Application of a New Method for Radiative Heat Transfer to Flat GlassTempering

Norbert Siedow and Teodor Grosan

Fraunhofer Institute for Industrial Mathematics, ITWM, Kaiserslautern, Germany

Dominique Lochegniesw and Eric Romero

Laboratory of Industrial and Human Automation, Mechanics and Computer Science, LAMIH, UMR CNRS 8530,Valenciennes, France

Temperature distributions within semi-transparent materialslike glass strongly determine their behavior. It is fundamentalto take radiation into account to determine exact temperaturedistributions. The Rosseland Approximation for radiation isusually an appropriate method, but is only valid with an opti-cally thick glass. In the present paper, we propose an improvedapproximation that is both efficient and sufficiently accurateeven for the semi-transparent region. This new radiation modelis used to determine the temperature along the flat glass thick-ness during tempering. The impact of temperature evaluation onresidual stresses is discussed.

I. Introduction

THE temperature of glass and its evolution over time are ofprimary importance in the success of glass forming and

glass tempering. In glass forming, it determines the correct glassflow and the final glass topology. In glass tempering, with theresidual stresses induced inside the glass, the temperatures are atthe origin of the glass mechanical behavior. As in industry,modeling of glass forming and tempering needs a correct eval-uation of the temperatures inside the glass.

In addition to glass conductivity being dependent on temper-ature and the convective heat transfer between the glass andsurroundings, the internal radiation of glass is also importantfor the evaluation of the temperature.

The intensified interest in the treatment of radiative heattransfer in semi-transparent media has encouraged the develop-ment of a number of methods devoted to this subject. Thesemethods can be divided into three types: methods based on aspherical harmonic expansion (PN approximations), methodsusing a finite discretization of the direction (discrete ordinatemethod (DOM), ray-tracing method), or diffusion approxi-mations (the Rosseland Approximation). A review of the gov-erning equations for radiative heat transfer and computationalmethods can be found in Siegel.1 The first two types of methodsare generally CPU time consuming, so for industrial purposes,the Rosseland Approximation is often used and is available ingeneral commercial software packages.

The current research into temperature computations in glassconcerns first of all the development of alternative methods is-sued from the three types, the main objective being to reduce theCPU time and maintain a correct accuracy level. Algebraic ab-

straction of the ray-tracing method2 and preconditioningtechnique for the radiative heat transfer equations using P1

approximations3 are examples of recent investigations.The second aspect covered by the current research is the im-

pact of radiative heat transfer and temperature level on glassprocessing. As radiative effects are more effective at high tem-peratures, the glass forming is the first concerned with a tem-perature range of [800, 1400 K]. Hohne et al.4 analyzed the heattransfer at the glass/mold throughout the interface and the effectof radiation on it, using both experiments in laboratory andnumerical thermal modeling. Van der Linden et al.5 used theirheat transfer equations to analyze in which step of the press-and-blow process radiation has to be taken into account, andwhich is the most suitable radiative heat transfer model. Potze6

developed his alternative heat transfer method for the temper-ature analysis of axisymmetric quartz tubes. In all these cases,because of the long CPU time of the heat transfer methods, nodirect coupling with forming modeling is realized, and analysisof the forming is only based on thermal analysis.

In our present industrial tempering process, the temperaturesare lower around the transition range and in the [300 K, 950 K]range. In the modeling of tempering with commercial softwarepackages, the current approaches are found:

(1) not taking radiation into account7,8 or including it byintroducing the concept of radiative conductivity.9 In this way,the phonon thermal conductivity of glass is increased with theradiative conductivity,

(2) using the Rosseland Approximation10 or extending it.11

Here, the idea is to split the radiative flux into parts, with a de-composition of the total emissivity of glass into a volume emis-sivity defined for the spectral field where glass is semi-transparent, and a surface emissivity defined for the spectralfield where glass is opaque.

More recently, Khaleel et al.12 were the first to apply the CPUtime-consuming DOM to estimate the temperatures in the sim-ulation of glass forming and tempering. Coupling thermal res-olution with radiation and forming/tempering resolution ispossible because the authors are developing two dedicated soft-ware packages.

