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Finite Elements in Analysis and Design 42 (2006) 10871096

www.elsevier.com/locate/finel

Inverse heat conduction analysis of quenching process using finite-elementand optimization method

Li Huiping, Zhao Guoqun, Niu Shanting, Luan YiguoMould and Die Engineering Technology Research Center, Shandong University, Jinan, Shandong, 250061, PR China

Received 23 November 2004; received in revised form 6 March 2006; accepted 18 April 2006

Available online 9 June 2006

Abstract

The calculation of surface heat transfer coefficient during quenching process is one of the inverse heat conduction problems, and it is anonlinear and ill-posed problem. A new method to calculate the temperature-dependent surface heat transfer coefficient during quenching

process is presented, which applies finite-element method (FEM), advanceretreat method and golden section method to the inverse heat

conduction problem, and can calculate the surface heat transfer coefficient according to the temperature curve gained by experiment. In order

to apply the advanceretreat method to the inverse heat conduction problem during quenching process, the arithmetic is improved, so that

the searching interval of optimization can be gained by the improved advanceretreat method. The optimum values of surface heat transfer

coefficient can be easily obtained in the searching interval by golden section method. During the calculation process, the phase-transformation

volume and phase-transformation latent heat of every element in every time interval can be calculated easily by FEM. The temperature and

phase-transformation volume of every element are calculated with the coupling calculation of phase-transformation latent heat.

2006 Elsevier B.V. All rights reserved.

Keywords: Quenching; Inverse heat conduction; Finite-element method; Optimization

1. Introduction

The heat exchange problem with a known initial condition

and boundary condition is a well-posed problem. It can be

solved by mathematical analysis. But if the boundary condition

of heat exchange problem has to be ascertained by measuring

some temperatures in the interior or the surface of the domain,

it can be defined as an inverse heat conduction problem (IHCP).

IHCP has numerous important applications in various sciences

and engineering. For example, the temperature of a very hot

surface is not easily measured directly with sensors. Usuallysensors are placed beneath the surface and the temperature of

the hot surface is estimated by inverse analysis. Other examples

of IHCP are the estimation of unknown temperature-dependent

thermo-physical parameters of materials from the temperature

recordings at the boundary surfaces of the domain [1].

Corresponding author. Tel./fax: +86 531 8395811.E-mail addresses: [email protected] (L. Huiping),

[email protected] (Z. Guoqun).

0168-874X/$- see front matter 2006 Elsevier B.V. All rights reserved.

doi:10.1016/j.finel.2006.04.002

IHCP is an ill-posed problem, which is more difficult to solve

than the normal heat exchange problem. Many researchers

have studied the methods of evaluating temperature-dependent

surface heat transfer coefficient. Based on the KarhunenLove

Galerkin procedure, Park and Chung proposed a method for

the solution of inverse problem of estimating the time-varying

strength of a heat source in a two-dimensional heat conduction

system [2]. Osman and Beck [3] treated the problem of esti-

mating the temperature-dependent heat transfer coefficient in

the quenching of a sphere as a nonlinear parameter estimation

problem, and used the sequential function specification methodto estimate the unknown heat transfer coefficients one by one.

According to the temperaturetime data measured at subsurface

locations, Naylor and Osthuizen [4] determined the heat trans-

fer coefficient in a force convective flow over a square prism

using an iterative algorithm. Taler and Zima [5] used the con-

trol volume methods to solve multi-dimensional inverse heat

conduction problem. By this method, the partial heat conduc-

tion equation is replaced by a system of ordinary differential

equations in time, which are then solved sequentially, the pro-

cedure is started at a partial node where sensor is located and
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1088 L. Huiping et al. / Finite Elements in Analysis and Design 42 (2006) 10871096

Nomenclature

thermal conductivity, W/m Ccp constant pressure specific heat, J/kg

CT temperature, Cj partial differential operator

Hk convection coefficient, W/m2 CTw temperature of boundary,

C boundary of object

W weight function

s curve boundary

N transient thermal capacity matrix

V volume fraction of phase-transformation

density of material, kg/m3

t time, s

qv latent heat of phase transformation, J/m3

x , y , z rectangular coordinates

Hs radiation coefficient, W/m2 CTc temperature of ambience,

Cr radius

J function

K stiffness matrix

H enthalpies, J/m3

sequentialy marches through space to the surface node.

