thermal and chemical behavior of glass forming batches

Thermal and chemical behavior of glass forming batches

Verheijen, O.S.

DOI:10.6100/IR565202

Published: 01/01/2003

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Citation for published version (APA):Verheijen, O. S. (2003). Thermal and chemical behavior of glass forming batches Eindhoven: TechnischeUniversiteit Eindhoven DOI: 10.6100/IR565202

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Thermal and chemical behavior

of glass forming batches

Thermal and chemical behavior

of glass forming batches

PROEFSCHRIFT

ter verkrijging van de graad van doctoraan de Technische Universiteit Eindhoven,

op gezag van de Rector Magnificus, prof.dr. R.A. van Santen,voor een commissie aangewezen door het College voor Promoties

in het openbaar te verdedigen op woensdag 25 juni 2003 om 16.00 uur

door

Oscar Silvester Verheijen

geboren te Arnhem

Dit proefschrift is goedgekeurd door de promotoren:

prof.dr.ir. R.G.C. Beerkensenprof.dr.rer.nat. R. Conradt

CIP-DATA LIBRARY TECHNISCHE UNIVERSITEIT EINDHOVEN

Verheijen, Oscar S.

Thermal and chemical behavior of glass forming batches / by Oscar S. Verheijen. -Eindhoven : Technische Universiteit Eindhoven, 2003.Proefschrift. - ISBN 90-386-2555-3NUR 913Trefwoorden: glasfabricage ; glasovens / glassmelten / warmteoverdracht /fysisch-chemische simulatie en modellering / reactiekinetiekSubject headings: glass technology ; glass furnaces / glass melting / heattransfer / physicochemical simulation and modeling / reaction kinetics

Printed by: Universiteitsdrukkerij Technische Universiteit Eindhoven.

Voor Roelie

Summary

The quality of the glass melt produced in a glass furnace is to a large extent determined bythe temperature-time trajectory of the freshly molten glass melt in the melting tank of the glassfurnace. The residence time and the temperature of the glass melt need to be sufficient forcomplete dissolution of raw materials, the release of dissolved gases from the glass melt andthe homogenization of the glass melt. To understand, control and optimize glass melt qualityand to improve glass furnace design, since the early 1980’s, mathematical simulation models,describing melting and heat transfer in glass furnaces, have been used. These mathematicalsimulation models show that thickness, length, reactivity and thermal properties of glass form-ing batches have a large impact on the temperature distribution and glass melt flow patterns ina glass furnace. Therefore, detailed models describing the thermal and chemical behavior ofglass forming batches are required in mathematical simulation models.

The melting of glass forming batches is a complex process involving different reaction typessuch as dehydration reactions, crystalline inversions, solid-state reactions between different rawmaterial grains, decomposition reactions, melt forming reactions and dissolution processes. Theheating of the three-phase (reacting) glass forming batch is determined by both heat transferfrom the combustion space above and the glass melt underneath the glass batch and by heattransport in the interior of the glass batch. Because of the complexity of the melting and heat-ing process of glass batches and due to the lack of sensors and accurate analyzing techniquesto measure and monitor the progress of glass batch melting, no quantitative description of thesimultaneous heating and reactive melting of glass forming batches is known so far.

The objective of this study is to obtain a quantitative description of the heating process ofa glass forming batches and the conversion rate of the glass forming batch into the glass melt.The heat transfer in the interior of a glass forming batch can be described by:

∂(ρm cp,m Tm)

∂t= −∇ · (ε ρg cp,g vg,z Tg)−∇ · (λeff∇Tm)+∇ ·−→q r,eff +(1− ε)

∂Hchem

∂t. (1)

The left-hand-side term in the energy equation describes the local accumulation of heat. Thefirst, second and third right-hand-side terms in the energy equation indicate convective, con-ductive and radiative heat transfer, respectively. The fourth right-hand-side term in the energyequation indicates the energy consumption by chemical reactions. A detailed description of theenergy equation of the glass forming batch requires values for the temperature dependent glassforming batch properties. In this study, the temperature dependent chemical energy demandand the heat conductivity of glass forming batches are determined. To describe the conversionrate of a glass forming batch into a glass melt, the dissolution rate of sand grains, which is con-sidered as the most significant criterion for the conversion of glass forming batches, has beenstudied.

vii

viii

The glass forming batch properties have been determined by the following steps:

1. The identification of the main batch reactions in a typical float and TV-panel glass batchduring heating by phase analysis on quenched samples of heat-treated glass batch mix-tures.

2. The development of both experimental and mathematical techniques to determine quanti-tatively the kinetics of batch reactions and the heat penetration in typical industrial glassbatches.

3. The measurement and modelling of the temperature dependent chemical energy demand,the effective heat conductivity and reaction kinetic parameters.

For a float glass batch, containing silica sand, soda ash and dolomite, it was demonstratedthat the thermal calcination of dolomite and limestone and the reactive calcination of soda ashoccur (almost) independently from each other. The reactive calcination of soda ash with sanddetermines the onset for batch-to-melt conversion of the float glass batch, which starts at about1070 K. The formation of this primary formed melt phase enhances the dissolution of sand,(at least part of) MgO and intermediately formed binary and ternary silicates. In contrast tothe float glass batch, the rate of the calcination reactions in TV-panel glass batches are depen-dent on other batch reactions such as the dissolution of intermediate formed crystalline silicates.

The chemical energy demand of a glass forming batch is mainly dependent on the energyrequired for calcination reactions in the glass batch. Combination of measured kinetics of calci-nation reactions in a float glass batch with the enthalpies for these calcination reactions, resultsin an expression for the chemical energy required for complete calcination of the float glassbatch as function of time, temperature and partial CO2-pressure in the float glass batch. Todetermine the chemical energy demand of a float glass batch, the kinetics of the thermal decom-position of dolomite and limestone and the kinetics of the reactive calcination of soda ash havebeen measured.

The heat transport in the interior of a glass batch is determined by a combination of threemodes of heat transfer. To study the contribution of these three modes, viz. convection, con-duction and radiation, an experimental set-up was developed to measure the temperatures in aglass batch as function of time and position. A numerical-experimental technique was used toderive heat conductivity data from the measured time- and position dependent temperatures.

With the numerical-experimental technique, the temperature dependent heat conductivity ofsolid particle mixtures of individual glass batch components and mixtures of glass batch com-ponents were estimated. It is shown that the heat flow through a glass batch in the solid-stateregime can be regarded as a combination of a serial and parallel connection of thermal resis-tances.

The prediction of the effective heat conductivity of mixtures of glass batch componentsbased on the glass batch composition and the porosity of the glass batch, has become pos-sible by regarding the glass batch as only a parallel connection of thermal resistances usingapparent values for the thermal heat conductivity of the glass batch components instead of theintrinsic values for the thermal heat conductivity. Using this approach, the difference in es-timated and predicted effective heat conductivity of a three-component mixture was less than

ix

0.01 W m−1 K−1 for a value of the heat conductivity between 0.15 en 0.40 W m−1 K−1.The effect of the presence of cullet in a glass batch on the net effective heat conductivity

of the glass batch in the temperature range up to 1200 K has been studied with the numerical-experimental technique. The calculation of the contribution of radiative heat transport on thenet effective heat conductivity of a glass batch shows that the expected increase in effective heatconductivity for a cullet containing glass batch is only observed (in case of a cullet particle sizebetween 4 and 8 mm) in case the cullet fraction exceeds 50 %.

The heating of glass forming batch floating on top of a glass melt is not only determinedby the heat penetration through the glass forming batch, but also by the heat transfer from thecombustion space above the glass forming batch to the top layer of the glass forming batch. Itis shown that the heat transfer towards the glass batch is mainly determined by radiative heattransfer and less by free and forced convective flow of hot gasses over the glass forming batch.

In general, the degree of dissolution of sand grains during heating of glass forming batchesis considered as the most significant criterion for glass batch conversion. The mathematicalformulation of this mainly diffusion governed process requires a description of phenomena atmicro scale, such as the wetting of single sand grains by low viscous melt phases, in combina-tion with complex time-dependent boundary conditions. For the use of glass tank simulationmodels as tool in the glass industry, it is desired to have a simple expression describing the time-and temperature dependent dissolution of sand grains in glass batches instead of using complexdissolution models on micro scale. A prerequisite of the use of such a simple expression is thatthis expression should reflect the practical observed dissolution behavior of sand grains as func-tion of e.g. particle size, heating rate and cullet fraction in the glass batch. It is a widespreadpractice to use simple approximate theoretical models to describe complex processes similar tothe dissolution of sand grains.

It is shown that the best available approximate model describing the kinetics of a threedimensional diffusion governed process, such as the dissolution of sand grains embedded in aliquid reactive phase during heating of glass forming batches, is the Ginstling-Brounstein model(GB-model). The applicability of the GB-model for describing the dissolution of sand grainsduring the heating of glass batches was studied by comparing the results of the GB-model withresults of a more detailed first principle numerical model. Under certain conditions, it appearedto be possible to predict the dissolution of sand grains as function of time and temperature witha modified GB-model. This is the case for large sand grains dissolving in a thick glass meltlayer surrounding the sand grain. For very thin layers, the prediction of the modified GB-modelappears to be rather inaccurate.

The use of the GB-model for the description of the dissolution of sand grains during heatingof glass batches has experimentally been validated by analysis of the conversion rate of sandgrains in float glass batches. The degree of sand grain conversion as function of time and tem-perature has been analyzed by quantitative phase analysis using X-ray diffraction. For the floatglass batch, the GB-model parameters have been determined as function of sand grain size,heating rate and cullet fraction.

Contents

1 Introduction 11.1 Description of the industrial glass melting process . . . . . . . . . . . . . . . . 11.2 The chemical behavior of glass forming batches . . . . . . . . . . . . . . . . . 41.3 Modelling the thermal and chemical behavior of glass forming batches . . . . . 61.4 Objectives and outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.5 Nomenclature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101.6 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2 Energy demand of glass forming batches 132.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.2 Energy demand during heating of glass forming batches . . . . . . . . . . . . . 152.3 Description of the kinetics of batch reactions . . . . . . . . . . . . . . . . . . . 21

2.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212.3.2 Kinetics of homogeneous reactions . . . . . . . . . . . . . . . . . . . 212.3.3 Kinetics of heterogeneous reactions . . . . . . . . . . . . . . . . . . . 232.3.4 Heterogeneous reaction kinetics with participation of melt phases . . . 262.3.5 Characteristics of calcination reactions . . . . . . . . . . . . . . . . . 29

2.4 Experimental techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312.4.1 Thermogravimetric analysis . . . . . . . . . . . . . . . . . . . . . . . 312.4.2 Differential thermal analysis . . . . . . . . . . . . . . . . . . . . . . . 342.4.3 Phase analysis on partly reacted glass batches . . . . . . . . . . . . . . 34

2.5 Calcination of a float glass batch . . . . . . . . . . . . . . . . . . . . . . . . . 352.5.1 Calcination of dolomite . . . . . . . . . . . . . . . . . . . . . . . . . 362.5.2 Calcination of limestone . . . . . . . . . . . . . . . . . . . . . . . . . 422.5.3 Calcination of soda ash . . . . . . . . . . . . . . . . . . . . . . . . . . 492.5.4 Calcination of a mixture of silica sand, soda ash and limestone . . . . . 622.5.5 Reaction mechanism of a float glass batch . . . . . . . . . . . . . . . . 642.5.6 Chemical energy demand of a float glass batch . . . . . . . . . . . . . 67

2.6 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 712.7 Nomenclature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 722.8 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

3 Dissolution of sand grains during heating of glass forming batches 773.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 773.2 Mathematical and experimental descriptions of the sand grain dissolution process 803.3 Evaluation of the application of the Ginstling-Brounstein model . . . . . . . . 92

xi

xii Contents

3.3.1 Derivation of the Ginstling-Brounstein model . . . . . . . . . . . . . . 923.3.2 Modelling of the dissolution of a single sand grain in a sodium silicate

melt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 983.3.3 Applicability of the Ginstling-Brounstein model . . . . . . . . . . . . 103

3.4 Quantitative phase analysis with X-ray diffraction . . . . . . . . . . . . . . . . 1093.5 Experimental determination of the apparent GB-model parameters . . . . . . . 116

3.5.1 Apparent GB-model parameters as function of the sand grain particle size1163.5.2 Apparent GB-model parameters as function of the cullet fraction . . . . 120


4 Heat conductivity of glass forming batches 1294.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1294.2 Phonon and photon conductivity . . . . . . . . . . . . . . . . . . . . . . . . . 132

4.2.1 Intrinsic phonon conduction of solid species . . . . . . . . . . . . . . . 1324.2.2 Intrinsic photon conduction of solid species . . . . . . . . . . . . . . . 1334.2.3 Phonon conduction of mixtures of solid species . . . . . . . . . . . . . 1354.2.4 Photon conduction in mixtures of solid species . . . . . . . . . . . . . 137

4.3 Estimation of the heat conductivity from heat penetration experiments . . . . . 1394.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1394.3.2 Parameter estimation without a-priori knowledge of λ = f (T ) . . . . . 1414.3.3 Parameter estimation with a-priori knowledge of λ = f (T ) . . . . . . . 143

4.4 Experimental set-up for measuring the heat penetration in solid particle mixtures 1564.5 Experimental determination of heat conductivity in a particle bed . . . . . . . . 160

4.5.1 Heat conductivity of a silica sand batch . . . . . . . . . . . . . . . . . 1614.5.2 Heat conductivity for multi-component mixtures . . . . . . . . . . . . 1644.5.3 Heat conductivity of particle beds containing cullet . . . . . . . . . . . 166


5 Complete simulation model for the heating of glass forming batches 1775.1 Energy conservation equation and temperature dependent glass batch properties 1775.2 Heat transfer towards a batch blanket . . . . . . . . . . . . . . . . . . . . . . . 180

5.2.1 Forced convective heat transfer . . . . . . . . . . . . . . . . . . . . . . 1805.2.2 Free convective heat transfer coefficient . . . . . . . . . . . . . . . . . 1835.2.3 Radiative heat transfer coefficient . . . . . . . . . . . . . . . . . . . . 184

5.3 Nomenclature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1865.4 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188

A Calcination of a TV-panel glass batch 189A.1 Description of the calcination mechanism of a TV-panel glass forming batch . . 189A.2 Calcination of SrCO3- and BaCO3-grains . . . . . . . . . . . . . . . . . . . . 192A.3 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196

Contents xiii

B Estimation of model parameters by a least squares approach 197B.1 Description of the least squares approach for parameter estimation . . . . . . . 197B.2 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199

C Derivation of 2nd order derivative of position dependent temperature 201

Samenvatting 203

Dankwoord 207

Curriculum Vitae 209

xiv Contents

Chapter 1

Introduction

1.1 Description of the industrial glass melting process

Nowadays, glass is mainly produced in continuously operating glass melting furnaces. In gen-eral, the glass melt flows through three segments of the furnace, i.e. the melting tank, theworking end and the feeder section. Figure 1.1 shows a schematic view of the melting tank andthe superstructure of an industrial glass furnace with a so-called throat.

Batch blanket

Throat

SuperstructureCombustion room

Spring zone

First circulation loop

Short cut flow

Doghouse

Bac

k w

all

Fron

t wal

l

Glass melt

Second circulation loop

Burner

Figure 1.1: Scheme of the melting tank and the superstructure of an industrial glass furnace.

In the doghouse of the melting tank, a glass forming batch1 is charged on top of the hot glassmelt and forms the so-called batch blanket or batch rolls (see figure 1.2) floating on top of theglass melt. The batch blanket is mainly heated by radiative heat transport from the flames andsuperstructure in the combustion chamber above the batch blanket and by conductive and ra-

1The glass forming batch is the mixture of raw material components, which forms a glass melt during heating.Further in this thesis, a glass forming batch is denoted as glass batch.

1

2 Chapter 1. Introduction

diative2 heat transport from the hot glass melt underneath the batch blanket. During heating ofthe glass batch, up to temperatures of even 1850 K, a complex process of chemical reactionstakes place transforming the solid raw material batch components into a liquid melt phase. Thefreshly formed glass melt, still containing undissolved raw materials and gas bubbles, enters thebulk glass melt underneath the batch blanket and follows the flow pattern(s) through the meltingtank towards the throat.

Figure 1.2: Batch rolls floating on top of the glass melt in an end-port fired container glass furnace

The flow pattern of the glass melt in the melting tank is the result of both forced and freeconvection driven flow. Forced convective flow is imposed on the glass melt by the pull rate3

of the furnace. Due to an unequal fuel distribution over the burners above the glass melt andthe cooling effect of the relative cold batch blanket in the doghouse, temperature gradients arepresent at the glass melt surface. These temperature gradients at the surface of the glass meltlead to temperature gradients deeper in the glass melt by conductive, convective and radiativeheat transport. Free convective flow of the glass melt is caused by density gradients in the glassmelt as a consequence of the temperature gradients in the glass melt.

The combination of furnace geometry, pull rate and fuel distribution over the burners, mayin the ideal case cause two glass melt circulation loops in the melting tank. In the first and oftenlargest circulation loop, the glass melt flows from the spring zone4 in the direction of the batchblanket and returns via the back wall (at the doghouse side) towards the throat. In the second

2The contribution of radiative heat transport to the overall heat transport from the hot glass melt to the batchblanket is dependent on the emissivity of the glass that is produced.

3Here, the pull rate of the furnace is defined as the amount of glass melt per period of time withdrawn from thefurnace for producing glass products.

4The position at the surface of the glass melt at which the glass melt flow splits up in a forward (towards thethroat) and backward (towards the charging end) flow. In general, the spring zone is positioned close to the hotspot of the glass melt, at which the position with the highest glass melt temperature is located.

1.1. Description of the industrial glass melting process 3

loop, the glass melt flows from the spring zone to the front wall and then downwards towardsthe throat. A part of the glass melt enters the throat and a part follows the second circulationloop towards the spring zone. In case this return flow of glass melt reaches the bottom of themelting tank, a shortcut flow of glass melt from the batch blanket directly into the throat isprevented. Now, the glass melt is forced to flow over the glass melt surface at which the highestglass melt temperatures exist.

Next to the melting of the raw materials, also homogenization of the freshly formed non-homogeneous melt phases is required, and fining and refining of the glass melt should take placein the melting tank. During the fining process, bubbles which initially mainly contain CO2 andair, and dissolved gases, are removed from the glass melt. At elevated temperatures, deliber-ately added fining agents such as sodium sulfate and antimony oxide decompose, producingSO2 and/or O2. As an example, the thermal decomposition of chemically dissolved SO2−

4 inoxidized glass melts is given by

SO2−4 (m)

T↑−→ SO2(g)+12

O2(g)+O2−(m). (1.1)

By the formation of the fining gases, the existing gas bubbles expand resulting in an en-hanced rise of the bubbles towards the surface of the glass melt. At the surface of the glassmelt, the bubbles release their gases into the combustion space or may form a foam layer float-ing on top of the glass melt. During the take up of the fining gases in the gas bubbles, the initialbatch gases, which are enclosed in the bubbles, are diluted by the fining gases. The decrease inthe partial pressure of these gases in the bubbles is followed by diffusion of the dissolved gasesfrom the glass melt into the gas bubbles. In this way, dissolved gases are stripped from the glassmelt.

During the subsequent refining process, residual small gas bubbles are resorbed in the glassmelt due the shift of reaction 1.1 towards the left-side-side as a consequence of the temperaturedecrease in later stages of the process. Via the throat of the melting tank, the glass melt entersthe working end and subsequently the feeder section(s), in which the glass melt is slowly cooleddown to the temperature required for the forming process of the final glass products.

In order to understand, control and optimize the glass melting process, since the early1980’s, mathematical simulation models, based on computational fluid dynamics, have beenused. These models describe the physical and chemical processes in industrial glass meltingfurnaces [1]. The basic equations of these simulation models are the conservation laws for mass,energy and momentum. With these equations glass melt temperatures and velocity vectors inindustrial glass furnaces are calculated. For describing typical features of the glass meltingprocess, additional models are added to the conservation laws such as a separate batch blanketmodel. The batch blanket model describes the thermal, chemical and rheological behavior ofglass batches floating on top of a glass melt.

In industrial glass furnaces, the batch blanket has a direct impact on both the energy trans-port in the glass furnace and on the flow patterns of the glass melt in the melting tank. A batchblanket can be regarded as a layer with a low heat conductivity and high heat reflecting proper-ties at the upper surface. Due to these properties, the batch blanket acts as an insulation layerblocking direct heat input from the combustion room into the glass melt. Consequently, thedirect energy input from the combustion room in the glass melt itself is mainly limited to thearea of the glass melt, which is not covered with the batch blanket. The energy input per amountof produced glass for given combustion conditions, is dependent on the batch blanket size and


shape.The batch blanket has also an impact on the flow pattern of the glass melt in the melting

tank. The intensity of the first circulation loop in the melting tank is determined by the tem-perature difference between the hot spot of the glass melt and the temperatures under the batchblanket. An increase of this temperature difference results in a more intense circulation loop,i.e. higher glass melt velocities, intensifying the bottom flow of glass melt from the back walltowards the throat. In case the intensity of the second circulation loop is not sufficient, a shortcut flow of glass melt over the bottom from the first circulation loop directly into the throat ispossible.

Despite the large impact of the batch blanket on the temperatures and glass melt flow in glassfurnaces, the current batch blanket models still represent a too much simplified description ofthe physical and chemical processes occurring during heating glass batches. Improvement of theglass tank simulation models requires a thorough investigation of the melting and heat transferprocesses in the glass batch layers, which is the main subject of the current study.

1.2 The chemical behavior of glass forming batchesThe composition of a glass batch is dependent on the glass type to be produced. A glassbatch producing float glass5 typically contains silica sand (SiO2), soda ash (Na2CO3), dolomite(MgCO3 ·CaCO3) and/or limestone (CaCO3), sodium sulfate (Na2SO4) as fining agent and areducing agent such as carbon or the steel slag calumite. As mentioned by Hrma [2], the melt-ing of glass batches is a complex process involving different reaction types such as dehydrationreactions, crystalline inversions, solid-state reactions between the different raw material grains,decomposition reactions, melt forming reactions and dissolution processes. Below, a short de-scription of the characteristics of these different reaction types is given.

• Dehydration reactions: In the glass batch, water may be present as either molecular wa-ter and chemically bonded water (e.g. Na2CO3 ·H2O or water bonded in clays). Molec-ular water is added to the glass batch to lower the dusting of fine batch particles duringtransport towards the glass furnace and to prevent extensive carry-over of loose batchparticles in the glass furnace. Molecular water, with a fraction of about 1.5-4 wt.%in the glass batch, is released at temperatures below 373 K. The temperature at whichchemically bonded water is released, is dependent on the bonding strength of the watermolecule(s) to the batch constituents and may last up to temperatures of 880 K. The im-pact of dehydration reactions on the melting process of the glass batches is mainly theconsumption of extra energy during water evaporation.

• Crystalline inversions: During melting of glass batches, structural modifications of somecrystalline batch constituents occur. An example of a crystalline inversion in sand con-taining glass batches is the transformation of the α-modification of quartz into the β-modification of quartz at 846 K. Crystalline inversions have no significant impact on themelting process of the glass batch.

• Decomposition reactions: The main decomposition reactions during melting are the cal-cination reactions. The typical weight loss for a float batch, without the presence of

5Float glass is flat glass, e.g. window glass, that is produced via the so-called ’float process’.

1.2. The chemical behavior of glass forming batches 5

cullet6, due to these calcination reactions equals about 15-20 wt.%. Alkaline (Na2CO3,K2CO3) and earth-alkaline carbonates (MgCO3, CaCO3, SrCO3 and BaCO3) either de-compose due to temperature increase, i.e. thermal calcination above a certain temperaturelevel, or via a reaction with other batch constituents, i.e. reactive calcination. As an ex-ample, the thermal calcination of limestone is given by

CaCO3(s)T↑−→ CaO(s)+CO2(g). (1.2)

The calcination temperature7 for the earth-alkaline carbonates, which is described inchapter 2, is dependent on the bonding strength of CO2 with the earth-alkaline oxideand equals 687 K, 1173 K, 1513 K and 1646 K for MgCO3, CaCO3, SrCO3 and BaCO3,respectively.An example of reactive calcination of a carbonate is the calcination of soda ash via a solid-state reaction with silica sand resulting in the formation of a crystalline sodium silicatephase. The formation of sodium metasilicate (Na2O ·SiO2) due to reactive calcination ofsoda ash with silica sand is given by

Na2CO3(s)+SiO2(s) −→ Na2O ·SiO2(s)+CO2(g). (1.3)

Although the formation of Na2O ·SiO2 is already thermodynamically favorable at tem-peratures below 620 K (see chapter 2), the kinetics of these solid-state reactions are loweven at temperatures of 1000 K. These low reaction kinetics are caused by slow solid-state diffusion of reacting species through the formed crystalline binary silicates towardsthe interface of the reacting grains. Enhanced reactive calcination is observed at the pres-ence of a melt phase [3] by which the transport of reactants and reactions products isacceleration due to the higher diffusion coefficient of the formed oxides in the liquid statecompared with the solid-state.

• Melt forming reactions: Melt phases are either formed by the melting of pure substances(e.g. the melting of soda ash at 1129 K) or by eutectic melting of two solid species. Anexample of eutectic melting is the melting of a mixture of Na2O ·SiO2 and Na2O ·2SiO2 at1110 K. The formation of these so-called primary melt phases is regarded as the measur-able onset for both reactive calcination of carbonates and the dissolution of solid oxidesin the glass batch such as e.g. silica (SiO2), zircon (ZrO2 ·SiO2) and alumina (Al2O3).

• Dissolution reactions: According to Conradt et al. [4], the dissolution rate of the oxidessuch as silica is dependent on primary melt phase properties such as viscosity, surfacetension and composition. Next to these properties, the particle size of the oxide grainshas a large impact on the dissolution rate of the oxides in the glass melt [5]. Because themain fraction of commercial glass batches is occupied by silica, the dissolution processof silica is regarded as the most significant criterion for the degree to which the meltingof a glass batch based on silica sand has advanced [6].

Concerning the melting of glass batches, two reaction paths are distinguished for soda-lime-silicate glass batches [2, 4], i.e. the carbonate route and the silicate route (see also chapter 2).

6Recycled glass which acts as alternative raw material for glass production7The temperature at which the thermodynamic equilibrium CO2-pressure of the carbonate equals 1 bar.


The carbonate route is characterized by reactive dissolution of silica sand with a binary meltphase of soda ash and limestone at temperatures below 1170 K. Soda ash and limestone mayform a double carbonate by a solid-state reactions according to

Na2CO3(s)+CaCO3(s) −→ Na2Ca(CO3)2(s). (1.4)

At 1090 K, the double carbonate forms a low viscous melt phase, with which silica grains mayform a sodium calcium silica melt:

Na2Ca(CO3)2(l)+ν SiO2(s) −→ NCS(m)8. (1.5)

For the silicate route, the (eutectic) melting of crystalline sodium silicates is regarded as theonset for batch melting. The eutectic melting of soda disilicate with silica at 1072 K is given by

Na2O ·2SiO2(s)+SiO2(s) −→ NS(m) (1.6)

The carbonate route is assumed to be predominant at rapid heating rates in the order of 200 Kmin−1, whereas the silicate route is predominant at heating rates in the order of 10 K min−1. Thepresence of different reaction mechanisms as function of heating rate was clearly demonstratedby Riedel [7], who studied the reactivity of mixtures of glass batch components with hot-stagemicroscopy. The mechanism of the conversion of a typical float and TV-panel batch as functionof temperature is discussed in more detail in chapter 2.

1.3 Modelling the thermal and chemical behavior of glassforming batches

A glass batch is a mixture of a gas phase and solid particles. During heating of the glass batch,the two-phase mixture transforms into a three-phase mixture also containing the melt phases,which are formed by (reactive) melting of the glass batch. An accurate prediction of the energybalance of a reacting glass batch requires knowledge of intrinsic properties of the solid phase,the melt phase and the gas phase, such as e.g. heat capacity and heat conductivity. Also theheat exchange between the different phases in the glass batch has to be known. Ungan andViskanta [9] assumed that the solid phase and the melt phase are, locally, in thermal equilib-rium, which simplifies the reacting glass batch to a two-phase mixture.

Wakao and Kaguei [8] presented different heat transfer models for gas-solid mixtures undertransient conditions. These heat transfer models, which could also be applied to glass batchescomposed of a condensed phase and a gas phase, consist of the fundamental energy equationsfor both phases in the two-phase mixture. The form of these idealized energy equations is de-pendent on the arrangement of the two phases in the two-phase mixture, the intrinsic propertiesof both phases and the flow conditions of the gas phase through the porous solid phase. How-ever, all energy equations used in these heat transfer models are approximations of the energyequation taking into account all modes of heat transfer (conductive, convective and radiativeheat transfer) in the individual phases and the heat exchange between the two phases. As (ex-perimental) discrimination between the temperature of both the gas phase and the condensedphase in glass batches is very difficult or even impossible at the moment, in general a mean

8NCS(l) denotes a glass melt containing the oxides Na2O (N), CaO (C) and SiO2 (S).

1.3. Modelling the thermal and chemical behavior of glass forming batches 7

temperature of the glass batch is obtained when measuring the temperature in the glass batch.The overall energy balance of a glass batch during heating, with respect to a fixed frame ofreference, can be described by

∂∂t

(

ρ cp T)

mn = −∇ ·[

(1− εp) ρc cp,c−→vc Tc + εp ρg cp,g

−→vg Tg]

−∇ · (λeff ∇ Tmn)+∇ ·−→q r,eff +(1− εp)∂Hchem

∂t. (1.7)

in which∂∂t

(

ρ cp T)

mn =∂∂t

[

(1− εp) ρc cp,c Tc + εp ρg cp,g Tg]

. (1.8)

The subscripts c and g denote the condensed phase and the gas phase, εp is the porosity of theglass batch, ρ is the intrinsic density, cp is the heat capacity, T is temperature, t is time, −→v is thethree-dimensional velocity, λeff represents the effective heat conductivity of the glass batch witha mean temperature Tmn, −→q r,eff represents the effective three-dimensional radiative heat flux inthe glass batch and Hchem is the energy per unit volume of the glass batch required for batchreactions. Assuming that the horizontal velocity of the condensed phase and the gas phase aresimilar [9], the Lagrangian description of the energy balance of a glass batch is given by

∂∂t

(

ρ cp T)

mn = −∇ ·(

εp ρg cp,g vg,z Tg)

−∇ · (λeff∇Tmn)+∇ ·−→q r,eff +(1− εp)∂Hchem

∂t,

(1.9)in which vg,z is the vertical velocity of the gas phase relative to the condensed phase. The heattransport in the interior of the glass batch is determined by a combination of three modes ofheat transfer:

• convective heat transport by the ascending gas phase (characterized by the first right-hand-side term in equation 1.9),

• conductive heat transport by mutual contact between solid particles and between solidparticles and liquid phases (characterized by the second right-hand-side term in equation1.9),

• radiative heat transport through the transparent phases (characterized by the third right-hand-side term in equation 1.9).

The mean and effective properties of the glass batch are dependent on the time- and temperaturedependent composition of the glass batch. Therewith, these glass batch properties are also time-and temperature dependent.

The melting of the initial solid glass batch producing a particle free glass melt proceeds viaa series of serial and parallel batch reactions. To describe the complete chemical conversionprocess of a glass batch, for each reactant and (intermediate) reaction product, a single massconservation equation has to solved. In case accurate analyzing techniques and sensors wouldbe present to measure the kinetics of all individual batch reactions occurring simultaneouslyduring heating of a glass batch, still the complete description of the kinetics of all batch reactionsremains a very complex and time-consuming task. Therefore, first the kinetics of the mostimportant batch reactions have to be determined. As mentioned in section 1.2, the dissolutionprocess of sand grains is regarded as the most significant criterion for the degree to which themelting of a glass batch based on silica sand has advanced. In chapter 3, the dissolution of sandgrains during glass batch melting is studied.


1.4 Objectives and outlineThe objective of this study is to obtain a quantitative description of the heating process of a glassbatches and the conversion rate of the glass batch into the glass melt. To meet the objectives ofthis study, the following activities are distinguished:

1. The identification of the main batch reactions in a typical float and TV-panel glass batchduring heating.

2. The development of both experimental and mathematical techniques to determine quanti-tatively the kinetics of batch reactions and the heat penetration in typical industrial glassbatches.

3. The measurement and modelling of the temperature dependent chemical energy demandHchem, the effective heat conductivity λeff and reaction kinetic parameters using the in thisstudy developed techniques.

A better quantitative understanding of these glass batch properties may lead to the improvementof the performance of glass tank simulation models. The implementation of the glass batchproperties, which are determined during this study, in glass tank simulation models will resultin improved predictions of both the energy balance of the glass batch and the dimensions (sizeand shape) of the batch blanket. These two aspects have a large impact on both the glass quality,by influencing the intensity of the first circulation loop in the melting tank (see section 1.1), andthe energy consumption of the glass tank.

In this study, the experimental determination of the glass batch properties mentioned above,are confined to a glass batch with a typical composition for the production of float glass. Beside,the conversion mechanism of a glass batch with a typical composition for the production of TV-panel glass is studied qualitatively. The melting experiments are performed with industrialraw materials and at conditions (i.e. temperatures and heating rates) that are typical for theindustrial glass melting process. However, not the full range of industrial conditions is covereddue to experimental limitations.

One of the glass batch parameters that is required for solving the energy equation of theglass batch is the source term Hchem representing the energy demand for chemical reactionsoccurring during glass batch heating. In chapter 2, the chemical energy demand of the glassbatch producing float glass is studied. According to Madivate et al. [10], the main energyconsuming batch reactions are the decomposition reactions during which batch gases such asCO2 is released. Therefore, the major part of the activities described in chapter 2 focus on theexperimental determination of the calcination rate of the raw material carbonates present in thefloat glass batch. These calcination reactions are studied by both:

• thermogravimetric analysis of (mixtures of) glass batch components, and by

• identification of (intermediate formed) crystalline species in cooled samples of (mixturesof) glass batch components, which have been imposed to a defined temperature program.