In this paper, we present a new improved approximation to esti-mate, with both accuracy and reduced CPU time, the temperatureinside glass. This new method is used to estimate the transient andresidual tempering stresses in flat glass and is then widely discussed.

II. Mathematical Methods for Radiation

The mathematical model of radiative heat transfer in one di-mensional (1D) and a semi-transparent material like glass isgiven by13,14

cpðTÞrgqTqtðx; tÞ ¼ q

qxkhðTÞ

qTqxðx; tÞ

� �� qqqxðxÞ (1)

2181

JournalJ. Am. Ceram. Soc., 88 [8] 2181–2187 (2005)

DOI: 10.1111/j.1551-2916.2005.00402.x

G. W. Scherer—contributing editor

This work was Supported by the Fraunhofer Institute for Industrial Mathematics, theLaboratory of Industrial and Human Automation, Mechanics and Computer Science, theCNRS and the Ministere de l’Education Nationale et de la Recherche.

wAuthor to whom all correspondence should be addressed. e-mail: Dominique.Lochegnies@univ-valenciennes.fr

Manuscript No. 20127. Received September 9, 2004; approved February 7, 2005.

for 0 � x � ‘ and 0 � t � t� with boundary conditions

khðTÞqTqxð0; tÞ ¼ hðTð0; tÞ � TaÞ þ ep

�Zopaque

ðBaðTð0Þ; lÞ � BaðTa; lÞÞdl

(2)

�khðTÞqTqxð‘; tÞ ¼ hðTð‘; tÞ � TaÞ þ ep

�Zopaque

ðBaðTð‘Þ; lÞ � BaðTa; lÞÞdl

(3)

where x is the space position in meters and c is the length of thedomain. t denotes the current time in s, t� is the final time,T(x, t) is the temperature in K, and q(x) is the radiative flux inW/m2. At temperature T, cp(T) is the specific heat in J/kg/K,kh(T) is the thermal conductivity inW/m/K, and rg is the densityof glass in kg/m3. l denotes the wavelength in mm and Ba(T,l)the Planck function of air in J/m2. e is the mean hemisphericsurface emissivity. p is the constant number pi. ng is the refrac-tive index of the glass. h is the convective heat transfer coefficientin W/m2/K, and Ta is the temperature of the surroundings in K.

The initial condition at time t5 0 s is given by the equationTðx; 0Þ ¼ T0; 0 � x � ‘ where T0 is the initial uniform glasstemperature in K.

In Eq. (1), the heat loss because of radiation is described bythe divergence of the radiative flux

qðxÞ ¼ 2pZ 10

Z 1

�1mIðx; m; lÞ dm dl (4)

where m is the direction. I(x,m,l) is the spectral radiative inten-sity in J/m2 that needs the solution of the radiative transferequation14,15

mqIqxðx; m; lÞ þ kðlÞIðx; m; lÞ ¼ kðlÞBgðTðx; tÞ; lÞ (5)

for 0rxrc and with specular reflecting boundary conditions

Iðxg; m; lÞ ¼ rðmÞIðxg;�m; lÞ þ ð1� rðmÞÞBgðTa; lÞxg ¼ 0 form > 0 and xg ¼ ‘ for m < 0:

(6)

k(l) is the absorption coefficient and r(m) is the reflectivity of theglass. Bg(T,l) denotes the Planck function for glass.

Using Eq. (5), the radiative flux (Eq. (4)) can be calculated as

qqqxðxÞ ¼2p

Z 10

Z 1

�1kðlÞðBgðTðx; tÞ; lÞÞ

� Iðx; m; lÞdm dl(7)

Even for one space variable, the problem is 4D (space, direc-tion, wavelength, time) and highly non-linear. This is why ef-fective numerical methods are important for solving this systemof equations.

In the following section, we briefly describe three differentnumerical methods: a method based on the formal solution,which we assume to be exact, the Rosseland Approximation,which is widely used in industrial practice, and finally we developa newmethod based on an approximation of the formal solution.