Hernandez-Morales et al. [6] and Chantasiriwan [7] also stud-

ied the one-dimensional problem of estimating the transient

heat transfer coefficient at the surface of steel bars subjected to

quenching using the sequential function specification method.

According to the temperaturetime data of several interior lo-

cations in the quenching part measured by sensors, Gu et al.

[1,8] used the inverse heat conduction method to estimate the

heat transfer coefficients between quenching part and water or

oil. They did not consider the influence of phase-transformation

latent heat by using specific steel in their study, and supposed

it was one-dimensional heat conduction along the thickness of

part. Cheng et al. [9] and Chen et al. [10] also estimated the heat

transfer coefficients between quenching part and quenching

medium using the inverse heat conduction method. The results

gained by this method are consistent with the results gained byexperiment. Cheng et al. [11] used the finite-difference method

to solve the inverse-estimating problem of heat conduction

between steel 45 and quenching medium. They considered the

influence of phase-transformation latent heat in their study.

Their solving precision is not very satisfactory, even though

the process iteration efficiency is very high.

In this paper, a new method of estimating the temperature-

dependent surface heat transfer coefficient during quenching

process is presented, which applies finite-element method

(FEM), advanceretreat method and golden section method

to the inverse heat conduction problem, and can calculate

the surface heat transfer coefficient according to the temper-

ature curve obtained by experiment. In order to apply the

advanceretreat method to inverse heat conduction problem

during quenching process, the arithmetic of advanceretreat

method is improved, so that the searching interval of optimiza-

tion can be gained by the improved advanceretreat method,

and then the optimum values of surface heat transfer coeffi-

cient can be easily obtained in the searching interval by golden

section method. During the calculation process, the phase-

transformation volume and phase-transformation latent heat of

every element in every time interval can be calculated easily

by FEM. The temperature and phase-transformation volume

of every element are calculated with the coupling calculation

of phase-transformation latent heat.

2. FEM modeling of heat conduction and phase

transformation

2.1. FEM modeling of heat conduction

2.1.1. Basic equation

According to the Fourier law, the Fourier heat conduction

equation of transient problem with the phase-transformation

latent heat can be achieved by using conservation of energy in a

rectangular coordinates system. The equation can be written as

j

jx

jT

jx

+ j

jy

jT

jy

+ j

jz

jT

jz

+ qv = cp

jT

jt, (1)

where is the thermal conductivity, T the temperature of

quenching part, qv the latent heat of phase transformation,

the density of material, cp the constant pressure specific heatand t time.

2.1.2. Initial condition

Initial condition is the initial temperature of quenching part.

It is the starting point of calculation. The initial condition at

time t = 0 can be described asT|t=0 = T0(x,y,z), (2)where T0(x,y,z) is the function of initial temperature.

2.1.3. Boundary condition

Boundary condition is the way of heat exchange betweenquenching part and ambience. The boundary condition of

quenching is the third-type condition, and it is the mixed heat

exchange boundary of convection and radiation. It can be

written as

jTjn

= Hk(Tw Tc) + Hs

T4w T4c

= H (Tw Tc),(3)

where n is the outer normal of boundary surface, Hk the con-

vection coefficient, Hs the radiation coefficient, Tw the temper-

ature of boundary and Tc the temperature of ambience. H is

the total heat transfer coefficient. It is the estimating goal of

finite-element analysis for inverse heat conduction.
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L. Huiping et al. / Finite Elements in Analysis and Design 42 (2006) 1087 1096 1089

2.1.4. Discretization of temperature field

Applying the Galerkin principle, the transient heat conduc-

tion partial differential equation with the latent heat of phase-

transformation process can be written as

D

Wl r j2T

jx

2

+rj2T

jr

2

+

jT

jr+ qvr

cprjT

jt

dx dr = 0 (l = 1, 2, . . . , n ), (4)

where Wis the weight function, D the whole region of integral,

l the serial number of finite-element node, dthe full differential

operator and r the radius.