Also in this chapter, a qualitative description of the conversion of a TV-panel batch is discussed.Because the dissolution process of silica sand is regarded as the most significant criterion

for the degree to which the melting of a glass batch based on silica sand has advanced, themechanism and the rate of the sand grain dissolution process in a glass batch is studied (chapter

1.4. Objectives and outline 9

3). To measure the degree of sand grain dissolution in a glass batch as function of time andtemperature, a quantitative analyzing technique is required. This technique should be able todetermine the residual amount of crystalline sand in the partly molten glass batches. In thisstudy, the method of quantitative phase analysis with X-ray diffraction is applied as analyz-ing technique for measuring the crystalline silica content in partly molten glass batches. Next,an approximate analytical model describing the sand grain dissolution during heating of glassbatches is proposed. The applicability of this approximate model is studied by comparing re-sults of this approximate model with the results obtained with a more detailed numerical model.Finally, the parameters required for this approximate model are determined for the float glassbatch as function of the particle size of the sand grains and the cullet fraction in the float glassbatch.

In chapter 4, the experimental determination of the effective heat conductivity of glassbatches is studied. To measure the heat penetration in a glass batch, an experimental set-upwas developed, with which at different positions in the glass batch the temperature as functionof time is monitored. Because the effective heat conductivity of a glass batch is temperaturedependent, the value for the effective heat conductivity of the glass batch cannot be derivedexplicitly from the heat penetration experiments. To calculate the effective heat conductivityfrom the measured heat penetration in a glass batch, a numerical-experimental technique is ap-plied. With this technique, for different glass batch mixtures, the effective heat conductivity isdetermined.

Finally, in chapter 5.1 the results of this study are summarized. To complete the descriptionof the heating process of glass batches, information of the heat transfer towards the boundariesof glass batches is required. In section 5.2, a description is given of the heat transfer process tothe top layer of the batch blanket and the major mode of heat transfer is identified.


1.5 NomenclatureLatin symbols

cp heat capacity [J kg−1 K−1]g gas phaseHchem chemical energy demand [J m−3]l liquid phasem dissolved in the melt phaseq heat flux [W m−2]s solid phaset time [s]T temperature [K]v velocity [m s−1]

Greek symbols

εp porosity [-]λ heat conductivity [W m−1 K−1]ν stoichiometric reaction coefficientρ density [kg m−3]

Subscripts

c condensed phaseeff effectivemn meanr radiativez vertical direction

1.6. Bibliography 11

1.6 Bibliography[1] R. Viskanta. Review of three-dimensional mathematical modeling of glass melting. J.

Non-Cryst. Sol., 177:347, 1994.

[2] P. Hrma. Complexities of batch melting. In: Proc. of the 1st International Conference onAdvances in the Fusion of Glass, pages 10.1–10.18, Alfred University, Alfred, New York,June 14-17, 1988.

[3] C. Kroger. Gemengereaktionen und Glasschmelze. Glastech. Ber., 25(10):307–324, 1952.

[4] R. Conradt, P. Suwannathada, and P. Pimkhaokham. Local temperature distribution andprimary melt formation in a melting batch heap. Glastech. Ber. Glass Sci. Technol.,67(5):103–113, 1994.

[5] R.G.C. Beerkens, H.P.H. Muijsenberg, and Heijden van der T. Modelling of sand graindissolution in industrial glass melting tanks. Glastech. Ber. Glass Sci. Technol., 67(7):179–188, 1994.

[6] L. Bodalbhai and P. Hrma. The dissolution of silica grains in isothermally heated batchesof sodium carbonate and silica sand. Glass Technology, 27(2):72–78, 1986.

[7] L. Riedel. Die Benetzung von Kalk und Quarz durch schmelzende Soda - Eine phanome-nologische Studie. Glastechn. Ber., 35(1):53–56, 1962.

[8] N. Wakao and S. Kaguei. Topics in chemical engineering: Vol. 1 - Heat and mass transferin packed beds. Gordon and Breach, Science Publishers, Inc., New York-London-Paris,1st edition, 1982.

[9] A. Ungan and R Viskanta. Melting behavior of continuously charged loose batch blanketsin glass melting furnaces. Glastech. Ber., 59(10):279–291, 1986.

[10] C. Madivate, F. Muller, and W. Wilsmann. Thermochemistry of the glass melting process- energy requirement in melting soda-lime-silica glasses from cullet-containing batches.Glastech. Ber. Glass Sci. Technol., 69(6):167–178, 1996.

12 Bibliography

Chapter 2

Energy demand of glass forming batches

2.1 IntroductionAs mentioned in chapter 1, a glass forming batch can be regarded as a mixture of a condensedphase, which encloses the solid batch particles and the formed melt phases, and a gas phase.The Lagrangian description of the energy equation of the two-phase glass batch is given by

∂∂t

(

ρ cp T)

mn = −∇ ·(



∂t,

(2.1)in which

(

ρ cp T)

mn represents the mean value of the enthalpy of the glass batch, ρg and cp,gare the density and heat capacity of the gas phase, Tmn is the mean temperature of the glassbatch, t is the time, εp is the porosity of the glass batch, vg,z is the vertical velocity of the gasphase relative to the condensed phase1, λeff is the effective heat conductivity of the glass batch,−→q r,eff is the effective radiative heat flux through the glass batch, and Hchem is the temperaturedependent energy per unit volume of the glass batch required for batch reactions. For a com-plete description of the heating of glass batches, values for the parameters εp, ρg, cp,g, vg,z, λeff,−→q r,eff, and Hchem are required.

In this chapter, the time- and temperature dependent glass batch property Hchem is deter-mined for a glass batch producing float glass. Madivate et al. [2] defined the net chemicalenergy demand of a glass batch as the enthalpy difference of the glass batch at 298 K and theformed melt phase with the released gases after cooling from the batch melting temperaturedown to 298 K. For a cullet-free glass batch, the net chemical energy demand for commercialcontainer glass batches was estimated to be approximately 15 % of the total net energy requiredfor batch melting. The residual 85 % is used for heating of the unreacted glass batch, the formedmelt phase and the released gases. Next, Madivate et al. [2] distinguished two types of batchreactions:

1. Gas-releasing decomposition reactions, during which H2O, CO2, NOx, SOx and/or O2 isreleased and during which solid oxides are formed, such as the calcination of limestonegiven by CaCO3(s) CaO(s) + CO2(g).

2. Mixing and fusion reactions of the solid oxides forming a glass melt.

1It is assumed that the horizontal velocity of the gas phase is similar to the horizontal velocity of the condensedphase [1].

13

14 Chapter 2. Energy demand of glass forming batches

The first type of batch reactions are endothermic reactions and require approximately 40 % ofthe total energy demand of a glass batch, whereas the latter type of batch reactions are exother-mic reactions, which compensate about 55 % of the energy required for the endothermic batchreactions. In practice, the batch reactions may already start at about 300 K and may range up toabout 1700 K for the dissolution of oxides. The chemical energy required for batch reactions istemperature dependent. A detailed description of the chemical energy demand of a glass batchcontaining silica sand, soda ash, limestone and dolomite is given in section 2.2.

The major part of the gas-releasing batch reactions are the calcination reactions duringwhich CO2 is released. Therefore, the temperature dependent net chemical energy demandduring melting of glass batches is to a large extent determined by the rate of the calcinationreactions. The net rate of consumption of chemical energy per unit volume of the glass batchfor all calcination reactions is given by

∂Hchem

∂t=

nc

∑i=1

(

Hr,i w0i ρc

Mi

∂ξi

∂t

)

, (2.2)

in which nc is the total number of carbonates in the glass batch, Hr,i is the enthalpy requiredfor the calcination of carbonate i expressed in J mole−1, w0

i is the initial weight fraction ofcarbonate i in the condensed phase, ρc is the density of the condensed phase, Mi is the molarmass of carbonate i, ξi is the degree of conversion2 of carbonate i, and t is the time. To determinethe time- and temperature dependent energy demand for the calcination reactions, the rate ofthe calcination reactions expressed by ∂ξi

∂t has to be determined. In section 2.3, the generalapproach for describing the reaction rate of homogeneous and heterogeneous reactions withand without the presence of a non-ideal liquid phases such as glass melts is discussed. Based onthe equations presented in this section, the rate of the calcination reactions which occur duringheating of a typical float glass batch is studied in section 2.5.

The identification of the calcination reactions in the float glass batch and the determinationof the calcination rate of these decomposition reactions are studied by

• analysis of the (intermediate formed) crystalline phases which are present in cooled heat-treated float glass batch samples, and

• thermogravimetric analysis.

The backgrounds and procedure for these two experimental techniques is discussed in section2.4.

The calcination behavior of dolomite, limestone and soda ash is discussed in sections 2.5.1,2.5.2 and 2.5.3, respectively. The calcination behavior of the complete float glass batch con-taining these batch components is discussed in section 2.5.4. In section 2.5.5, the conversionmechanism of the float glass batch, based on literature data and on the results of the calcinationexperiments described in this chapter, is described. The temperature dependent energy demandof the float glass batch, calculated from the measured rate of the calcination reactions and thecalcination enthalpies of these decomposition reactions, is presented in section 2.5.6. In section2.6 concluding remarks are presented concerning the reaction mechanism, the calcination rateand the chemical energy demand of the float glass batch are given. Also a few comments aregiven to the calcination behavior of a TV-panel glass batch.

2The degree of conversion of a carbonate is defined as the ratio of the weight loss with respect to the totalweight loss when the carbonate has completely been dissociated.

2.2. Energy demand during heating of glass forming batches 15PSfrag replacements

T0 Tr,onset Tr,end Tfinal

H(T0)

H(Tr,onset)

H(Tr,end)

H(Tfinal)

A

B

C

D

E

r2,1

r2,2r2,3

∆H1

∆H2

∆H3

r1

r2

r3

Figure 2.1: Schematic representation of the energy demand of a glass batch as function of temperature. Aindicates the enthalpy of the glass batch at the initial glass batch temperature T0. E indicatesthe enthalpy of the formed glass melt and gas phase after heating and melting the glass batchup to Tfinal. B, C and D are intermediate enthalpy states of the reacting glass batch. Tr,onsetand Tr,end are the onset and end temperature for the batch reactions. The routes along whichthe energy demand of a glass batch may run are indicated by r1, r2, r2,1 or r2,2 or r2,3 and r3.It is remarked that the energy required for evaporation of water, which is in general presentin glass batches, is not indicated in this figure.

2.2 Energy demand during heating of glass forming batchesIn this section, a general description is given of the energy demand of a glass batch duringheating. Next, the temperature dependent energy demand of a glass batch producing float glassis calculated based on literature data. It is shown that for an accurate description of the tem-perature dependent energy demand of a glass batch, the onset temperature and the kinetics ofthe (mainly) energy consuming batch reactions is required.

The energy demand of a glass batch is determined by the energy required for heating the rawmaterials, the formed melt phases and the released gases, and the energy required for the en-dothermic and exothermic batch reactions. The melting of a glass batch can be described by

(1+α)batch(T0) ∆H−→ 1 melt(Tfinal)+α gas(Tfinal), (2.3)


in which T0 is the initial temperature of the glass batch, Tfinal is the temperature to which theglass batch is heated3, ∆H is the energy demand of heating and melting the glass batch, and αis the mass of gas that is released during heating of a glass batch with mass (1+α).

With respect to the energy demand during the melting process of glass batches, three tem-perature ranges (see figure 2.1) are distinguished:

1. The temperature range between the initial temperature of the glass batch T0 and the onsettemperature for the batch reactions Tr,onset. The reaction occurring in this temperaturerange is given by

(1+α)batch(T0)∆H1−→ (1+α)batch(Tr,onset). (2.4)

During heating of the glass batch in this temperature range, the enthalpy of the glass batchruns via route r1 from A towards B in figure 2.1. The energy demand of this reaction,expressed in J per (1+α) kg of batch, is given by

∆H1 = H(Tr,onset)−H(T0) =∫ Tr,onset

T0

nb

∑i=1

mi cp,i dT, (2.5)

in which nb is the number of batch components, mi is the mass of batch component iper (1 + α) kg of batch, and cp,i is the temperature dependent heat capacity of batchcomponent i.

2. The temperature range between the onset for batch reactions Tr,onset and Tr,end, duringwhich the endo- and exothermic batch reactions occur, as described by

(1+α)batch(Tr,onset)∆H2−→ 1 melt(Tr,end)+α gas(Tr,end). (2.6)

In case no batch reactions occur in this temperature regime, the enthalpy change of theglass batch runs via route r2 towards C. However, because of the occurrence of (mainly)energy consuming batch reactions, the enthalpy change runs via the arbitrarily drawnroutes r2,1, r2,2 or r2,3 from B towards D. The route that is followed is dependent onthe temperature and time dependent chemical energy consumption during batch melting.The enthalpy change in this temperature range, which is dependent on the kinetics of theenergy consuming and producing batch reactions, expressed in J per (1 + α) kg of batchis described by

∆H2 = Hr,end −Hr,onset = Hr +∫ Tr,end

Tr,onset

nb

∑i=1

mi cp,i dT, (2.7)

in which Hr is the chemical energy demand for the batch reactions, nb is the number ofbatch components, mi is the temperature and time dependent mass of batch componenti per (1 + α) kg of batch and cp,i is the temperature dependent heat capacity of batchcomponent i, which is either a solid specie, a gaseous specie or a melt phase. The mixtureof all species i at temperature T and time t represents the composition of the reactingglass batch at temperature T and time t.

3Here it is assumed that the final heating temperature of the generated melt phase and the released gases isidentical.

2.2. Energy demand during heating of glass forming batches 17

3. The temperature range between Tr,end and the final heating temperature Tfinal, duringwhich the formed melt phases and the released gases are heated as described by

1 melt(Tr,end)+α gas(Tr,end)∆H3−→ 1 melt(Tfinal)+α gas(Tfinal). (2.8)

This equation describes the enthalpy change via route r3 from D towards E. The totalenergy demand of this reaction is given by

∆H3 =∫ Tfinal

Tr,end

mmelt cp,melt dT +∫ Tfinal

Tr,end

ngc

∑g=1

mg cp,g dT, (2.9)

in which ngc is the number of gaseous components, mmelt is the mass of melt phase pro-duced during melting of (1 + α) kg of batch, cp,melt is the temperature dependent heatcapacity of the formed melt phase, mg is the temperature dependent mass of gaseousspecie g per (1 + α) kg of batch, and cp,g is the temperature dependent heat capacity ofgaseous component g. The first right-hand-side term in equation 2.9 represents the dif-ference in sensible heat of the melt phase at T = Tfinal and T = Tr,end, whereas the secondright-hand-side term in equation 2.9 represents the difference in sensible heat of the gasphase at T = Tfinal and T = Tr,end.

According to equation 2.7, for the determination of the chemical energy demand of the glassbatch ∆Hr, the value for ∆H2 and information of the temperature dependent composition of thereacting glass batch are required. According to figure 2.1, the value for ∆H2 can be estimatedfrom

∆H2 = (H(Tfinal)−H(T0))− (∆H1 +∆H3) = ∆Ht − (∆H1 +∆H3) . (2.10)

The value for ∆H3 is deduced from

∆H3 = ∆Hmelt +α ∆Hgas, (2.11)

in which α is the mass of the gas phase released per unit mass of glass melt and ∆Hmelt and∆Hgas are the differences in sensible heat between Tr,end and Tfinal of the formed glass melt andthe released gases, respectively. Combining equations 2.7, 2.10 and 2.11 results in an expressionfor the chemical energy demand of the glass batch:

Hr = ∆Ht −(

∆H1 +∆Hmelt +α ∆Hgas)

−∫ Tr,end

Tr,onset

nb

∑i=1

mi cp,i dT. (2.12)

For a given onset and end temperature for batch reactions, the values for ∆H1 and ∆Hgas canbe calculated from thermodynamic tables [3]. The value for ∆Hmelt can either be derived froman empirical formula describing the temperature dependent heat capacity of melt phases [4] orby thermodynamic models [5]. Another possibility is to determine ∆Hmelt from the measuredsensible heat of the glass melt as function of temperature [2]. The total energy demand forheating and melting of the glass batch Ht requires knowledge of the formation enthalpies

• of the glass batch at T0,

• the formed melt phase at Tfinal, and


PSfrag replacements

T0 Tr,onset Tr,end Tfinal

H(T0)

H(Tr,onset)

H(Tr,end)

H(Tfinal)

A

B

C

D

E

Reaction zone

r2,1

r2,2

r2,3

r2,4

∆H1

∆H2

∆H3

Hr(Tonset)

Hr(Tend)

r1

r2

r3

Figure 2.2: Schematic representation of the energy demand of a glass batch as function of temperature.

• the released gases at Tfinal.

The enthalpy of both the glass batch and the released gases can be derived from thermodynamictables [3]. The formation enthalpy of the glass melt can be estimated from models describingthe thermodynamic behavior of glass melts [5–8] or it can be measured directly. Now, the onlyremaining unknown parameter required for calculation of the chemical energy demand of aglass batch by equation 2.12 is the sensible heat of the glass batch components between Tonsetand Tend.

In figure 2.2, the temperature dependent heat capacity of the unreacted glass batch in thereaction zone is characterized by the slope of line r2,3, whereas the heat capacity of the formedglass melt and the released gases in the reaction zone is characterized by the slope of line r2,2. Incase these slopes are identical, the chemical energy demand of the glass batch is independent onthe reaction path followed in the reaction zone. This means that ∆Hr(Tonset) equals ∆Hr(Tend).However, since these heat capacities are in general not identical, the chemical energy demandof a glass batch depends on the temperature range during which the batch reactions occur. Thedependency of the chemical energy demand ∆Hr on the temperature range in which the batchreactions take place is shown by the following example. For the ease of the calculation, itis assumed that the glass batch reacts at a distinct reaction temperature Tr and not during atemperature range. This simplifies equation 2.12 because the last right-hand-side term becomeszero.

Madivate et al. [2] measured the total energy demand of a glass batch, with the compositionlisted in table 2.1. For this glass batch, the value for the mass of gas that is released duringheating of a glass batch with mass (1+α) equals 0.212. The total energy demand of this glass

2.2. Energy demand during heating of glass forming batches 19

Table 2.1: Glass batch composition for which the chemical energy demand is calculated.

Batch component Concentration [wt. %]

SiO2 58.21Na2CO3 18.76CaCO3 5.24MgCO3 ·CaCO3 15.89

batch with respect to 298 K is given by

∆Ht = −514.9+1.818 T, (2.13)

in which the total energy demand is expressed in kJ per kg of glass and the temperature isexpressed in K. The sensible heat of the glass melt is given by

∆Hmelt = −614.0+1.373 T. (2.14)

Both ∆H1 and ∆Hgas are determined from thermodynamic tables [3]. Figure 2.3 shows thechemical energy demand of the glass batch expressed in J per kg of glass as function of thebatch reaction temperature. The chemical energy demand at 298 K, which equals 471 kJ kg−1

melt,is almost equal to the values presented by Conradt and Pimkhaokham [7] and Madivate et al. [2].

From this figure it is clearly seen that the chemical energy demand of the glass batch de-creases with increasing reaction temperature. The reason for this is the larger heat capacity ofthe unreacted glass batch components compared to the heat capacity of the mixture of the meltphase and the released gases. This means that the slope of line r2,3 in figure 2.2 is steeper thanthe slope of line r2,2.

Figure 2.4 shows the total energy demand during heating and melting of the glass batch forboth a reaction temperature of 1080 K (solid line) and a reaction temperature of 1400 K (dottedline). Line 1 indicates the total energy demand of the glass batch during heating from the initialtemperature up to 1080 K. Line 2 describes the instantaneous chemical energy consumption at1080 K, which is followed by line 3, which describes total energy demand during heating ofthe formed melt phase and the released gases. In case the reaction temperature equals 1400 Kinstead of 1080 K, the energy consumption during heating of the glass batch runs via line 1and followed by line 4. At 1400 K, the chemical energy is consumed instantaneously and theenthalpy change runs along line 5.

Resuming, an accurate prediction of the net chemical energy demand of a glass batch asfunction of time and temperature requires knowledge of the onset temperature and the kineticsof the energy consuming and producing batch reactions. As mentioned in the previous section,the main energy consuming batch reactions are the calcination reactions. In section 2.5, thekinetics of the calcination reactions occurring during heating of a float glass batch is studied.The determination of the calcination rate of these batch reactions is based on kinetic equationswhich are discussed in the next section.


400 600 800 1000 1200 1400 0

1x10 5

2x10 5

3x10 5

4x10 5

5x10 5

PSfrag replacementsReaction temperature [K]

Che

mic

alen

ergy

dem

and

[Jkg

−1

mel

t]

Figure 2.3: Chemical energy demand of the glass batch as function of the reaction temperature for thecomposition given in table 2.1 and with respect to 298 K.

200 400 600 800 1000 1200 1400 1600 0.0

5.0x10 5

1.0x10 6

1.5x10 6

2.0x10 6

2.5x10 6

PSfrag replacements

Temperature [K]

Tota

lene

rgy

dem

and

[Jkg

−1

mel

t]

1

2

3

4

5

Hr(1080 K)

Hr(1400 K)

Figure 2.4: Total energy demand of the glass batch as function of temperature for both a batch reactiontemperature of 1080 K (solid line) and 1400 K (dotted line).

2.3. Description of the kinetics of batch reactions 21

2.3 Description of the kinetics of batch reactionsIn this section, the kinetics of both homogeneous and heterogeneous chemical reactions, withand without the presence of a non-ideal liquid phase such as a glass melt is discussed. Basedon the equations presented in this section, the kinetics of calcination reactions occurring in afloat glass batch are studied in section 2.5.

2.3.1 IntroductionDuring heating and melting of glass batches, different types of reactions occur such as solid-state reactions, gas-solid reactions, gas-liquid reactions and solid-liquid reactions. In general,the overall rate of each of these heterogeneous batch reactions is dependent on the rate of threeseparate process steps, viz.:

• the transport of reactants to the reaction interface,

• the reaction rate at the reaction interface, and

• the transport of reaction products away from the reaction interface.

The overall rate of chemical reactions is governed by the process step with the lowest rate. Ingeneral, three reaction types are distinguished, i.e. reactions that are governed by mass transferprocesses, reactions that are governed by reaction kinetics and reactions that are governed bythermodynamic driving forces. To describe mass transfer governed processes knowledge is re-quired about the diffusive and convective transport of reactants and reaction products towardsand from the reaction interface. In general, for a mass transfer governed process, it is assumedthat the reaction rate is fast with respect to the rate of mass transfer. This means that for masstransfer governed processes it is assumed that at the reaction interface thermodynamic equilib-rium exists between the reactants and the reaction products.

Reaction kinetic governed processes require knowledge of reaction kinetic parameters. Forheterogeneous reactions, such as the main glass batches, these reaction kinetic parameters aredescribed in section 2.3.3. The description of kinetics of these heterogeneous reactions is basedon the description of reaction governed homogeneous reactions such as gas phase reactions.Although homogeneous batch reactions are not expected to contribute to a large extend to batchmelting, a description of homogenous reaction kinetics is given in section 2.3.2, because someaspects of the description of homogeneous reaction kinetics are applied for the description ofheterogeneous reaction kinetics. Section 2.3.4 describes the effect of the presence of non-idealliquid phases, such as glass melts, on the kinetic description of the rate of a chemical reaction.

2.3.2 Kinetics of homogeneous reactionsThe chemical conversion of reactants A and B into the reaction products C and D is given by

νA A+νB Bk f

kb

νC C +νD D, (2.15)

in which k f and kb are the rate constants of the forward and the backward reaction, and νA,νB, νC and νD are the stoichiometric reaction coefficients. Assuming that reaction 2.15 is a


reversible and elementary reaction4, the overall reaction rate of reaction 2.15 is given by

r = r f − rb = k f aνAA aνB

B − kb aνCC aνD

D , (2.16)

in which aA, aB, aC and aD are the activities of the species A, B, C and D. The rates of theforward and the backward reaction are indicated by r f and rb, respectively. For pure condensedphases, such as solids and pure liquids, the activity is defined as unity. The activity of a com-ponent i in a gas phase mixture, which is known as the fugacity f of component i, is givenby

fi = γi pi, (2.17)

in which γi is the activity (or fugacity) coefficient of component i and pi is the partial pressureof component i in the gas phase. For ideal gases, the fugacity or activity coefficient for eachspecie equals unity and the fugacity of component i equals the partial pressure of component i.

Similar to a gas phase, the activity of a component i in a liquid solution is given by

ai = γi xi, (2.18)

in which γi is the activity coefficient of component i and xi is the mole fraction of componenti in the liquid solution. For ideal liquid solutions, the activity coefficient of each specie equalsunity and the activity of component i equals the mole fraction of component i. However, fornon-ideal solutions such as glass melts, the activity coefficient deviates significantly from unity.For example, the activity coefficient for Na2O in a binary melt phase with a mole fraction ofNa2O equal to 0.25 and a mole fraction of SiO2 equal to 0.75 equals 2.4 ·10−11 at 1173 K [5].

Usually, a chemical reaction is the result of a number of successive elementary reactionsteps. In case reaction 2.15 is the overall reaction of the two successive elementary reactions,i.e.

νA A+νB Bk f

kb

νI I, (2.19)

and

νI Ik f

kb

νC C +νD D, (2.20)

in which I indicates an intermediate reaction product, two reaction rates, r2.19 and r2.20, areobtained, viz.

r2.19 = k f ,2.19 aνAA aνB

B − kb,2.19 aνII , (2.21)

andr2.20 = k f ,2.20 aνI

I − kb,2.20 aνCC aνD

D . (2.22)

The overall reaction rate of the reaction steps 2.19 and 2.20 is now governed by the rate of therate governing reaction step, which is either r2.19 or r2.20. Therefore, the determination of thekinetics of chemical reactions starts with the identification of the rate governing reaction step.

At thermodynamic equilibrium, the reaction rate of a chemical reaction equals zero, becausethe rate of the forward reaction equals the rate of the backward reaction. The ratio of the reaction

4A chemical reaction which proceeds exactly as expressed by the stoichiometric reaction equation [9]. How-ever, usually a chemical reaction is the result of a number of successive elementary reactions as will be described byequations 2.19 and 2.20. Than the chemical conversion proceeds via (a number of) intermediate reaction products.


rate constant of the backward and the forward reaction equals the reaction equilibrium constantKeq. The reaction equilibrium constant for reaction 2.15 is defined by

rb,2.15

r f ,2.15= 1 −→ Keq,2.15 =

k f ,2.15

kb,2.15=

aνCC aνD

D

aνAA aνB

B. (2.23)

Combining equations 2.16 and 2.23 results in a general expression for the reaction rate of areversible elementary chemical reaction as function of the rate constant of the forward reactionk f , the reaction equilibrium constant Keq, the activities of nr participating reactants and theactivities of nrp participating reaction products:

r = k f

(

nr

∏i=1

aνii − 1

Keq

nrp

∏j=1

aν jj

)

. (2.24)

The reaction equilibrium constant Keq is obtained from the standard chemical potential µ0 ofthe reactants and the reaction products according to

lnKeq = − 1R T

(

nrp

∑j=1

ν j µ0j −

nr

∑i=1

νi µ0i

)

= −∆G0r

R T, (2.25)

in which R is the gas constant, T is the absolute temperature and ∆G0r is the standard Gibbs free

energy of reaction. The values for the standard standard chemical potential for a large numberof species are tabulated in thermodynamic tables [3].

For gas phase reactions, the rate constants k f and kb are temperature dependent accordingto the so-called Arrhenius equation, which is given by

k = A e−Ea

R T , (2.26)

in which A is the pre-exponential factor and Ea is the reaction activation energy. The exponentialtemperature dependency of the gas phase rate constants is based on the exponential energydistribution over the gas molecules in a system at a specific temperature, which is known as theBoltzmann-distribution.

Resuming, the rate of a reversible homogeneous chemical reaction is described by equation2.24, in which the rate constant is given by an Arrhenius equation. In the next section, thekinetics of heterogeneous reactions, which is the major reaction type occurring during meltingof glass batches, is discussed.

2.3.3 Kinetics of heterogeneous reactions

The description of heterogeneous reactions originates from the kinetic description of solid-statereactions, which are either diffusion or reaction governed processes. Both for diffusion and re-action kinetics governed solid-state reactions, the overall rate of the reaction is dependent on thegeometry of the reacting species. Dependent on the size and the shape of the reacting species,four different processes are generally distinguished, viz. a one-, two- or three-dimensional dif-fusion governed process or a reaction governed process. Therefore, for the description of the


kinetics of solid-state reactions, different geometrical models have been developed. In general,the rate of a heterogeneous reaction is described by

∂ξi

∂t= k f (ξi) , (2.27)

in which ξi is the conversion of specie i 5, k is the reaction rate constant, and f (ξi) is theso-called reaction mechanism function for specie i. Table 3.1 lists the main generally appliedreaction types and their function f (ξ) derived from the reaction mechanism.

Table 2.2: Code, reaction type and reaction mechanism function (see for an overview of these reactionmechanism functions [10]).

Code Reaction type f (ξ)

Fn nroth order reaction (1−ξ)nro

D1 One-dimensional diffusion 12ξ

D2 Two-dimensional diffusion −1ln(1−ξ)

D3 Three-dimensional diffusion (Jander’s type) 1.5(1−ξ)2/3

1−(1−ξ)1/3

D4 Three-dimensional diffusion (Ginstling-Brounstein type) 1.5(1−ξ)−1/3−1

R2 Two-dimensional phase boundary reaction 2(1−ξ)1/2

R3 Three-dimensional phase boundary reaction 3(1−ξ)2/3

Similar to the homogeneous gas phase reaction kinetics, the reaction rate constant k is as-sumed to be temperature dependent according to the Arrhenius equation. In contrast to gasphase reactions, the energy distribution amongst the, in this case, immobilized constituents of(crystalline) species is not represented by the Boltzmann-distribution. For this reason, the ap-plicability of the Arrhenius equation, describing the exponential temperature dependency of therate constants for reactions at which solids participate, has been questioned for a long time byGalwey and Brown [11]. In solids, energy transfer is determined by either phonon or photontransfer. Recently, Galwey and Brown [11] reported that the energy distributions of these twoenergy transfer modes also show an exponential temperature dependency, which allows the useof the Arrhenius equation for describing the exponential temperature dependency of the rateconstants for solid-state reactions.

When applying equation 2.27, the reaction activation energy Ea and the pre-exponential fac-tor A for a chemical reaction can be derived by plotting the left-hand-side term of equation 2.28

5The conversion ξi of a specie i is defined as the ratio of the actual amount of specie i with respect to the initialamount of specie i.


versus the reciprocal absolute temperature.

ln(

∂ξ∂t

)

− ln f (ξ) = lnA− Ea

R T(2.28)

The intercept and the slope of this plot provide the values for lnA and −Ea/R, respectively. Ingeneral, the selection of the reaction mechanism function f (ξ) for a specific reaction is basedon the correlation coefficient describing the accuracy of fit of the reaction mechanism functionwith the measured conversion data. The reaction mechanism function f (ξ) providing the largestvalue for the correlation coefficient is identified as the reaction mechanism function describingthe conversion mechanism for the reaction that is investigated. According to Opfermann [12],the selection of the reaction mechanism function based on the correlation coefficient does notalways seem to be statistically well-founded. The difference in correlation coefficients for thedifferent reaction mechanism functions is often insignificant. It is also mentioned that the re-action mechanism based on this selection approach is likely to provide no accurate informationabout the real reaction mechanism. For this reason, the reaction kinetic parameters are not ex-pected to have a physical meaning. Therefore, the reaction activation energy determined by thisapproach will be denoted throughout this chapter as an apparent reaction activation energy.

Equation 2.27 can be regarded as a kinetic equation describing the reaction rate of a het-erogeneous reaction which occurs far from thermodynamic equilibrium. In order to predict thekinetic behavior of a heterogeneous reaction close to thermodynamic equilibrium, the reactionequilibrium has to be taken into account in equation 2.27. Pokol [13] discussed different ap-proaches to take thermodynamic equilibrium into account. Similar to homogeneous reactions,the reaction rate of a heterogeneous reaction can be described by an equation similar to equa-tion 2.24. Equation 2.29 describes the rate of a heterogeneous reaction both far and close tothermodynamic equilibrium in which Keq is the reaction equilibrium constant and Ka, givenby equation 2.30, describes the ratio of the actual activities of the reactants and the reactionproducts.

r = k f f (ξ)nr

∏i=1

aνii

(

1− Ka

Keq

)

(2.29)

Ka =∏

nrpj=1 aν j

j

∏nri=1 aνi

i(2.30)

The forward reaction of a reversible reaction is favored in case Ka < Keq, whereas the backwardreaction is favored in case Ka > Keq. For example, for the reversible calcination of limestone,which is given by CaCO3(s) ⇐⇒ CaO(s) + CO2(g), equation 2.29 simplifies to equation 2.31,because the activities of solid species equal unity by definition. The partial CO2-pressure inthe atmosphere surrounding the limestone particle is given by pCO2,a, whereas the tempera-ture dependent equilibrium partial CO2-pressure during limestone decomposition is given bypCO2,eq.

r = k f f (ξ)

(

1− pCO2,a

pCO2,eq

)

(2.31)

In case the actual partial CO2-pressure in the atmosphere surrounding the limestone particlesexceeds the thermodynamic equilibrium partial CO2-pressure, the carbonization reaction, i.e.CaO(s) + CO2(g) ⇐⇒ CaCO3(s), is favored, whereas in case pCO2,a is lower than pCO2,eq, thecalcination reaction, CaCO3(s) ⇐⇒ CaO(s) + CO2(g), is favored.


Resuming, the kinetics of reversible heterogeneous reactions is described by equation 2.29.Up to now, the kinetics of chemical reactions at which reactants participate with a fixed stoi-chiometry are discussed. However, melt phases formed during heating of glass batches have nofixed composition. In the next section, the effect of melt phases on the kinetics of heterogeneousreactions is discussed.

2.3.4 Heterogeneous reaction kinetics with participation of melt phasesThe reactive decomposition of soda ash at the presence of a binary sodium silicate melt is givenby

y Na2CO3(s)+xNa2O · (1-x)SiO2(l) ⇐⇒

(1+ y)[ (

x+ y1+ y

)

Na2O ·(

1− x1+ y

)

SiO2

]

(l)+ y CO2(g) (2.32)

Similar to equation 2.29, the kinetics of the soda ash calcination can be described by

r = k f f (ξ) amelt1(l)

(

1− Ka

Keq

)

, (2.33)

with:Keq = e

−∆G0r

R T , (2.34)

and

Ka =a1+y

melt2 pyCO2(g)

amelt1, (2.35)

in which amelt1 is the ’activity’ of the reacting melt phase and amelt2 is the ’activity’ of theformed melt phase. The calculation of the activity of the reacting and formed melt phases is nottrivial. In order to explain the effect of melt phases on the thermodynamics of batch reactions,the approach for describing the thermodynamics of molten oxide systems, such as glass meltsystems, presented by Shakhmatkin et al. [8] is discussed.