First of all, the absorption coefficient k(l) is approximated bya piecewise constant function:

kðlÞ ¼ kk ¼ const (8)

with l 2 ½lk; lkþ1� and k ¼ 1; 2; . . . ;Mk

Introducing the notations

Ikðx; mÞ ¼Z lkþ1

lk

Iðx; m; lÞ dl (9)

and

BkgðTðxÞÞ ¼

Z lkþ1

lk

BgðTðxÞ; lÞ dl;

BkgðTaÞ ¼ n2gB

kaðTaÞ

(10)

instead of Eq. (5), one has to solveMk equations—one equationfor each wavelength band.

(1) The Exact Numerical Solution

Using the linearity of the radiative transfer equation, the ra-diative intensity of the kth band [lk,lk11] is divided into twoparts:

Ikðx; mÞ ¼ Ik1 ðx; mÞ þ Ik2 ðx; mÞ (11)

where Ik1 ðx; mÞis the solution of the homogeneous equationwith inhomogeneous (reflecting) boundary conditions andIk2 ðx; mÞ is the solution of the inhomogeneous equation withhomogeneous (Dirichlet) boundary conditions. Because of thesymmetry of the problem with respect to x ¼ ‘=2 for the firstproblem, for m40,

Ik1 ðx; mÞ ¼e�kkdðx;mÞ

1� re�kkdð‘;mÞ

frI2ð0;�mÞ þ ð1� rðmÞÞn2gBkaðTaÞg

(12)

and for mo0,

Ik1 ðx; mÞ ¼e�kkdðx;mÞ

1� re�kkdð0;mÞ

frI2ð‘;�mÞ þ ð1� rðmÞÞn2gBkaðTaÞg

(13)

Ik2 ðx; mÞ can be calculated numerically using the formal solu-tion15

Ik2 ðx; mÞ ¼ kk

Z dðx;mÞ

0

BkgðTðx� smÞÞe�kks ds (14)

The distance d(x,m) in m is dðx; mÞ ¼ x=m for m40 anddðx; mÞ ¼ x�‘

m for mo0.

(2) The Rosseland Approximation

The Rosseland Approximation15 is widely used in industrialpractice and describes the radiation as a correction of the heatconductivity:

cprqTqtðx; tÞ ¼ q

qx

kh þ

4p3

Z 10

1

kðlÞdBg

dTðTðxÞ; lÞdl

!

� qTqxðx; tÞ

!(15)

This simple kind of diffusion approximation can be used in allcommercial software packages that are able to solve heat trans-fer problems. This method is very fast but it is only accurate foroptically thick glass.

2182 Journal of the American Ceramic Society—Siedow et al. Vol. 88, No. 8

(3) The New Method Developed by ITWM

Using the Taylor series for Bg(T(x�sm)), the formal solution(Eq. (14)) for Ik2 ðx; mÞ is approximated by

Ik2 ðx; mÞ � BkgðTðxÞÞ 1� e�kkdðx;mÞ

� �� mkk

1� ð1þ kkdðx; mÞÞe�kkdðx;mÞh i

�dBk

g

dTðTðxÞÞ dT

dxðxÞ

(16)

and Ik1 ðx; mÞ is given by Eqs. (12) and (13).The divergence of the radiative flux (Eq. (7)) becomes

qqqxðxÞ � 2p

XMk

k¼1kk 2Bk

gðTðxÞÞ �Z 1

�1Ik2 ðx; mÞdm

�

�Z 1

�1Ik1 ðx; mÞdm

� (17)

The first two terms on the right-hand side describe the radiationwith homogeneous boundary conditions whereas the third termis the correction because of the reflection at the boundaries. Theintegrals are numerically calculated using a discrete ordinateapproximation.

III. Application to Flat Glass Tempering

Thermal glass tempering consists of first heating of the glass at atemperature superior to the transition temperature Tg followedby a very rapid cooling. This thermal treatment gives a bettermechanical and thermal strengthening to the glass by way of theresidual stresses generated along the thickness. The residualstress state is in tension inside the glass and in compressionnear the surfaces.

The present analysis concerns the calculation of the evolutionof the temperature in the thickness (‘5 6.10� 10�3 m) of soda-lime silicate flat glass (Table I) in order to estimate the temperingtransient and residual stresses.