Applying the Green equation to Eq. (4), the following equa-

tion can be obtained:

jJD

jTl=

H rWl

jT

jnds

D

r

jT

jx

jWl

jx+ jT

jr

jWl

jr

qvrWl

+rcpWl

jT

jt dx dr = 0, (5)

where is the curve integral of close boundary. For the quad-

rangular finite-element, Eq. (5) can be written as a variational

format of element, and described as

jJe

jTl=

e

r

jT

jx

jWl

jx+ jT

jr

jWl

jr

qvrWl

+ rcpWljT

jt

dx dr

e

H rWljT

jnds

= K e{T}e + NejT

jt

e {p}e, (6)

where n is the normal direction of boundary, s the curve bound-ary, Ke the stiffness matrix of finite-element, Ne the transient

thermal capacity matrix of finite-element, {p}e includes boththe phase-transformation latent heat and boundary conditions

and e is the number of finite-elements.

In the calculation of temperature, the CrankNicolson

method is used [12].

2.1.5. Phase-transformation latent heat

During the cooling process of heat treatment, when there

are phase transformations in the quenching part, phase-

transformation latent heat will be produced. During the heating

process of heat treatment, when there are phase transfor-mations, phase-transformation latent heat will be absorbed.

Although the latent heat produced in the process of solids

transformation (one solid to another solid) is smaller than the

latent heat produced in the process of solidification or melt,

it is an important factor that cannot be neglected. From the

point of view of mathematics, the basic equation (1) becomes

a highly nonlinear equation due to the latent heat, and it is

more difficult to solve this equation.

In this paper, the material involved is steel, the quenching

process only involves cooling, and the heat source term only

derives from latent heat corresponding to transformations of

austenite during cooling. During the simulation process, the

phase-transformation latent heat produced in the finite-elements

are regarded as the inner heat sources of corresponding ele-

ments, and the correct values of phase-transformation latent

heat are gained by iterative method in every simulating step.

During the quenching process, the inner heat source item qvof the basic (1) is produced due to austenitic transformation,

the enthalpy H of austenitic decomposed into pearlite is

1.56 109

1.50 106

T J/m3

(where T is temperature in C[13]; the enthalpies H (J/m3) of austenitic decomposed into

ferrite, bainite and martensite correspond to 5.9108, 6.2108and 6.4 108, respectively [14,15].

Due to austenitic transformation relating to temperature, the

phase-transformation latent heat is also a linear function of

temperature. The equation for calculating latent heat can be

described as

qv = Hfn+1 fn

t= H f, (7)

where fn and fn+1 correspond to the phase-transformation vol-ume at time tn and time tn

+1, respectively and f is the phase-

transformation volume for a unit time.

2.2. Mathematic modeling of phase transformation

The phase transformations during quenching process is clas-

sified into diffusion-type and non diffusion-type transforma-

tions. The transformations of austenite decomposed into ferrite,

pearlite and bainite are controlled by temperature history. The

JohnsonMehlAvrami equation [16], one of the most popu-

lar kinetic equations, is applied for evaluating volume fractions

of ferrite, pearlite and bainite in the paper. The equation of

diffusion-type transformations can be described as

V = 1 exp(btn), n =ln

ln(1 V1)ln(1 V2)

ln

t1

t2

,

b = ln(1 V1)tn1

, (8)

where t1 and t2 are the isothermal times of certain tem-

perature and V1 and V2 are the volume fractions of phase-

transformations of certain temperature. All of them depend on

Timetemperature transformation (TTT) curve of steel, which

varies with the type of steel.The JohnsonMehlAvrami equation is valid only for isother-

mal transformations. In order to calculate the phase transfor-

mations by JMA equation and TTT curve, cooling curves are

regarded as a series of small isothermal time steps connected by

instantaneous temperature changes following constant volume

fraction lines. The transformed volume fractions are then calcu-

lated isothermally during each time step. Every phase transfor-

mation is assumed to occur between certain temperature limits.