In general, the energy state of a solid, liquid or gas phase mixture is given by the Gibbs freeenergy of the mixture according to

G =nms

∑i=1

xi µ0i +R T

nms

∑i=1

xi lnxi +GE , (2.36)

in which G is the molar Gibbs free energy of the mixture, nms is the number of species in themixture, xi is the mole fraction of specie i in the mixture, µ0

i is the standard chemical potential ofspecie i, R is the universal gas constant, T is the absolute temperature and GE is the excess Gibbsfree energy. For ideal mixtures, such as gas phase mixtures, the excess Gibbs free energy equalszero and the Gibbs free energy of the mixture can be calculated from the tabulated values of thestandard chemical potential of the mixture components and their mole fraction in the mixture asgiven by equation 2.36. However, glass melts do not behave as an ideal mixture of the individualoxides. The Gibbs free energy of a glass melt is given by

G =nms

∑i=1

xi µ0i +R T

nms

∑i=1

xi lnxi +R Tnms

∑i=1

xi lnγi, (2.37)


in which γi is the activity coefficient of oxide i in the glass melt. For species in an ideal mixture,the activity coefficient equals unity. The activity coefficient of Na2O in the non-ideal binarymelt phase with a mole fraction of Na2O equal to 0.25 and a mole fraction of SiO2 equal to 0.75equals 2.4 ·10−11 at 1473 K [5].

To describe the thermodynamics of glass melts, an expression for the activity coefficient ofall species in the glass melt as function of glass melt composition and temperature is required,i.e. lnγi = f (x1, ..,xi, ..,xn,T ). The main problem with non-ideal solutions is that setting-up amodel describing the interaction between solution components and the determination of theseinteraction parameters is a complex and laborious task. In general, thermodynamic modelsdescribing the deviation from ideal behavior of a solution formulate expressions for the ac-tivity coefficients of each specie i in the non-ideal solution as function of the composition ofthe non-ideal solution and the temperature, i.e. lnγi = f (x1, ..,xi, ..,xn,T ). For a glass melt,these expressions are based on assumed structural models of the glass melt, such as the quasi-chemical model [5]. The parameters of these models describing the interaction between theoxides in the glass melt are derived from phase diagrams of (subsystems of) the complete glassmelt and from measured thermodynamic properties of the glass melt such as for example partialvapor pressures of the glass melt oxides.

Shakhmatkin et al. [8] developed a thermodynamic model, with which the activity coef-ficients γ of glass melt oxides can be predicted based on thermodynamic properties of purecomponents, which are tabulated in several references such as Knacke et al. [3]. According toShakhmatkin et al. [8], a glass melt can be considered as an ideal solution of so-called stoichio-metric compounds in the vitreous state. These stoichiometric compounds have a stoichiometry,which is similar to the crystalline compounds existing in the phase diagrams of the subsys-tems of the glass melt. For example, for a binary melt phase containing the oxides SiO2 andNa2O, the main important stoichiometric compounds of the melt phase are SiO2, Na2O ·2SiO2,Na2O ·SiO2, 2Na2O ·SiO2 and Na2O (see figure 2.5). The Gibbs free energy of a glass melt isnow described by

G =nms

∑i=1

xsc,i µ0sc,i +R T

nms

∑sc,i=1

xsc,i lnxsc,i, (2.38)

in which sc denotes the stoichiometric compound in the vitreous state6. Minimization of theGibbs free energy with respect to the mole fractions of the stoichiometric compounds and takinginto account the element balances of the oxides in the glass melt, results in the thermodynamicmost stable composition of the glass melt expressed in the mole fractions of the stoichiometriccompounds.

Conradt proposed a similar model [15], in which the composition of a glass melt can be ap-proximated by an ideal mixture of the so-called ’nearest stoichiometric compounds’7, which arein thermodynamic equilibrium with the pure oxides of the glass melt. For example, for a sodiumsilicate melt phase with a molar SiO2 fraction of 0.75, the glass melt mainly contains the near-est stoichiometric compounds SiO2 (in which xSiO2=1) and Na2O ·2SiO2 (in which xSiO2=0.67).Now, the glass melt in thermodynamic equilibrium can be described by an equilibrium between

6In general, Shakhmatkin et al. use the crystalline state of the stoichiometric compounds as reference stateinstead of the vitreous state.

7Conradt uses the vitreous state of the stoichiometric compounds as reference state. Below the liquidus tem-perature of a glass melt, Conradt estimates the Gibbs free energy of the vitreous state.


Figure 2.5: Binary phase diagram of SiO2 and Na2O [14]. The temperature is expressed in C.

the stoichiometric compounds according to

Na2O∗(l)+2 SiO∗2(l) ⇐⇒ Na2O ·2SiO∗

2(l), (2.39)

in which Na2O∗(l), SiO∗2(l) and Na2O ·2SiO∗

2(l) represent the stoichiometric compounds in thevitreous state. The reactive calcination of solid soda ash with a binary melt phase in which themole fraction of SiO2 equals 0.75 can now be described by

Na2CO3(s)+2SiO∗2(l) ⇐⇒ Na2O ·2SiO∗

2(l)+CO2(g). (2.40)

Using equation 2.29, the calcination rate of soda ash is given by

r = k f f (ξ) x2SiO∗

2

(

1− Ka

Keq

)

, (2.41)

in whichKa =

xNa2O ·2SiO∗2

pCO2

x2SiO∗

2

, (2.42)


and xSiO∗2

and xNa2O ·2SiO∗2

are the mole fractions of the stoichiometric compounds SiO2 andNa2O ·2SiO2 in the reacting glass melt. Resuming, using the approach for modelling the ther-mochemistry of glass melts proposed by Conradt and Shakhmatkin et al., the kinetics of ther-modynamic driving force governed batch reactions at which melt phases participate can bedescribed.

Next to the presence of oxides in a glass melt, also dissolved gases can be present. A gaseousspecie can either be physically or chemically dissolved in the glass melt [16, 17]. The physicaldissolution of gases in a glass melt concerns the settling of gas atoms or molecules in holes inthe network of the glass melt matrix. On the other hand, gases can dissolve in the glass melt viaa reaction with glass melt components. For example, CO2 may react with free oxygen presentin the glass melt according to

CO2(g)+O2−(m) ⇐⇒ CO2−3 (m). (2.43)

The thermodynamic model presented by Shakhmatkin et al. and Conradt, incorporate the chem-ically dissolved CO2 by allowing the stoichiometric compound Na2CO3 to be present in theglass melt. The description of reactive dissolution of Na2O in the binary glass melt accordingto equation 2.32, requires both the thermodynamic equilibrium data of equation 2.40 and of

Na2CO3(s) ⇐⇒ Na2CO∗3(m), (2.44)

andNa2CO∗

3(m) ⇐⇒ Na2O∗(m)+CO2(g), (2.45)

in which Na2CO∗3(m) is the stoichiometric compound Na2CO3 in the melt phase, which repre-

sents the chemical dissolved CO2. Now, the calcination rate of soda ash is not only given byequation 2.41, which describes the reactive calcination of soda ash with the melt phase, but alsoby

r = k f f (ξ) x∗Na2CO3

(

1− Ka

Keq

)

, (2.46)

which describes the thermal calcination of chemically dissolved Na2CO3 in the glass melt. Thechemically solubility of Na2CO3 is given by x∗Na2CO3

and Ka is given by

Ka =xNa2O∗(m) pCO2(g)

xNa2CO∗3(m)

. (2.47)

2.3.5 Characteristics of calcination reactionsCalcination reactions, such as the decomposition of dolomite (MgCO3 ·CaCO3) and limestone(CaCO3) during melting of float and container glass batches and the decomposition of SrCO3and BaCO3 during melting of TV-panel batches, are the major decomposition reactions oc-curring during glass batch melting. According to Paulik and Paulik [18] and Young [19], theonset for calcination reactions is the formation of nucleı from which CO2 can be released. Thecalcination reaction leaves a porous layer, containing the initially formed decarbonated oxide.Through this porous layer, the CO2 resulting from the calcination reaction can easily escape.However, structural ordering of the formed oxides may result in a second reaction layer (seefigure 2.6) in which recrystallized oxides are present with a lower porosity. This low porositylayer hinders the CO2 transport from the reaction interface through the oxide layer and this may


retard the overall calcination rate.The rate of calcination reactions is dependent on three processes (see figure 2.6 for the

calcination of MCO3, in which M represents an alkaline-earth atom such as Ca or Mg), viz.:

• the heat transport q, which is required for the endothermic calcination reaction, from theouter radius r0, through the reaction layer towards the reaction interface ri,

• the calcination at the reaction interface ri, and

• the (diffusive) CO2 gas flux JCO2 from the reaction interface ri through the porous reactionlayer into the surrounding atmosphere.

PSfrag replacements

ri

ro

q JCO2

MCO3 grain

porous layer

MCO3(s) ⇐⇒MO(s)+CO2(g)

MC

O3

grai

n

Non

crys

talli

zed

MO

Cry

stal

lized

MO

Surr

ound

ing

Figure 2.6: Schematic representation of the different processes occurring during calcination of a MCO3particle, at which M represents an alkaline-earth atom such as Ca or Mg. The initial radiusof the MCO3 particle is indicated by r0, whereas the radius of the unreacted MCO3 particleis indicated by ri. The heat flux through the oxide layer containing MO is indicated by q,whereas the CO2 flux from the reaction interface at r = ri through the oxide layer is indicatedby JCO2 . In general, the oxide layer is composed of two layers. The first layer contains theinitial formed MO, whereas the second layer contains the recrystallized MO.

Because calcination reactions are accompanied with weight loss, a suitable technique for thedetermination of the kinetics of the calcination reactions is thermogravimetric analysis (TGA),which is discussed in section 2.4.1. The reaction kinetic parameters Ai and Ea,i describing thecalcination of carbonate i can be derived from transient TGA with a constant heating rate βusing equation 2.28, which results in:

ln(

∂ξi

∂T

)

+ lnβ− ln f (ξi) = lnAi −Ea,i

R T(2.48)

The intercept and the slope of this plot provide the values for lnAi and −Ea,i/R, respectively.To determine the rate of a calcination reaction, the experimental conditions during TGA have

2.4. Experimental techniques 31

to be selected such that additional mass and heat transfer problems are avoided as much aspossible. Accurate and reliable determination of the kinetic parameters requires a low heatingrate, a small sample volume and a vacuum technique for a quick removal of the released gases.A good alternative for vacuum is the use of a carrier gas which is purged over the sample with ahigh flow rate for a fast removal of the released gases from the sample. However, accumulationof gases in the small pores in the reaction zone will not be prevented.

2.4 Experimental techniquesThis section describes the experimental techniques and procedures, which are used for the iden-tification of the calcination reactions in a float glass batch and the determination of the calcina-tion rate of float glass batch. Sections 2.4.1 and 2.4.2 describe the backgrounds of thermogravi-metric and differential thermal analysis, whereas in section 2.4.3 the identification procedurefor analysis of (intermediate formed) crystalline phases in quenched heat-treated float glassbatch samples is discussed.

2.4.1 Thermogravimetric analysisDuring thermogravimetric analysis8, the weight change of a single component or a mixture ofcomponents during heating is recorded. As an example, figure 2.7 shows the measured normal-ized mass of a single CaCO3 particle during ramp heating with 10 K min−1 and applying a 100ml min−1 N2-flow in a TGA apparatus9. The onset temperature for calcination reactions derivedfrom TGA is defined as the intersection of the tangent of the weight loss curve with the zeropercent weight loss line. The tangent of the weight loss curve is determined at the temperatureat which the weight loss rate is maximal.In figure 2.7, the onset temperature for CaCO3 calcination equals about 953 K, although weightloss is observed already at 823 K. Complete calcination of CaCO3 is achieved at 1073 K.

Although TGA is the most common technique for studying decomposition reactions, theapplicability and results of TGA-experiments are influenced by the following encountered phe-nomena:

• The self-cooling effect:Because decomposition reactions are in general strong endothermic reactions, cooling ofthe reactants and reaction products during decomposition is possible. Due to cooling,the decomposition kinetics decrease, which results in a wrong estimation of the kineticdecomposition parameters determined with equation 2.48 [20].

• The Topley-Smith effect:The heating of a sample in a TGA-apparatus is caused by convective and radiative heattransport from a hot purge gas flow. At high temperatures, the sample in the TGA-apparatus is mainly heated by radiative heat transfer. Topley and Smith discovered anoma-lous variation of the dehydration rate with increasing partial water vapor pressure. The

8Further in this thesis, ’thermogravimetric analysis’ will be denoted as TGA.9All TGA-experiments in this study are performed with the SDT 2960 apparatus from TA Instruments com-

bined with the Universal Analysis software package. An empty Platinum crucible was used as reference, whereasthe amount of sample to be investigated varied between 0.5 mg up to 50 mg.


800 900 1000 1100 1200 0.0

0.2

0.4

0.6

0.8

1.0

PSfrag replacementsTemperature [K]

Nor

mal

ized

mas

sof

alim

esto

nepa

rtic

le[-

]

Figure 2.7: Measured weight loss during heating of CaCO3 when ramp heated with 10 K min−1 in a 100ml min−1 N2-flow.

dehydration rate decreases with increasing partial water vapor pressure on one hand. Onthe other hand, the dehydration rate increases due to enhanced heating of the sample byimproved radiative heat transfer with increasing partial water vapor pressure. The pres-ence of the Topley-Smith effect also results in less accurate estimation of the reactionmechanism and kinetic decomposition parameters. The Topley-Smith effect has not onlybeen observed with dehydration reactions, but also during calcination reactions, like thedecomposition of CaCO3 [20].

• The kinetic compensation effect:The shape of a TG-curve is not only dependent on the kinetics of the decomposition reac-tion, but also on the experimental conditions at which the TGA-experiment is performed.Gallagher and Johnson [21] measured a dependency of the apparent reaction activationenergy for the calcination of CaCO3 on both sample weight and heating rate. An increasein sample weight and heating rate resulted in a decrease of the apparent reaction activa-tion energy. Also a decrease of the apparent reaction activation energy was observed incase of a decrease in partial CO2-pressure in the atmosphere surrounding the CaCO3 par-ticle. The experimental conditions influence the shape of the TG-curve and therewith theapparent reaction activation energy deduced from it. In general, the steeper a TG-curve,the higher the value for the calculated apparent reaction activation energy. Because ofthe large variety in experimental conditions at which the calcination of CaCO3 is studied,also a large variation in apparent reaction activation energies are presented in literature

2.4. Experimental techniques 33

for the calcination of CaCO3. However, it appeared that plotting the natural logarithmof the pre-exponential factor versus the apparent reaction activation energy resulted in alinear relation. This linear relation is known as the kinetic compensation effect, which isdescribed by

lnA = ckc1 + ckc2 Ea, (2.49)

in which ckc1 and ckc2 are constants.An increase in the apparent reaction activation energy is compensated by an increase inthe value for the pre-exponential factor A. Combining equation 2.49 with the Arrheniusequation describing the temperature dependency of the reaction rate constant, results in

lnA = lnk +1

R TEa, (2.50)

in which k is the reaction rate constant. From the slope of the plot of the natural logarithmof the pre-exponential factor versus the apparent reaction activation energy, a value for Tis obtained. This temperature T reflects a reasonable value for the average temperature ofthe temperature range from which the kinetic compensation plot is made. The interceptof the plot reflects the value for the reaction rate at the average temperature.The dependency of the apparent reaction activation energy on experimental conditionssuch as sample size, heating rate and partial CO2-pressure indicates that the reaction ofinterest is governed by either heat transfer or mass transfer limitations. The reaction ki-netic parameters appear to be strongly dependent on the experimental conditions at whichthe thermal analysis are performed. Plotting the natural logarithm of the pre-exponentialfactor versus the apparent reaction activation energy provides information about the re-action mechanism as function of experimental conditions. In case the reaction kineticparameters for different experimental conditions satisfy the kinetic compensation equa-tion given by equation 2.50, it can be assumed that the reaction mechanism is independenton the experimental conditions. The kinetic compensation effect reflects the effect of ex-perimental conditions on the reaction kinetic parameters and can be used as a check forevaluation the reaction mechanism as function of experimental conditions. The real re-action kinetic parameters are the values for the apparent reaction activation energy andthe pre-exponential factor at experimental conditions at which both mass and heat trans-fer limitations are excluded as much as possible. These conditions are small amounts ofsample, slow heating rates and high purge gas flows.

In section 2.5.2, the reaction activation energy for limestone decomposition is determined asfunction of heating rate assuming a first order reaction mechanism. It will be shown that thereis no significant dependency of the reaction activation energy on the heating rate. Therefore,the presence of the kinetic compensation effect during the study can be neglected. The heatingrate and the sample size are chosen such that the heat transport towards the reaction interface isnot the rate limiting step.

To evaluate if the flow rate of the carrier gas has an impact on the calcination rate of lime-stone, the onset temperature for calcination as function of flow rate has been evaluated. Increas-ing the flow rate from the standard value of 100 ml min−1 did not show a significant change inonset temperature for calcination. Therefore it can be assumed that no gases are accumulatedin the crucible containing the sample. However, this study does not provide information aboutaccumulation of gases in the interior of the particle because the diffusive transport of CO2 isalso dependent on the porosity of the reaction layer.


2.4.2 Differential thermal analysis

Next to TGA-experiments, differential thermal analysis (DTA) are performed. From DTA-experiments the occurrence of endo- and exothermic reactions are observed. The temperature atwhich these DTA-peaks occur provide information about the mechanism of the reaction which isinvestigated. The TGA- and DTA-experiments are performed at the same time in the SDT2960apparatus from TA Instruments.

2.4.3 Phase analysis on partly reacted glass batches

Next to the use of thermal analysis, the mechanism of calcination reactions is supported byphase analysis on quenched samples of partly reacted mixtures of glass batch components. Be-low, both a description is given of the melting experiments for the preparation of the samples forphase analysis and the experimental conditions at which the phase analysis with X-ray diffrac-tion were performed.

For the batch melting experiments, about 5 grams of a mixture of the glass batch componentsis prepared by hand mixing of the batch components10. The mixed batch sample is brought in aconic Pt-crucible with a bottom diameter of 30 mm, a top diameter of 55 mm and a height of 55mm. Charging the 5 grams of batch in the Pt-crucible results in a thickness of the batch sampleof about 5 mm. The Pt-crucible, containing the batch sample, is positioned at room temperaturein an electrically heated laboratory furnace (Nabertherm HT16/17). The furnace temperatureis controlled by a thermocouple, which is located about 120 mm above the Pt-crucible. Thefurnace temperature is programmed with a defined heating rate up to a final heating tempera-ture. The temperature of the batch sample itself is monitored by an S-type (Pt/95%Pt-5%Rh)thermocouple, from which the joint is completely immersed in the batch sample during heating.Throughout this chapter, the indicated temperature of the batch sample is the temperature mea-sured by this S-type thermocouple. The set-up for the batch melting experiments is depicted infigure 2.8.

However, despite the small amount of batch used during the current study, temperature dif-ferences have been measured throughout the batch sample. Above 1100 K, the temperaturedifference in radial direction between the center of the batch sample and the crucible is lessthan 14 K. This temperature difference will be denoted as the uncertainty in the batch sampletemperature. After ramp heating of the batch sample up to a specified temperature, the Pt-crucible containing the heated batch sample is withdrawn from the furnace and directly cooledby dipping the Pt-crucible in a water bath at room temperature. Direct contact of the heatedbatch sample with the cooling water is prevented. The cooling rate during the quenching periodis estimated to be approximately 20 K s−1. Because of this high cooling rate, it is assumed thatonly the pure molten substances, such as soda ash or non-silicate salts are capable of recrystal-lizing. Directly after quenching, the residual weight of the partly molten glass forming batchsample is measured. The value of the weight loss during heating, which is mainly caused bythe release of CO2 during heating, is required for quantitative analysis of the total amount ofcrystalline quartz in the thermal treated glass batch sample.

10Repeated melting experiments and analysis of the fraction crystalline quartz for glass batches with identicalcompositions showed that the differences in measured fraction crystalline quartz were less than 1 % absolute.This indicates that the preparation of the glass forming batch samples by hand mixing provides a reproduciblehomogeneity.

2.5. Calcination of a float glass batch 35

Heating elements

Batch sample

Pt-crucible

5 mm

55 mm

55 mm

280 mm

260 mm

Closed control thermocouple

S-type thermocouple

TIC TI Insulation bricks

30 mm

Figure 2.8: Front view of the experimental set-up for batch melting experiments.

After quenching, the batch sample is crushed with a tungsten carbide mortar resulting ina residual particle size of the crushed batch sample below 10 µm. From this crushed sample,a tablet is pressed, which is used for qualitative phase analysis with X-ray diffraction. Iden-tification of the phases present in a mixture of crystalline species is achieved by comparisonof the X-ray diffraction patterns from the unknown sample with an internationally recognizeddatabase containing a large amount of reference patterns. The diffraction patterns are collectedusing a Philips X’Pert X-ray diffractometer using Cu-Kα radiation. In general, the diffractiondata are collected from 2θ equals 50 up to 800 in steps of 0.020 using a counting time of 1.0s ·step−1.

2.5 Calcination of a float glass batch

In this section, the calcination behavior of both single float batch components and mixturesof a complete float glass batch is studied. The raw material components of the float glassbatch are silica sand (SiO2), soda ash (Na2CO3) and dolomite (MgCO3 ·CaCO3). Section2.5.1 describes the calcination behavior of the MgCO3-part of dolomite. In section 2.5.2, thecalcination behavior of limestone is discussed. The decomposition of soda ash is described insection 2.5.3, whereas the calcination behavior of the complete float glass batch is discussedin section 2.5.4. The mechanism of melting the float glass batch is discussed in section 2.5.5.Finally, the temperature dependent chemical energy demand during heating of the float glassbatch is described in section 2.5.6.


2.5.1 Calcination of dolomiteAccording to Ozao et al. [22], mineral dolomite contains both MgCO3 ·CaCO3 and a smallamount of free MgCO3 present as magnesite11. The thermal decomposition of mineral dolomiteis regarded as a three-step decomposition process:

1. The thermal decomposition of MgCO3 present as magnesite:

MgCO3(s)MgO(s)+CO2(g) (2.51)

2. The thermal decomposition of the MgCO3-part of the dolomite:

MgCO3 ·CaCO3(s)MgO(s)+CO2(g)+CaCO3(s) (2.52)

3. The thermal decomposition of the CaCO3-part of the dolomite:

CaCO3(s)CaO(s)+CO2(g) (2.53)

The general form of the Gibbs free energy of reaction is given by

∆Gr =

(

nrp

∑j=1

νi µ0i −

nr

∑i=1

ν j µ0j

)

+R T lnKa = ∆G0r +R T lnKa, (2.54)

in which m is the number of reactants, n is the number of reaction products, νi is the stoichio-metric reaction coefficient of specie i, µ0

i and µ0j are the standard molar Gibbs free energy of

species i and j, Ka is the ratio of the actual activities of the reactants and reaction products(see equation 2.30) and ∆G0

r is the standard Gibbs free energy of reaction. At thermodynamicequilibrium, Ka is given by Keq and the Gibbs free energy of reaction equals zero. Equation2.54 simplifies to:

∆Gr = 0 = ∆G0r +R T lnKeq. (2.55)

Because the activities of solid species equal unity by definition, the reaction equilibrium con-stant Keq for the three calcination reactions described above, equals the partial equilibriumCO2-pressure. Now, the equilibrium partial CO2-pressure surrounding the carbonates at ther-modynamic equilibrium is given by

pCO2,eq = e−∆G0

rR T . (2.56)

In case the equilibrium partial CO2-pressure surrounding the (partly decomposed) dolomite,calculated with equation 2.56, exceeds the partial CO2-pressure in the atmosphere above thedolomite particle, dolomite starts to decompose. Figure 2.9 shows both the standard Gibbs freeenergy of reaction ∆G0

r for these three decomposition reactions and isobars for CO2. The cal-cination reactions occur in the case that the partial CO2-pressure surrounding the carbonate,characterized by the isobars, is positioned above the line describing the standard Gibbs freeenergy of reaction. The decomposition temperatures for MgCO3, MgCO3 ·CaCO3 and CaCO3,in a 1 bar CO2-atmosphere equal 576 K, 687 K and 1170 K, respectively. Above this decompo-sition temperature the carbonates decompose completely.

11Mineralogic name of free MgCO3


600 700 800 900 1000 1100 1200 1300 -6.0x10 4

-4.0x10 4

-2.0x10 4

0.0

2.0x10 4

4.0x10 4

6.0x10 4

8.0x10 4

1.0x10 5

PSfrag replacements

CaCO3 (s) ⇐⇒ CaO(s)+CO

2 (g)

MgCO3 ·CaCO

3 (s) ⇐⇒MgO(s)+CO

2 (g)+CaCO3 (s)

MgCO3 (s) ⇐⇒

MgO(s)+CO2 (g)

Calcination is favored

Carbonization is favored

Temperature [K]

Stan

dard

Gib

bsfr

eeen

ergy

ofre

actio

n[J

mol

−1 ]

10−4

10−3

10−2

10−1

10−0.3

100

100.7

101

102

Figure 2.9: Standard Gibbs free energy of decomposition reactions of MgCO3, MgCO3 ·CaCO3 andCaCO3. The dashed lines are isobars for the partial CO2-pressure in bar surrounding thecarbonates.

Thermal calcination of the MgCO3-part of dolomiteThe thermal behavior of dolomite was studied by thermogravimetric analysis. Figure 2.10shows the degree of dolomite calcination12 as function of temperature in both a N2- and a 1bar CO2-atmosphere.

It is remarked that during these TGA of dolomite no calcination peak of free MgCO3 isobserved. However, in general free magnesite is present in dolomite as is shown in figure2.11. In the CO2-atmosphere, a two-step decomposition process is observed. Point A in fig-ure 2.10 marks the measured onset temperature for CaCO3 calcination, which is almost similarto the thermodynamic onset temperature for calcination of individual limestone. In the N2-atmosphere, complete calcination is already observed at 1030 K, which indicates that the cal-cination of CaCO3 is dependent on the partial CO2-pressure. The small temperature range inwhich the calcination of limestone is completed in a CO2-atmosphere indicates high calcinationkinetics, which will be discussed in section 2.5.2.

12At this position the degree of dolomite calcination is defined as the ratio of the current weight loss with respectto the weight loss after complete calcination of dolomite in MgO and CaO.


800 900 1000 1100 1200 1300 0.0

0.2

0.4

0.6

0.8

1.0

PSfrag replacements

Temperature [K]

Deg

ree

ofdo

lom

iteca

lcin

atio

n[-

]

A

1

2

CaC

O3(

s)⇐⇒

CaO

(s)+

CO

2(g)

MgCO3 ·CaCO3(s) ⇐⇒MgO(s)+CO2(g)+CaCO3(s)

MgCO3 ·CaCO3(s) ⇐⇒MgO(s)+CaO(s)+2CO2(g)

Figure 2.10: Measured degree of dolomite calcination (MgCO3 ·CaCO3(s) −→ MgO(s) + CaO(s) +2 CO2(g)) as function of temperature in both a N2-atmosphere (1) and a 1 bar CO2-atmosphere (2). The applied heating rate equals 10 K min−1.

850 900 950 1000 1050 1100 0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.000

0.005

0.010

0.015

0.020

0.025

PSfrag replacements

Temperature [K]

Deg

ree

ofM

gCO

3ca

lcin

atio

n[-

]

Dol

omite

calc

inat

ion

rate

[K−

1 ]

Calcination of free MgCO3Calcination of MgCO3

part of dolomite

Figure 2.11: Measured degree of calcination of MgCO3 in dolomite and calcination rate as function oftemperature in a CO2-atmosphere.


Although the thermodynamic calcination temperature for the MgCO3-part of the dolomitein a 1 bar CO2-atmosphere equals 687 K, the onset temperature for calcination of dolomite isobserved to be at about 800 K. The fact that this measured calcination onset temperature is farabove the thermodynamic calcination temperature and the observation of a broad temperaturerange during which calcination of dolomite occurs, is indicative for a reaction kinetic or masstransfer governed process. Although accumulation of CO2 near the dolomite grains is not ex-pected13, accumulation of CO2 in the pores of the partly calcinated MgCO3-part of dolomitemay be present. The diffusive transport of CO2 through the pores may govern the calcinationof the MgCO3-part of dolomite.

During heating of dolomite usually a third calcination peak is observed. Figure 2.11 showsthe calcination of MgCO3 present in dolomite in a CO2-atmosphere (The calcination of CaCO3in dolomite is not observed in this figure, because the onset for CaCO3 calcination in a 1 barCO2 atmosphere equals 1173 K). The first broad weight loss peak is assumed to be attributed tothe calcination of the free MgCO3 in the dolomite (see equation 2.51), whereas the second calci-nation peak is attributed to the calcination of the MgCO3-part of dolomite (see equation 2.52).For the determination of the reaction kinetic parameters for the calcination of the MgCO3-part of dolomite, a first-order reaction mechanism F1 is assumed. The calcination rate of theMgCO3-part of dolomite is given by

r = A e−EaR T(

1−ξMgCO3

)

(

1− Ka

Keq

)

, (2.57)

in which ξMgCO3 is the degree of calcination of the MgCO3-part of the dolomite. Becausethe calcination of MgCO3 occurs far from thermodynamic equilibrium, i.e. Ka << Keq, thecalcination of the MgCO3-part of the dolomite is given by

r = A e−EaR T(

1−ξMgCO3

)

. (2.58)

The apparent reaction kinetic parameters A and Ea are determined from TGA of dolomiteheated in a 1 bar CO2-atmosphere, because a pure CO2-atmosphere facilitates the discrimi-nation between the calcination of the MgCO3-part of dolomite and the residual CaCO3. Fornon-isothermal calcination of dolomite, the values for the apparent activation energy and thepre-exponential factor are derived by plotting [lnr + lnβ− ln(1−ξ)] versus the reciprocal ab-solute temperature according to

lnr + lnβ− ln(1−ξ) = lnA− Ea

R T, (2.59)

in which β is the applied constant heating rate. Figure 2.12 shows the plot of[lnr + lnβ− ln(1−ξ)] versus the reciprocal of the absolute temperature for heating rates of 10K min−1 and 20 K min−1. According to equation 2.59, the intercept of the plot equals thenatural logarithm of the pre-exponential factor A and the apparent activation energy can be cal-culated from the slope of the plot.

From figure 2.12, it can be seen that the apparent reaction activation energy is different forboth heating rates. In case that the effect of the heating rate on the values for the reaction kinetic

13In section 2.4.1, it was shown that the N2 gas flow during TGA did not effect the calcination kinetics oflimestone. It was concluded that this indicates that no CO2, which is produced during calcination, accumulatesduring the TGA.


0.00096 0.00097 0.00098 0.00099 0.00100 0.00101

-6.4

-6.0

-5.6

-5.2

-4.8

-4.4

-4.0

PSfrag replacements

Reciprocal temperature [K−1]

lnr+

lnβ−

ln(1

−ξ)

[ln

s−1 ] β=10 K min−1:

lnA=51.6, Ea=513 kJ mol−1, r2 =0.9943

β=20 K min−1:lnA=56.8, Ea=560 kJ mol−1, r2 =0.9836

Figure 2.12: [lnr + lnβ− ln(1−ξ)] versus the reciprocal absolute temperature describing the calcinationof the MgCO3-part of dolomite. Line 1 and line 2 indicate [lnr + lnβ− ln(1−ξ)] in caseof a heating rate of 10 K min−1 and 20 K min−1, respectively.

parameters is caused by the kinetic-compensating effect, which was discussed in section 2.4.1,it is expected that the apparent reaction activation energy increases with decreasing heating rate.However, figure 2.12 shows the opposite relation. It can be concluded that the non-isothermalcalcination curves do not provide sufficient information for unambiguous determination of thekinetic parameters.

Olszak-Humienik and Mozejko [23] selected the one-dimensional diffusion type D1 as reac-tion mechanism for the calcination of the MgCO3-part in dolomite. The value for the apparentreaction activation energy determined by these authors with the D1 model (see table 3.1) equals219 kJ mol−1, which is already much smaller than the apparent reaction activation energy deter-mined from figure 2.12 (590 kJ mol−1). However, describing the MgCO3 calcination measuredin this study with the D1 model, instead of the F1 model, results in a value of 254 kJ mol−1,which is much closer to the value presented by Olszak-Humienik and Mozejko. This exampleindicates that the values of derived reaction kinetic parameters is strongly dependent on the se-lected reaction type.

For technological purposes, the description of the MgCO3 calcination with a first order re-action type is sufficient in case this reaction type describes the calcination reaction accurately.Figure 2.13 shows the measured and the simulated calcination of the MgCO3-part of dolomitefor a heating rate of 10 K min−1 and the two sets of kinetic parameters (which were determinedat two different heating rates). It can be seen that the maximum difference in degree of calcina-tion is less than 0.05 for the set of kinetic parameters determined from the measured calcinationduring ramp heating with 10 K min−1. Applying the set of kinetic parameters determined fromthe measured calcination during ramp heating with 20 K min−1 results in a maximum differencein degree of calcination of 0.10.


980 1000 1020 1040 1060 1080 0.0

0.2

0.4

0.6

0.8

1.0

PSfrag replacements

Temperature [K]

Deg

ree

ofdo

lom

iteca

lcin

atio

n[-

]

12

3

Figure 2.13: Measured (1) versus simulated (2, 3) calcination of MgCO3 for a heating rate of 10 Kmin−1. Line 2 describes the calcination with lnA =55.7 and Ea=513 kJ mol−1, whereas line3 the calcination describes with lnA =60.9 and Ea=560 kJ mol−1.

0.0 0.2 0.4 0.6 0.8 1.0 1040

1060

1080

1100

1120

1140

1160

1180

1200

1220

PSfrag replacementsPartial CO2-pressure [bar]

Cal

cina

tion

onse

ttem

pera

ture

[K]

Figure 2.14: Measured (•) and calculated () calcination onset temperatures of limestone as function ofthe partial CO2-pressure in the carrier gas during TGA.


Conradt [24] studied the effect of dolomite, limestone and soda ash on the formation of aninterconnected melt phase during heating of glass batches containing flint cullet. The formationof an interconnected melt phase in melting glass batches is regarded as the onset of primarymelt phase formation. In the primary melt phase, the mobility of reactants and reaction prod-ucts, which are formed during solid-state reactions, increases. This increased mobility enhancesthe, in general, mass transfer limited solid-state batch reactions.