(1) Calculation of Temperatures

The temperatures in the glass thickness are calculated for fourmodels:

(1) without radiation (denoted No Rad),(2) with the DOM of Section II(1) with Mk 5 30 wave-

length bands (denoted Exact solution),

(3) with the Rosseland Approximation of Section II(2) (de-noted Ross App), and

(4) with the new approximation of Section II(3) (denotedITWM).

For the computations, the evolution of the specific heat cp(T)and the conductivity kh(T) function of the temperature are rep-resented in Table II.16 For the radiation properties, the absorp-tion coefficient is represented in Fig. 1. The refractive index ofthe glass is equal to 1.46, and the value of the refractive index ofthe surrounding is 1.17,18 The values of the heat transfer coef-ficient h, the ambient temperature Ta of the surroundings, andthe initial glass temperature T0 are 280.5 W/m2/K, 298.15 and948.15 K, respectively. These values are representative of theindustrial tempering practice.

(2) Calculation of Stresses

The behavior of the glass quickly changes in the vicinity of theglass transition temperature Tg between the solid and the liquidglass states. A numerical model for the calculation of transientand residual stresses in glass during the tempering process, in-cluding both structural relaxation and viscous stress relaxation,has been developed by Narayanaswamy.19 Details are given inthe Appendix A.

Using Narayanaswamy’s model (Eqs. (A-8) to (A-11)), tran-sient and residual stresses are computed along the glass thick-ness. All the glass mechanical properties are given in Tables IIand III.

(3) Numerical Modeling

The numerical modeling is based on symmetrical tempering, andthe analysis is limited to the half-thickness.

A decoupled calculation is performed. Firstly, the tempera-tures along the glass thickness are computed during the temper-ing duration (for each of the four thermal models described inSection II). Secondly, the non-linear mechanical problem issolved using the previous temperature evolutions.

To calculate stresses, a 1D finite-element model is used totake Narayanaswamy’s model into account. Eight points arepositioned along the glass thickness (Fig. 2). The coordinates(� 10�3 in m) are 0.00, 0.34, 0.83, 1.54, 2.11, 2.53, 2.82, and3.05, respectively. This discrete model is finer near the surfaceand near the mid-plane. It is a good compromise between thetime of calculation, the size of result files, and the accuracy ofthe residual stresses. In fact, using a finer grid with 50 points

Fig. 1. Absorption coefficient on the spectral zone of a float glass.17

Table I. Chemical Composition of Soda-Lime Silicate Glass(%)

Si O2 Ca O Na2 O Mg O Fe2 O3 Others oxides

71.05 9.90 13.60 4.00 0.09 1.35

Table II. Thermal and Mechanical Properties

Tg (K) 824.4kh (W/m/K) 0.97518.58� 10�4� (T�273.15)cp (J/kg/C) ToTg: 89310.4�T2–1.8� 107/T 2;

T4Tg: 143316.5� 10�3Tag (K

�1) 9� 10�6

al (K�1) 25� 10�6

rg (kg/m3) 2550

n 0.22E (Pa) ToTg: 7.1� 1010–4.916� 106T;

T4Tg: 9.67� 1010–3.611� 107T

Table III. Typical Viscoelastic Properties

Wi Ti (s) Properties Values

0.0427 19 H/R (K) 150 000/1.98720.0596 291.9 X 0.50.0877 1843 Tr (K) 763.660.2454 11800 Ke/Kg 0.30.2901 494900.2498 171 700

August 2005 Application of a New Method for Radiative Heat Transfer 2183

along the thickness multiplies the CPU time by a factor 10 for arelative variation of the surface and mid-plane stresses lowerthan 1%.

The temperature evolution for the four thermal modelsis calculated using a much finer grid to ensure convergence for300 s of the tempering of the glass. Between two consecutivepoints of the 1D finite-element model in Fig. 2, 10 points arelocated in the thermal grid. Stresses are then computed at eachof the eight points of the 1D finite-element model using thetemperatures obtained at the same location in the thermal grid.