For the diffusion-type transformation, the incubation period and

the phase-transformation volume fractions can be determined

according to Scheils additivity method [17,18].

The volume fraction of non diffusion-type transformation de-

pends only on temperature, and is not controlled by temperature
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history. Not considering the effect of stresses to non diffusion-

type transformation, Koistinen and Marburger [19] described

the equation of non diffusion-type transformation as the func-

tion of temperature:

V = 1 exp[(Ms T )], (9)

where Ms is the temperature of martensite beginning to trans-

form. is a constant, whose value varies with the type of steel.

It can be obtained by calculation in TTT curve. For steel P20,

= 0.023.

3. Optimizing model and method

3.1. Optimizing model

In this research, only the heat transfer coefficient is involved,

and other parameters are not involved. For the evaluating

process of inverse heat conduction problem using optimizing

method and FEM, the iterative processes are necessary. Inorder to keep the evaluation process continuously and achieve

the desired evaluation precision, it is important to construct a

correct convergence criterion or optimization model. Accord-

ing to the character of evaluating the temperature-dependent

surface heat transfer coefficient during quenching process, the

criterion of convergence is constructed with the measuring

values of temperature field and the computing values of every

iteration, which can be described as

min f(x) = min {T} = min

N

i=1

(Ti Ti )2, (10)

Hi /Hmin 10,Hi /Hmax 10,where f(x) is the standard deviation of the computing values

of temperature field, N the total number of target points and

min f(x) the target function. When the value of f(x) reaches

the desired precision, the correct surface heat transfer coeffi-

cient corresponding to certain temperature is gained. In this

study, the minimum value of f(x) is set as 1 104. Hi isthe surface heat transfer coefficient ofith step, Ti the measured

value of temperature corresponding to ith step, Ti the com-puted value of temperature corresponding to ith step and Hminand Hmax the lower and upper limits of optimizing interval.

3.2. Searching the optimizing interval

A simple method of searching interval is advanceretreat

method. The principle of this method is stated as follows: Start

to search the interval from certain point along one direction,

and search three points which function values show up-down-

up trend according to certain searching step size; if it is not

successful along this direction, then retreat the starting point

and search the interval along other direction (along an inverse

direction) [20]. During the process of searching interval us-

ing advanceretreat method, as soon as a boundary point is

ascertained, the value of this point will be regarded as the

surface heat transfer coefficient and used in the simulation

program of temperature and phase-transformation fields to eval-

uate the temperatures. The surface heat transfer coefficients be-

tween quenching part and quenching agent are different from

the surface temperatures of quenching part, and the difference

range is very large. For example, the surface heat transfer co-efficient is about 135 W/m2 C between 20 C water and thequenching part of 860 C; the surface heat transfer coefficientis about 13,491W/m2 C between 20 C water and the quench-ing part of 430 C; the surface heat transfer coefficient is about4350 W/m2 C between 20 C water and the quenching partwhose surface temperature is close to room temperature. There-

fore the searching time of an appropriate interval strongly de-

pends on the value of initial step size s. If the value of s is

bigger, the optimization iterative time of golden section method

is increased largely although the searching time of appropriate

interval is decreased; if the value of s is smaller, the search-

ing time of appropriate interval is increased largely although

the optimization iterative time of golden section method is de-

creased. In order to decrease the searching time of appropri-

ate interval when s keeps the same value, the advanceretreat

method is improved. The following error function is applied to

evaluate the result.

E() =

Ni=1(Ti T

i )

2,

N

i=1Ti

Ni=1

T

i 0

,

Ni=1

(Ti Ti )2,

Ni=1

Ti N

i=1T

i < 0

,

(11)

where is the surface heat transfer coefficient and Ti the com-

puting values of finite-element node i. Ti is the objective val-ues of node i. Its value can be gotten by experiment or other

methods.