Conradt observed that these glass batch components affected the formation of the intercon-nected melt phase layer. This seems to indicate that (flint) cullet is not inert to these glassbatch components. However, thermogravimetric analysis of mixtures of float glass cullet anddolomite, which have been performed in this study, did not show a change in the onset temper-ature for MgCO3 calcination. This indicates that float cullet has no or only a minor effect onthe calcination of dolomite.

Further throughout this thesis, the calcination of MgCO3 in a float glass batch is regardedas a single thermal decomposition reaction.

2.5.2 Calcination of limestoneIn contrast to MgCO3, the calcination of limestone is dependent on the partial CO2-pressure.Figure 2.14 shows both the measured14 and the calculated onset temperature for calcination ofindividual limestone grains as function of the partial CO2-pressure.

According to Gallagher et al. [25], most studies concerning the determination of the mech-anism and the values for the kinetic parameters of the thermal decomposition of limestonewere, in general, accompanied with mass and/or heat transfer problems within the experimentalequipment for thermogravimetric analysis. Dependencies of the kinetics parameter values withsample size, heating rate and furnace geometry were observed. Criado et al. [26] found that, atexperimental conditions at which mass and/or heat transfer problems were limited, the thermaldecomposition of limestone follows 1st order kinetics. They also observed an influence of theparticle size of the limestone grains on the thermal decomposition of limestone. The valuesfor the apparent reaction activation energy increased with particle size and ranged from 142 kJmol−1 (25 µm< dlimestone <50µm) to 163 kJ mol−1 (100 µm< dlimestone <160 µm).

Kinetic parameters describing the thermal calcination of limestoneThe reaction kinetic parameters for the calcination of limestone are determined assuming a firstorder reaction type. The limestone grains were ramp heated with a heating rate of 20 K min−1

in a 100 ml min−1 N2-flow. Figure 2.15 shows [lnr + lnβ− ln(1−ξ)] versus the reciprocalabsolute temperature describing the thermal calcination of limestone.Table 2.3 shows the calculated apparent reaction activation energy and the pre-exponential fac-tor for limestone calcination in a N2-atmosphere at three different heating rates. It is observedthat the heating rate does not affect the derived apparent reaction activation energy, which indi-cates the absence of the kinetic-compensation effect.Figure 2.16 shows both the measured and the simulated thermal calcination of limestone forthe three different heating rates. The maximum difference in temperature for the calculated andsimulated calcination level equals 14 K in the conversion range from 0 up to 0.9.

14The onset temperature for calcination is determined by the approach described in section 2.4.1. However, theonset for weight loss commences already at lower temperatures, which explains the difference in measured andcalculated calcination onset temperatures


0.00090 0.00095 0.00100 0.00105 0.00110 -9

-8

-7

-6

-5

-4

PSfrag replacements

Reciprocal absolute temperature [K−1]

lnr+

lnβ−

ln(1

−ξ)

[ln

s−1 ]

lnA=15.9, Ea=190 kJ mol−1

r2 =0.9902

Figure 2.15: [lnr + lnβ− ln(1−ξ)] versus the reciprocal absolute temperature characterizing the cal-cination kinetics of limestone during ramp heating with 20 K min−1 in a 100 ml min−1

N2-flow.

The kinetic equation describing the thermal calcination of solid limestone during which solidCaO is formed is described by

r = k f (1−ξCaCO3)

(

1− pCO2,a

pCO2,eq

)

, (2.60)

in which pCO2,a is the partial CO2-pressure in the atmosphere surrounding the limestone grains.At thermodynamic equilibrium, the standard Gibbs free energy of the thermal calcination reac-tion is given by

∆G0r = ∆H0

r −T ∆S0r = −R T lnKeq = −R T ln pCO2,eq, (2.61)

in which ∆H0r is the standard reaction enthalpy and ∆S0

r is the standard reaction entropy. Theonset temperature for thermal calcination of limestone as function of the actual partial CO2-pressure surrounding the limestone grains pCO2,a is given by

Tonset =∆H0

r

∆S0r −R ln(pCO2,a)

. (2.62)

The average value for both the reaction enthalpy and entropy in the temperature range between975 K and 1175 K, which are determined from thermodynamic tables [3], are ∆H 0

r = 175.3


Table 2.3: Apparent reaction activation energy and pre-exponential factor for limestone calcination.

Heating rate [K min−1] Ea [kJ mol−1] lnA [ln s−1]

5 188.1 16.5810 190.8 16.7520 187.7 16.5020 187.2 16.2320 190.6 17.05

kJ mol −1 and ∆S0r = 150.7 J mol −1 K−1, respectively. The thermal calcination kinetics of

limestone at a given partial CO2-pressure is now given by

T ≤ Tonset : r = 0 (2.63)

T > Tonset : r = k f (1−ξCaCO3)

(

1− pCO2,a

pCO2,eq

)

. (2.64)

Reactive calcination of limestoneThe general equation for limestone calcination, assuming a first-order reaction mechanism, isgiven by

r = k f (1−ξCaCO3) aCaCO3

(

1− Ka

Keq

)

;with Ka =aCaO pCO2,a

aCaCO3

. (2.65)

As mentioned in section 2.3.3, the calcination reaction is favored over the carbonisation reac-tion in case Ka < Keq. As can be seen from equation 2.65, at a constant partial CO2-pressureand a constant CaO activity, the ratio of Ka relative to Keq increases with a decreasing limestoneactivity. This indicates that in case of a limestone activity less than unity, it is expected thatthe calcination of limestone retards. On the other hand, in case the activity of the formed oxideCaO is less than unity, and the limestone activity remains constant, the value of the ratio ofKa relative to Keq decreases, which results in an increase of the calcination rate of limestone.Below, it is studied if either of these two cases is likely to be present in industrial glass batchesduring heating.

Limestone calcination in the presence of soda ashIn a mixture containing both limestone and soda ash, the double carbonate Na2Ca(CO3)2 maybe formed via a solid-state reaction [27]:

Na2CO3(s)+CaCO3(s) → Na2Ca(CO3)2 (2.66)

The double carbonate melts at 1090 K [14], whereas a liquid phase may already be formedat 1058 K by eutectic melting of the double carbonate with an excess of soda ash. The onsettemperature for limestone calcination in a 1 bar CO2-atmosphere as function of the limestoneactivity in the liquid solution of limestone and soda ash is given by

Tonset =∆H0

r

∆S0r +R ln(aCaCO3)

, (2.67)

in which in which ∆H0r is the standard reaction enthalpy and ∆S0

r is the standard reaction entropyof the calcination of limestone.


850 900 950 1000 1050 1100 0.0

0.2

0.4

0.6

0.8

1.0

PSfrag replacements

Temperature [K]

Deg

ree

ofth

erm

allim

esto

neca

lcin

atio

n[-

]

β=5

Km

in−1

β=10

Km

in−1

β=20

Km

in−1

Figure 2.16: Measured (solid lines) and simulated (dotted lines) thermal calcination of limestone as func-tion of temperature for three different heating rates β in a N2- atmosphere.

Figure 2.17 shows that the calculated onset temperature for the limestone calcination, whichis 1175 K for the calcination of a pure solid limestone grain, equals 1423 K for a limestoneactivity of 0.05 in the liquid solution. This indicates that the presence of soda ash in a glassbatch may retard the calcination of limestone.Sheckler and Dinger [28] studied the reaction mechanism of glass batches containing silicasand, soda ash and limestone qualitatively. Both TGA and DTA were performed on the separatebatch components and on the complete glass batch as function of particle sizes of the batchcomponents. With X-ray diffraction, the formation of intermediate crystalline phases and thedisappearance of the initial crystalline batch components was followed. The presence of thecrystalline double carbonate was observed during heating a mixture of soda ash and limestonewith particle sizes below 53 µm. The calcination behavior of limestone in this binary mixturewas measured with TGA and compared with the calcination behavior of limestone in a mixtureof limestone with silica. In the mixture of soda ash and limestone, the calcination reactionlasted up to 1153 K, whereas the calcination of limestone in the mixture with silica was alreadycompleted at about 1063 K. The slower limestone calcination in the mixture of limestone withsoda ash can be explained by a limestone activity less than unity in the liquid double carbonatesolution.

Although the retarding effect of the double carbonate on the calcination of limestone is


0.0 0.1 0.2 0.3 0.4 0.5 1200

1250

1300

1350

1400

1450

PSfrag replacementsCaCO3 activity [-]

Cal

cina

tion

onse

ttem

pera

ture

[K]

Figure 2.17: Calculated calcination onset temperature of CaCO3 in a liquid mixture of Na2CO3 andCaCO3 in case pCO2 equals 1 bar as function of the CaCO3 activity.

observed, this effect may not be present in practice, because the double carbonate is only formedat special conditions. Kautz and Stromburg [27] identified the crystalline phases present in apartly reacted glass batch containing silica, soda ash, limestone and dolomite which was meltedin a CO2-atmosphere. It was observed that the double carbonate formation depends on threeconditions:

1. The particle size of both limestone and dolomite. For a mixture of soda ash with limestoneit was observed that the crystalline double carbonate was identified in case the limestoneparticle size was below 100 µm. Also during experiments with ’coarse’ dolomite insteadof limestone no double carbonate was identified.

2. The type of soda ash (i.e. heavy or light soda). The formation of the double carbonatewas observed to occur at about 30 K lower temperatures in case heavy soda was appliedinstead of light soda. This was explained by the larger reactive surface area of heavy sodacompared to light soda due to the smaller crystals in heavy soda.

3. The water content of the glass batch. The presence of the double carbonate was observedto be more pronounced in case limestone was humidified.

Kautz and Stromburg [27] explained these observations by describing the formation of the dou-ble carbonate as a surface reaction of soda ash with fine distributed limestone. Therefore, theformation of the double carbonate is dependent on the contact area and the contact time betweensoda ash and limestone or dolomite.

Up to now, the effect of the partial CO2-pressure on the formation of the double carbonate


has not been discussed. The formation of the double carbonate is dependent on the presenceof both soda ash and limestone. Because the presence of limestone is dependent on the partialCO2-pressure, it is expected that the formation of the double carbonate also depends on thepartial CO2-pressure. The effect of the CO2-pressure on the formation of the double carbonatehas been evaluated by three experiments, viz.:

• combined TGA and DTA of soda ash heated in a 1 bar CO2-atmosphere,

• combined TGA and DTA of a mixture of soda ash and limestone heated in a N2-atmosphere,and

• combined TGA and DTA of a mixture of soda ash and limestone heated in a 1 bar CO2-atmosphere.

The DTA-signals of these three experiments are shown in figure 2.18. Curve 1 shows the DTA-signal of soda ash heated in a CO2-atmosphere. At 1129 K, an endothermic peak is observed,which characterizes the melting of soda ash. Curve 2 shows the DTA-signal for the mixture ofsoda ash and limestone heated in a CO2-atmosphere. Already at 1058 K, an endothermic peakis observed, which coincides with the eutectic melting temperature of soda ash and limestone.However, during heating of a mixture of soda ash and limestone in a N2-atmosphere, the en-dothermic peak of soda ash melting at 1129 K is observed instead of the endothermic peak at1058 K, which indicates that the double carbonate is not formed in the N2-atmosphere. This canbe explained by the observation (see section 2.5.2) that in a N2-atmosphere limestone alreadystarts to decompose below 923 K. The early release of CO2 prevents the formation of the doublecarbonate. Therefore, the formation of the double carbonate is more pronounced in a CO2- thanin a N2-atmosphere.

During reheating of the mixture of soda ash and limestone, which was heated in a 1 bar CO2-atmosphere, two endothermic peaks were observed at 659 K and 703 K, respectively. Theseendothermic peaks correspond to two crystalline inversions of the double carbonate, which wasobserved by Wilburn et al. [29] and Taylor and Rowan [30]. During the TGA of the mixtureof limestone with soda ash heated in a 1 bar CO2-atmosphere up to 1273 K, no calcinationwas observed. This supports the observation that the double carbonate was formed, which mayretard the calcination temperature of limestone up to temperatures above the thermodynamiccalcination temperature in a CO2-atmosphere. However, for the formation of the double car-bonate, also soda ash is required. As will be shown in the next section, the calcination of sodaash is already almost completed at about 1133 K. Therefore, it is expected that the retardedcalcination of limestone due to the formation of the double carbonate is not expected in glassforming batches.

Reactive limestone calcination in the presence of float glass culletAs mentioned at the beginning of this section, the calcination reaction of limestone is favoredabove the carbonization reaction of CaO in case the activity of the formed CaO is less than unity.The CaO activity in a glass melt containing 75 wt.% SiO2, 15 wt.% Na2O and 10 wt.% CaOequals 5.7 ·10−6 at 1473 K. Therefore, it is expected that the onset temperature for calcinationof solid limestone with simultaneous dissolution of CaO in a present glass melt shifts to lowertemperatures. The effect of the presence of a glass melt on the calcination of limestone is stud-ied by TGA of a mixture of limestone with float glass cullet. Figure 2.19 shows the measured


1000 1050 1100 1150 1200

PSfrag replacements

Temperature [K]

DTA

sign

al1

2

3

∆T =80 K

Figure 2.18: DTA signal during heating of Na2CO3 in a 1 bar CO2-atmosphere (1), a mixture of Na2CO3and CaCO3 in a 1 bar CO2-atmosphere (2) and a mixture of Na2CO3 and CaCO3 in a N2-atmosphere (3).

calcination rate of limestone in the mixture of limestone and crushed float cullet ramp heatedwith 10 K min−1 in a 100 ml min−1 CO2-flow.

It can be seen that the calcination of limestone already starts at about 1020 K15, which isabout 160 K lower than the thermal calcination of limestone in a 1 bar CO2-atmosphere. Atabout 1053 K, the calcination of limestone stops. At this temperature, the presence of a crys-talline sodium calcium silicate, 2Na2O ·CaO ·3SiO2, was identified with X-ray diffraction. Thisternary crystalline silicate is formed in case the CaO-content in the sodium silicate melt phaseexceeds the maximum CaO-solubility in the sodium silicate melt phase. The formation of thecrystalline silicate may prevent direct contact between the limestone grain and a melt phase witha low CaO-content, by which the reactive calcination of limestone slows down or even stops.

Now, the reactive calcination of limestone during further heating of the glass batch dependson

• the supply of a melt phase with a low CaO-content in which the crystalline sodium cal-cium silicate is able to dissolve, and

• the temperature dependent CaO-solubility in the sodium silicate melt phase.

15Similar to a mixture of dolomite and flint cullet, Conradt [24] observed that limestone affects the formation ofan interconnected melt phase during heating flint cullet. This may indicate that flint cullet is not inert to limestoneas is shown in figure 2.19.


1000 1050 1100 1150 1200 1250 0.000

0.002

0.004

0.006

0.008

0.010

0.012

0.014

PSfrag replacements

Temperature [K]

Cal

cina

tion

rate

oflim

esto

ne[K

−1 ] Thermal calcination

of limestone

Reactive calcinationof limestone

Figure 2.19: Calcination of CaCO3 at the presence of float glass cullet in a CO2-atmosphere.

At 1173 K, the regular thermal calcination of limestone in a CO2-atmosphere is observed. Thisindicates that limestone is mainly decomposed via thermal dissociation. At this point, it isconcluded that the presence of a melt phase does not influence the calcination behavior oflimestone significantly.

2.5.3 Calcination of soda ash

In this section, the calcination behavior of individual soda ash grains and soda ash grainsin mixtures with other glass batch components is studied in both a N2- and a 1 bar CO2-atmosphere. From thermogravimetric analysis, the kinetics of reactive soda ash calcinationis determined.

Dissociative calcination of individual soda ash grainsFigure 2.20 shows both the degree of soda ash calcination16 and the DTA-signal measured dur-ing heating of soda ash grains in both a 100 ml min−1 N2-flow and a 100 ml min−1 CO2-flow.It is observed that up to the melting temperature of soda ash, which equals 1128 K, hardlyany calcination of soda ash is observed in both atmospheres. Above the melting temperatureof soda ash, a slow weight loss is observed in the N2-atmosphere. The partial CO2-pressure,which is in equilibrium with the soda ash, is calculated using thermodynamic tables [3] andequals 1.8 · 10−7 bar at 1200 K. This indicates that thermal calcination of soda ash is not ex-

16The degree of soda ash calcination is defined as the ratio of the actual weight loss of soda ash with respect tothe weight loss of soda ash after complete calcination.


950 1000 1050 1100 1150 1200 1250 -0.01

0.00

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.10

-0.30

-0.25

-0.20

-0.15

-0.10

-0.05

0.00

0.05

0.10

PSfrag replacements 1

2

3

4

A

Temperature [K]

Deg

ree

ofso

daas

hca

lcin

atio

n[-

]

DTA

-sig

nal[

a.u.

]

Figure 2.20: Measured degree of soda ash calcination and DTA-signals during heating of soda ash in a100 ml min−1 N2-flow and in a 100 ml min−1 CO2-flow. Line 1 indicates the degree ofsoda ash calcination during heating in the N2-atmosphere, line 2 indicates the degree ofsoda ash calcination in the CO2-atmosphere, line 3 indicates the DTA-signal of soda ashheated in the N2-atmosphere and line 4 indicates the DTA-signal of soda ash heated in theCO2-atmosphere. Point A marks a small endothermic peak located at 963 K.

pected during melting of glass batches. The thermal calcination of soda ash is not observedin the CO2-atmosphere, which is likely because the CO2-pressure in the atmosphere abovethe soda ash grains in the TGA/DTA-apparatus exceeds the thermodynamic equilibrium partialCO2-pressure.

In the N2-atmosphere, a small endothermic peak is observed at 963 K, which is not ob-served during heating of soda ash in the CO2-atmosphere. However, additional experimentswith mixtures of soda ash and silica sand heated in a CO2-atmosphere did show the presenceof this small endothermic peak, whereas during heating of these mixtures in a N2-atmospherethis endothermic peak was not observed. During the quantitative determination of the reactivecalcination of soda ash, which is described later in this section, it seems that this endothermicpeak is related to accelerated calcination of soda ash. A more intense endothermic peak at 963K resulted in a shift of the onset temperature for reactive calcination of soda ash with silica sandtowards 973 K. The presence of the endothermic peak at 963 K seems to be more pronouncedin a CO2-atmosphere than in a N2-atmosphere. However, because the focus in this section ison the quantitative determination of the kinetics of soda ash calcination, without identificationof the detailed calcination mechanism, the cause of this endothermic peak is not investigated inmore detail.


Reactive calcination of soda ash grainsTGA-experiments of soda ash combined with dolomite, limestone or silica sand showed thatsoda ash calcination only occurs via reactive dissociation with silica sand. During the reactivecalcination of soda ash with silica, several crystalline sodium silicate phases can be formed be-low 1072 K as can be seen from the binary phase diagram of Na2O and SiO2 (see figure 2.5).The predominant sodium silicates in the binary Na2O-SiO2 phase diagram are sodium disilicate(2Na2O ·SiO2), sodium metasilicate (Na2O ·SiO2) and sodium orthosilicate (Na2O ·2SiO2).Figure 2.21 shows the standard Gibbs free energy of the calcination reactions of soda ash bysilica sand resulting in the formation of these sodium silicates. It can be seen from this figurethat the calcination temperatures17 are 556 K, 632 K and 1011 K for the reactions 3.18, 3.19and 3.20, respectively.

Na2CO3(s)+2 SiO2(s) Na2O.2SiO2(s)+CO2(g) (2.68)

Na2CO3(s)+SiO2(s) Na2O.SiO2(s)+CO2(g) (2.69)

Na2CO3(s)+0.5 SiO2(s) 0.5 2Na2O.SiO2(s)+CO2(g) (2.70)

Figure 2.22 shows the degree of soda ash calcination and the DTA-signals in case of heatingbinary mixtures of soda ash and silica sand in both a N2- and a CO2-atmosphere. For thecomposition of these binary mixtures and for the experimental conditions at which the com-bined TGA/DTA-experiments are performed, is referred to table 2.4. Lines 1 and 2 indicatethe degree of soda ash calcination in the N2- and the CO2-atmosphere, respectively. Lines 3and 4 indicate the DTA-signals during heating of the binary mixtures in the N2- and the CO2-atmosphere, respectively. It is observed that the onset for reactive soda ash calcination in theCO2-atmosphere occurs at lower temperatures than in the N2-atmosphere. It is also observedthat during heating of the binary mixture in the CO2-atmosphere, the endothermic peak at 963K is present, which is absent in the N2-atmosphere.Based on the measured degree of soda ash calcination, it is concluded that reactive soda ashcalcination hardly occurs below 973 K, which is significantly higher than the thermodynamiconset temperatures for reactive calcination of soda ash with silica sand according to figure 2.21.The onset temperature for calcination of soda ash in a CO2- and a N2-atmosphere appearedto be approximately 973 K and 1073 K, respectively. The latter finding was also observed byKroger [31], who studied isothermally the reaction kinetics of binary mixtures of silica and sodaash by measuring the release of CO2 during the heating of mixtures of silica sand and soda ashwith particle sizes ranging from 150 µm to 250 µm. During his experiments, Kroger made thefollowing observations:

1. The reactive calcination of soda ash with silica sand steeply increased between 1098 Kand 1127 K for SiO2:Na2CO3-ratio’s in the binary mixtures of 1:1 and 2:1.

2. The calcination rate of soda ash increased with increasing fraction of silica sand in theinitial mixture of soda ash and silica sand.

3. The calcination rate of soda ash decreased with increasing partial CO2-pressure.

17Here, the calcination temperature is defined as the temperature at which the thermodynamic equilibrium partialCO2-pressure equals 1 bar.


500 600 700 800 900 1000 -6.0x10 4

-4.0x10 4

-2.0x10 4

0.0

2.0x10 4

4.0x10 4

6.0x10 4

8.0x10 4

1.0x10 5

PSfrag replacements

Na2CO3(s)+0.5 SiO2(s) ⇐⇒ 0.5 [2Na2O ·SiO2 ](s)+CO2(g)Na2CO3(s) + SiO2(s) ⇐⇒ Na2O ·SiO2(s)+CO2(g)

Na2CO3(s) + 2 SiO2(s) ⇐⇒ Na2O ·2SiO2(s)+CO2(g)

Temperature [K]

Stan

dard

Gib

bsfr

eeen

ergy

ofre

actio

n[J

mol

−1 ]

10−4

10−3

10−2

10−1

10−0.3

100

100.7

101

102

Figure 2.21: Standard Gibbs free energy of the calcination reactions of soda ash with silica sand. Thedashed lines are isobars, expressed in bar, for the partial CO2-pressure above the carbonates.

4. During isothermal calcination of soda ash at temperatures above 1138 K, a part of thesoda ash was not decomposed even after long isothermal periods. The amount of the sodaash that was not decomposed appeared to be dependent on temperature.

In the temperature range between 1098 K and 1127 K, primary melt phase formation is pos-sible due to eutectic melting of sodium metasilicate with sodium disilicate at 1108 K, meltingof sodium disilicate at 1129 K and melting of soda ash at 1128 K. The reason for the absence,or very slow, reactive calcination of soda ash at temperatures below 973 K are likely the lowsolid-state diffusion coefficients of the reactants and reaction products. The rate of solid-statereactions is namely governed by the slow diffusion of reactants and reaction products from thereaction interface. Because the liquid-state diffusion coefficients of the SiO2 and Na2O are sev-eral orders of magnitude larger than their solid-state diffusion coefficients, it is expected thatthe calcination of soda ash is enhanced at the temperature at which the primary melt phase isformed.

The second, third and fourth observation of the above mentioned observations that weremade by Kroger, can be explained as follows. According to equation 2.41, the reactive calci-nation rate of soda ash is proportional to the square of the mole fraction of the stoichiometriccompound SiO2 in the melt phase, which increases with increasing mole fraction of silica sand


900 950 1000 1050 1100 1150 1200 1250 0.0

0.2

0.4

0.6

0.8

1.0

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

PSfrag replacements

Temperature [K]

Deg

ree

ofso

daas

hca

lcin

atio

n[-

]

DTA

-sig

nal[

a.u.

]

12

3

4

Figure 2.22: Degree of soda ash calcination and DTA-signals in case of heating soda ash with 10 Kmin −1 in a 100 ml min−1 N2-flow and in a 100 ml min−1 CO2-flow. Line 1 indicates thedegree of conversion of soda ash heated in the N2-atmosphere, line 2 indicates the degreeof conversion of soda ash in the CO2-atmosphere, line 3 indicates the DTA-signal of sodaash heated in the N2-atmosphere and line 4 indicates the DTA-signal of soda ash heated inthe CO2-atmosphere.

in the initial mixture of soda ash and silica sand. Therefore, the calcination rate of soda ash isexpected to be increased with increasing silica sand content in the binary mixture.

Assuming that the chemically dissolved CO2 in the glass melt is characterized by the stoi-chiometric compound Na2CO∗

3(m) (see section 2.3.4), the mole fraction of this stoichiometriccompound is given by

xNa2CO3(m) =aNa2O(m) pCO2(g)

K, (2.71)

in which K is the thermodynamic reaction equilibrium constant. It is seen that the mole fractionof the stoichiometric compound Na2CO∗

3(m), and therewith the amount of CO2 dissolved inthe glass melt, increases with increasing partial CO2-pressure above the sodium silicate melt,which was observed by Kroger. The temperature dependent amount of soda ash, which was notdecomposed during an isothermal reaction of a mixture of soda ash and silica sand is given bythe temperature dependent chemical solubility of CO2 in the binary glass melt.

As mentioned above, the onset temperature for soda ash calcination in a 1 bar CO2-atmosphereappeared to be lower than the onset temperature for calcination in a N2-atmosphere. This low-ered calcination temperature can not be explained by the thermodynamic description of thecalcination of soda ash in the mixture of soda ash and silica sand. At this moment, the cause ofthe lowered calcination temperature of soda ash is not clarified.

During thermogravimetrical analysis of mixtures of soda ash and silica sand, above 1123


1120 1140 1160 1180 1200 1220 1240 1260 0.00

0.05

0.10

0.15

0.20

0.25


Con

cent

ratio

nof

Na 2

O·S

iO2

[a.u

]

Figure 2.23: Measured concentration of crystalline Na2O ·SiO2 in a partly molten mixture of SiO2 andNa2CO3.

K a decrease of the calcination rate of soda ash was observed. This temperature is close to theeutectic melting temperature of Na2O ·2SiO2 with Na2O ·SiO2 (i.e. 1108 K). According to thebinary phase diagram of Na2O and SiO2, beyond this temperature crystalline sodium metasil-icate (Na2O ·SiO2(s)) can be formed by a reaction of soda ash with the eutectic melt phase.The presence of crystalline sodium metasilicate in quenched samples of partly molten mixturesof soda ash and silica sand was identified during phase analysis with X-ray diffraction. Figure2.23 shows the measured semi-quantitative concentration of crystalline Na2O ·SiO2, in arbitraryunits, in a partly molten mixture of SiO2 and Na2CO3.

Similar to the system of limestone reacting with float cullet, the presence of the crystallinespecie surrounding the soda ash grains may act as a barrier preventing contact between soda ashand silica sand. This hampered contact may result in a decrease of the calcination rate of sodaash, which was observed from the TGA-experiments above 1123 K.

Resuming, the calcination of soda ash by a reaction with silica sand, mainly occurs abovethe onset temperature for primary melt phase formation. In the following, the kinetic parame-ters describing the reactive calcination of soda ash by reaction with silica sand are determinedfrom TGA-experiments of binary mixtures of soda ash and silica sand as function of

• the ratio of the amount of soda ash with respect to the amount of silica sand,

• the particle size of the soda ash and silica sand grains, and

• the atmosphere composition in which the binary mixtures are heated.


Determination of the kinetic parameters of reactive calcination of soda ashAs mentioned above, the reactive calcination of soda ash with silica sand mainly occurs incase a sodium silicate melt phase is present. According to the binary phase diagram of SiO2and Na2O (see figure 2.5), the temperature at which the first sodium silicate melt is formedequals 1053 K, at which silica sand eutectic melts with sodium disilicate. The first formed meltphase is regarded as a mixture of the stoichiometric compounds SiO2 and sodium disilicate(Na2O ·2SiO2). Now, the reactive calcination of soda ash in contact with the first formed meltphase, neglecting the chemical dissolution of CO2 in the melt phase, is described by

Na2CO3(s)+2 SiO2∗(m) Na2O.2SiO2

∗(m)+CO2(g), (2.72)

or byNa2CO3(s)+Na2O.2SiO2

∗(m) 2 Na2O.SiO2∗(m)+CO2(g) (2.73)

in which the superscript ∗ denotes the stoichiometric compound. The general equation describ-ing the kinetics of heterogeneous reactions is given by (see section 2.3.3)

r = k f f (ξ)nr

∏i=1

aνii

(

1− Ka

Keq

)

. (2.74)

Because the calcination temperature for both reactions 2.72 and 2.73 are far below 1053 K,the calcination of soda ash occurs far from thermodynamic equilibrium, which indicates thatKa < Keq. The reactive calcination of solid soda ash is now described by

r = k f f (ξ) aνrr , (2.75)

in which ar is the activity of the reactant with which soda ash reacts, which is either silicaor sodium disilicate. Estimation of the reaction kinetic parameters for the reactive calcina-tion of soda ash requires knowledge of the temperature and composition dependent activity ofthe stoichiometric compounds SiO2 and Na2O ·2SiO2 during the reactive calcination reaction.However, from the measured calcination rate of soda ash with TGA, the apparent soda ashcalcination rate, as defined by equation 2.76 is measured.

∂ξ∂t

=r

aνrr

= r∗ = k f f (ξ) , (2.76)

The thermogravimetrical analysis does not provide information on the activities of the stoi-chiometric compounds during the calcination reaction.

The reaction mechanism function, which provides the best fit with the measured data ofthe reactive calcination of soda ash is determined by plotting [lnr∗ + lnβ− ln f (ξ)] versus thereciprocal absolute temperature for different reaction mechanism functions. Figure 2.24 showsthis plot in the temperature range from 1068 K up to 1139 K for the reaction mechanism func-tions F0, F1, D3, D4, R2 and R3 for the binary mixture nr. 4 listed in table 2.4.

It can be seen that the pattern for all reaction mechanism functions is similar. Between1083 K and 1123 K, an almost linear relation between [lnr∗ + lnβ− ln f (ξ)] and the reciprocalabsolute temperature is observed. However, above 1123 K the linear relation is absent. Thisindicates that above 1123 K the calcination of soda ash is governed by an other process thanin the temperature range between 1083 K and 1123 K. As mentioned before, the formation ofcrystalline sodium metasilicate is likely the cause of this change in calcination rate of soda ash.


0.88 0.90 0.92 0.94 -11

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

PSfrag replacements

Reciprocal absolute temperature [ ·103 K−1]

[lnr∗

+ln

β−

lnf(

ξ)]

[ln

min

−1 ] 1

1 22

34 5

6

1123 K 1083 K

Figure 2.24: Plot of [lnr∗ + lnβ− ln f (ξ)] versus the reciprocal absolute temperature for reactive calci-nation of soda ash with silica sand in the temperature range from 1068 K up to 1139 K asfunction of the reaction mechanism functions for the binary mixture nr. 4 from table 2.4.Line 1 describes the F0-mechanism function, line 2 describes the F1-mechanism function,line 3 describes the D3-mechanism function, line 4 describes the D4-mechanism function,line 5 describes the R2-mechanism function and line 6 describes the R3-mechanism func-tion.

Table 2.4: Experimental conditions for TGA of mixtures of silica sand and soda ash.

Nr. Ratio silica: dSiO2 dNa2CO3 Atmosphere Heating ratesoda ash [µm] [µm] [K min−1]

1 1:0.225 160 - 250 250 - 500 N2 202 1:0.285 160 - 250 250 - 500 N2 203 1:0.347 160 - 250 250 - 500 N2 204 1:0.295 63 - 125 250 - 500 N2 205 1:0.281 160 - 250 106 - 125 N2 206 1:0.214 160 - 250 106 - 125 CO2 207 1:0.170 160 - 250 250 - 500 CO2 20


Figure 2.25 shows a detailed part of figure 2.24 in the temperature range from 1083 Kup to 1123 K. For this temperature range, the correlation coefficient of the linear relation of[lnr∗ + lnβ− ln f (ξ)] versus the reciprocal absolute temperature is determined for the six re-action mechanism functions. The correlation coefficients for these six reaction mechanismfunctions are listed in table 2.5.

0.895 0.900 0.905 0.910 0.915 0.920 0.925

-7

-6

-5

-4

-3

-2

-1

0

PSfrag replacements


[lnr∗

+ln

β−

lnf(

ξ)]

[ln

min

−1 ]

1

1 2

3

4

5

6

Figure 2.25: Plot of [ln r∗ + lnβ− ln f (ξ)] versus the reciprocal absolute temperature for reactive calci-nation of soda ash with silica sand in the temperature range from 1083 K up to 1123 K asfunction of the reaction mechanism functions for the binary mixture nr. 4 from table 2.4.Line 1 describes the F0-mechanism function, line 2 describes the F1-mechanism function,line 3 describes the D3-mechanism function, line 4 describes the D4-mechanism function,line 5 describes the R2-mechanism function and line 6 describes the R3-mechanism func-tion.

It can be seen that the reaction mechanism functions D3 and D4 show the highest value forthe correlation coefficient. The reaction mechanism function D3, which is known as the Janderequation, was used by Kroger [31] to describe the reactive calcination of soda ash by a reactionwith silica. However, Kroger mentioned that the experimental data for reactive calcination ofsoda ash could only be described by the Jander equation in a limited conversion range. Thisindicates that it is likely that reactive calcination of soda ash does not proceed via only onereaction.

As mentioned in section 2.3.3, the approach for selecting the most accurate reaction mecha-nism function based on the highest value for the correlation coefficient is arbitrarily and does notnecessary provide information about the reaction mechanism. This is supported by figure 2.26,which describes the simulated calcination of soda ash as function of the reaction mechanismfunctions with the kinetic reaction parameters as presented in table 2.5.


Table 2.5: Correlation coefficients for six reaction mechanism functions describing the reactive calcina-tion of soda ash by silica sand in the temperature range from 1083 K up to 1123 K.

Reaction mechanism function Correlation coefficient Ea lnAkJ mol−1 ln s−1

F0 0.9939 657.4 66.57F1 0.9846 939.6 97.76D3 0.9972 1699.6 177.72D4 0.9988 1605.5 167.32R2 0.9936 798.5 81.48R3 0.9911 845.5 86.27

1080 1090 1100 1110 1120 0.0

0.2

0.4

0.6

0.8

1.0


Deg

ree

ofso

daas

hca

lcin

atio

n[-

]

Figure 2.26: Simulated reactive calcination of soda ash with silica sand when ramp heated with 20 Kmin−1 in a N2-atmosphere and as function of the reaction mechanism function. Becausesimulated curves are very close to each other, the individual lines are not labelled with theirreaction mechanism function.