IV. Results and Discussions

Results and discussions concern the temperatures and stressesalong the glass thickness computed from the four models ofSection III(2) (Exact solution, No Rad, Ross App, andITWM). The analysis is divided into two parts: the first oneconcerns the analysis of the transient temperatures inside glass(Section IV(1)), and the second one concerns the analysis of thetransient and residual stresses (Section IV(2)). In the followingsection, the results of No Rad, Ross App, and ITWM modelsare systematically compared with the Exact solution ones interms of temperature and stresses.

(1) Temperature Results in the Glass Thickness

The first temperature evolution in Fig. 3 is given for the Exactsolution at two specific locations in the glass thickness: the mid-plane (point 1 in Fig. 2) and the surface (point 8 in Fig. 2). Asignificant difference between the surface and the mid-plane isobserved during the first 100 s with a maximum value of 140 K.After 100 s, the difference between the mid-plane and the surfaceis lower than 15 K. At this moment of tempering, the radiationis no longer effective.

The comparison between the three models (No Rad, RossApp, and ITWM) and the Exact solution is first expressed by thetemperature difference in the mid-plane (Fig. 4). In the first 30 sof the tempering, the maximum differences with the Exact so-lution are 2, 7, and 27 K, respectively, for ITWM, Ross App,

and No Rad models. Afterwards, the ITWM model gives theexact value of the temperature in the mid-plane.

A similar temperature comparison is then carried out on theglass surface (Fig. 5). In this case, the lowest maximum differ-ence with the Exact solution is also obtained with the ITWMmodel (4 K) in comparison with the Ross App model (9 K) andNo Rad model (24 K). Afterwards, the temperature evolutioncomputed from the ITWM model is similar to the Exact solu-tion one.

As the temperature difference between the glass surface andthe mid-plane is one of the main factors in the creation of thetempering residual stresses, the first difference Tsurf�Tmid be-tween surface and mid-plane temperatures is calculated for thefour models. The evolution of the difference between(Tsurf�Tmid)Exact and (Tsurf�Tmid)Case, where Case is one ofthe three models No Rad, Ross App, and ITWM models, isgiven in Fig. 6. The minimum difference with the Exact solutionis given by the ITWMmodel with a maximum difference of 1 Kin the first 20 s and almost zero after 50 s of tempering. In thefirst 100 s of the tempering, when radiation is most significant(Fig. 3), the Rosseland Approximation produces the maximumerror, up to 4 K, in the difference (Tsurf�Tmid)RossApp�(Tsurf�Tmid)Exact.

Table IV shows the CPU time (computations have been madeon a Pentium II, 400 MHz, Ionfos, Kaiserslautern, Germany)for the four thermal models. The ITWM model is about 100times faster than the Exact solution but 10 times slower than theNo Rad model, which ignores the radiation. Taking both accu-racy and related computer time into account, the ITWM modelis a practical way for solving the radiative heat transfer prob-lems inside glass.

(2) Stress Results in the Glass Thickness

The following section concerns the analysis of the transient andresidual tempering stresses along the glass thickness.

Fig. 3. Evolution over time of the temperatures T in the mid-plane andon the glass surface for the Exact solution and of the difference DT be-tween mid-plane and surface temperatures.

Fig. 4. Evolution over time of the mid-plane temperature differencebetween the Exact solution and Case (where Case is one of No Rad,Ross App, or ITWM models).

Fig. 2. One-dimensional finite-element model in the half-thickness ofthe glass plate.

Fig. 5. Evolution over time of the surface temperature difference be-tween the Exact solution and Case (where Case is one of No Rad, RossApp, or ITWM models).

2184 Journal of the American Ceramic Society—Siedow et al. Vol. 88, No. 8

Figure 7 gives the evolution over time of the tempering stress-es in the mid-plane (point 1 in Fig. 2) and on the glass surface(point 8 in Fig. 2) for the reference Exact solution. The inversionfrom compression to tensile states on the glass surface takes3.9 s. During this period, the maximum tensile stress on the glasssurface is equal to 9.7 MPa at 2.7 s of tempering. When the glasssurface is in the compression state, the main increase in the sur-face stresses occurs less than 100 s after the tempering hasbegun. Finally, the residual stresses are tensile ones equalto 57.7 MPa in the mid-plane and compression ones equal to122.1 MPa on the glass surface. The absolute ratio ssurf/smid is2.1 for the Exact solution.

