Jtam.dvi


JOURNAL OF THEORETICAL

AND APPLIED MECHANICS

49, 4, pp. 1019-1028, Warsaw 2011

IDENTIFICATION OF ALLOY LATENT HEAT USING THE

DATA OF THERMAL AND DIFFERENTIAL ANALYSIS

Bohdan Mochnacki

Romuald Szopa

Czestochowa University of Technology, Częstochowa, Poland

e-mail: moch@imi.pcz.pl

Thermal and differential analysis (TDA) is often used as a tool for qu-
antitative estimation of solidification parameters of alloys (e.g. tempe-
ratures corresponding to the beginning and the end of phase change,
kinetics of latent heat evolution, etc). TDA system offers a possibility of
observation of the cooling (heating) rate, which means that the course
of derivative ∂T/∂t can be analyzed. In this paper, the identification of
alloy latent heat on the basis of additional information resulting from
TDA measurements is discussed. At the stage of numerical modelling,
the finite difference method (FDM) is used, the examples of computa-
tions are also shown.

Key words: solidification process, latent heat identification, numerical
methods

1. Introduction

A typical TDA system consists of the following elements (Fig.1):

• sample casting with thermocouple (1),

• amplifier and derivative creator (2),

• recording system and data presentation (3).

The geometry of typical sample casting is close to a cylindrical one (Moch-
nacki and Suchy, 1995), but in this place the other real shape of casting can
also be considered, and the thermocouples can be located at the optional set
of points from the domain considered. In Fig.2, an example of TDAmeasure-
ments is presented (cast iron) (Kapturkiewicz, 2003). One can see, the course
of TDA curves (cooling curve and its derivative) is a smooth one, and it is a



1020 B. Mochnacki, R. Szopa

Fig. 1. TDA system

Fig. 2. Example of measured TDA curves

result of using additional numerical procedures at the stage of creation of the
cooling curve and its derivative.
The position of characteristic points A, B, C,... allows one to predict dif-

ferent thermal andmechanical features of the castingmaterial, but these pro-
blems will not be discussed here. The aim of considerations presented here is
the identification of the alloy latent heat on the basis of information resulting
fromTDAmeasurements.

2. Mathematical description of casting solidification

The following energy equation is considered (Chang et al., 1992; Kapturkie-
wicz, 2003; Majchrzak et al., 2008; Mochnacki and Suchy, 1995)

c(T)
∂T(x,t)

∂t
=∇· [λ(T)∇T(x,t)]+L

∂fS(x,t)

∂t
(2.1)

where λ(T) is the thermal conductivity, c(T) is the heat capacity, L is the
volumetric latent heat, fS is the volumetric solid state fraction of a metal,



Identification of alloy latent heat... 1021

T(x,t), x, t denote temperature, spatial co-ordinates and time, respective-
ly. One can see that only conductional heat transfer is taken into account
(the convection is neglected) – it results from the geometrical features and di-
mensions of sample casting (Fig.2). In the case of more complex thick-walled
castings and considerable rates of pouringmoulds (e.g. continuous casting), it
is possible to apply equation (2.1), but in the place of real thermal conducti-
vity of a molten metal, the so-called effective thermal conductivity should be
introduced (see: Mochnacki and Suchy, 1995).

Assuming the knowledge of function fS = fS(T) for the interval of tem-
peratures [TS,TL] corresponding to the mushy zone sub-domain, one has

∂fS(x,t)

∂t
= f ′S(T)

∂T(x,t)

∂t
(2.2)

Additionally, for T > TL : fS = 0 and for T < TS : fS = 1, it results
from the definition of the function discussed. Introducing (2.2) into (2.1) one
obtains

[c(T)−Lf ′S]
∂T(x,t)

∂t
=∇· [λ(T)∇T(x,t)] (2.3)

or

C(T)
∂T(x,t)

∂t
=∇· [λ(T)∇T(x,t)] (2.4)

where C(T) is the substitute thermal capacity (STC) of the alloy (Kaptur-
kiewicz, 2003; Majchrzak et al., 2008; Mochnacki and Suchy, 1995). The soli-
dification model based on equation (2.4) is called “a one-domain approach”,
because the same equation describes the thermal processes proceeding in the
whole, conventionally homogeneous casting domain. One can see that for the
molten metal and solid state, the derivative dfS/dT = 0 and the substitute
thermal capacity directly correspond to the volumetric specific heats of these
sub-domains.