It is noticed that the difference in the simulated versus the measured degree of soda ashcalcination is less than 0.02. This indicates that the choice of the reaction mechanism functionfor describing the reactive calcination of soda ash is arbitrarily. For the description of thereactive calcination of soda ash with silica sand, the F1 reaction mechanism function is applied.The reaction kinetic parameters have been determined for 7 binary mixtures. The compositionof these mixtures and the experimental conditions at which the TGA-experiments are performedare listed in table 2.4. The values for the apparent reaction activation energy and the naturallogarithm of the pre-exponential factor are listed in table 2.6 for the F1 reaction mechanismfunction.


Table 2.6: Apparent reaction activation energy, pre-exponential factor, correlation coefficient in the tem-perature range from 1083 K up to 1123 K for seven calcination experiments of soda ashassuming a F1 reaction mechanism function.

Nr. Ea lnA r2

[kJ mol−1] [ln s−1]

1 953.0 99.1 0.98082 1028.6 107.6 0.99683 919.0 95.5 0.98574 939.6 97.8 0.98465 780.0 80.3 0.99156 499.4 49.9 0.98287 512.7 51.5 0.9762

200 400 600 800 1000

20

40

60

80

100

120

PSfrag replacements

Apparent reaction activation energy [kJ mol−1]

Nat

ural

loga

rith

mof

A[l

nm

in−

1 ]

A

B

C

D

Figure 2.27: Kinetic compensation effect: plot of the natural logarithm of the pre-exponential factorversus the reaction activation energy for the reactive calcination of soda ash (A indicatesthe experiment 1, 2, 3 and 4; B indicates experiment 5; C indicates the experiments 6 and7) and the calcination of limestone indicated by D. According to 2.50, the slope of theline equals (R T )−1 and the intercept equals lnA. The values for both slope and interceptare 1.09 ·10−4 mole J−1 and 0.58, respectively. The unit of A is min−1. The correlationcoefficient of this line equals 0.9995.


Figure 2.27 plots the relation between the apparent reaction activation energy versus thenatural logarithm of the pre-exponential factor as is described by equation 2.49, which describesthe kinetic compensation effect.

The reaction kinetic parameters derived from the TGA-experiments performed in a N2-atmosphere are indicated by the dots in the square indicated by A, representing the values forthe binary mixtures 1, 2, 3 and 4, and by the dot indicated by B, representing TGA-experiment5. From this figure it seems that the particle size of the silica grains does not have a significantimpact on the calcination of soda ash. However, the particle size of the soda ash seems to affectthe reactive calcination of soda ash by silica sand.

Figure 2.28 shows the simulated reactive calcination of soda ash with silica sand whenramp heated with 20 K min−1 in both a N2- and a 1 bar CO2-atmosphere and as function of thereaction kinetic parameters presented in table 2.6. Although it seemed that the particle size ofsoda ash has an impact on the reaction kinetic parameters, the simulated reactive calcination ofsoda ash for experiment 5 is almost similar to the simulated reactive calcination of soda ash forthe experiments 1, 2, 3 and 4.

1040 1060 1080 1100 1120 1140 0.0

0.2

0.4

0.6

0.8

1.0

PSfrag replacements

Temperature [K]

Deg

ree

ofca

lcin

atio

nof

soda

ash

[-]

A

B

C

D

Figure 2.28: Simulated reactive calcination of soda ash with silica sand when ramp heated with 20 Kmin−1 in both a N2- and a 1 bar CO2-atmosphere and as function of the reaction kineticparameters presented in table 2.6. A indicates the simulated reactive calcination of soda ashfor the experiments 1, 2, 3 and 4. B indicates the simulated reactive calcination of sodaash for experiment 5. C indicates the simulated reactive calcination of soda ash for theexperiments 6 and 7.


According to figure 2.27, the presence of CO2 in the atmosphere surrounding the binarymixture shows the lowest value for the apparent reaction activation energy. The different re-action kinetic parameters for reactive calcination of soda ash in a CO2-atmosphere show a sig-nificant difference with the calcination of soda ash in the N2-atmosphere. From the slope ofthe kinetic compensation curve presented in figure 2.27, a temperature value can be calculatedwhich, according to Gallagher and Johnson [21], represents the average temperature over whichthe reaction kinetic parameters are derived. For the reactive calcination of soda ash this valueequals 1103 K. However, in case the reaction kinetic parameters for the calcination of lime-stone are added to figure 2.27, the slope of the kinetic compensation curve does not change,whereas the limestone calcination in a N2-atmosphere occurs, dependent on the heating rate,in the temperature range from 873 K up to 1073 K (see figure 2.16). This indicates that thekinetic compensation curve is quite insensitive for changes in reaction temperatures of 100-200K. Therefore, the explanation of different reaction activation energies presented in literature forthe same chemical reaction by the kinetic-compensation effect is questioned.

Because it is expected that the partial CO2-pressure is high during melting of glass batchescontaining carbonated raw materials, the calcination of soda ash is described by

r∗ =

(

1β

)

(1−ξ) e(54− 500000R T ), (2.77)

in which the pre-exponential factor and the apparent reaction activation energy describe thereactive calcination of soda ash with silica sand in a CO2-atmosphere. This equation doesnot include the calcination of soda ash at temperatures above 1123 K at which the soda ashcalcination is retarded. However, because the main part of the soda ash reacts below 1123 K,equation 2.77 is used as a first approximation of the reactive calcination of soda ash by silicasand in a CO2 containing atmosphere.


2.5.4 Calcination of a mixture of silica sand, soda ash and limestoneAs described in the previous sections, the calcination of dolomite, limestone and soda are de-scribed as separate reactions. In this section, the measured calcination18 of a mixture of silicasand, soda ash and limestone is compared with the simulated calcination of this mixture.

Table 2.7 lists the batch composition and the experimental conditions during heating of fourglass batches.

Table 2.7: Experimental conditions for TGA of mixtures of silica sand, soda ash and limestone.

Nr. Ratio silica: dSiO2 dNa2CO3 dCaCO3 Atmosphere Heating ratesoda ash: limestone [µm] [µm] [K min−1]

1 1:0.254:0.452 250 - 500 106 - 125 coarse N2 202 1:0.294:0.197 250 - 500 106 - 125 coarse N2 203 1:0.256:0.224 250 - 500 106 - 125 coarse N2 204 1:0.295:0.264 250 - 500 106 - 125 coarse CO2 20

The values for the apparent reaction activation energy and the natural logarithm of the pre-exponential factor for the calcination of limestone and for the calcination of soda ash in both aN2- and a 1 bar CO2-atmosphere are listed in table 2.8. These values are only valid for describ-ing the calcination of limestone and soda ash by a first-order reaction mechanism.

Table 2.8: Reaction activation energy and natural logarithm of the pre-exponential factor for calcinationof limestone and calcination of soda ash in both a N2- and CO2-atmosphere.

Specie Atmosphere Ea lnAkJ mol−1 lns−1

limestone N2 and CO2 188.9 16.6soda ash N2 953.0 98.9soda ash CO2 500.0 49.9

Figure 2.29 shows the measured (dotted line) and the calculated (solid line) degree of calci-nation of the ternary mixture nr. 1 heated in a N2-atmosphere. It can be seen that the maximumtemperature difference between both curves equals 20 K.

Figure 2.30 shows the simulated versus the measured degree of calcination for the experi-ments 1, 2 and 3. It can be seen that the maximum difference between simulated and measureddegree of calcination, which is observed for experiment 3, is less than 0.20. This is caused by amaximum temperature difference between both curves of 25 K. Figure 2.31 shows the measured(dotted line) and the calculated (solid line) degree of calcination of the ternary mixture nr. 4heated in a 1 bar CO2-atmosphere.

Resuming, with the reaction kinetic parameters describing the calcination rate of the indi-vidual carbonates, the calcination of a float glass batch can be predicted with an uncertainty inthe degree of calcination of the complete float glass batch less than 0.20.

18Here the degree of calcination is defined as the ratio of the actual weight loss with respect to the weight lossafter complete calcination of limestone and soda ash


900 950 1000 1050 1100 1150 0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0


Deg

ree

ofca

lcin

atio

n[-

]

Figure 2.29: Measured (dotted line) and simulated (solid line) degree of calcination as function of tem-perature for glass batch mixture 1 from table 2.7.

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

PSfrag replacements

Measured degree of calcination [-]

Cal

cula

ted

degr

eeof

calc

inat

ion

[-]

1

2

3

Figure 2.30: Simulated versus the measured degree of calcination of the glass batch mixture 1, 2 and 3from table 2.7.


900 950 1000 1050 1100 1150 1200 1250 0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0


Deg

ree

ofca

lcin

atio

n[-

]

Figure 2.31: Measured (solid line) and simulated (dotted line) degree of calcination as function of tem-perature for glass batch mixture 5 from table 2.7.

2.5.5 Reaction mechanism of a float glass batchAccording to Conradt et al. [32], the melting of commercial glass batches, with as major com-ponents silica sand, soda ash, limestone and/or dolomite, proceeds via either the carbonate orthe silicate route (see also Riedel [33]). During the carbonate route, the formation of the pri-mary melt phase is governed by the melting of the double carbonate of soda ash and limestone,Na2Ca(CO3)2, or by eutectic melting of a mixture of soda ash and limestone (see section 2.5.2).During the silicate route, the formation of the primary melt phase is governed by eutectic melt-ing of sodium disilicate with silica.

This section describes, based on experimental studies performed by Kautz and Stromburg[27], Sheckler and Dinger [28] and Savard and Speyer [34], the conditions for the occurrenceof these two reaction routes. Additionally, the reaction mechanism of the float glass batch, forwhich the calcination kinetics was studied in the previous sections, is determined by phase anal-ysis on quenched float glass batch samples, which were heated up to temperatures in the rangeof 973 K to 1673 K.

Study on the reaction mechanism of soda-lime-silica glass by Kautz and Stromburg [27]Kautz and Stromburg [27] identified the crystalline phases in quenched glass batch samples af-ter heating the subsystems and the complete mixture of a glass batch composed of sand, sodaash and limestone or dolomite. The mixture of the glass batch components was ramp heatedwith 10 K min−1 in a CO2-atmosphere. Both during heating of a mixture of soda ash andlimestone and during heating of the complete glass batch, the presence of the double carbonate


Na2Ca(CO3)2 was observed in the temperature range from 773 K up to 1113 K.Similar as was mentioned in section 2.5.2, the double carbonate was only formed in case

of small particle sizes (d< 100 µm) of the soda ash grains and the limestone grains and in casethe glass batch was humidified. The observed presence of the double carbonate in a pure CO2-atmosphere is supported by the own observation presented in section 2.5.2, which showed theeffect of the partial CO2-pressure on the double carbonate formation.

During heating a of mixture of silica sand, soda ash and limestone, crystalline CaO wasidentified at temperatures above 1173 K, which equals the thermodynamic onset temperaturefor limestone calcination in a 1 bar CO2-atmosphere. The presence of crystalline CaO at 1173K indicates that limestone is not (completely) decomposed by reactive calcination with the pri-mary formed sodium silicate melt phase, which was also observed in section 2.5.2.

Phase analysis on quenched samples after both heating the complete glass batch and a mix-ture of soda ash and silica, showed the presence of crystalline sodium metasilicate in the temper-ature range from 1123 K up to 1223 K. The presence of sodium metasilicate was also observedby own measurements presented in figure 2.23.

At about 1073 K, the presence of the ternary crystalline silicate 2Na2O ·CaO ·3SiO2 wasidentified. This ternary silicate remained in the partly molten glass batch up to 1373 K, whereasthe maximum concentration of the ternary silicate was observed at 1163 K. Using fine lime-stone and soda ash particles (d< 100 µm) instead of coarser particles (d> 100 µm), showed thatthe ternary silicate was present, but sodium metasilicate could not be identified. CrystallineMgO was observed to remain present in the partly molten batch up to temperatures of 1373 K,whereas crystalline cristobalite was observed starting at about 1273 K.

Study on the reaction mechanism of soda-lime-silica glass Sheckler and Dinger [28]Sheckler and Dinger [28] identified in the temperature range from 1048 K up to 1138 K

the crystalline phases present in quenched samples after heating of a mixture of silica, sodaash and limestone. The particle sizes of both the soda ash and the limestone were less than 53µm. Three different particle size distributions for the silica have been tested, viz. d<53 µm,125 µm<d<180 µm and 425 µm<d<500 µm. In all mixtures the presence of the double car-bonate was observed, which was expected due to the small particle sizes of both soda ash andlimestone. However, in case of fine silica particles, only a very slight amount of the doublecarbonate was identified.

Starting from 1078 K, different ternary silicates were observed. The Na2O ·2CaO ·3SiO2phase was identified at 1093 K and was more pronounced with smaller particle size of the sil-ica sand grains. At slight higher temperatures, the presence of crystalline 2Na2O ·CaO ·3SiO2was observed for the two coarser silica particle distributions. Reactive dissolution of limestoneresulted in the formation of crystalline ternary silicates with relative high silica content suchas 2Na2O ·CaO ·3SiO2 and Na2O ·CaO ·3SiO2. Reactive decomposition of limestone even re-sulted in the formation of wollastonite.

Study on the reaction mechanism of soda-lime-silica glass Savard and Speyer [34]Savard and Speyer [34] identified the crystalline phases of quenched glass batch samples afterramp heating with 10 K min−1 in an ambient atmosphere. The glass batch mixture containedsand, soda ash, calcite, dolomite and feldspar. The chemical behavior of two different particlesize distributions were evaluated, viz. 125 µm<d<250 µm and d<45 µm. In case of the coarserbatch, mainly sodium metasilicate was identified and also a slight amount of sodium disilicate.


With decreasing particle size, the presence of sodium disilicate became more pronounced andsodium metasilicate disappeared. Next to these crystalline silicates no other binary or ternarysilicates were identified.

Conditions for occurrence of the carbonate and the silicate routeBased on the observations presented by Kautz and Stromburg, Sheckler and Dinger and Savardand Speyer, the conditions required for following either the carbonate route or the silicate routeduring melting of glass batches containing as major components silica sand, soda ash and lime-stone and/or dolomite are:

• The carbonate route:The carbonate route requires the formation of the double carbonate Na2Ca(CO3)2, whichis favored in case of a small particle size of limestone and/or dolomite, improved wettingof the glass batch mixture and a high partial CO2-pressure. At the presence of the doublecarbonate, a primary melt phase composed of Na2CO3 and CaCO3, can be formed from1053 K. From this mixture, limestone decomposes via reactive calcination of this pri-mary formed melt phase with silica grains, which results in the formation of a crystallineternary silicate, e.g. Na2O ·2CaO ·3SiO2. In contrast to the silicate route, the formationof crystalline sodium metasilicate is not observed or only as a minor component. Withdecreasing particle size of the silica particles, the onset for melt formation shift from thecarbonate melt towards the sodium silicate melt.

• The silicate route:In case ’large’ limestone grains are present in the glass batch mixture, it is expected tono double carbonate is formed. The primary melt phase formed during heating of theseglass batch mixtures is likely to be caused by eutectic melting of sodium disilicate withsilica starting at about 1053 K. The sodium disilicate is formed by reactive calcination ofsoda ash with silica sand grains. Upon further heating, crystalline sodium metasilicate isformed above the eutectic melting temperature of sodium disilicate and sodium metasili-cate. Because of the large particle size of the limestone, reactive calcination of limestoneis only moderate, as can be seen from figure 2.19 for a mixture of limestone and float cul-let. The ternary crystalline silicate will only be present as minor component. The mainpart of the limestone decomposes above the thermodynamic calcination temperature oflimestone, which is dependent on the partial CO2-pressure.

Identified crystalline phases during heating of a float glass batchFigure 2.32 shows the identified crystalline phases as function of temperature. The particle sizesof both soda ash and dolomite ranged between 63 µm and 150 µm, while the particle size of thesilica sand was between 212 µm and 300 µm. Phase analysis of the quenched batch samplesdid not show the presence of the double carbonate. The absence of the double carbonate duringmelting the float glass batch indicates that either the dolomite grains are too large or that therelease of CO2 caused by the low temperature thermal calcination of dolomite prevents directcontact of the CaCO3 grains with the soda ash grains. Because the glass batch is heated in aN2-atmosphere, it is expected that both MgO and CaO are present at temperatures of 973 K.Because the melting of this glass batch at the melting conditions indicate that the melting ofthe glass batch proceeds via the silicate route, it is expected that sodium disilicate is formed.


However, no sodium disilicate is identified. The main intermediate crystalline phase that isobserved is sodium metasilicate which is present at about 1123 K. Starting at about 1173 K, theternary silicate Na2O ·2CaO ·3SiO2 is observed. The formation of the ternary silicate at highertemperatures indicate the reaction of (decomposed) limestone with the primary formed sodiumsilicate melt. MgO remains in the melt phase till about 1423 K, which was also observed Kautzand Stromburg. Cristobalite is formed starting at about 1373 K.

1000 1100 1200 1300 1400 1500 1600 1700

PSfrag replacements

Temperature [K]

SiO2(q)

SiO2(cr)

Na2CO3

MgCO3 ·CaCO3

CaCO3

CaO

MgO

Na2O ·SiO2

Na2O ·2CaO ·3SiO2

Figure 2.32: Identified crystalline species during heating of a float glass batch.

2.5.6 Chemical energy demand of a float glass batchIn the previous sections, the calcination rate of dolomite, limestone and soda ash is determined.Combining the equation describing the rate of these calcination reactions with the energies re-quired for these calcination reactions provide expressions for the temperature dependent energydemand of the calcination reaction for carbonate i, ∂Hi

∂T , expressed in kJ K−1 mole−1i . The total

temperature dependent energy demand for the calcination of a glass batch is given by(

∂H∂T

)

=n

∑i=1

xi

(

∂Hr,i

∂T

)

, (2.78)

in which xi is the amount of moles of carbonate i per kg of glass batch. However, next tothe calcination reactions, the total temperature dependent chemical energy demand of a glassforming batch is also dependent on the endo- and exothermic reactions for the dissolution ofthe oxides CaO, MgO and SiO2 is the primary formed melt phases. In this section, the kineticsof these reactions are not evaluated, because:


• the main part of the energy required for chemical reactions is necessary for the calcinationreactions,

• quantitative analysis of the amount of undissolved oxides in the primary melt phases isnot trivial and requires advanced analyzing techniques, and

• the dissolution energy for the oxides is both temperature and composition dependent,which requires a sophisticated thermodynamic model.

In the following, the temperature dependent energy demand for the calcination reactions for theglass batch with the composition given in table 2.1 is calculated. The temperature dependentcalcination energy of the glass forming batch is compared with the total temperature dependentchemical energy demand as was presented in figure 2.3.

Chemical energy demand for the thermal calcination of the MgCO3-part of dolomiteFrom figure 2.13 it can be seen that calcination of the MgCO3-part of dolomite occurs in thetemperature range between 873 K and 1073 K. The average calcination enthalpy in this temper-ature range, which is derived from thermodynamic tables, equals 128.8 kJ mol−1. The energyrequired for calcination of the MgCO3-part of dolomite, expressed in kJ K−1 mole−1

MgCO3, is

given by(

∂H∂T

)

MgCO3

=

(

128.8β

)

(

1−ξMgCO3

)

e(55.7− 513000R T ). (2.79)

Chemical energy demand for the thermal calcination of the CaCO3From figure 2.16 it can be seen that the thermal calcination of the CaCO3 occurs in the tempera-ture range between 873 K and 1173 K. The average calcination enthalpy, which is derived fromthermodynamic tables, in this temperature range equals 171.8 kJ mol−1. The energy requiredfor CaCO3 calcination, expressed in kJ K−1 mole−1

CaCO3, is given by

(

∂H∂T

)

CaCO3

=

(

171.8β

)

(1−ξCaCO3) e(20.8− 190800R T ). (2.80)

Chemical energy demand for the reactive calcination of Na2CO3In contrast to the thermal calcination of both MgCO3 and CaCO3, the explicit reaction equationdescribing the reactive calcination of soda ash can not be given. The reason for this is that sodaash mainly reacts with the primary formed melt phase, which composition is dependent on bothtime and temperature. In order to describe the energy required for reactive calcination of sodaash, it is assumed that the reactive calcination of soda ash can be described by

Na2CO3(s)+2 SiO2(s) −→ Na2O ·2SiO2(l)+CO2(g). (2.81)

For the determination of the calcination energy of the calcination reaction, this reaction is de-scribed by two serial reactions, viz. the formation of crystalline sodium disilicate

Na2CO3(s)+2 SiO2(s) −→ Na2O ·2SiO2(s)+CO2(g), (2.82)

and the melting of the crystalline sodium disilicate according to

Na2O ·2SiO2(s) −→ Na2O ·2SiO2(l). (2.83)


The reaction enthalpy of reaction 2.82 in the temperature range from 973 K up to 1123 K equals82.8 kJ per mol Na2CO3, whereas the melting of the crystalline sodium disilicate requires 35.6kJ per mol Na2CO3. The energy required for reactive calcination of soda ash is now given by

(

∂H∂T

)

Na2CO3

=

(

118.3β

)

(1−ξNa2CO3) e(54.0− 500000R T ), (2.84)

assuming that the reactive calcination of soda ash occurs in a CO2-atmosphere.

Total energy demand for the calcination of the float glass batchFigure 2.33 shows the simulated temperature dependent total energy demand as function of tem-perature for the float batch listed in table 2.1. In contrast to figure 2.4, the total energy demandis expressed in J kg−1

batch instead of J kg−1melt. The energy demand of the float batch is simulated

for a heating rate 20 K min−1 in both 0 bar CO2-atmosphere and an 1 bar CO2-atmosphere.During the simulation it is assumed that the dissolution of the oxides CaO, MgO and SiO2 willoccur after the (reactive) calcination of the float batch is finished.

400 600 800 1000 1200 1400 0.0

5.0x10 5

1.0x10 6

1.5x10 6

2.0x10 6

PSfrag replacements

Temperature [K]

Tota

lene

rgy

dem

and

[Jkg

−1

batc

h]

pCO2=0 bar

pCO2=1 bar

1A B

Onset temperature forcalcination reactions

C

Figure 2.33: Simulated total energy demand as function of temperature during ramp heating with 20K min−1 of the float batch with the composition as listed in table 2.1 in case pCO2=0 barand pCO2=1 bar. A indicates the end temperature for the calcination of the float batch, Bindicates the enthalpy of the completely molten float batch according to Madivate et al. [2]and C indicates the enthalpy of the float batch in case the dissolution of the oxides CaO,MgO and SiO2 in the sodium silicate melt phase is not taken into account.


Up to a temperature of 973 K, the energy demand of the float batch is determined by theheat capacity of the raw material components of the float batch. The onset temperature forcalcination of the float batch equals 973 K at which the calcination of the MgCO3-part of thedolomite starts. The curved lines show the change in energy demand of the float batch in thetemperature range from 973 K up to 1183 K due to heating of the glass batch componentsand the endothermic calcination reactions. During the simulations, at 1183 K the glass batchcontains a binary sodium silicate melt phase, crystalline MgO, CaO and SiO2 and the CO2which is released during the calcination reactions. The total energy demand of a completelymolten glass melt at 1443 K is fixed (point B) and is derived from [2].

In case the solid oxides would not dissolve in the sodium silicate melt, the energy demandof the float batch would follow line 1 up to point C. The difference in energy demand of point Cand point D is the net energy which is released during the dissolution of the oxides and mixingthe glass melt, which equals about 0.1 ·106 J kg−1

batch.It can be seen from figure 2.33 that the atmosphere composition does not have a large impact

on the temperature dependent energy demand of the float glass batch. Also the effect of theheating rate of the float batch on the temperature dependent energy demand of the float batch(see figure 2.34) is only small.

400 600 800 1000 1200 1400 0.0

5.0x10 5

1.0x10 6

1.5x10 6

2.0x10 6

PSfrag replacements

Temperature [K]

Tota

lene

rgy

dem

and

[Jkg

−1

batc

h]

5 K min−1

20 K min−1

1A B

Onset temperature forcalcination reactions

C

Figure 2.34: Simulated total energy demand as function of temperature during ramp heating of the floatbatch with the composition as listed in table 2.1 in case pCO2=1 bar for both a heating rate of5 K min−1 and 20 K min−1. A indicates the end temperature for the calcination of the floatbatch, B indicates the enthalpy of the completely molten float batch according to Madivateet al. [2] and C indicates the enthalpy of the float batch in case the dissolution of the oxidesCaO, MgO and SiO2 in the sodium silicate melt phase is not taken into account.

2.6. Concluding remarks 71

2.6 Concluding remarksAs discussed in section 2.1, the chemical energy demand of glass batches is to a large extentdependent on the energy required for decomposition reactions, such as calcination reactions.For the estimation of the time and temperature dependent chemical energy demand of a glassbatch, the kinetics of the different decomposition reactions are required. For the float glassbatch, which was discussed in section 2.4, the calcination of dolomite, limestone and soda ashoccurred (almost) independently from each other. This allows the determination of the kineticsof the individual calcination reactions to describe the calcination behavior of the complete floatglass batch. Combination of the measured kinetics of the individual calcination reactions withthe calcination enthalpies, which were derived from thermodynamic tables [3], resulted in anexpression for the chemical energy required for complete calcination of the float glass batch asfunction of time, temperature and partial CO2-pressure.

Kramer [35] and Kawachi et al. [36] measured the release of batch and fining gases as func-tion of temperature for a TV-panel glass batch. It was observed that the release of CO2 occursin a broad temperature range from 700 K up to 1500 K with a maximum CO2 release between1100 K and 1200 K. To determine the chemical energy required for complete calcination ofa TV-panel glass batch as function of time, temperature and partial CO2-pressure, similar tothe float glass batch, the kinetics of the different calcination reactions occurring during heatingof the TV-panel glass batch need to be determined. In appendix A, the calcination behaviorof a TV-panel glass batch, with as major constituents silica sand, soda ash, potash (K2CO3),potassium nitrate (KNO3), nepheline (Na2O ·Al2O3 ·SiO2), strontium carbonate (SrCO3), bar-ium carbonate (BaCO3), and zircon silicate (ZrO2 ·SiO2), is investigated.

In contrast to the float glass batch, at which only soda ash decomposed via reactive calcina-tion, during heating of a TV-panel batch, it appeared that soda ash, potash, strontium carbonateand barium carbonate decompose via reactive calcination. These reactive calcination reactionsdoes not allow the easy determination of the kinetics of the individual calcination reactions inthe TV-panel glass batch. Therefore, in contrast to the float glass batch, an expression for thechemical energy required for complete calcination of the TV-panel glass batch as function oftime, temperature and partial CO2-pressure cannot easily be determined.

Because the detailed analysis of the calcination rate of the complete TV-panel batch is time-consuming, the chemical energy demand of the TV-panel batch is not easy to determine. Beforethe kinetics of the individual calcination reactions in the TV-panel glass batch is studied in de-tail, it should be known how important a detailed description of the chemical energy demand isfor an accurate prediction of the heating process of a glass batch.

For the determination of the impact of the chemical energy demand on glass batch heating,information of other glass batch properties which affect the heating of a glass batch is required.In chapter 4, the heat conductivity of the glass batch will be discussed.



a activity [-]A pre-exponential factor [s−1]cp heat capacity [J kg−1 K−1]ckc1 kinetic compensation constant (equation 2.49)ckc2 kinetic compensation constant (equation 2.49)d diameter [m]Ea (apparent) reaction activation energy [J mol−1]f fugacity [Pa]f (ξ) reaction mechanism function [-]G Gibbs free energy [J mol−3]GE excess Gibbs free energy [J mol−3]H enthalpy [J mol−3]Hchem chemical energy demand [J m−3]Hr reaction enthalpy [J mol−1]g, i, j indicatorsk reaction rate constant [s−1]Ka ratio of the activities of reaction products and

reactants given by equation 2.30 [-]Keq reaction equilibrium constant [-]l liquid phasem mass [kg]M molar weight [kg mol−1]nro reaction ordernb number of glass batch componentsnc number of carbonates in a glass batchngc number of gas phase componentsnms number of mixture speciesnr number of reactantsnrp number of reaction productsp pressure [Pa]q heat flux [W m−2]r reaction rate [s−1]r∗ apparent reaction rate [s−1]R universal gas constant [J K−1 mole−1]s solid phaset time [s]T temperature [K]T0 initial temperature of a glass batch [K]Tfinal final heating temperature of a glass batch and

the released gases[K]

Tr glass batch reaction temperature [K]Tr,onset onset temperature for batch reactions [K]

2.7. Nomenclature 73

Tr,end end temperature for batch reactions [K]v velocity [m s−1]w weight fraction [-]x mole fraction [-]y mole fraction [-]

Greek symbols

α mass of gas formed that is releasedper mass (1+α) of glass batch [-]

β heating rate [K s−1]∆Gr Gibbs free energy of reaction [J mol−1]∆G0

r standard Gibbs free energy of reaction [J mol−1]∆H enthalpy change [J kg−1]∆Ht total energy demand of a glass batch [J kg−1]∆H0 standard reaction enthalpy [J mol−1]∆S0 standard reaction entropy [J mol−1]εp porosity [-]γ activity coefficient [-]λ heat (phonon) conductivity [W m−1 K−1]µ chemical potential [J mol−1]ρ density [kg m−3]ν stoichiometric reaction coefficientξ degree of conversion [-]

Sub- and superscripts

a actualb backwardc condensed phaseeff effectiveeq equilibriumf forwardg gas phasemn meanr radiativesc stoichiometric compoundz vertical direction0 initial

74 Bibliography

2.8 Bibliography[1] A. Ungan and R. Viskanta. Melting behavior of continuously charged loose batch blankets

in glass melting furnaces. Glastech. Ber., 59(10):279–291, 1986.


[3] O. Knacke, O. Kubaschewski, and K. Hesselmann. Thermochemical Properties of Inor-ganic Substances. Springer-Verlag Berlin, Heidelberg, Germany, 2nd edition, 1991.

[4] D.E. Sharp and L.B. Ginther. Effect of composition and temperature on the specific heatof glass. J. Am. Ceram. Soc., 34(9):260–271, 1951.

[5] Manual FactSage 5.1. Centre for Research in Computational Thermochemistry, EcolePolytechnique (Universite de Montreal), Montreal, Quebec, Canada.

[6] C. Kroger. Theoretischer Warmebedarf der Glasschmelzprozesse. Glastech. Ber.,26(7):202–214, 1953.

[7] R. Conradt and P. Pimkhaokham. An easy-to-apply method to estimate the heat demandfor melting technical silicate glasses. Glastech. Ber., 63K:134–143, 1990.

[8] B.A. Shakhmatkin, N.M. Vedishcheva, and C.A. Wright. Thermodynamic properties: Areliable instrument for predicting glass properties. In Proc. Int. Congr. Glass, volume 1,pages 52–60, Edinburgh, Scotland, 1-6 July 2001.

[9] R.A. Van Santen and J.W. Niemantsverdriet. Chemical kinetics and catalysis. Lecturenotes, Eindhoven University of Technology, Eindhoven, The Netherlands, June 1992.

[10] H. Salmang and H. Scholze. Die physikalischen und chemischen Grundlagen derKeramik. Springer-Verlag, Berlin/Heidelberg, Germany, 5 edition, 1968.

[11] A.K. Galwey and M.E. Brown. Application of the arrhenius equation to solid state kinet-ics: can this be justified? Thermochim. Acta, 386:91–98, 2002.

[12] J. Opfermann. Kinetic analysis using multivariate non-linear regression I. Basic concepts.J. Thermal Anal. Cal., 60:641–658, 2000.

[13] G. Pokol. The thermodynamic driving force in the kinetic evaluation of thermoanalyticalcurves. J. Thermal Anal. Cal., 60:879–886, 2000.

[14] E.M. Levin, C.R. Robbins, and H.F. McMurdie. Phase Diagrams for Ceramics. TheAmerican Ceramic Society, 1964.

[15] R. Conradt. A simplified procedure to estimate thermodynamic activities in multicompo-nent oxide melts. Molten Salt Forum, 5-6:155–162, 1998.

[16] M.L. Pearce. Solubility of carbon dioxide and variation of oxygen ion activity in soda-silica melts. J. Am. Ceram. Soc., 47(7):342–347, 1964.

Bibliography 75

[17] P.R. Laimbock. Foaming of glass melts. PhD thesis, Eindhoven University of Technology,1998.

[18] F. Paulik and J. Paulik. Investigations under quasi-isothermal and quasi-isobaric condi-tions by means of the derivatograph. J. Thermal Anal., 5:253–270, 1973.

[19] D.A. Young. The international encyclopedia of physical chemistry and chemical physics,Topic 21: Solid and surface kinetics, Volume 1: Decomposition of solids. Pergamon PressLTD., London, UK, 1st edition, 1966. Editor: P.C. Tomkins.

[20] B.V. L’vov. Mechanism and kinetics of thermal decomposition of carbonates. Ther-mochim. Acta, 386:1–16, 2002.

[21] P.K. Gallagher and D.W. Johnson. Kinetics of the thermal decomposition of CaCO3 inCO2 and some observations on the kinetic compensation effect. Thermochimica Acta,14:255–261, 1976.

[22] R. Ozao, M. Ochiai, A. Yamazaki, and R. Otsuka. Thermal analysis of ground dolomites.Thermochimica Acta, 183:183–198, 1991.

[23] M. Olszak-Humienik and J. Mozejko. Kinetics of thermal dolomite of dolomite. J. Ther-mal Anal. Cal., 56:829–833, 1999.

[24] R. Conradt. Melting behavior of batches containing ground cullets. In Fundamentals ofGlass Science and Technology, pages 290–297, Vaxjo, Sweden, June 9-12, 1997.

[25] P.K. Gallagher and D.W. Johnson. The effects of sample size and heating rate on thekinetics of the thermal decomposition of CaCO3. Thermochimica Acta, 6:67–83, 1973.

[26] J.M. Criado and A. Ortega. A study of the influence of particle size on the thermal de-composition of CaCO3 by means of constant rate thermal analysis. Thermochimica Acta,195:163–167, 1992.

[27] K. Kautz and Stromburg G. Untersuchungen der Vorgange beim Einschmelzen vonGlasgemengen im Gradientofen. Glastech. Ber., 42(7):309–317, 1969.

[28] C.A. Sheckler and D.R. Dinger. Effect of particle size distribution on the melting of soda-lime-silica glass. J. Am. Ceram. Soc., 73(1):24–30, 1990.

