coefficients (IHTCs) in the invest-ment casting of a solid blade have been investigated on the basisof an experimental study and an optimization method. A commer-cial software ProCAST and an optimization tool with globally con-vergent method (GCM) are employed.

Equivalent parameterized models of the IHTCs including a con-tinuous three-step evolution for the casting–mould IHTC and apower function of time for the mould–environment IHTC are pro-posed. Involved parameters in the model are resolved by the GCMoptimization method. Good agreements between the experimentaland the predicted temperatures are achieved with the maximumdeviation being less than 5.5%. Even with different starting condi-tions of design parameters, the convergence can be achievedefficiently.

Acknowledgement

This work was supported by a grant from National Science Fundfor Distinguished Young Scholars (No. 10925212).

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