In the mid-plane, the difference in transient and residualstresses between the three models (No Rad, Ross App, andITWM) and the Exact solution is presented in Fig. 8. Through-out the tempering duration, the ITWM model gives the closestresult for the transient and residual stresses; the difference fromthe Exact solution is always lower than 1.4 MPa. For the RossApp model, if the error is lower than 1.1 MPa in the first 10 s ofthe tempering, it reaches 5.1 MPa afterwards and remains con-stant until the end of the tempering. The No Rad model presentsa maximum stress error of 4.8 MPa at 14.0 s after the temperinghas begun but this error is reduced afterwards; a 0.8 MPa dif-ference with the Exact solution is obtained on the estimation ofthe residual tensile stresses in the mid-plane.

The glass surface is analyzed in the same way (Fig. 9). Forthe transient stresses in the first 30 s of the tempering, theNo Rad model produces the most significant error (7.2 MPa),while the ITWM model error is lower than 0.5 MPa. Consid-ering residual stresses, the ITWM prediction error is 1.0 Mpa,while 2.9 MPa is obtained for the No Rad model. The error withthe Ross App model on the surface compression stress is higher,equal to 8.9 MPa.

V. Conclusions

A newmathematical model is proposed to calculate the radiativeheat transfer in semi-transparent materials. 1D analysis of flatglass tempering is performed using four different methods tocompute the transient temperature inside glass: computationswithout taking radiation into account, exact solution with thediscrete ordinate model with Mk 5 30 wavelength bands, theclassical Rosseland Approximation and the new approximation.

For the temperature computations along the glass thickness,it turns out that the results obtained with the new method arethe closest to the Exact solution without increasing the CPU-time too much. The major error in the estimation of the thermalevolution in the glass thickness occurs when radiation is ignored.For the difference in temperatures between the glass surface andthe mid-plane, an important decision factor in the tempering,the maximum error is given by the Rosseland Approximation.

In terms of transient and residual tempering stresses, the clas-sification of the models is: the ITWM model is the best solutionto estimate the stresses along the glass thickness during thetempering with 0.8% difference with the Exact solution. Whenradiative heat transfer is ignored, the error goes up to 2.4%. TheRosseland Approximation, which is the most widely used in the

Fig. 6. Evolution over time of the surface and mid-plane temperaturedifference for the Exact solution compared with Case (where Case is oneof No Rad, Ross App, or ITWM models).

Fig. 7. Evolution over time of the stress evolution function of time forthe Exact solution in the mid-plane and on the glass surface.

Fig. 8. Evolution over time of the stress difference in the mid-planebetween the Exact solution and Case (where Case is one of No Rad,Ross App, or ITWM models).

Table IV. CPU Time in Seconds for the Four Models (SectionIII(2))

Exact solution model 806.50No Rad model 0.65Ross App model 1.65ITWM model 7.50

Fig. 9. Evolution over time of the stress difference on the glass surfacebetween the Exact solution and Case (where Case is one of No Rad,Ross App, or ITWM models).

August 2005 Application of a New Method for Radiative Heat Transfer 2185

modeling of industrial glass tempering, produces the maximumaverage error for transient stresses around 7.3%.

With reduced CPU time, this new method for radiative heattransfer is a solution for the calculation at a high level of accu-racy of residual stresses and especially transient stresses insidetempered glass. The optimization of glass tempering is based onthe control and limitation of these transient stresses that causeglass breakage during the processing.

Appendix A

Stress relaxation of glass is well modeled by the viscoelasticgeneralized Maxwell law using shear G(t) and bulk K(t) modulifunctions of time t, described with instantaneous (Gg,Kg) anddiffered moduli (Ge,Ke) expanded into Prony’s series of sixterms20:

GðtÞ ¼ 2Gg

X6i¼1

Wi exp�tti

� �(A-1)

KðtÞ ¼ 3Ke � 3Ke � 3Kg

� �X6i¼1

Wi exp�t6ti

� �(A-2)

where Wi and ti are, respectively, the weights and relaxationtimes assumed to be constant with temperature. Shear differedvalue Ge is considered equal to zero.