One of the most popular approximation of fS(T) is the function of the
form

fS(T)=
(TL−T(x,t)

TL−TS

)n

for T ∈ [TS,TL], n> 0 (2.5)

Formula (2.5) assures the compliance with requirements fS(TS) = 1,
fS(TL) = 0. Let us assume the linear form of function (2.5). Then, for con-
stant values of heat capacities cS, cL (liquid and solid states) one obtains an



1022 B. Mochnacki, R. Szopa

approximation of C(T) in the form of a stair-case function (see: definition of
STC – Eq. (2.3))

C(T)=



















cL for T >TL

cP +
L

TL−TS
for TS ¬T ¬TL

cS for T <TS

(2.6)

where cP = 0.5(cL + cS). The parameter L/(TL −TS) is called the spectral
latent heat. In this place, more complex formulas resulting from the general
form of equation (2.5) (for other values of the exponent n) can be considered,
but the introduction of well known approximation (2.6) leads to a simple

model determining the sensitivity coefficients W
f
i at the stage of solution of

the inverse problem (see next Section). A similar formula determining the
changes of casting thermal conductivity is assumed, which means

λ(T)=















λL for T >TL

λP for TS ¬T ¬TL

λS for T <TS

(2.7)

Thealloy solidificationandcoolingprocessproceed inthe interior of themould.
The transient heat transfer in this domain is described by the typical Fourier
equation

cM(T)
∂TM(x,t)

∂t
=∇· [λM(T)∇TM(x,t)] (2.8)

where λM(T) is the mould thermal conductivity, cM(T) is the heat capacity
of the mould. On the contact surface between the casting and mould, the
condition of ideal thermal contact is assumed (continuity of temperature and
heat fluxes)

−λ(T)
∂T(x,t)

∂n
=−λM(T)

∂TM(x,t)

∂n
T(x,t)=TM(x,t)

(2.9)

where ∂/∂n denotes a normal derivative.
On the external surface of the system, the boundary condition in a general

form

Φ
[

TM(x,t),
∂TM(x,t)

∂n

]

=0 (2.10)

is given.
The initial condition

t=0 : T(x,0)=T0(x), TM(x,0)=TM0(x) (2.11)

is also known.



Identification of alloy latent heat... 1023

3. Solution to the inverse problem

To solve the inverse problem discussed, the least squares criterion of the fol-
lowing form is applied (Kurpisz andNowak, 1995;Majchrzak andMochnacki,
2007; Majchrzak et al., 2007)

S(L)=
1

MF

M
∑

i=1

F
∑

f=1

(U
f
i −U

f
di
)2 =min (3.1)

where

U
f
i =
(∂T

∂t

)f

i
U
f
di
=
(∂T

∂t

)f

di
(3.2)

are the measured and estimated cooling rates, M is the number of sensors,
F is the number of time levels for which the function U is determined. The
estimated cooling rates are here obtained from the solution to the direct pro-
blem.
The least squares criterion used here can be treated as a special case of

the criterion

S(L)=
w

MF

M
∑

i=1

F
∑

f=1

(T
f
ri−T

f
rdi
)2+
1−w

MF

M
∑

i=1

F
∑

f=1

(U
f
ri−U

f
rdi
)2 =min (3.3)

where T
f
rdi
and T

f
ri = Tr(xi, t

f) are the measured and estimated dimension-

less temperatures, U
f
ri, U

f
rdi
are the measured and estimated dimensionless

cooling (heating) rates, w is a tapering function w ∈ [0,1]. Effectiveness of
this generalized approach will be a topic of the future research.
Using the Taylor formula, one has

U
f
i =(U

f
i )
k+(W

f
i )
k(Lk+1−Lk) (3.4)

where

(W
f
i )
k =
∂U
f
i

∂L

∣

∣

∣

∣

L=Lk
(3.5)

The necessary condition of optimum, aftermathematical manipulations, leads
to the formula

Lk+1 =Lk+

M
∑

i=1

F
∑

f=1

[U
f
di
− (U

f
i )
k](W

f
i )
k

M
∑

i=1

F
∑

f=1

[(W
f
i )
k]2

k=0,1,2, . . . ,K (3.6)



1024 B. Mochnacki, R. Szopa

where k is the number of iterations and L0 is the initial, arbitrarily assumed
value of L.

To determine the sensitivity coefficients appearing in the algorithm of the
inverse problem solution, one can use the method of differentiation of the
governing equations with respect to the unknown parameter (the direct ap-
proach of sensitivity analysis (Dems, 1999; Kleiber, 1997; Majchrzak andKa-
łuża, 2008)). In the case of the problem discussed, the sensitivity model ob-
tained in this way is rather complicated, though the approximation of C(T)
and λ(T) by the stair-case functions leads to essential simplifications of this
model.Amore practical approach consists in the application of differential qu-
otients (Szopa, 2006). The numerical solution of the basic model (Eqs. (2.4),
(2.8)-(2.11)) allows one to directly determine the temporal and local values
of the cooling (heating) rates. So, one can find the solutions to the basic pro-
blem corresponding to successive values of Lk and Lk+∆Lk, where ∆Lk is a
small increase of latent heat, and next to apply the differential quotients as an
approximation of the local and temporary derivatives ∂U/∂L (see Eq. (3.4)).

4. Example of computations

The symmetrical fragment of casting (steel frame) shown in Fig.3 is con-
sidered. The casting is produced in a typical sand mould. Thermophysical
parameters of the casting material are the following: cS = 4.875MJ/m

3K,

Fig. 3. Domain considered

cL = 5.9MJ/m
3K, cP = 5.3875MJ/m

3K, L = 1984.5MJ/m3 (this value is
identified), TS = 1470

◦C, TL = 1505
◦C, λS = 35W/mK, λL = 20W/mK,



Identification of alloy latent heat... 1025

λP = 27.5W/mK, while for the mould sub-domain cM = 1.75MJ/m
3K,

λM =1W/mK.The initial temperature of themoltenmetal equals to 1550
◦C,

initial temperature of the mould TM0 =20
◦C.