The computing values of Ti will vary with the heat transfer

coefficient in every time interval, and the error values due to the

different heat transfer coefficients can be calculated by Eq. (11).

There is only one appropriate heat transfer coefficient, which

will make the error E() small enough, and this heat transfer

coefficient is the objective. Except this appropriate heat transfer

coefficient, the error E() of other heat transfer coefficients will

be a negative or positive number. The first job of searching the

appropriate heat transfer coefficient is to ascertain an interval.

One endpoint of the interval can make the error E() to be anegative number, and other endpoint of the interval can make

the error E() to be a positive number. For every heat transfer

coefficient, the procedure will be applied to calculate the Ti ,

and then the error E() will be calculated by Eq. (11).

The computing procedures of searching an appropriate inter-

val using improved advanceretreat method are as follows:

(i) Select the value of initial searching step size s and the

value of initial surface heat transfer coefficient 3 (for the

first time, the value of3 is random; for the later times, the

value of3 is the surface heat transfer coefficient attained in

the former optimization). The value of3 is sent to the pro-

cedure of evaluating temperature and phase-transformation

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Fig. 1. SDL flow chart of improved advanceretreat method.

fields, and the computing values of some node tempera-

tures will be gained. Then the error E3 = E(3) betweencomputing values and objective values of temperature is

evaluated using Eq. (11). Let kk = 0, where kk is used totake count of searching times.

(ii) Let the surface heat transfer coefficient = 3 + s. Thevalue of is sent to the program of evaluating temperature

and phase-transformation fields, and the computing values

of some node temperatures will be attained. Then the error

E = E() between computing values and objective valuesof temperature is evaluated using Eq. (11). Let kk =kk +1.

(iii) IfE

E3 > 0, then compare Ewith E3. If

|E

| |E3|, then let s = s, and go to the procedure(ii). If |E| = |E3|, then go to the procedure (v).

(iv) IfE E30, then go to the procedure (v).(v) Let l = min{, 3}, r = max{, 3}, El = min{E, E3},

Er = min{E, E3}.(vi) Let =(1 +r)/2.0. The value of is sent to the program

of evaluating temperature and phase-transformation fields,

and the computing values of some node temperatures will

be gotten. Then the error E = E() between computingvalues and objective values of temperature is evaluated

using Eq. (11). If E E3 > 0, then let El = E, 1 = ,kk = kk 1. Otherwise let Er = E, r = , kk = kk 1.

(vii) Ifkk > 1, then go to the procedure (vi). Ifkk1, then stop,

and the appropriate interval attained is the interval [l, r].

The SDL flow chart of improved advanceretreat method is

shown in Fig. 1.

During the process of quenching, the surface heat transfer

coefficients have a wide range. If the general advanceretreat

method is used to search the appropriate interval, the searching

times will increase largely. Suppose the former surface tem-

perature is 400 C, for which the heat transfer coefficient is11,961W/m2 C; the current surface temperature is 430 C, forwhich heat transfer coefficient is 13,491 W/m2 C. The differ-ence between them is 1530 W/m2 C. If the step size of search-ing interval is 5W/m2 C, the times of searching the appropri-ate interval are 1530/5 + 1 = 307. Therefore the program willspend a lot of time to search the appropriate interval as it is

evaluating temperature and phase-transformation fields every

time.

If the improved advanceretreat method is used to search the

appropriate interval, the step size is not constant, and the later

step size is double the former step size. The interval length will

be (2n 2n1)s = 2n1s after n times of search. For this in-terval, the bisearch method is used to continue searching, and

the interval length of s will be found after n 1 times ofbisearch. The total times of searching the appropriate interval

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are 8 + (8 1) = 15 for the above-mentioned example. Com-paring the improved advanceretreat method with the general

advanceretreat method, it shows that the searching times of

improved advanceretreat method are reduced largely.