[29] F.W. Wilburn, S.A. Metcalfe, and R.S. Warburton. Differential thermal analysis, differ-ential thermogravimetric analysis, and high temperature microscopy of reactions betweenthe major components of a sheet glass batch. Glass Technology, 6(4):107–114, 1965.

[30] T.D. Taylor and K.C. Rowan. Melting reactions of soda-lime-silicate glasses containingsodium sulphate. Comm. Am. Ceram. Soc., pages C227–C228, 1983.

[31] C. Kroger. Gemengereaktionen und Glasschmelze. Glastech. Ber., 25(10):307–324, 1952.


76 Bibliography

[33] L. Riedel. Die Benetzung von Kalk und Quarz durch schmelzende Soda - Eine phanome-nologische Studie. Glastechn. Ber., 35(1):53–56, 1962.

[34] M.E. Savard and R.F. Speyer. Effect of particle size on the fusion of soda-lime-silicateglass containing NaCl. J. Am. Ceram. Soc., 76(3):671–677, 1993.

[35] F. Kramer. Gasprofilmessungen zur Bestimmung der Gasabgabe beimGlasschmelzprozeß. Glastechn. Ber., 53(7):177–188, 1980.

[36] S. Kawachi, M. Kato, and Y. Kawase. Evaluation of reaction rate of refining agents.Glastech. Ber. Glass Sci. Technol., 72(6):182–187, 1999.

Chapter 3

Dissolution of sand grains during heatingof glass forming batches

3.1 IntroductionAs mentioned by Hrma [1], the melting of glass batches is a complex process involving dif-ferent reaction types such as dehydration reactions, crystalline inversions, solid-state reactionsbetween the different raw material grains, decomposition reactions, melt forming reactions anddissolution processes. The dissolution of sand grains in the generated melts is regarded as themost significant criterion for the degree to which the melting of a glass batch based on silicasand has advanced [2]. In general, during the glass melting process, two dissolution stages ofsand grains are distinguished, viz.:

1. the dissolution of sand grains in the batch blanket, and

2. the dissolution of sand grains in the bulk of the glass melt.

The first dissolution stage concerns the dissolution of sand grains in the primary formed meltphases during batch heating1. Although the major part of a sand grain dissolves in the batchblanket, a minor part of the sand grain may dissolve in the glass melt underneath the batchblanket. The return flow of glass melt from the spring zone in the melting tank underneaththe bottom of the batch blanket (see figure 1.1 in chapter 1), may drag undissolved sand grainsfrom the batch blanket into the bulk of the glass melt. The major part of the experimental andmodelling studies of sand grain dissolution presented in literature, focusses on the descriptionof the dissolution of the sand grains in a large glass melt volume as function of time and tem-perature. These studies from literature represent the dissolution of sand grains in the bulk of theglass melt in industrial glass melting furnaces. Primarily, this dissolution stage is investigatedbecause undissolved sand grains in the bulk of the glass melt are potential causes for productreject in case the time-temperature trajectory of sand grains in the melting tank are insufficientfor complete or late dissolution. Because the objective of this thesis is to describe the behaviorof glass batches, this section focusses on the more complex description of the dissolution ofsand grains in the batch blanket.

1The primary formed melt phases are the first melt phases formed during heating of a glass batch. The onsettemperature for primary melt phase formation, which is dependent on the composition of the glass batch, equalsapproximately 1073 K for soda-lime-silica glass types (see section 2.5.3).

77

78 Chapter 3. Dissolution of sand grains during heating of glass forming batches

In section 3.2, a literature review of mathematical models is given describing the sand graindissolution during heating of glass batches. From the literature review, it is concluded that thesand grain dissolution process is a fairly complex process of different simultaneously occur-ring processes, which depend on a large variety of glass batch properties such as for examplehomogeneity of the glass batch, particle size (distribution) of the raw material components,viscosity and surface tension of the glass melts formed during glass batch heating. Because itis (almost) impossible to describe such a complex process, due to e.g. the lack of sensors tomonitor the processes or to measure different properties that determine the rate of this process,it is a widespread practice to use simple approximate theoretical models [3] for the descriptionof such a complex process. An approximate theoretical model is a simplification of a detailedfirst-principles model, which can provide a detailed description of the process that is studied.To derive an approximate theoretical model, simplifications in both the governing equations andthe boundary conditions of the detailed first-principles model are applied. The main advantageof using an approximate theoretical model above the detailed first-principles model for describ-ing the dissolution of sand grains during glass batch heating is that the approximate theoreticalmodel provides a far more simple expression for the dissolution rate of a sand grain. A disad-vantage of the approximate theoretical model is that, because of the assumptions that are madeto obtain the simple expression, the prediction of the dissolution rate of the sand grain is likelyto be less accurate than the predictions obtained with the numerical model.

The general form of an approximate theoretical model (see chapter 2) is given by

r∗ = k f (ξ), (3.1)

in which r∗ describes the rate of the process that is studied and k is the rate constant of theprocess, which is in general given by an Arrhenius type equation given by

k = A e−EaR T , (3.2)

in which A is a pre-exponential factor, Ea is an apparent reaction activation energy, R is theuniversal gas constant and T is the temperature in K. The parameter f (ξ) in equation 3.1 is theso-called reaction mechanism function. Table 3.1 lists the main generally applied reaction typesand their function f (ξ) derived from the reaction mechanism.

The reaction mechanism functions f (ξ) in table 3.1 characterize different types of processessuch as for example reaction kinetic limited reactions and mass transfer governed processes.The most suitable approximate model describing the dissolution of sand grains during heatingof glass batches depends on the rate governing step of the dissolution process. In section 3.2, adescription is given of the dissolution mechanism of sand grain in glass batches during heating.It appears that it is most likely that the dissolution of spherical sand grains2 can be regarded as athree dimensional diffusion governed process in the phase surrounding the sand grain. For thistype of process, two approximate models are listed in table 3.1, viz. the Jander model and theGinstling-Brounstein model3. According to Frade and Cable [3], the GB-model is favored overthe Jander model. The reason for this is that in the derivation of the Jander model, equationswhich are based on different shapes of the dissolving particle (planar and spherical) are com-bined to predict the degree of conversion of the spherical particle (see also Carter [5]). In this

2Although sand has different shapes, it is assumed for modelling of the sand grains dissolution behavior thatsand grains have a spherical shape.

3Further in this chapter, the Ginstling-Brounstein model will be denoted as the GB-model.

3.1. Introduction 79

Table 3.1: Code, reaction type and reaction mechanism function (see for an overview of these reactionmechanism functions [4]).

Code Reaction type f (ξ)

Fn nroth order reaction (1−ξ)nro

D1 One-dimensional diffusion 12ξ

D2 Two-dimensional diffusion −1ln(1−ξ)

D3 Three-dimensional diffusion (Jander’s type) 1.5(1−ξ)2/3

1−(1−ξ)1/3

D4 Three-dimensional diffusion (Ginstling-Brounstein type) 1.5(1−ξ)−1/3−1

R2 Two-dimensional phase boundary reaction 2(1−ξ)1/2

R3 Three-dimensional phase boundary reaction 3(1−ξ)2/3

chapter, the applicability of the approximate GB-model for the description of the dissolution ofsand grains during glass batch heating is studied.

The applicability of the GB-model for describing the dissolution of sand grains during glassbatch heating is studied by comparison of simulation results obtained with the GB-model withthe simulation results obtained with a more detailed numerical model describing the same pro-cess. In section 3.3, it is shown that the GB-model is not capable of predicting accurately thedegree of conversion of a sand grain dissolving in a surrounding medium in case the true val-ues for the physical and chemical properties used in the GB-model are applied. However, itis also observed that the shapes of the calculated conversion rate of the sand grain as functionof time with the GB-model and the more detailed numerical model are similar. This indicatesthat a modified GB-model can be used for the prediction of the sand grain dissolution process,although the values for the GB model parameters do not have a chemical and physical meaninganymore.

To measure the degree of sand grain conversion as function of time and temperature, a quan-titative analyzing technique is required. This technique should be able to determine the residualamount of crystalline silica in the partly molten glass batches. In section 3.4, the method ofquantitative phase analysis with X-ray diffraction, which is used as analyzing technique formeasuring the residual crystalline silica content in partly molten glass batches, is described.

In section 3.5, the experimental determination of the GB-model parameters as function ofthe initial particle size of the sand grain, and the cullet fraction in a float glass batch is dis-cussed. The float glass batch, with which the experiments are performed contain silica sand,soda ash and dolomite4. Although minor batch components such as sodium sulphate, calumite5

and water may have an impact on the sand grain dissolution process, the effect of these minorcomponents has not been studied in detail during this study.

4Because of confidentiality of the composition of the float glass batch, no detailed batch recipe is provided.5A steel slag often used as raw material during soda-lime-silica glass production.


3.2 Mathematical and experimental descriptions of the sandgrain dissolution process

In this section, a literature review is given of the mathematical description of sand grain dis-solution during heating of glass batches. First, the mathematical description presented byMuhlbauer and Nemec [6] and Beerkens et al. [7] is described, followed by the descriptiongiven by Hrma and co-authors [2, 8–10]. Next, a description is given of the dissolution mecha-nism of sand grains during the initial stage of glass batch heating. This description is based onliterature data and on own experimental results.

Dissolution models presented by Muhlbauer and Nemec and Beerkens et al.During heating of a glass batch, Muhlbauer and Nemec [6] and Beerkens et al. [7] distinguishedthree different dissolution stages in the reacting glass batch (see figure 3.1), viz.:

1. A reactive stage, during which the sand grains are consumed by a reaction with otherbatch components such as the reactive calcination of soda ash, which is given by

Na2CO3(s,l)+νSiO2(s,l) −→ Na2O ·νSiO2(s,l)+CO2(g). (3.3)

2. A transient stage, during which the dissolution of the sand grains is determined by thediffusion of SiO2 in the formed melt phase surrounding the sand grains. For this diffusiongoverned process, the shrinkage of the sand grains is determined by the SiO2 concentra-tion gradient and the inter SiO2 diffusion coefficient in the melt phase at the surface ofthe sand grain according to

ρs∂rs

∂t=

(

DSiO2

∂CSiO2

∂r

)

r=rs

, (3.4)

in which ρs is the constant density of the sand grain, rs is the radius of the sand grain,t is the time, DSiO2 is the temperature dependent inter diffusion coefficient of SiO2 inthe glass melt and CSiO2 is the time, temperature and position dependent concentration ofSiO2 expressed in kg m−3 in the melt phase. In the transient stage, the SiO2 concentrationgradient in the melt phase at the surface of the sand grain depends on time.

3. A quasi-stationary stage, during which the dissolution of the sand grains is also deter-mined by the SiO2 diffusion rate through the formed melt phase. However, in the quasi-stationary stage, the SiO2 concentration gradient in the melt phase at the surface of thesand grain is (almost) independent on time.

According to Beerkens et al. [7], the shrinkage of a sand grain during heating of a glass batch(see figure 3.2 for a schematic representation of the dissolution of SiO2 in a glass melt surround-ing the sand grain) is described by

ρs∂rs

∂t= −h

(

ρm,i wi −ρm,b wb)

, (3.5)

in which h is the time- and temperature dependent effective mass transfer coefficient, ρm,i andρm,b are the temperature dependent density of the glass melt at the interface of the sand grain

3.2. Mathematical and experimental descriptions of the sand grain dissolution process 81

PSfrag replacements

Reactive stage:

Transient andquasi-stationary

stage:

A

AA BB C

MCSiO2,rs

Reaction interface

t ↑Diffusionof SiO2

Figure 3.1: Schematic representation of different sand grain dissolution stages during heating of glassbatches containing sand grains and soda ash grains. The sand grain is represented by A,whereas the soda ash grain is represented by B. C represents the reaction product formedafter reactive dissolution of SiO2 according to equation 3.3. M represents a melt phase. Thecurved lines in the melt phase in the lower figure represent the SiO2 concentration profiles inthe melt phase as function of time. CSiO2,rs is the SiO2 concentration at the surface of the sandgrain rs, which is constant for a constant ratio of Na2O and CaO in the melt for a constanttemperature.

and in the bulk of the glass melt, respectively. The temperature dependent weight fraction ofSiO2 in the glass melt at the interface of the sand grain and in the bulk of the glass melt aregiven by wi and wb, respectively.

The reciprocal of the effective mass transfer coefficient h characterizes the total resistanceof mass transfer of SiO2 from the surface of the sand grain at r = rs towards the bulk of the glassmelt. Similar to Muhlbauer and Nemec [6], Beerkens et al. [7] described the total resistance ofmass transfer, in case that reaction kinetic limitations as well as SiO2 diffusion determine theoverall mass transfer of SiO2 into the bulk of the glass melt, as the sum of the resistance againstreactive sand dissolution at the interface of the sand grain and the resistance against diffusivetransport of SiO2 through the melt phase as is given by

1h

=1hr

+1hd

, (3.6)

in which 1/hr is the resistance for reactive sand grain dissolution and 1/hd is the resistance forSiO2 diffusion through the formed melt phase.

For the diffusion governed stage, both Muhlbauer and Nemec [6] and, Beerkens et al. [7]distinguished between a mass transfer coefficient for the transient stage and a mass transfer co-efficient for the quasi-stationary stage. Muhlbauer and Nemec [6] defined a characteristic time


PSfrag replacements

Sand grain Glass melt

Interface sand grain with melt phase

Bulk of the glass melt

ρs

ρm,i wi

ρm,b wb

ρ mw

rrs

Figure 3.2: Schematic representation of the dissolution of SiO2 in a glass melt surrounding a sand grain.

H for settling the quasi-stationary diffusion layer surrounding the silica grain and described thereciprocal overall mass transfer coefficient by

1h

=1hr

+1hd

(

1− e−H t)

. (3.7)

Beerkens et al. [7] described the mass transfer coefficient for the diffusion process by

hd = DSiO2

(

1rs

+1

√

π DSiO2 t

)

. (3.8)

The first right-hand-side term between brackets characterizes the quasi-stationary diffusionstage, whereas the transient stage is described by the second right-hand-side term betweenbrackets. Equation 3.8 is based on the analytical description of the diffusive transport of SiO2from the surface of a spherical sand grain into a semi-infinite glass melt volume described by

∂CSiO2

∂t=

DSiO2

r2∂∂r

(

r2 ∂CSiO2

∂r

)

, (3.9)


in which DSiO2 is the inter diffusion coefficient of SiO2 in the melt phase, which is taken inde-pendently from the melt phase composition. The derivation of this equation is given in section3.3.1.

According to Beerkens et al. [7], the dissolution of sand grains in a glass melt can be re-garded as a mass transfer process of SiO2 by diffusion enhanced by convective flow of the glassmelt. Free convective flow in the glass melt surrounding the sand grains is present in casethe glass melt density varies with the radial position in the melt phase. In general, glass meltdensity is dependent on both glass melt composition and temperature. Therefore, a radial de-pendent glass melt density can be caused by a radial temperature and/or composition gradientin the glass melt. The contribution of the free convective flow of the glass melt surrounding thesand grain on the mass transfer coefficient is described by the Sherwood number Sh accordingto

hd = DSiO2

Sh2

(

1rs

+1

√

π DSiO2 t

)

. (3.10)

The Sherwood number characterizes the effect of both flow conditions and properties of theglass melt on the mass transfer of SiO2 away from the sand grain interface. In case of masstransfer without convection, the Sherwood number equals the value 2 and equation 3.8 is re-tained. In case of free and forced convective flow of the glass melt surrounding the sand grain,the Sherwood number exceeds 2. The Sherwood number describing the contribution of free andforced convective flow on the mass transfer process from a spherical particle is given by

Sh =

(

2+0.89[

Re Sc+(Gr Sc3/4)]1/3

)

, (3.11)

in which Re, Sc and Gr are the Reynolds, Schmidt and Grasshof number, respectively. For adetailed description and derivation of these dimensionless numbers is referred to Bird et al. [11].

Next to the contribution of convective flow of the glass melt on the mass transfer process ofSiO2, Beerkens et al. [7] also take into account the effect of a so-called moving boundary onthe dissolution of sand grains6. In case a moving boundary is taken into account, the Sherwoodnumber is corrected according to Ready and Cooper [12], resulting in

Sh′ =Sh

(1− CSiO2,rsCSiO2,s

), (3.12)

in which CSiO2,rs is the SiO2 concentration in the glass melt at the interface of the sand grainand CSiO2,s is the SiO2 concentration in the sand grain. Combining equations 3.10, 3.11 and3.12 results in equation 3.13 describing the mass transfer coefficient for SiO2 transport in thediffusive governed stage taking into account both SiO2 diffusion, free and forced convectiveflow of the glass melt and the effect of a so-called moving boundary:

hd = DSiO2

(

2+0.89[

Re Sc+(Gr Sc3/4)]1/3

)

2 (1− CSiO2,iCSiO2,s

)

(

1rs

+1

√

π DSiO2 t

)

(3.13)

Combination of equations 3.5, 3.6 and 3.13 results in an expression describing the shrinkage ofa sand grain dependent on the free and forced convective flow conditions in the surrounding melt

6The phenomenon of the moving boundary is explained in section 3.3.2


phase, the melt phase properties and the resistance towards reactive dissolution of sand grainscharacterized by hr. Beerkens et al. [7] did not provide an explicit expression for the masstransfer coefficient resistance hr for reactive sand grain dissolution. Based on experimental dataof residual crystalline silica as function of time and temperature in a reacting batch, Beerkens etal. [7] proposed the empirically derived equation for the shrinkage of a sand grain in the glassbatch:

∂rs

∂t=

13

r3s,0

r2s

K1 eK2−T

K3 , (3.14)

in which rs,0 is the initial radius of the sand grain and K1, K2 and K3 are constants which dependon the composition of the glass batch.

Muhlbauer and Nemec [6] determined the values for hr, hd and H from experimental datafor a float glass batch. However, the procedure for the determination of these values is notgiven. It is also not known whether these values are dependent on glass batch properties suchas particle size of the raw materials.

Dissolution model presented by HrmaHrma [9] described the dissolution process/reaction of sand grains in mixtures of sand and sodaash during which five different stages are distinguished:

1. An initial stage, which is controlled by a surface reaction of sand with soda ash duringwhich a sodium silicate melt phase is formed, which surrounds the sand grain.

2. A transient stage, which is controlled by both the surface reaction and non steady-statediffusion of reactants and reaction products. During this stage, the thickness of the meltphase around the sand grain increases with time. The transport of reactants and reactionproducts in the melt phase is determined by diffusion and convective flow of the glassmelt surrounding the sand grain. The convection of the glass melt can either be causedby free (buoyancy) convection due to density gradients in the glass melt or by forcedconvection due to the effect of ascending gas bubbles on the melt.

3. A stationary stage, during diffusion and convection is the rate governing step. The SiO2concentration at the sand grain interface is in thermodynamic equilibrium with the SiO2in the sand grain.

4. A disappearance stage, during which the sand grain dissolution is enhanced by the reduc-ing sand grain size.

5. A homogenization stage, during which local variations in SiO2 concentration in the formedmelt phase are smoothed out.

For the initial stage, Hrma [8] described the silica dissolution during heating of a mixture ofsilica sand and soda ash based on observations and identification of intermediate formed crys-talline sodium silicates presented in literature. During the calcination of soda ash, Hrma as-sumes that molten soda ash reacts with silica sand grains according to

Na2CO3(l)+ν SiO2(s) −→ Na2O ·νSiO2(s,l)+CO2(g). (3.15)


The shrinkage of the silica grain during the calcination of molten soda ash can be derived fromthe CO2 release given by

∂rs

∂t= ϑsa ν J

V s

V CO2

, (3.16)

in which ϑsa is the silica grain surface fraction which is wetted by the molten soda ash, ν isthe stoichiometric reaction coefficient of reaction 3.15, J is the volume flux of evolving CO2per unit of grain surface which is wetted by the molten soda ash, V s is the molar volume ofsolid silica and V CO2 is the molar volume of the released CO2 at the reaction temperature.According to experimental studies (e.g. [13–15]), the predominant stoichiometric crystallinesodium silicate, which is formed in the temperature range between 973 K and 1173 K, is sodiummetasilicate (Na2O ·SiO2). Now, describing the shrinkage of the sand grain requires the rate ofthe calcination reaction of soda ash at the silica surface forming sodium metasilicate.

Finally, knowledge of the degree of wetting of the silica grains by molten sodium carbonateand the sodium silicate melt phase would complete the information necessary for describing theshrinkage of silica grains.

The wetting of sand grains by liquid phases is dependent on several parameters such asthe particle size of the soda ash particles, gas evolution, surface tension of the liquid phasesand humidity and homogeneity of the glass batch. The evolution of gases during heating ofglass batches may, on one hand, retard the dissolution of sand grains by preventing contactbetween the sand grain and the melt phase. On the other hand, gas evolution may also enhancethe dissolution of sand grains by improved micro mixing of the primary formed melt phasessurrounding the sand grains. This results in an enhanced transport of SiO2 away from the sandgrain surface. The wetting of the sand grains will also be improved in case that the surfacetension of the liquid phase formed during heating of the glass forming batch is lowered.

Humidification of the glass batch prior to heating may also enhance the dissolution rate ofsand grains by for example improving the contact of the sand grains with the soda ash layers.During humidification of a glass batch, part of the soda ash will dissolve in the water. Duringheating, the soda ash will recrystallize and thereby forming a finer distributed layer throughoutthe glass batch. Local variations in the composition of the glass batch are likely because of:

• segregation of batch components due to differences in the particle size(s) distribution ofthe different glass batch components, and

• the ascension of undissolved batch grains in the formed melt phases due to density differ-ences or due to the ascension of undissolved batch grains with attached bubbles.

Because of the processes mentioned above, it is not likely that an unambiguous prediction ofthe wetting behavior and the local homogeneity of the glass batch is possible.

According to Hrma, after complete calcination of the soda ash, which is given by equation3.15, the remaining species in the melting glass batch are the primary formed melt phase, theresidual silica sand grains and the intermediate crystalline sodium silicates. The dissolution ofsilica grains is now regarded as a diffusion governed process for which the shrinkage of thesilica grains is described by

∂rs

∂t= ϑs DSiO2

∆wSiO2

δ, (3.17)

in which ϑs is the silica grain surface fraction which is wetted by the liquid sodium silicatemelt, ∆wSiO2 is the difference in SiO2 weight fraction at the surface of the silica grain and the


the outer boundary of the sodium silicate (melt) phase, and δ is the Nernst boundary layer thick-ness between the silica grain and the surface of the temporary (liquid) sodium metasilicate layer.This equation is similar to the equation presented by Muhlbauer and Nemec and Beerkens et al.for the diffusion governed stage. Again, the wetting of the silica grains is an important param-eter in the equation describing the shrinkage of the silica grains, which can hardly be predictedbeforehand.

Discussion of the conversion mechanism of silica sand during heating of glass batchesAccording to the mathematical models presented above, the conversion of sand grains duringheating of glass batches is determined by a combination of a reaction kinetic governed processand a SiO2 diffusion governed process. In general, a chemical reaction can also be governed bythermodynamic driving forces. For the sand grain conversion in a glass batch, it is easily provedthat thermodynamics of the reactions do not govern this process.

Based on thermodynamics (see section 2.5.3), in a 1 bar CO2-atmosphere, the stoichiomet-ric sodium silicates Na2O ·SiO2 and Na2O ·2SiO2 are likely to be formed as solid phases attemperatures above 632 K by a reaction of soda ash with sand according to

Na2CO3(s)+2 SiO2(s) Na2O.2SiO2(s)+CO2(g), (3.18)

andNa2CO3(s)+SiO2(s) Na2O.SiO2(s)+CO2(g). (3.19)

At 1011 K, also the formation of sodium disilicate is thermodynamically favorable:

2 Na2CO3(s)+SiO2(s) 2Na2O.SiO2(s)+2 CO2(g). (3.20)

From thermogravimetrical analysis of mixtures of sand and soda ash, which are described insection 2.4, it was observed that the onset temperature for soda ash calcination7, during heatingof a mixture of sand and soda ash in a N2-atmosphere, is determined on approximately 1073K. This indicates that the calcination of soda ash occurs far from thermodynamic equilibrium.Therefore, it is concluded that this reaction is not determined by thermodynamics, but that ei-ther reaction kinetics and/or mass transfer limitations govern these conversions.

Kroger and co-authors performed an extensive experimental study on the (solid-state) kinet-ics of soda ash calcination in both binary mixtures with sand grains and in ternary and quater-nary systems at different temperatures [16–26]. Kroger observed that at temperatures at whichonly solid state reaction products are thermodynamically stable, the rate of the solid-state reac-tions provided by reactions 3.18, 3.19 and 3.20 are slow. An enhanced calcination rate of sodaash was observed after the formation of a liquid phase. Thermogravimetric analysis performedduring the current study (see section 2.5.3), showed a measured calcination onset temperaturefor soda ash of 1073 K, which is almost equal to the eutectic melting temperature of sodiumdisilicate (Na2O ·2SiO2) and SiO2 at 1072 K (see also figure 3.3). This agrees with Kroger’sobservation that the presence of a melt phase enhances the calcination of soda ash.

Assuming that the eutectic melting temperature of Na2O ·2SiO2 and SiO2 determines theonset for soda ash calcination, it is expected that crystalline sodium disilicate (Na2O ·2SiO2) ispresent as intermediate crystalline specie during heating of glass batches containing silica sand

7The onset temperature for soda ash calcination, which is determined by thermogravimetrical analysis, is char-acterized by the observed onset temperature for weight loss of the soda ash grain during heating.


0.5 0.6 0.7 0.8 0.9 1.0 1000

1100

1200

1300

1400

1500

1600

1700

1800

PSfrag replacements

SiO2 mole fraction [-]

Tem

pera

ture

[K]

L1

L2

L3

w1w2

Figure 3.3: Part of the binary phase diagram of Na2O and SiO2 for a mole fraction of SiO2 in the binarysystem larger than 0.5. This phase diagram is redesigned after [27]. L1, L2, and L3 are theliquidus lines of SiO2, Na2O ·2SiO2 and Na2O ·SiO2, respectively. w1 and w2 indicate thecomposition of the melt phases formed after eutectic melting of SiO2 with Na2O ·2SiO2 andNa2O ·2SiO2 with Na2O ·SiO2, respectively. The values for w1 and w2 are 0.735 and 0.62,respectively.

and soda ash. However, in the current study, sodium disilicate has neither been identified inquenched samples of partly molten mixtures of soda ash and sand grains nor in partly moltenfloat glass and TV-panel glass batches. Similar to this observation, during most melting exper-iments with glass batches containing sand and soda ash presented in literature (e.g. [13–15]),also no sodium disilicate was identified. Only during a few experimental studies (e.g. [28, 29]),crystalline sodium disilicate or both crystalline sodium disilicate and sodium metasilicate wereidentified at temperatures around 1073 K. From the results presented by Savard and Speyer [28],it seems that the preference of either of these two sodium silicates is dependent on the particlesize of the raw materials. The reason that no sodium disilicate is found during the current study,is probably that the conditions at which the melting experiments are performed (e.g. the particlesize of the raw material components) did not allow a sufficient amount of sodium disilicate tobe formed in order to be detected with X-ray diffraction. Phase analysis on heated samples ofmixtures of sand and soda ash with systematic variations of parameters such as particle size ofthe reactants and heating rate, combined with microscopic observations of the heated sampleswould reveal information on the detailed mechanism of interaction of sand and soda ash. How-ever, this detailed information of the microscopic reaction mechanism of silica sand and sodaash is not required to determine the rate of this reaction.


PSfrag replacements

wSiO2(r)

wNa2O(r)

wSiO2,L2

wNa2O,L2

wSiO2,L1

wNa2O,L1

NC SM

C

w = 1

w = 0

Figure 3.4: Simplified schematic representation of the different phases present during heating of a mix-ture containing silica sand and soda ash. NC, C, S and M indicate Na2CO3, the intermediatecrystalline sodium disilicate, SiO2 and a melt phase, respectively. wi,L1 and wi,L2 indicate theweight fraction of component i given by the liquidus lines L1 and L2, respectively (see figure3.3).

At temperatures above the measured onset temperature for soda ash calcination (1073 K),the primary formed melt phase separates the sand grain from a thin layer of intermediate formedcrystalline sodium disilicate. Assuming that at the surface of the silica sand grain and thecrystalline sodium disilicate phase thermodynamic equilibrium exist with the melt phase, thereacting mixture of silica sand and soda ash can schematically be represented by figure 3.4.The overall rate of the reaction of soda ash with sand grains is now determined by the followingthree process steps:

1. The dissolution of silica at the interface of the silica sand grain with the sodium silicatemelt phase.

2. The diffusion of SiO2 from the surface of the sand grain through the melt phase towardsthe intermediate crystalline sodium disilicate and the diffusion of Na2O in opposite direc-tion.

3. The dissolution of the crystalline sodium silicate in the melt phase, which is assumedto be instantaneously followed by the reactive calcination of soda ash at the interface ofcrystalline sodium silicate.


The conversion rate of the sand grain can now be determined by the measured calcinationrate of soda ash according to

∂ξSiO2

∂t=

MSiO2

MNa2CO3

m0Na2CO3

m0SiO2

wSiO2,m

1−wSiO2,mANa2CO3 e

−ENa2CO3R T (1−ξNa2CO3) , (3.21)

in which ξSiO2 is the degree of conversion of the sand grain, MNa2CO3 and MSiO2 are the molarweights of soda ash and silica sand, respectively. The initial composition of the binary com-ponent glass batch is given by m0

SiO2and m0

Na2CO3. The mean SiO2 weight fraction in the melt

phase formed above 1073 K, which is dependent on the SiO2 concentration profile is the meltphase, is given by wSiO2,m. The reaction kinetics parameters describing the reactive calcinationof soda ash are ANa2CO3 and ENa2CO3 , respectively (see section 2.5.3). The degree of calcinationof soda ash is given by ξNa2CO3 . The conversion of silica after complete calcination of soda ashis given by

ξSiO2 =m0

Na2CO3

m0SiO2

MSiO2

MNa2CO3

wSiO2,m

1−wSiO2,m. (3.22)

The mean SiO2 weight fraction in the melt phase is at most equal to the maximum SiO2solubility given by the liquidus line of silica. The minimum value for the mean SiO2 weightfraction in the melt phase is equal to the SiO2 solubility given by the liquidus compositions ofthe crystalline sodium silicates. The degree of conversion of silica sand in a float glass batch, inthe temperature range during which still soda ash is present, is evaluated by quantitative phaseanalysis of the amount of sand grains8 in partly molten float glass batch samples. Figure 3.5shows both the measured and the calculated conversion of the sand grains in the partly moltenfloat glass batch as function of temperature.

According to the phase diagram of SiO2 and Na2O, in the temperature range from 1072 Kup to 1110 K, the maximum and minimum SiO2 solubility in the melt phase is given by theliquidus composition of silica (wL1) and sodium disilicate (wL2), respectively. Above 1110 K,the minimum SiO2 solubility is given by the liquidus line of sodium metasilicate (wL3). Fromfigure 3.5, it is seen that the measured sand grain conversion at temperatures above 1110 K islower than is expected in case the sand grain conversion would have followed line C duringheating. This indicates that, assuming that recrystallization of SiO2 from the sodium silicatemelt phase is not likely to occur, at temperatures below 1110 K, a melt phase is present with alower SiO2 weight fraction as is expected based on the SiO2 and Na2O phase diagram.

Because the mean weight fraction of SiO2 in the melt phase is not equal to the SiO2 weightfractions given by one of the liquidus lines, it is concluded that sand grain dissolution duringmelting of glass batches is at least partly determined by the diffusion of silica through the pri-mary formed melt phase, which is supported by Rottenbacher and Engelke [30].

Based on the experimental evaluation of the mechanism of the sand grain dissolution processin glass batches and based on literature, the dissolution process of a sand grain in a surroundingglass melt is described as a diffusion governed process of SiO2 from the surface of the sandgrain through the melt phase surrounding the sand grain. Assuming that the sand grain has aspherical shape and that the sand grain is completely wetted by the melt phase, figure 3.6 showsa schematic representation of the sand grain dissolution process in a primary formed melt phase.

8For the description of the quantitative analysis of the amount of sand grains in partly molten glass batches isreferred to section 3.4


1080 1100 1120 1140 1160 1180 0.0

0.1

0.2

0.3

0.4

0.5

PSfrag replacements

Temperature [K]

Deg

ree

ofsa

ndgr

ain

conv

ersi

on[-

] A

B

C

Figure 3.5: Measured (blacks dots) and simulated (solid lines) conversion of the sand grains during heat-ing a float glass batch as function of temperature. A indicates the simulated sand grain con-version in case the average SiO2 weight fraction in the melt phase equals wL1(T ), B indicatesthe simulated sand grain conversion in case the average SiO2 weight fraction in the meltphase equals wL3(T ) and C indicates the simulated sand grain conversion in case the SiO2weight fraction in the melt phase equals wL2(T ).

The situations represented by A, B and C describe the dissolution of the sand grain at the pres-ence of still undecomposed soda ash. Throughout the dissolution process of the sand grain, it isassumed that at the surface of the sand grain, the SiO2 concentration is given by the temperaturedependent maximum solubility of SiO2 in the sodium silicate melt phase CSiO2,L1(T). As long asthe soda ash is not completely decomposed, the SiO2 concentration at the surface of the sodaash grain is given by the temperature dependent solubility of SiO2 CSiO2,L2(T). At the surfaceof the sand grain, SiO2 dissolves in the sodium silicate melt phase and the sand grain size rsdecreases. At the surface of the soda ash grain, soda ash reacts with the sodium silicate meltphase during which CO2 is released. Because of mass loss during the release of CO2, the radiusof the total system rt decreases. After complete calcination of soda ash, the radius of the totalsystem remains constant. Because a melting glass batch can be regarded as a three dimensionalarrangement of sand grains dissolving in a surrounding glass melt as is indicated in figure 3.6,the SiO2 concentration at the outer radius of the system can be approximated by a zero gradientboundary condition. The concentration profiles in the sodium silicate melt phase flattens withincreasing time.