These relaxation functions are known as the reference tem-perature Tr and can be determined at any temperature as glassbehaves in the temperature range like a Thermo RheologicalSimple material (T.R.S.). Variation of temperature is then re-placed by variation of time, reduced time x(x, t):

xðx; tÞ ¼Z t

0

exp �H

R

1

Tðx; t0Þ �1

Tr

� �� �dt0 (A-3)

is introduced. In Eq. (A-3), R is the ideal gas constant, H is theactivation energy, and t0 is the integration variable.

Narayanaswamy19 has established a model of structural re-laxation with the so-called fictive temperature that takes bothmultiple relaxation mechanisms (memory) and non-linearityinto account. Using Tf(x, t), the deviation of the glass fromthe equilibrium state because of structural relaxation can be es-timated. Structural relaxation in glass is monitored by measur-ing changes in some macroscopic properties. The responseMp(x, t) of any property p(t) to a step change in temperaturefrom T1 to T2 in the transition region is time dependent and maybe written as

Mpðx; tÞ ¼pðtÞ � plð1Þpgð0Þ � plð1Þ

¼ Tf x; tð Þ � T2

T1 � T2(A-4)

The subscripts 0 and N denote the instantaneous initialand long-time value of p. pg(0) is the property value imme-diately following the temperature change and afterwards, theproperty relaxes in time toward the equilibrium liquid valuepl(N).

The formulation of the structural model is completed with thefollowing definition of fictive temperature. The fictive tempera-ture Tf(x, t) can be calculated from Eq. (A-4), by giving the formof the relaxation function as a function of time and temperatureusing the Boltzmann superposition principle and integrationover the thermal history:

Tfðx; tÞ ¼Tðx; tÞ �Z t

0

Mp½xðx; tÞ � xðx; t0Þ�

� dTðx; t0Þdt0

dt0

(A-5)

Reduced time x(x, t) depends on Tf(x, t):

xðx; tÞ ¼Z t

0

exp �H

R

X

Tðx; t0Þ þ1� X

Tf ðx; t0Þ� 1

Tr

� �� �dt0

(A-6)

where X is a constant parameter; exp denotes the exponentialfunction.

By analogy with the viscous relaxation (Eq. (A-1)), the re-laxation function is represented as a series of 6 exponential re-laxation functions:

Mpðxðx; tÞÞ ¼X6i¼1

Wiexp�xðx; tÞ

9ti

� �(A-7)

The algorithm of Markowsky and Soules21 (Eqs. (A-8) to (A-11)) is used to calculate the fictive temperature Tf(x,t):

tf i ðx; 0Þ ¼ T0 (A-8)

Tific ¼ 9ti expH

R

X

Tðx; tÞ þ1� X

Tf ðx; t� DtÞ �1

Tr

� �� �(A-9)

Tf i ðx; tÞ ¼Tf i ðx; t� DtÞ þ Tðx;tÞDt

tific

� �1þ Dt

tific

� � (A-10)

Tfðx; tÞ ¼X6i¼1

WiTf i ðx; tÞ (A-11)

The response functionMp(x, t) (Eq. (A-4)) is expressed as a sumof exponential relaxations using the differential form of Eq. (A-5). Initial values of partial fictive temperature Tf i ðx; tÞ are as-signed to T0 and numerically integrated during the transientanalysis using an implicit scheme. Dt is an increment of time of astep change in temperature.

Finally, thermal strains eth(x, t) induced by tempering of theglass are given by

ethðx; tÞ ¼ ag½Tðx; tÞ � T0� þ as½Tðx; tÞ � Tfðx; tÞ� (A-12)

where ag is the thermal expansion coefficient of the glass in thesolid state. as is the structural expansivity responding toas 5al�ag, with al being the thermal expansion coefficient ofthe liquid glass.

Acknowledgments

The authors gratefully acknowledge the support of the institutions.

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