In order to search the appropriate interval, program of evalu-

ating temperature and phase-transformation fields is used time

after time in the searching procedure. The temperature andphase-transformation fields are not given new values after ev-

ery repeating use. It is just to find out the values of tempera-

ture field corresponding to the heat transfer coefficient which

value is the endpoint of interval after cooling for definite time.

So the values of temperature and phase-transformation fields

should be resumed to the original values after every repeat-

ing use. Then the error between computing values and objec-

tive values of temperature is evaluated using Eq. (11), and the

searching direction and the convergent state can be ascertained

according to the error. Only after the appropriate heat transfer

coefficient corresponding to certain objective values of temper-

ature is found out, the temperature and phase-transformation

fields are given new values, and then continue to search the heat

transfer coefficient corresponding to other objective values of

temperature.

3.3. Ascertaining the optimum value of heat transfer coefficient

Golden section method is one of the methods used for eval-

uating the minimum value in the interval of unimodal distribu-

tion function. The principle is that: for reducing the searching

interval which includes the optimum value, the function val-

ues are compared by accepting different tentative points, and

the approximative value of function minimum point will be

achieved when the searching interval is reduced to a prescriptive

value.

Firstly, two tentative points are selected in the interval [a, b].Suppose the left tentative point is

l = a + (1 )(b a),

the right tentative point is

r = a + (b a),

where are the solution of binomial equation 2 + 1 = 0.Secondly, evaluate two function values of left and right ten-

tative points according to Eq. (11)

l = E(l), r = E(r).

According to the characteristics of unimodal distribution func-

tion, if 1 r, there is nominimum value in the interval [a, l]. The interval [a, l] canbe neglected, and let a

=l. The new interval

[l, b

]can be

attained.

The computing procedures of ascertaining optimum value

of heat transfer coefficient using golden section method are as

follows:

(i) Suppose the appropriate interval searched by improved

advanceretreat method is [a, b], the prescriptive iterativeprecision is . The left and right tentative points are

l = a + (1 )(b a) and r = a + (b a),

where = (

5 1)/2.0. According to Eq. (11), the valuesof error are

l = E(l) and r = E(r).

(ii) Ifl

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Fig. 2. SDL flow chart of gold section method.

(2) Ifl >r, then the interval [a, l] will be neglected, andthe new left tentative point can be described by

l

=a

+(1

)(b

a

)

=l

+(1

)(b

l),

Because values are the solutions of binomial equation 2 + 1 = 0, the new left tentative point can also be described by

l = a + (1 )(b a) + (1 )(b a)= a + (1 2)(b a) = a + (b a) = r.

The new left tentative point is just the right tentative point of

old interval.

That is to say, if values are equal to the solutions of binomial

equation 2 +1=0, only one tentative point and its functionvalue need be calculated in every iterative calculation except

the first iterative calculation. So the workload of calculationcan be decreased largely, and the time of optimization also can

be reduced largely.

4. Example of evaluating surface heat transfer coefficient

A software is programmed using FEM technology,

advanceretreat method and golden section method. The soft-

ware can evaluate the temperature field, phase-transformation

field and surface heat transfer coefficient with the coupling

calculation of phase-transformation latent heat. In order to

verify the software, the temperature curves of several positions

in the quenching part are calculated using the FEM software

packageMARC and user subroutine programmed to evaluate

the phase transformation. During the process of evaluating, the

surface heat transfer coefficient is gained from Ref. [21], andthe

thermal physical parameters of material are shown in Fig. 3.The

size of simulation specimen is 40 200 mm, and the materialis P20 steel. The material is heated to 850 C. Suppose that thistemperature is kept for sufficient time to ensure the material

transforms into austenite fully. Then the specimen is quenched

in 20 C water. The FEM model includes 640 elements and 729nodes. The circumference of specimen has adiabatic boundary

condition, except the quenched end. The cooling curves at 0,

30, 60, 100, 150 and 200 mm from the quenched end are shown

in Fig. 4.