Concluding remarksBecause of the complex dissolution mechanism of sand grains in glass batches, Frade andCable [3] mentioned that a model with the correct mass balances together and the appropri-ate time-dependent boundary conditions, describing the time- and temperature-dependent sand


PSfrag replacements

00

00

rsrs

rsrs

roro

roro rtrt

II

II

IIII

IIII IIIIII

CSiO2,L1CSiO2,L1

CSiO2,L1CSiO2,L1

CSiO2,L2CSiO2,L2

CSiO2,L2

CSiO2,sCSiO2,s

CSiO2,sCSiO2,s

A B

C D

t↑

Figure 3.6: Schematic representation of the sand grain dissolution process in a surrounding melt phase.The x-axis represents the radial distance from the center of the sand grain and the y-axisrepresents the SiO2 concentration, The sand grain, the melt phase and the soda ash grainare indicated by I, II and III, respectively. The SiO2 concentration in the sand grain, at thesurface of the sand grain and at the surface of the soda ash grain is given by CSiO2,s, CSiO2,L1 ,and CSiO2,L2 , respectively. Here, it is assumed that the soda ash grain is covered with a thinlayer of crystalline sodium disilicate. The sand grain size, the outer radius of the melt phaseand the outer radius of the total system are given by rs, ro and rt , respectively.

grain dissolution process is fairly complex. Because of the large number of parameters and pro-cesses, which influence the sand grain dissolution process, a generally applicable microscopicmodel describing the sand grain dissolution is at the moment not feasible. For the descriptionof such complex processes and phenomena, it is a widespread practice to use simple approxi-mate theoretical models [3]. Table 3.1 listed the main generally applied approximate theoreticalmodels. The approximate models which agree the best with the dissolution mechanism of sandgrains described above, are the Jander and the Ginstling-Brounstein model, because these mod-els are based on three dimensional diffusion governed mass transfer processes similar to thedissolution of sand grains. According to Frade and Cable [3], the GB-model is favored over theJander model. The reason for this is that in the derivation of the different Jander model equa-tions, which are based on different shapes of the dissolving particle (planar and spherical), arecombined to predict the degree of conversion of the spherical particle. Therefore, the Ginstling-


Brounstein is used as approximate model for the description of the dissolution of sand grainsduring glass batch.

However, the Ginstling-Brounstein model contains several approximations, which are likelyto be invalid for the overall description of the dissolution process of sand grains during heatingof glass batches. In the next section, it is studied whether and to which extent, the Ginstling-Brounstein is able to describe the sand grain dissolution process. The applicability of thismodel is studied by comparison of simulation results of the GB-model with simulation resultsobtained with a more detailed model. In the next section, first the derivation and assumptionsof the Ginstling-Brounstein model are described. Section 3.3.2 describes the governing equa-tions of the numerical sand grain dissolution model. In section 3.3.3, the applicability of theGB-model is studied by comparison of the results of the GB-model with results from a moredetailed numerical model and with the results of experiments.

3.3 Evaluation of the application of the Ginstling-Brounsteinmodel

In this section, the applicability of the Ginstling-Brounstein model for the description of thedissolution of sand grains during the melting of glass batches is studied. To evaluate andinterpret the applicability of the Ginstling-Brounstein model, the dissolution of a sand grain ina surrounding finite melt phase volume is simulated with two mathematical models, viz.:

• the Ginstling-Brounstein model, with which an approximate description of the sand graindissolution process can be simulated, and

• a numerical model, describing the sand grain dissolution process with more complex(real) melt properties and more realistic boundary conditions.

Section 3.3.1 describes the derivation of the Ginstling-Brounstein model. Section 3.3.2 de-scribes the governing equations of the numerical sand grain dissolution model. The applica-bility of the Ginstling-Brounstein model for the description of the sand grain dissolution duringheating of glass batches is evaluated in section 3.3.3 by comparison of results of the numericalmodel and the results of the Ginstling-Brounstein model and by the interpretation of the founddifferences.

3.3.1 Derivation of the Ginstling-Brounstein modelThe Ginstling-Brounstein model was developed for describing the kinetics of diffusion gov-erned processes such as solid-state reactions [3,13,31]. The GB-model describes the shrinkageof a spherical particle by dissolution in a infinite surrounding phase. At the interface of thespherical particle, a reaction layer is formed separating the spherical particle and the surround-ing phase. The growth of the reaction layer is assumed to be governed by diffusion of reactantsand reaction products through the formed diffusion layer. Figure 3.7 gives a schematic rep-resentation of this diffusion governed process. The radius of the spherical particle is denotedby rs,0, whereas the outer radius of the reaction zone is given by rs. With time, the sphericalparticle shrinks accompanied by an increase of the thickness of the reaction layer.With the GB-model, a quasi steady-state solution for the diffusion governed process in a phase

3.3. Evaluation of the application of the Ginstling-Brounstein model 93

PSfrag replacements

Spherical particlecomposed of specie i

Diffusion layer

Infinite phase

rs

rs

rs,0

rs,0

t ↑ t ↑

t = 0t = t

Ci,0

CSiO2,rs

CSiO2,rt

Ci,rs,0

Figure 3.7: Schematic representation of the diffusion governed dissolution of a spherical sand grain in asurrounding glass melt.

surrounding a spherical particle is calculated. In this section, the GB-model will be applied fordescribing the dissolution rate of a sand grain in a finite shell of melt phase surrounding the sandgrain. With the description of this diffusion governed process, the dissolution of sand grains dur-ing the melting of glass batches will be simulated. The main advantage of the GB-model abovethe numerical model presented in the next section, is that the GB-model will provide a far moresimple expression for the dissolution rate of a sand grain. A disadvantage of the GB-model maybe that, because of the assumptions that are made to obtain the simple expression, the predictionof the dissolution rate of the sand grain is likely to be less accurate than the predictions obtainedwith the numerical model.

The mathematical starting point of the GB-model, when applied to the dissolution of a sandgrain in a glass melt surrounding the sand grain, is the equation describing the transient diffusionof SiO2 through the surrounding glass melt given by

∂CSiO2

∂t= DSiO2

∂2CSiO2

∂r2 +2DSiO2

r∂CSiO2

∂r, (3.23)

in which CSiO2 is the weight concentration of SiO2 in the glass melt, t is the time, DSiO2 is theinter diffusion coefficient of SiO2 and r is the radial position in the glass melt. The derivationof the GB-model based on equation 3.23, is based on the following assumptions:


• For the determination of the SiO2 concentration at the boundary of thesand grain, which governs the dissolution rate of the sand grain, it isassumed that the radius of the sand grain is constant with time.

• The outer radius of the melt phase, rs,0, is constant.• The SiO2 concentrations at the inner and outer radius of the reaction zone

CSiO2,rs and CSiO2,rs,0 remain constant.• The initial concentration of SiO2 in the melt phase, CSiO2,0, is constant

throughout the melt phase.• The inter SiO2 diffusion coefficient in the melt phase is independent on

the local composition of the melt phase.• The density of the sand grain is similar to the density of the melt phase.

The analytical solution, describing the radial dependent weight concentration of SiO2 in theglass melt layer as function of time and taking into account the assumptions listed above, isgiven by Carslaw and Jager [32]:

CSiO2 =rs CSiO2,rs

r+

(rs,0 CSiO2,rb − rs CSiO2,rs) (r− rs)

r (rs,0 − rs)+

+2πr

∞

∑n=1

rs,0 (−1)n CSiO2,rs,0 − rs CSiO2,rs

nsin(

nπ(r− rs)

rs,0 − rs

)

e−n2π2τ + (3.24)

+2πr

∞

∑n=1

−rs,0 (−1)nCSiO2,0 + rs CSiO2,0

nsin(

nπ(r− rs)

rs,0 − rs

)

e−n2π2τ.

The symbol τ is the so-called Fourier number:

τ =DSiO2 t

(rs,0 − rs)2 . (3.25)

The term DSiO2

(rs,0−rs)2 is the reciprocal process time needed for a noticeable change in SiO2 con-centration at the outer radius of the spherical glass melt shell rs,0. If the time t is less than theprocess time DSiO2

(rs,0−rs)2 , the glass melt surrounding the sand grain can be regarded as an infiniteglass melt shell.

The equation describing the shrinkage of a sand grain in a finite melt phase surrounding thesand grain, which is derived in the next section, is given by

∂rs

∂t=

DSiO2

(

∂CSiO2

∂r

)

r=rs

CSiO2,s −CSiO2,rs

(3.26)

in which CSiO2,s is the SiO2 concentration in the sand grain. The shrinkage of the sand grain isdependent on the radial concentration gradient of SiO2 at the surface of the sand grain. Althoughequation 3.24 is only valid for non-shrinking cores, the GB-model uses this equation for thedetermination of the radial concentration gradient of SiO2 at the surface of the shrinking sand


grain, which is given by(

∂CSiO2

∂r

)

r=rs

=rs,0 (CSiO2,rs,0 −CSiO2,rs)

rs (rs,0 − rs)+ (3.27)

2rs (rs,0 − rs)

∞

∑n=1

[

rs,0 (−1)n (CSiO2,rs,0 −CSiO2,0)+ rs (CSiO2,0 −CSiO2,rs)]

e−n2π2τ.

In case the SiO2 concentration at the outer boundary of the glass melt equals the initial con-centration of SiO2 throughout the glass melt, i.e. CSiO2,rs,0 = CSiO2,0, equation 3.27 simplifiesto

(

∂CSiO2

∂r

)

r=rs

=(CSiO2,rs,0 −CSiO2,rs)

rs

[

2rs

(rs,0 − rs)

∞

∑n=1

e−n2π2τ +rs,0

(rs,0 − rs)

]

. (3.28)

The time dependent radial concentration gradient of SiO2 at the surface of the sand grain isdetermined by two terms characterizing different diffusion stages. The first term between thesquare brackets in equation 3.28 characterizes the transient stage, which was mentioned byboth Muhlbauer and Nemec [6] and Beerkens et al. [7]. The latter term represents the quasi-stationary stage during which the SiO2 concentration gradient in the melt phase at the surface ofthe sand grain is (almost) independent on time. This indicates that for sufficient large values oft, the time itself has no significant effect on the concentration profile of SiO2 at the sand grainboundary. During this stage, the radial concentration gradient of SiO2 in the glass melt at thesurface of the sand grain is mainly determined by the size of the residual sand grain.

Figure 3.8 shows the term[

2rs(rs,0−rs)

∑∞n=1 e−n2π2τ +

rs,0(rs,0−rs)

]

versus the reciprocal value ofthe square root of the time for the dissolution of a sand grain, with a grain size of 50 µm, inthe surrounding glass melt layer with a thickness of 950 µm. The inter diffusion coefficient isset on 1 ·10−12 m2 s−1. The time period during which the quasi-stationary stage predominatesthe dissolution process is indicated by A. The transition period between the quasi-stationarystage and the transient stage is indicated by B, whereas the time period during which the tran-sient stage predominates the dissolution process is indicated by C. In the transient stage of thedissolution process (C), the radial concentration gradient in the melt phase at the surface of thesand grain is proportional to the reciprocal of the square root of the time. This is similar to thetransient term presented by Beerkens et al. [7]. In case the quasi-stationary stage predominatesthe dissolution process (A), the radial concentration gradient in the glass melt at the surface ofthe silica sand grain is independent of time.

To evaluate which dissolution stage predominates during heating of glass batches, the endtime of the transient stage as function of particle size and melt phase thickness is determined.Defining the end time of the transient stage by the time at which the transient term in equation3.28 is less than 1 % of the quasi-stationary term, the Fourier number at the end of the transientstage, τte , satisfies

∞

∑n=1

e−n2 π2 τte =1

200rs,0

rs. (3.29)

The time required for reaching the end of the transient stage is calculated from equation 3.25and is given by

te =τte (rs,0 − rs)

2

DSiO2

. (3.30)


0.000 0.002 0.004 0.006 0.008 0.010 0.012

1.05

1.10

1.15

1.20

1.25

1.30

1.35

PSfrag replacements

t−0.5 [s−0.5]

[

2rs

(rs,

0−r s

)∑

∞ n=1

e−n2 π2 τ

+r s

,0(r

s,0−

r s)

]

[-]

A B C

Figure 3.8:[

2rs(rs,0−rs)

∑∞n=1 e−n2π2τ +

rs,0(rs,0−rs)

]

versus the reciprocal value of the square root of the time.A indicates the quasi-stationary stage, B indicates the transition period and C indicates thetransient range for sand grain dissolution in a surrounding melt phase in case the radius ofthe sand grain equals 50 µm, the thickness of the surrounding melt phase equals 950 µm andthe inter diffusion coefficient equals 1 ·10−12 m2 s−1.

Figure 3.9 shows the time required for reaching the end of the transient stage as function of thethickness of the glass melt rs,0 − rs surrounding the sand grain for both a radius of 50 µm and100 µm. The value for te is calculated from equation 3.25, with a constant diffusion coefficientof 1 ·10−12 m2 s−1. It is clearly seen that the the duration time of the transient stage increaseswith increasing thickness of the melt phase surrounding the sand grain. In case of thin glassmelt layers surrounding the sand grain, the dissolution process of sand grains mainly occurs inthe quasi-stationary stage. For a thickness of the surrounding melt phase of 25 µm and 50 µmsurrounding the sand grain with a radius of 50 µm, the duration time of the transient stage equals5 min and 20 min, respectively. This indicates that the duration time of the transient stage islinear proportional to the square of the thickness of the melt phase.

Resuming, in case convection of the melt phase surrounding the sand grains is absent, ac-cording to the GB-model the dissolution of sand grains in a large volume of glass melt mainlyoccurs in the transient stage. The dissolution of sand grains in the primary formed thin layersof melt phase during melting of a glass batch, mainly occurs in the quasi-stationary stage. In

general, for the quasi-stationary stage, t >(rs,0−rs)

2

DSiO2, equation 3.28 simplifies to

(

∂CSiO2

∂r

)

r=rs


rs

rs,0

(rs,0 − rs). (3.31)

In case of a thick glass melt phase surrounding the silica sand grain and a very long residence


0 100 200 300 400 500 0

200

400

600

800

1000

1200

PSfrag replacements

Thickness of melt phase rs,0 − rs [µm]

Dur

atio

ntim

eof

the

tran

sien

tsta

get e

[min

.]

Dissolution process inthe transient stage

Dissolution process inthe quasi-stationary stage

Figure 3.9: The time required for reaching the end of the transient stage te as function of the thicknessof the melt phase rs,0 − rs surrounding the sand grain for both a radius of 50 µm (solid line)and a radius of 100 µm (dotted line). The value for te is calculated from equation 3.25, witha constant diffusion coefficient of 1 ·10−12 m2 s−1.

time of the silica grain in the surrounding glass melt, equation 3.31 further simplifies to(

∂CSiO2

∂r

)

r=rs


rs. (3.32)

Then the concentration gradient in the melt phase at the surface of the sand grain is only depen-dent on the actual size of the silica sand grain, which was also described by Beerkens et al. (seeequation 3.8).

The GB-model describes the conversion rate of the silica sand grain during dissolution in afinite glass melt surrounding the silica sand grain in the quasi-stationary stage. Combination ofequations 3.26 and 3.31 results in

∂rs

∂t= −φ DSiO2

1rs

rs,o

rs,o − rs, (3.33)

in which φ is given by

φ =CSiO2,rs −CSiO2,rs,0

CSiO2,s −CSiO2,rs

. (3.34)

Integration of equation 3.33 results in

r2s − r2

s,0

2− r3

s3 rs,0

+r3

s,0

3 rs,0= −φ D t. (3.35)


Expressing the degree of silica conversion by

ξ = 1−(

rs

rs,0

)3

, (3.36)

and combination of equations 3.35 and 3.36 results in the GB-model equation:

1− 23

ξ− (1−ξ)2/3 =2 φ DSiO2

r2s,0

t (3.37)

Resuming, the GB-model is an approximate model describing the conversion of diffusion gov-erned solid-state reactions in a finite layer surrounding the spherical particle. In the GB-model,the effect of the grains size on the SiO2 concentration profiles is neglected. Then, the conver-sion of sand grains depends on the initial sand grain size, the inter SiO2 diffusion coefficientand the SiO2 solubility characterized by φ. To evaluate whether and when this model is capa-ble of describing the dissolution of sand grains during heating of glass batches, the results ofsimulations with the GB-model will be compared with the results of simulations with a morerealistic numerical model described in the next section.

3.3.2 Modelling of the dissolution of a single sand grain in a sodium sili-cate melt

For the mathematical description of the diffusion governed dissolution of sand grains duringheating of glass batches, the dissolution of a single spherical sand grain, which is completelywetted by a finite shell of a sodium silicate melt, is considered. In this section, the governingequations describing this dissolution process and the numerical solution method for these equa-tions by the commercial finite element code SEPRAN is described.

Figure 3.10 shows a schematic representation of a sand grain surrounded by a sodium silicatemelt9. The dissolution rate of the sand grain in the NS-melt, characterized by the time depen-dent weight change of the sand grain is, for a diffusion governed dissolution rate, determinedby the transient SiO2-concentration gradient at the interface of the sand grain with the NS-meltaccording to

∂mSiO2

∂t= As

(

DSiO2

∂CSiO2

∂r

)

r=rs

, (3.38)

in which mSiO2 is the mass of the spherical sand grain, t is the time, As is the surface area of thesand grain, which is assumed to be completely wetted with the NS-melt, DSiO2 is the time- andtemperature dependent inter diffusion coefficient of SiO2 in the NS-melt expressed in m2 s−1,CSiO2 is the time- and position dependent SiO2-concentration in the NS-melt and r is the radialdistance from the center of the sand grain.In figure 3.10, the SiO2 concentration in the sand grain is denoted by CSiO2,s and equals thedensity of the sand grain. At the interface of the sand grain and the binary glass melt at positionr = rs, it is assumed that thermodynamic equilibrium exists between the sand grain and theNS-melt. This means that the SiO2 concentration at r = rs is given by

CSiO2,rs = ρm,rs wL1 . (3.39)

9Further in this section, the sodium silicate melt is denoted as NS-melt.


PSfrag replacements

Sand grainSodiumsilicate

melt

rs rs,0CSi

O2

rrs rt

δ

CSiO2,r=rs

CSiO2,s

∂CSiO2∂r = 0

Figure 3.10: Schematic representation of the diffusion governed process of silica dissolution in a sodiumsilicate melt phase.

in which ρm,rs is the density of the NS-melt at r = rs and wL1 is the SiO2 weight fraction givenby the liquidus line of SiO2 in the NS-melt. In this model, it is also assumed that the glassbatch can be regarded as a three dimensional arrangement of spherical sand grains surroundedby a melt phase. Then, the radial gradient of the SiO2 concentration at the outer radius of theNS-melt equals zero:

(

∂CSiO2

∂r

)

r=rt

= 0. (3.40)

The concentration gradient of SiO2 at the surface of the sand grain can be determined by solvingthe continuity equation for SiO2 in the NS-melt given by

∂CSiO2

∂t+(∇ ·−→n SiO2) = 0, (3.41)

in which −→n SiO2 is the mass flux of SiO2 through the NS-melt expressed in kg m−2 s−1. Themass flux vector of SiO2 is given by

−→n SiO2 = CSiO2−→v SiO2 , (3.42)


in which −→v SiO2 is the average velocity vector of SiO2 in the NS-melt. The velocity of SiO2through the NS-melt is composed of a convective and a diffusive contribution according to

−→v SiO2 =

−→j SiO2

CSiO2

+−→v , (3.43)

in which−→j SiO2 is the SiO2 mass flux relative to the mass-average velocity −→v of the NS-melt,

and describes the diffusion of SiO2 through the NS-melt. Convection of the NS-melt, which isindicated by the mass-average velocity −→v , is caused by melt expansion. Melt expansion resultsfrom three phenomena, viz.:

• a difference in specific volume of SiO2 in the silica grain and in the melt phase,

• a difference in specific volume of dissolved SiO2 and dissolved Na2O, which results in aradial density gradient in the melt phase, and

• an inhomogeneous temperature distribution during non-isothermal dissolution.

Although the glass melt density is likely to be dependent on the glass melt composition (see e.g.Shartsis et al. [33]), in the numerical model the glass melt density is set constant. A constantvalue for the glass melt density in the numerical model is required to be able to validate theGB-model in which the glass melt density is set constant as well .

In the following, the dissolution behavior of a sand grain in a NS-melt phase at constanttemperature is evaluated by numerical simulation of the diffusion of SiO2 through the NS-melt following an instantaneous reaction of dissolution of silica at the silica interface. Becauseonly the isothermal dissolution of silica is investigated by numerical simulations, the effect ofmelt phase expansion by temperature change on the convection of the NS-melt is excluded.According to Bird [11], the diffusive mass flux of SiO2 through the binary melt phase is givenby

−→j SiO2 = −ρm DSiO2 ∇

(

CSiO2

ρ

)

, (3.44)

in which ρm is the density of the melt phase. Because the density of the binary melt phase isdefined to be independent on the composition of the binary melt phase, equation 3.44 simplifiesto

−→j SiO2 = −DSiO2 ∇CSiO2 . (3.45)

Combination of equations 3.41, 3.42, 3.43 and 3.45 results in

∂CSiO2

∂t= −(∇ ·CSiO2

−→v )+(∇ ·DSiO2 ∇CSiO2) . (3.46)

For a spherical silica grain and in the case that no concentration gradients of SiO2 in the θ- andφ-direction are present, the transient concentration profile of SiO2 in the binary melt phase isgiven by

∂CSiO2

∂t= − 1

r2∂∂r

(

CSiO2 r2 vr)

+1r2

∂∂r

(

r2 DSiO2

∂CSiO2

∂r

)

, (3.47)


in which vr is the radial convective flow of the binary melt phase. The convective radial velocityof the NS-melt is derived from a mass balance of the system of both the sand grain and the NS-melt surrounding the sand grain. This mass balance is given by

∂(ρsVs +ρmVm,r)

∂t= 0, (3.48)

in which ρs is the constant density of the silica grain, Vs is the volume of the silica grain, ρmis the constant density of the melt phase and Vm,r is the volume of the melt phase at position r.Because both the density of the solid silica grain and the binary melt phase are assumed to beconstant during isothermal dissolution, equation 3.49 describes the expansion rate of the meltphase as a function of the shrinkage of the silica particle during isothermal dissolution of silicain the binary melt phase.

vr =

(

ρm −ρs

ρm

) (

rs

rt

)2 ∂rs

∂t(3.49)

The shrinkage rate of the sand grain is calculated by setting up a mass balance over the sandgrain as is shown in figure 3.11. The dissolution of the sand grain is given by equation 3.38,which describes the change in mass of the sand grain as function of the radial concentrationgradient of SiO2 in the NS-melt at the interface of the sand grain and the NS-melt, the interdiffusion coefficient of SiO2 in the NS-melt and the surface area of the sand grain. For thecurrent model, it is assumed that, for a small time-step, a fresh thin NS-melt layer is formedwith a constant composition (Area B in figure 3.11). This composition equals the liquiduscomposition of SiO2 in the sodium silicate melt phase CSiO2,rs . This implies that the amount ofSiO2, which diffuses from the surface of the silica grain into the binary melt phase is representedby Area A in figure 3.11.The mass balance of SiO2 is now given by

(

∂mSiO2

∂t

)

grain+

(

∂mSiO2

∂t

)

nfm= As DSiO2

(

∂CSiO2

∂r

)

r=rs

, (3.50)

in which the subscript nfm indicates the new formed melt phase. Rewriting equation 3.50 resultsin

4 π r2s CSiO2,s

(

∂rs

∂t

)

+4 π r2s CSiO2,wL1

(

∂rs

∂t

)

= 4 π r2s DSiO2

(

∂CSiO2

∂r

)

r=rs

. (3.51)

From equation 3.51, an expression for the shrinkage of the silica grain is derived:

∂rs

∂t=

DSiO2

(

∂CSiO2

∂r

)

r=rs

CSiO2,s −CSiO2,wL1

(3.52)

The simulation of the dissolution process of the silica grain in the sodium silicate melt phase,by solving equations 3.47, 3.49 and 3.52, requires values for the model parameters DSiO2 , ρs,ρm, CSiO2,s and CSiO2,wL1

. Similar to the glass melt density, the inter diffusion coefficient is de-pendent on the composition of the glass melt [14,34]. Because in the GB-model a constant interSiO2 diffusion coefficient is used, in the numerical model the inter SiO2 diffusion coefficientis set constant. The temperature dependent SiO2 solubility in the NS-melt is derived from the


PSfrag replacements

Silica grain

CSi

O2

rs(t = t)rs(t = t +∆t) ro

CSiO2(r)

CSiO2,s

CSiO2,rs B: Amount of silica whichforms a melt with

CSiO2=CSiO2,rs .

A: Amount of silica whichdissolves in the melt

at r > rs(t).

0

(

∂CSiO2

∂r

)

r=rs

Figure 3.11: Schematic representation of dissolution of SiO2 in a sodium silicate melt phase. The con-centration of SiO2 in the silica grain is CSiO2,s. The concentration of SiO2 at the interface ofthe silica grain and binary melt phase is CSiO2,s.

commercial thermodynamic software package FACT [35] and is fitted with a 2nd order poly-nomial. The concentration of SiO2 in the sand grain equals the density of the sand grain andis constant. Assuming that the density of the silica grain equals the density of the melt phase,ρs = ρm = ρ, equation 3.47 further simplifies to

∂CSiO2

∂t=

1r2

∂∂r

(

r2 DSiO2

∂CSiO2

∂r

)

. (3.53)

Resuming, the isothermal dissolution of silica in the sodium silicate melt phase is simulated bysolving equation 3.53. The shrinkage of the silica grain is given by equation 3.26. The boundaryconditions are given by equations 3.54 and 3.55, whereas the initial condition given by equation3.56.


CSiO2(rs(t), t) = ρ wSiO2,L1 (3.54)

(

∂CSiO2

∂r

)

(rt , t) = 0 (3.55)

CSiO2(r,0) =

CSiO2,L2 ,r = rs,0CSiO2,0 ,rs,0 < r ≤ rt

(3.56)

The equations listed above are implemented in the finite element code SEPRAN. Equation 3.53is solved for this domain using a grid consisting of 200 nodes. The time-step for solving thisequation equals 10−3 s.

3.3.3 Applicability of the Ginstling-Brounstein modelIn the previous section, the GB-model describing the dissolution of a sand grain in a finite glassmelt layer surrounding the sand grain, under the consideration of several assumptions, wasderived. Because the assumptions of the GB-model are not generally valid for the dissolutionof sand grains during the melting of glass batches, the applicability of the GB-model for pre-dicting the sand grain dissolution rate is questioned.

In this section, the applicability of the GB-model is evaluated by comparison of the resultsof the numerical simulation of the sand grain dissolution process with the results derived fromthe GB-model.

Table 3.2 lists the values for the model parameters of the numerical and the GB-model, viz.the initial radius of the sand grain rs,o, the SiO2 concentration in the sand grain CSiO2,s, the SiO2concentration in the melt phase at the interface with the sand grain CSiO2,rs , the initial SiO2 con-centration throughout the melt phase CSiO2,0, the inter diffusion coefficient of SiO2 in the meltphase DSiO2 and the parameter φ, which are used during simulation studies.

Table 3.2: Values for the model parameters of the numerical and the GB-model.

Parameter Unit Value

rs,0 [µm] 35CSiO2,s [kg m−3] 2200CSiO2,rs [kg m−3] 1668CSiO2,0 [kg m−3] 1440

D [m2 s−1] 1 ·10−12

φ [-] 0.43

As an example of a result obtained from a numerical simulation of the sand grain dissolutionprocess, figure 3.12 shows both the radius and the degree of conversion of a sand grain with aninitial radius of 35 µm surrounded by a melt phase with an initial thickness of 25 µm as functionof time. It is observed that the sand grain completely dissolves in the surrounding melt phasewithin 1625 s.


0 250 500 750 1000 1250 1500 1750 2000 0

5

10

15

20

25

30

35

0.0

0.2

0.4

0.6

0.8

1.0

PSfrag replacements

Time [s]

Rad

ius

ofth

esa

ndgr

ain

r s[µ

m]

Deg

ree

ofco

nver

sion

ofsa

ndgr

ain

ξ[-

]

Figure 3.12: Calculated radius and degree of conversion of a sand grain with an initial radius of 35µm surrounded by a melt phase with an initial thickness of 25 µm (For the other modelparameters is referred to table 3.2) as function of time.

According to the GB-model, the plot of[

1− 23ξ− (1−ξ)2/3

]

versus time would give a constant

slope equal to[

2 φ D r−2s,0

]

(see equation 3.37). To study the applicability of the GB-model fordescribing the sand grain dissolution process during heating of glass batches, the results ob-tained from numerical simulations studies are reported by the plot of

[

1− 23ξ− (1−ξ)2/3

]

ver-sus time. In figure 3.13, the results of numerical simulations, using the model parameter valueslisted in table 3.2, for five different values of the initial thickness of the melt phase surroundingthe sand grain (10 µm (case 1), 20 µm (case 2), 25 µm (case 3), 465 µm (case 4) and 965 µm(case 5)) are shown. Also in figure 3.13, the result of the simulation using the GB-model isshown.At t = 0, the value for

[

1− 23ξ− (1−ξ)2/3

]

equals zero. Complete dissolution of the sand

grain is represented by ξ = 1 and results in a value for[

1− 23ξ− (1−ξ)2/3

]

equal to 1/3. Itis observed that for case 1, no complete dissolution of the sand grain in the surrounding meltphase is reached, i.e. ξ 6= 1/3. For this case, the initial amount of melt phase surrounding the 35µm sand grain is insufficient for complete dissolution of the sand grain. Complete dissolutionof the sand grains is observed for the cases 2, 3, 4 and 5, which indicates that for these cases theinitial amount of melt phase surrounding the sand grain is sufficient for complete dissolution.

It is also observed that an increase of the thickness of the melt phase surrounding the sandgrain from 20 µm to 465 µm results in an increase in the dissolution rate of the sand grains.However, a larger value for the initial melt phase thickness than 465 µm is not accompaniedwith a further increase in dissolution rate. The reason for this is explained as follows:


0 200 400 600 800 1000 1200 1400 1600 1800 2000 0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

PSfrag replacements

Time [s]

[

1−

2 3ξ−

(1−

ξ)2/

3]

[-]

Case 1

Case 3Case 4, 5

Case 2

BA

GB-model

[

1− 23 ξ− (1−ξ)2/3

]

in case ξ = 1

Figure 3.13: Plot of[

1− 23 ξ− (1−ξ)2/3

]

versus time representing the calculated conversion of a sandgrain with an initial radius of 35 µm during dissolution in a surrounding glass melt accordingto the Ginstling-Brounstein model and the numerical model. The initial thickness of themelt phase surrounding the sand grain δ equals 10 µm (case 1), 20 µm (case 2), 25 µm (case3), 465 µm (case 4) and 965 µm (case 5). A and B indicate the end time for the transientstage for case 1 and case 2.

Figure 3.14 shows a schematic representation of the SiO2 concentration profiles in the glassmelt surrounding a sand grain with an initial radius of 45 µm as function of time. These con-centration profiles are calculated with the numerical model. At t = 0, the surface of the sandgrain is given by rs(t = 0), which equals 45 µm. From the surface of the sand grain, at whichthe SiO2 concentration is constant and given by CSiO2,rs=1668 kg m−3, SiO2 diffuses throughthe melt phase towards the outer boundary of the melt phase at r = rt , which equals 53 µmin this example case. The initial SiO2 concentration in the glass melt CSiO2,0 equals 1100 kgm−3. With time, the SiO2 concentration profiles in the glass melt follow the profiles indicatedin figure 3.14. The concentration profile in the glass melt in case the time equals the Fouriernumber DSiO2

(rt−rs)2 , which is also referred as the end time of the transient stage te. When the time tis smaller than the Fourier number, no change in the SiO2 concentration at the outer boundaryof the glass melt is observed, whereas at t > te, SiO2 accumulates at the boundary of the meltphase. As long as t < te, the initial thickness of the melt phase does not affect the dissolutionprocess. For both an initial glass melt thickness of 465 µm and 965 µm, the dissolution rate cal-culated with the numerical model is identical, which indicates that for both cases the dissolutiontime t is less than te and that the surrounding melt phase can be regarded as a semi-infinite meltphase.


3.4 3.6 3.8 4 4.2 4.4 4.6 4.8 5 5.2 5.4

x 10−5

1100

1200

1300

1400

1500

1600

1700

PSfrag replacements CSi

O2

[kg

m−

3 ]

Radial position [µm]

CSiO2,rs

t ↑

te

rt

CSiO2,0

rs,0

Figure 3.14: Schematic representation of the SiO2 concentration profiles in the glass melt as functionof time. The parameter te represents the end time of the transient dissolution stage. Incase t < te, the SiO2 concentration at r = rt is constant, whereas in case t > te, the SiO2concentration at r = rt increases.

In section 3.3.1, the equations with which the duration time of the transient stage te can becalculated are given. The duration time for the transient stage for the different values for themelt phase thickness are 3 min. (δ=20 µm), 5 min. (δ=25 µm), 964 min. (δ=465 µm) and 3064min. (δ=965 µm), respectively. The high values for the duration time of the transient stagelonger than the total dissolution time agree with the assumption that t < te for the cases 4 and5. For the cases 2 and 3, the duration time are plotted in figure 3.13 and marked by A and B,respectively. At the end of the transient stage, the SiO2 concentration profiles start to deviatefrom the curves for cases 4 and 5.

In figure 3.13, it is clearly seen that the results of the simulations obtained from the numer-ical model deviate significantly from the simulation result obtained from the GB-model. Thereason for these large deviations is caused by the difference in SiO2 concentration gradient atthe surface of the sand grain as function of time for the two models. As can be seen from figure3.14, the concentration profiles in the glass melt surrounding the sand grain flattens with time.The flattening of the concentration profiles is caused by two effects:

1. The stretching of the concentration profile at the surface of the sand grain due to theshrinkage of the sand grain.

2. The accumulation of SiO2 at the outer boundary of the glass melt.

In contrast to the numerical model, in the GB-model it is assumed that both boundaries of themelt phase, rs and rt , remain constant during the dissolution process. It is also assumed that the


SiO2 concentration at these boundaries are constant. Therefore, the SiO2 concentration profilesat the boundary of the sand grain will be steeper in the GB-model compared to the numericalmodel. Combining equations 3.31 and 3.36 results in an expression for the SiO2 gradient at thesurface of the sand grain for the GB-model:

(

∂CSiO2

∂r

)

rs

=CSiO2,rs,0 −CSiO2,rs

rs,0 (1−ξ)−1/3(

1− (1−ξ)−1/3) (3.57)

Figure 3.15 shows the concentration gradient at the surface of a sand grain with an initial grainsize of 35 µm surrounded by a melt phase calculated with both the numerical model and theGB-model during the first second of the dissolution process. The values for CSiO2,s, CSiO2,rs ,CSiO2,0, D and φ are listed in table 3.2. It is observed that the SiO2 concentration gradient cal-culated with the GB-model is steeper than the SiO2 concentration gradient calculated with thenumerical model. This explain the deviation of the degree of conversion of the sand grain cal-culated with the numerical model relative to the result of the GB-model (see figure 3.13).