Both the basic and sensitivity problem have been solved using the explicit
scheme of FDM for non-linear parabolic equations. Details concerning this
approach to simulation of the solidificationproblemcanbe found inMochnacki
andSuchy (1995). The casting-moulddomainhas beendiscretized bya regular
mesh containing 900 nodes, with the time step equal to 0.1s.

The solution to the direct problem corresponding to the above presented
input data at the point marked in Fig.3 is shown in Fig.4a (cooling curves)
and Fig.4b (cooling rate) – the results obtained were treated as the results of
“measurements”.

Fig. 4. (a)Cooling curve; (b) undisturbed cooling rates

The results of latent heat identification using the iteration procedure re-
sulting from Eq. (3.6) for undisturbed input data are presented in Fig.5.

Fig. 5. Identification of L – exact data



1026 B. Mochnacki, R. Szopa

TDA system creates, as a rule, the time derivative in form of a smooth
curve (see Fig.2b), but the inverse problem discussed has been also solved for
the case of adisturbedcourse of this function.The randomlydisturbedcooling
rate at the point corresponding to the “sensor” position is shown in Fig.6.

Fig. 6. Disturbed cooling rate

The identification process for the disturbed input data is presented in
Fig.7.

Fig. 7. Identification of L – disturbed data

5. Final remarks

The concept of cooling rate application to the solution of inverse problems
results from the capability of TDA equipment. In a such case, the typical le-
ast squares criterion characteristic for gradientmethods is connected with the
differences betweenmeasured and calculated cooling or heating rates. The cri-
terion assuring the optimal value of identified parameter can be generalized by



Identification of alloy latent heat... 1027

the introduction of additional information concerning the temperature history
(then the dimensionless temperatures and derivatives should be considered).
This approach will be a subject of further research. It should be pointed out
that the iteration procedure resulting fromapplication of the gradientmethod
is quickly convergent both in the case of undisturbedanddisturbed inputdata
– even when the distance between the starting point and the real value of the
unknown parameter is considerable.

Acknowledgement

This work is a part of Research Project BS-1-105-301/99/S.

References

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id/solid and eutectoid transformation in spherical graphite cast iron,Metallur-
gical Transactions A, 23A, 1333

2. DemsK., Rousselet B., 1999, Sensitivity analysis for transient heat conduc-
tion in a solid body, Structural Optimization, 17, 36-45

3. Kapturkiewicz W., 2003, Modelling of Cast Iron Crystallization, Cracow:
AKAPIT

4. Kleiber M., 1997,Parameter Sensitivity, J.Wiley & Sons Ltd., Chichester

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ston: ComputationalMechanics Publications

6. Majchrzak E., Kałuża G., 2008, Explicit and implicit approach of sen-
sitivity analysis in numerical modelling of solidification, Archives of Foundry
Engineering, 8, 1, 187-192

7. Majchrzak E., Mochnacki B., 2007, Identification of thermal properties of
the system casting – mould,Materials Science Forum, 539-543, 2491-2496

8. Majchrzak E., Mochnacki B., Suchy J.S., 2007, Kinetics of casting soli-
dification – an inverse approach, Scientific Research of the Institute of Mathe-
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stochowa, 1, 6, 169-178

9. Majchrzak E., Mochnacki B., Suchy J.S., 2008, Identification of substi-
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1028 B. Mochnacki, R. Szopa

10. Mochnacki B., Suchy J.S., 1995, Numerical Methods in Computations of
Foundry Processes, PFTA, Cracow

11. Szopa R., 2006, Sensitivity Analysis and Inverse Problems in the Thermal
Theory of FoundryProcesses, Publ. ofCzest.Univ. ofTechnology,Częstochowa

Identyfikacja utajonego ciepła krzepnięcia stopów na podstawie wyników

analizy termiczno-derywacyjnej

Streszczenie

Analiza termiczno-derywacyjna (ATD) stanowi efektywne narzędzie ilościowej
ocenyparametrówkrzepnięcia stopówodlewniczych (np. temperaturypoczątku i koń-
ca krzepnięcia, kinetyki wydzielania się ciepła przemiany, itd.). UrządzenieATDdaje
możliwość obserwacji krzywych stygnięcia (nagrzewania) oraz szybkości tych proce-
sów, czylimożna również obserwowaćzmianypochodnej temperaturywzględemczasu
(∂T/∂t).W niniejszej pracy rozpatruje się zadanie dotyczące identyfikacji utajonego
ciepła krzepnięcia stopów na podstawie informacji wynikających z pomiarów realizo-
wanych urządzeniem ATD. Na etapie obliczeń numerycznych wykorzystano metodę
różnic skończonychdla nieliniowychzadańnieustalonegoprzewodzenia ciepła.Wkoń-
cowej części pracy przedstawiono wyniki identyfikacji utajonego ciepła krzepnięcia
staliwa.

Manuscript received November 17, 2010; accepted for print December 16, 2010