According to the temperature curves gained from MARC

software package, the temperature-dependent surface heat

transfer coefficient is calculated using the software that is pro-

grammed in the paper. The FEM model of evaluating surfaceheat transfer coefficient is same as the following FEM model.

During the process of evaluating, the required length of interval

is 5 W/m2 C, and the evaluating precision is 0.01 W/m2 C.The flow chart of calculating heat transfer coefficient is shown

in Fig. 5.

According the temperature curves in Fig. 4, the temperature-

dependent surface heat transfer coefficient is calculated. During

the process of calculating, the iterative times of searching the

appropriate interval using advanceretreat method are shown

in Fig. 6 (searching interval). After the appropriate interval is

found out, the iterative times of evaluating the heat transfer

coefficient in the appropriate interval using gold section method

are shown in Fig. 6 (optimization). The curves of heat transfer

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Fig. 3. Thermophysical properties of different phases in steel P20 as a function of temperature: (a) specific heat; (b) density; (c) thermal conductivity.

(Mmartensite; Bbainite; Ppearlite; Fferrite; Aaustenite).

Fig. 4. Cooling curves at different distances (mm) from quenched end.

coefficients and temperatures in the quenched end are shown

in Fig. 7.

During the calculation process, the phase-transformation vol-

ume and phase-transformation latent heat of every element

in every time interval can be calculated easily by FEM. The

temperature and phase-transformation volume of every ele-

ment are calculated with the coupling calculation of phase-

transformation latent heat. The heat transfer coefficients calcu-

lated using FEM and the heat transfer coefficients in Ref. [21]

are shown in Fig. 8.

Fig. 5. Flow chart of calculating heat transfer coefficient.

Fig. 8 shows that the heat transfer coefficients gained using

FEM and optimization method correspond well with the results

of Ref. [21]. So the method of evaluating heat transfer coeffi-

cients represented in the paper is accurate and dependable. The

method can calculate easily the heat transfer coefficients with

the coupling calculation of phase-transformation latent heat.

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Fig. 6. Iterative times of searching interval and heat transfer coefficient

optimization.

Fig. 7. Heat transfer coefficient and temperature of end (0 mm) gained by

FEM.

The cooling speed is very rapid in the process of bubble boil,

and the temperature difference can be more than 4050 C in aperiod of 0.1 s. But the cooling speed is slow in the process of

convection and film boil. The slowest temperature difference is

0.010.1 C in a period of 0.1 s. Due to these reasons, the tem-perature interval is very large in the process of evaluating heat

transfer coefficients, and the difference between the heat trans-

fer coefficients gained using IHCP method and the results of

Ref. [21] is large (Fig. 8(a)). In order to avoid this difference,

a self-adaptable time step is used, which computes the time

step according to the size of temperature difference, and can

improve the precision of evaluating heat transfer coefficients

(Fig. 8(b)).

5. Conclusions

From the temperature curves, the temperature-dependent sur-

face heat transfer coefficients of inverse heat conduction prob-

lem with the phase-transformation latent heat are evaluated us-

ing FEM, the improved advanceretreat method and the golden

section method. During the process of calculation, the phase-

transformation latent heat is coupled with temperature and

Fig. 8. Heat transfer coefficient gained by optimization and reference.

(a) Invariable time step. (b) Self-adaptable time step.

phase-transformation. The heat transfer coefficients gained us-

ing FEM and optimization method are compared with the re-

sults of reference. It shows that the precision of the method

given in this paper is satisfactory, and the convergence speed

of iteration is very rapid.

The temperature curves are obtained using FEM soft-

ware package and special user subroutine developed in

this paper. According to these temperature curves, the

temperature-dependent surface heat transfer coefficients can

be evaluated. If the temperature curves of several posi-

tions in the quenched part are measured by experiment, the

temperature-dependent surface heat transfer coefficients can

also be evaluated using the software programmed in this

paper.

Acknowledgment

This research work has been supported by the National Sci-

ence Foundation for Distinguished Young Scholars of China

50425517 and Foundation of New Century Excellent Talents

Plan of the Education Ministry of China.

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