0.0 0.2 0.4 0.6 0.8 1.0

-5.0x10 9

-4.5x10 9

-4.0x10 9

-3.5x10 9

-3.0x10 9

-2.5x10 9

-2.0x10 9

-1.5x10 9

-1.0x10 9

-5.0x10 8

0.0

PSfrag replacements

Time [s]

(

∂CSi

O2

∂r

)

r s[k

gm

−4 ]

Numerical model

GB-model

Figure 3.15: Concentration gradient at the surface of a sand grain with an initial grains size of 35 µmsurrounded by a melt phase calculated with both the numerical model and the GB-modelduring the first second of the dissolution process. The values for CSiO2,s, CSiO2,rs , CSiO2,0, Dand φ are listed in table 3.2.

From figure 3.13, it is observed that during the dissolution of sand grains in large glass meltvolumes, which is for example characterized by case 4 and case 5,

[

1− 23ξ− (1−ξ)2/3

]

is(almost) linear proportional to time, which is similar to the GB-model. Also for case 3, analmost linear relation is observed. The proportionality constant for the GB-model, relating


0.000 0.002 0.004 0.006 0.008 0.010 0.0000

0.0005

0.0010

0.0015

0.0020

0.0025

0.0030

0.0035

PSfrag replacements

kGB =[

2 φ Drs,0

]

[-]

k nm

[-]

r2s,0 ↓

D ↑φ ↑

knm = kGB

Figure 3.16: knm versus kGB as function of φ, D and rs,0 calculated for case 3 (see figure 3.13).

[

1− 23ξ− (1−ξ)2/3

]

versus time and defined as kGB, equals[

2 φ D r−2s0

]

, whereas the propor-tionality constant for the numerical model, defined as knm, is given by the slope of the linesshown in figure 3.13. In case the difference of kGB with respect to knm is independent on thevalues for φ, D and rs,0, a modified GB-model can be defined, with which the sand grain dis-solution process, simulated with the numerical model can be approximated. In this modifiedGB-model, the proportionality constant kGB is multiplied by the constant knm/kGB. However,for sand grains dissolving in a very thin glass melt layer, the ratio of knm/kGB is non-linear,which can for example be seen from the curved line for case 2 in figure 3.13. In case theamount of glass melt surrounding the sand grain is insufficient for complete dissolution of thesand grain (see case 1 in figure 3.13), the GB-model is not able to predict the dissolution rate ofthe sand grain.

To evaluate whether the modified GB-model is able to predict the sand grain dissolutionas function of the GB model parameters φ, D and rs,0, the effect of these parameters on thevalue for knm as function of kGB is studied for case 3. Figure 3.16 shows the results of a sys-tematic variation of these model parameters. The used values for these model parameters areD=1 ·10−12 m2 s−1, 2 ·10−12 m2 s−1 and 5 ·10−12 m2 s−1, φ=0.43, 2.83 and 5.60, and rs,0=10µm, 15 µm, 25 µm and 35 µm.

It is observed that the absolute difference between kGB and knm decreases with increasinginitial sand grain radius, decreasing diffusion coefficient, decreasing value for φ, i.e. with de-creasing SiO2 solubility in the glass melt. It is also observed that knm/kGB is independent of

3.4. Quantitative phase analysis with X-ray diffraction 109

D and r2s,0. Only for very high (and non-realistic values for the glass melting process10) values

for φ, no linear relation between knm and kGB is observed. Excluding high values for φ, thelinear relation between knm versus kGB indicates that the sand grain dissolution calculated withthe modified GB-model can be used to estimate the sand grain dissolution process as calculatedwith the numerical model. However, the ratio of knm/kGB will be a function of the ratio of thethickness of the melt phase surrounding the sand grain, δ/rs,o, which becomes non-linear forlow ratios of δ/rs,0. In practice, the value for δ/rs,0 will depend on a large series of parameterssuch as the bulk composition of the glass batch, the homogeneity of the glass batch, the wettingof the sand grains by the primary formed melt phase, the presence of gas bubbles preventingcontact between glass melt and sand grain. Because of this complexity, the applicability of themodified GB-model for the description of sand grain dissolution in industrial glass batches isquestioned.

In the section 3.5, it is studied whether the modified GB-model can still be used for the de-scription of the dissolution of sand grains in a multi-component glass batch. The modified GBmodel proportionality constant kGB is determined from non-isothermal dissolution experimentsof sand grains in a float glass batch. The non-isothermal dissolution of sand grains as functionof temperature is described by

∂ξ∂T

=kGB

βf (ξ) . (3.58)

The reaction mechanism function f (ξ) for the GB-model is derived by differentiation of equa-tion 3.37, which results in

f (ξ) =1.5

(1−ξ)−1/3 −1. (3.59)

Assuming that the temperature dependency of the GB model proportionality constant kGB isgiven by an Arrhenius type equation, the apparent Arrhenius parameters AGB and EGB are de-rived from

ln(

∂ξ∂T

)

+ lnβ− ln f (ξ) = lnAGB −EGB

R T. (3.60)

To measure the degree of sand grain fusion in glass batches as function of time and temperature,a quantitative analyzing technique is required. This technique should be able to determine theresidual amount of crystalline silica in partly molten glass batch samples. In the next section,the method of quantitative phase analysis with X-ray diffraction is described.

3.4 Quantitative phase analysis with X-ray diffraction

To determine the fusion rate of sand grains in glass batches during heating, the content of resid-ual crystalline silica in glass batch samples, which have been imposed to a defined time/temperature-program, is measured. For the preparation of the heated glass batch samples, about5 grams of a mixture of glass batch components is heated in an electrical laboratory furnace.After ramp heating of the batch sample up to a specified temperature, the heated glass batch

10A value for φ equal to 5.60 indicates a SiO2 weight fraction in the glass melt of 0.95 in case the initial SiO2weight fraction in the melt equals 0.65.


sample is withdrawn from the furnace and rapidly cooled11. After cooling, the glass batchsample is crushed with a tungsten carbide mortar resulting in a powder mixture with a residualparticle size below 10 µm. This small particle size is required for accurate quantitative phaseanalysis with X-ray diffraction. To the crushed batch sample, a known amount of an internalstandard, which is discussed later in this section, is added. From the mixture of the crushedglass batch sample and the internal standard, a tablet is pressed, which is used for quantitativephase analysis by X-ray diffraction.

Figure 3.17 shows the diffraction pattern of silica sand in the quartz structure for 50<2θ<800.This diffraction pattern is collected using a Philips X’Pert X-ray diffractometer using Cu-Kαradiation. The data are collected from 50 up to 800 in steps of 0.020 using a counting time of1.0 s step−1. These diffractometer settings are used for all qualitative and quantitative phaseanalysis during the current study.

Identification of the phases present in a mixture of crystalline species is achieved by com-parison of the X-ray diffraction patterns from the unknown sample with an internationally rec-ognized database containing a large amount of reference patterns.

Next to qualitative phase analysis with X-ray diffraction, also quantification of the amount ofcrystalline species is possible with X-ray diffraction. As mentioned by Madsen et al. [36], thereare several methods based on X-ray diffraction, which can be applied for quantitative phaseanalysis. The method used for the quantitative determination of the amount of crystalline SiO2in this study, is the internal standard Reference Intensity Ratio (RIR) method. As mentionedabove, each crystalline specie can be characterized by its typical diffraction pattern. Figure3.17 depicts the diffraction pattern of silica sand, which shows the intensities of the differentdiffraction lines as a function of the diffraction angle 2θ.The intensity of the diffraction line i of quartz particles in a multi-component mixture is de-scribed by

Ii,SiO2(q) =

(

Ke Ki,SiO2(q)

µs

)

(

ρs

ρSiO2(q)

)

xSiO2(q), (3.61)

in which Ke is a constant for the particular X-ray diffraction system, Ki,SiO2(q) is a constantfor diffraction line i for quartz, µs is the mass absorption coefficient of X-rays in the multi-component mixture, ρs is the density of the multi-component mixture, ρSiO2(q) is the densityof quartz and xSiO2(q) is the weight fraction of quartz in the mixture. For the determination ofthe required weight fraction of quartz (xSiO2(q)) in the multi-component mixture, the so-calledmass attenuation coefficient of the mixture sample, (µs/ρs), has to be known. The wavelengthdependent mass attenuation coefficient is given by

(

µρ

)

s=

n

∑i=1

(

µρ

)

ixi, (3.62)

in which n equals the total number of different species in the multi-component mixture. Thus,determination of the weight fraction of quartz in the multi-component mixture requires informa-tion of the composition of the mixture, which is not known a-priori. To overcome this problem,several approaches have been presented in literature, from which the so-called Internal StandardMethod is applied during the current study. This method is based on the elimination of the massattenuation coefficient of the multi-component mixture in equation 3.61 by relating the intensity

11For a detailed description of the melting procedure of the glass batch samples is referred to section 2.4.3.


10 20 30 40 50 60 70 80 0

5000

10000

15000

20000

25000

30000

PSfrag replacementsDiffraction angle 2θ [o]

Peak

inte

nsity

[cou

nts

s−1 ]

Figure 3.17: XRD-pattern for silica sand for 50<2θ<800. The different lines indicate the differentdiffraction lines for quartz.

of the diffraction lines of quartz to the intensity of the diffraction lines of an internal standard12,which has been added to the multi-component mixture in a known amount. In this study, puresilicon (Si) powder is used as internal standard material.

The ratio of the intensity of diffraction line i of quartz is now related to the intensity of thediffraction line j of the internal standard Si according to

Ii,SiO2

I j,Si=

Ki,SiO2

K j,Si

ρSi

ρSiO2

xSiO2

xSi= K′ xSiO2

xSi= RIRSiO2(q),Si

xSiO2

xSi, (3.63)

in which the constant K ′ is the Reference Intensity Ratio (RIRSiO2(q),Si) of SiO2(q) with respectto Si. A mixture containing quartz to which a known amount of the internal standard is addedpermits quantitative analysis of the amount of quartz in this multi-component mixture by mea-suring the peak intensities of SiO2(q) and Si and applying equation 3.63. The RIR of SiO2(q)with respect to Si, RIRSiO2(q),Si, is derived from mixtures of SiO2 and Si with known composi-tions.

The accuracy of the determined Reference Intensity Ratio is dependent on the accuracy ofthe measured intensities of the diffraction lines and on the accuracy of the weight fractions ofSiO2(q) and Si in the binary standard mixtures. According to Jenkins and Snyder [37], there arefour types of factors that affect the measured intensity of a diffraction line:

1. Crystal structure-sensitive factors.

2. Instrument-sensitive factors.12A compound with a well known X-ray diffraction pattern.


3. Sample-sensitive factors.

4. Measurement-sensitive factors.

The crystal structure-sensitive factors are specific for each diffraction line i for each specie sand are described by the diffraction line constant Ki,s as given in equation 3.61. In case no crys-tal structural changes occurs in quartz during heating of the glass batch samples, the relativeintensities of the diffraction lines of quartz remain unaltered and structure-sensitive factors donot contribute to errors in the calculated weight fraction of quartz.

The main instrument-sensitive factors are the intensity of the incident beam and the wave-length of the X-ray beams. These factors affect the absolute diffraction line intensities, whichmeans that the intensities of the diffraction lines of both SiO2(q) and Si are modified to thesame extent. Because for quantitative phase analysis of quartz, the diffraction line intensitiesof SiO2(q) are always measured with respect to the intensities of the diffraction lines of Si, thecontribution of instrument-sensitive factors to an inaccurate measurement of the diffraction lineintensities can also be neglected.

Sample-sensitive factors are able to influence both absolute and relative diffraction line in-tensities. The main sample-sensitive factors are preferred orientation, extinction and micro-absorption. Preferred orientation is the phenomenon that a crystal exhibits a preferred orien-tation of crystal planes within the sample during preparation of the tablet for X-ray diffractionanalysis. There are sample preparation methods and instrumental techniques to reduce preferredorientation. During the current study, preferred orientation is opposed by spinning the sampletablet during the diffraction measurement.

Extinction is the phenomenon that the intensity of the most intense diffraction lines of acrystal decreases by interference of diffraction X-rays with reflected X-rays from upper latticeplanes. Extinction mainly occurs in perfect crystals. In glass batches containing industrial rawmaterials, extinction is not expected to have a significant effect on the X-ray diffraction lines ofthermal treated glass batch samples, because the glass batch components are not perfect crys-tals.

Micro-absorption of X-rays in a sample occurs

• in case a large difference in absorption coefficient of the different phases in the sampleare present, and

• in case a large difference in crystal size of the different phases in the sample are present.

Micro-absorption can be reduced by grinding the sample as fine as possible, but should at leastbe down to a grain size below 10 µm.

The main important measurement-sensitive factors are the definition of the onset of thediffraction line (i.e. A in figure 3.18 for diffraction line 2 for silica sand), the end of the diffrac-tion line (i.e. B in figure 3.18 for diffraction line 2 for silica sand) and the baseline of thediffraction line, which gives the area to be integrated (i.e. C in figure 3.18 for diffraction line 2for silica sand).

The error in the weight fraction quartz calculated from X-ray diffraction patterns by usingequation 3.63, is dependent on both systematic and random errors in the different parameterslisted in equation 3.63. The presence of random errors is prevented as much as possible byusing the same instrument settings, sample preparation method and peak integration procedurefor all diffraction measurements. The accuracy of the weight fraction of quartz in the glass


20 30 40

10 1

10 2

10 3

10 4

PSfrag replacements

Diffraction angle 2θ [o]

Peak

inte

nsity

[cou

nts

s−1 ]

1

2

3 45

6

A B

C

Figure 3.18: XRD-pattern for silica sand for 150<2θ<450. A is the onset of diffraction line 2, B is theend of diffraction line 2, C is the baseline of diffraction line 2.

batch sample is mainly dependent on the random errors induced by sample- and measurement-sensitive factors. According to Coleman and Steele [38], in case a parameter Y is dependent onn independent parameters X , the uncertainty or error in the parameter Y due to random errors inthe n parameters X is given by

σY =

√

√

√

√

n

∑i=1

[

(

∂Y∂Xi

)2

σX2i

]

, (3.64)

in which σY is the uncertainty or error in the parameter Y and σXi is the uncertainty in theparameter Xi. The uncertainty in the parameter Xi is given by the value of the standard deviationin the parameter Xi.

To calculate with equation 3.63 the weight fraction of quartz in heated glass batches towhich a known amount of internal standard is added, the RIRSiO2(q),Si has to be known. Inorder to reduce the uncertainty in RIRSiO2(q),Si, it is recommended to use multiple diffractionlines of both quartz and the internal standard Si. For each diffraction line of quartz and theinternal standard Si, the relative peak intensity has been determined from the pure individualcomponents. The relative peak intensity of a diffraction line i is the peak intensity of thisdiffraction line with respect to the most intense diffraction line for that substance. For quartz themost intense diffraction peak is positioned at 2θ = 26.6o, whereas the most intense diffractionpeak for internal standard Si is positioned at 2θ = 28.4o. The relative diffraction peak intensitiesfor quartz and Si are listed in table 3.3 and table 3.4, respectively. The standard deviation of therelative peak intensity of a diffraction line, σIrel

i,SiO2(q)or σIrel

j,Si, is determined from the measured


Table 3.3: Relative peak intensities for quartz.

Diffraction line 2θ Ireli σIrel

i[%]

1 20.87 0.150 0.12 26.62 1.000 -3 36.51 0.097 0.24 39.43 0.085 0.25 40.26 0.050 0.1

Table 3.4: Relative peak intensities for the internal standard Si

Diffraction line 2θ Irelj σIrel

j[%]

1 28.44 1.00 -2 47.30 0.98 0.83 56.12 0.65 0.7

peak intensities of the diffraction line for ten different samples of either silica sand or the internalstandard Si.The Reference Intensity Ratio for quartz with respect to the internal standard Si can now becalculated from a quartz tablet to which a known amount of the internal standard Si is added:

RIRSiO2(q),Si =

[

1n

n

∑i=1

(

Ii,SiO2(q)

Ireli,SiO2(q)

)] [

1m

m

∑j=1

(

I j,Si

Irelj,Si

)]−1 (XSi

XSiO2(q)

)

, (3.65)

in which n and m are the numbers of diffraction lines of quartz and the internal standard Si,which are used for the determination of the RIRSiO2(q),Si. Irel

i,SiO2(q) and Irelj,Si are the relative peak

intensities of line i of quartz and line j of the internal standard Si with respect to the mostintense diffraction line and xSiO2(q) and xSi are the weight fractions of quartz and the internalstandard in the mixtures of quartz and Si with known composition.

The RIRSiO2(q),Si has been determined by plotting the quotient of the average weighted in-tensities of the quartz and Si diffraction lines versus the ratio of the weight fractions of quartzand Si for three binary mixtures of quartz and Si according to

RIRSiO2(q),Si =ISiO2(q)

ISi

(

XSi

XSiO2(q)

)

, (3.66)

in which ISiO2(q) equals[

1n ∑n

i=1

(

Ii,SiO2(q)

Ireli,SiO2(q)

)]

and ISi equals[

1m ∑m

j=1

(

I j,Si

Irelj,Si

)]

. The results are

plotted in figure 3.19. The slope of the line, which equals RIRSiO2(q),Si, is 0.89, with a regressioncoefficient of 0.9988. In a similar way as presented in equation 3.64, the standard deviation inthe RIRSiO2(q),Si is determined, which resulted in a value of 0.04. So, the Reference IntensityRatio RIRSiO2(q),Si equals 0.89±0.04.

From equation 3.63, the weight fraction of quartz in the batch sample diluted with the in-ternal standard can be calculated. Equation 3.67 provides the weight fraction of quartz in thebatch sample before dilution with the internal standard and in case multiple diffraction lines are


1 2 3 4 5 6 7 8 9 10 1

2

3

4

5

6

7

8

9

PSfrag replacementsxSiO2(q)/xSi [-]

I SiO

2(q)

/ISi

[-]

Figure 3.19: The ratio of the average diffraction line intensities versus the ratio of the weight fractionsfor quartz and Si in three binary mixtures.

used for both quartz and the internal standard.

xSiO2(q) =1

RIRSiO2(q),Si

ISiO2(q)

ISi

xSi

1− xSi(3.67)

For three non-heated mixtures of silica sand, soda ash and dolomite, the weight fraction ofquartz in these mixtures is determined with X-ray diffraction and compared with the knownweight fraction of quartz in the ternary mixtures. Figure 3.20 shows the calculated weightfraction of quartz, the 95 % confidence limits of these values and the known weight fraction ofquartz. The calculated weight fractions of quartz fall clearly within the confidence limits whichhas been calculated with equation 3.64.

The standard deviation in the calculated quartz content equals 5 % relative. This value willbe taken as the uncertainty in the calculated SiO2 content with X-ray diffraction. The degree offusion of the silica sand in the heated glass batch samples can now be calculated with

ξSiO2(q) = 1−xSiO2(q)

x0SiO2(q)

m0b

mb, (3.68)

in which x0SiO2(q) is the initial weight fraction of quartz in the glass batch sample, m0

b is the initialmass of the glass batch sample and mb is the actual mass of the glass batch sample.

As was shown in section 2.5.5, starting at about 1300 K, SiO2 is not only present in thequartz modification, but also in the cristobalite modification. To determine the total contentof crystalline SiO2 in heat-treated glass batch samples, also the Reference Intensity Ratio ofSiO2(cr) with respect to Si has been determined.


45 50 55 60 65 70 75

50

55

60

65

70

75

80

PSfrag replacementsCalculated weight fraction of quartz [-]

Kno

wn

wei

ghtf

ract

ion

ofqu

artz

[-]

Figure 3.20: Calculated SiO2 weight fraction versus real SiO2 weight fraction in a glass batch sampleconsisting of silica sand, soda ash and dolomite.

3.5 Experimental determination of the apparent GB-modelparameters

In this section, the apparent Ginstling-Brounstein model parameters AGB and EGB are deter-mined for the dissolution of silica in a glass batch producing float glass. The GB model pa-rameters are easily determined from measured conversion of silica during heating of the floatglass batch. The values for the GB model parameters for the float glass batch are determinedas function of the initial particle size of the sand grains in the float batch and the cullet fractionin the float batch.

3.5.1 Apparent GB-model parameters as function of the sand grain par-ticle size

Figure 3.21 shows the measured degree of conversion of silica sand in the float glass batch asfunction of temperature during ramp heating with 25 K min−1 in an ambient atmosphere. Line1 and line 2 indicate the measured degree of silica conversion for particle sizes of the silicagrains between 212 µm and 300 µm and between between 63 µm and 150 µm, respectively. Theonset for silica conversion in the float glass batch is observed at approximately 1023 K, whichis almost similar to the onset temperature for reactive soda ash calcination (see section 2.5.3).Complete conversion of the silica sand grains is observed at 1723 K.

3.5. Experimental determination of the apparent GB-model parameters 117

1000 1100 1200 1300 1400 1500 1600 1700 1800 0.0

0.2

0.4

0.6

0.8

1.0

PSfrag replacements

Temperature [K]

Deg

ree

ofsi

lica

conv

ersi

on[-

]1

2

Figure 3.21: Measured degree of silica conversion during ramp heating of a float batch with 25 K min−1

in an ambient atmosphere. Line 1 and line 2 present the measured silica conversion asfunction of temperature for particle sizes of the silica grains between 212 µm and 300 µmand between between 63 µm and 150 µm, respectively.

0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95

-8

-7

-6

-5

-4

-3

PSfrag replacements


ln(

∂ξ ∂T

)

+ln

β−

lnf(

ξ)[l

nm

in−

1 ]

1073 K

1173 K

1673 K

Figure 3.22: ln(

∂ξ∂T

)

+ lnβ − ln f (ξ) characterizing the experimentally derived conversion of silicagrains in the float glass batch as function of temperature during ramp heating with 25 Kmin−1 in an ambient atmosphere and in case the particle size of the silica grains variesbetween 212 µm and 300 µm.


It is observed that a decrease of the particle size of the sand grains results in a slightly higherconversion rate of the sand grains in the float batch. The reason for this is the larger surface areaper unit volume of the sand grains, in case of smaller sand grains, which promotes the rate ofthe diffusion controlled dissolution of the sand grains in the primary formed melt phases duringheating of the float batch.

From the combination of equations 3.58 and 3.59, the non-isothermal conversion of silicasand during heating of glass batches can be derived

∂ξ∂T

=1β

AGB eEGBR T

1.5

(1−ξ)−1/3 −1, (3.69)

in which ξ is the degree of silica conversion, T is the temperature in K, β is the applied constantheating rate and AGB and EGB are the apparent GB model parameters. Figure 3.22 shows theplot of

[

ln(

∂ξ∂T

)

+ lnβ− ln f (ξ)]

, with f (ξ) given by 1.5(1−ξ)−1/3−1

, versus the reciprocal abso-

lute temperature in the temperature regime between 1073 K and 1723 K. It is observed thatabove 1173 K, an (almost) linear relation between

[

ln(

∂ξ∂T

)

+ lnβ− ln f (ξ)]

and the reciprocalabsolute temperature is present, which allows the calculation of the apparent GB model param-eters.

0.55 0.60 0.65 0.70 0.75 0.80 0.85 -6.0

-5.5

-5.0

-4.5

-4.0

-3.5

-3.0

PSfrag replacements


ln(

∂ξ ∂T

)

+ln

β−

lnf(

ξ)[l

nm

in−

1 ]

lnAGB = 0.583;−EGB/R = 7608;r2 = 0.9749

lnAGB = 0.324;−EGB/R = 6256;r2 = 0.9861

Figure 3.23: ln(

∂ξ∂T

)

+ lnβ − ln f (ξ) characterizing the experimentally derived conversion of silicagrains versus the reciprocal absolute temperature for two different silica grain particle sizes.The and the • indicate ln

(

∂ξ∂T

)

+ lnβ− ln f (ξ) in case 212 µm < dSiO2 < 300 µm and

63 µm < dSiO2 < 150 µm, respectively. The heating rate equals 25 K min−1.

The apparent GB model parameters AGB and EGB, are determined with equation 3.60 for the


two float glass batches with different particle size of the silica grains. Figure 3.23 shows theplot of ln

(

∂ξ∂T

)

+ lnβ− ln f (ξ) versus the reciprocal absolute temperature starting from 1173K. The intercept and the slope of the lines in figure 3.23 provide the values for lnAGB and−EGB/R, respectively. The experimental derived values for the apparent GB model parametersare 3.0 ·10−2 s−1 and 63.3 kJ mol−1 in case 212 µm < dSiO2 < 300 µm and 2.3 ·10−2 s−1 and52.0 kJ mol−1 in case 63 µm < dSiO2 < 150 µm.

In case the modified GB-model is valid for the description of the sand grain dissolution pro-cess in the float glass batch, knm would be proportional to

[

D φ r−2s,0

]

. Because the compositionof both glass batches is identical, both the inter SiO2 diffusion coefficient D and the parameter φare assumed constant. Now, it is expected that knm is proportional to the square of the reciprocalinitial sand grain radius rs,0:

ln(

kGB,ds

kGB,dl

)

= ln(

rs,0,dl

rs,0,ds

)2

, (3.70)

in which kGB,ds is the proportionality constant for the float glass batch with the small particles,kGB,dl is the proportionality constant for the float glass batch with the large particles, rs,0,dl is theaverage initial radius of the large sand grains and rs,0,ds is the average initial radius of the smallsand grains. However, because of the broad particle size distribution, no unambiguous valuefor the initial radius of the sand grains in both float glass batches can be given. The measured

difference in ln(

kGB,dskGB,dl

)

equals 0.7. The value for ln(

rs,0,dlrs,0,ds

)2is between 0.7 and 3.2. Based on

these results, the applicability of the GB-model for the description of the sand grain dissolutionprocess in glass batches is not proved unambiguously. To test whether the modified GB-modelis able to predict the effect of particle size on the dissolution rate of sand grains, it is recom-mended to perform melting experiments with a more narrow particle size distribution despitethat fact that industrial glass batches may contain a broad particle size distribution for the sandgrains.

The accuracy of the prediction of the degree of silica conversion in the float batch for dif-ferent heating rates is evaluated by comparison of the measured and the predicted (calculatedwith equation 3.69) silica conversion at 1323 K as function of the heating rate of the float batchcontaining sand grains with 212 µm < dSiO2 < 300 µm and in case the values for AGB and EGBare 3.0 ·10−2 s−1 s−1 and 63.3 kJ mol−1, respectively. The measured degree of silica conver-sion at 1223 K for the different heating rates is defined as the starting point for the simulationof the silica conversion at 1323 K. The measured values for the silica conversion at 1223 K, themeasured values for the silica conversion at 1323 K and the predicted values, from the modifiedGB-model and the measured degree of conversion at 1323 K, for the silica conversion at 1323K are listed in table 3.5.

The predicted and the measured degree of silica conversion at 1323 K in case of a heatingrate of 25 K min−1 are identical, because the apparent GB model parameters describing thedissolution rate of the sand grains in the float batch are derived from the measured conversionof silica in the float glass batch when ramp heated with 25 K min−1. It is observed that at lowheating rates the GB-model overestimates the silica conversion, whereas at high heating rates,the silica conversion is underestimated. Similar to the prediction of the effect of the particle sizeof the sand grains described above, the effect of heating rate on the dissolution of sand grains isnot accurately described by the modified GB-model.


Table 3.5: The measured silica conversion at 1223 K and 1323 K and the predicted silica conversion at1323 K as function of the heating rate of the float glass batch.

Heating rate Measured ξ Measured ξ Calculated ξK min−1 at 1223 K [-] at 1323 K [-] at 1323 K [-]

5 0.41 0.63 0.7810 0.34 0.56 0.6225 0.33 0.48 0.4835 0.32 0.46 0.44

3.5.2 Apparent GB-model parameters as function of the cullet fractionFigure 3.24 shows the measured degree of conversion of silica sand during ramp heating ofthe float glass batch with 25 K min−1 in an ambient atmosphere as function of the content ofcrushed cullet in the float glass batch.

It is observed that the onset for silica conversion in the float glass batch increases withincreasing cullet content. The reason for the shift in onset temperature for silica conversion tohigher values is the lower additional solubility of SiO2 in the float glass cullet compared to theprimary formed melt phase. The liquidus temperature, by the crystallization of SiO2, of theglass melted from the float glass batch is calculated with FACT [35] and equals 1309 K. Thisindicates that up to 1309 K, no silica will dissolve in the float glass cullet.

1000 1100 1200 1300 1400 1500 1600 1700 1800 0.0

0.2

0.4

0.6

0.8

1.0

PSfrag replacements

Temperature [K]

Deg

ree

ofsi

lica

conv

ersi

on[-

]

1

2

34

Figure 3.24: Measured degree of silica conversion during ramp heating a float glass batch with 25 Kmin−1 in an ambient atmosphere. Lines 1, 2, 3 and 4 present the degree of silica conversionin case the cullet content in the float glass batch equals 0 wt.%, 25 wt.%, 50 wt.% and 75wt.%, respectively. The silica grain particle size in the float glass batch is between 212 µmand 300 µm.


Table 3.6: The apparent GB model parameters AGB and EGB for the float batch as function of the floatcullet content.

Percent cullet AGB EGB r2

in float batch [%] [s−1] [kJ mol−1] [-]

0 3.0 ·10−2 63 0.974925 8.5 ·100 120 0.999550 3.2 ·101 144 0.994075 1.3 ·105 245 0.9929

Figure 3.25 shows the plot of ln(

∂ξ∂T

)

+ lnβ− ln f (ξ) versus the reciprocal absolute temper-ature for the four float batches with different float cullet content itself. The apparent GB modelparameters AGB and EGB for the float batch as function of the float cullet content are listed intable 3.6.

0.55 0.60 0.65 0.70 0.75 0.80 0.85 -8.0

-7.5

-7.0

-6.5

-6.0

-5.5

-5.0

-4.5

-4.0

-3.5

-3.0

PSfrag replacements


ln(

∂ξ ∂T

)

+ln

β−

lnf(

ξ)[l

nm

in−

1 ]

1

2

34

Figure 3.25: ln(

∂ξ∂T

)

+ lnβ− ln f (ξ) characterizing the dissolution of silica grains versus the reciprocalabsolute temperature for two different cullet fractions. Line 1, line 2, line 3 and line 4present ln

(

∂ξ∂T

)

+ lnβ in case the cullet content of the float batch equals 0 wt.%, 25 wt.%,50 wt.% and 75 wt.%, respectively. The silica grain particle size was between 212 µm and300 µm. The applied heating rate was 25 K min−1

Figure 3.26 shows the relation of the apparent GB model parameters EGB and lnAGB asfunction of the cullet fraction. With these relations, the dissolution rate of sand grains, withthe same sand grain particle size and about similar heating rates, as function of the content ofcrushed cullet can be simulated.


0 10 20 30 40 50 60 70 80 0

50

100

150

200

250

0

2

4

6

8

10

12

14

16

18

20

22

24

PSfrag replacements

Cullet fraction xc [-]

EG

B[k

Jm

ol−

1 ]

lnA

GB

[ln

min

−1 ]

EGB=2.28 xc+57.5; r2=0.9372

ln AGB=0.19 xc+0.48; r2=0.9310

Figure 3.26: Apparent GB model parameters EGB and lnAGB describing the dissolution of sand grains asfunction of the cullet fraction in a float glass batch. The apparent GB model parameters aredetermined for sand grains with a average particle size of 250 µm and for a heating rate of25 K min−1.

3.6 Concluding remarksIn this chapter, the applicability of the relative simple Ginstling-Brounstein model for the pre-diction of the sand grain dissolution process in soda rich glass batches is studied. The accuracyof the prediction of the sand grain dissolution process with the GB-model is determined bycomparison of simulation results obtained from the GB-model with simulation results obtainedfrom a more detailed numerical model. It appeared that with the GB-model, no quanitativeprediction of the sand grain dissolution process can be performed in case the true chemical andphysical properties of the sand grain and the surrounding thin melt phases are used. However,for sand grains surrounded by a large volume of glass melt, the sand grain dissolution processsimulated with the numerical model can be approximated by a modified GB-model. In the mod-ified GB-model, the proportionality constant kGB is multiplied with a constant less than unity.For a small amount of glass melt surrounding a sand grain, the ratio between the constants kGBand knm is not dependent on kGB. For these systems, the GB-model is not suitable for describingthe sand grain dissolution process.

The main important parameter which determines the linearity between the proportionalityconstants kGB and knm, is the ratio of the melt phase thickness surrounding the sand grain δand the initial sand grain radius rs,0. In the numerical model, an idealized situation of a com-pletely and uniform wetted sand grain is modelled. In practice, the ideal situation is disturbedby e.g. incomplete wetting of the sand grains, an inhomogeneous glass batch, the presence of


gas bubbles separating the sand grain and the glass melt. Therefore, it was also questioned towhich extent the modified GB-model is capable of describing the sand grain dissolution processin industrial glass melts. The applicability of the modified GB-model for describing the sandgrain dissolution process in industrial glass batches is studied by comparison of the measuredGB proportionality constant kGB from a float glass batch with different sand grain sizes withthe expected proportionality constant. From the experiments, the applicability of the GB-modelcould not be proved unambiguously. Although the conversion rate of sand grains in the floatglass batch could be described well with the GB-model resulting in apparent values for GBmodel parameters EGB and AGB, the prediction of the dissolution rate of sand grains as functionof particle size and as function of the heating rate, using the GB-model and the derived EGB- andAGB-values, show relative large differences with the measured dissolution rates. The reason forthis is likely the complexity of the glass melting process and the broad particle size distributionpresent in the float glass batch.

By the experiments which are performed in this chapter, it is possible to predict, for a givenparticle size and heating rate (for which EGB and AGB are determined experimentally), the tem-perature dependent conversion rate. Also the presence of (crushed) cullet in the float glass batchwith a defined particle size of the sand grains can be simulated. To evaluate whether the modi-fied GB-model is capable of describing the sand grain dissolution, additional experiments withwell-defined and narrow particle size distributions are required. It is recommended to start withsimple glass batches such as binary systems before extending the melting experiments to multi-component batches. This approach will provide more detailed information of the dissolutionmechanism of sand grains in glass batches.

Another option to simulate the sand grain dissolution process in glass batches is to developa model based on a large series of experiments. This model does not need to be based on first-principles like the GB-model.



As surface area [m2]A pre-exponential factor [s−1]CSiO2 SiO2 concentration [kg m−3]DSiO2 inter diffusion coefficient of SiO2 [m2 s−1]Ea (apparent) activation energy [J mol−1]f (ξ) reaction mechanism function [-]g gas phaseGr Grasshof number [-]h mass transfer coefficient [m s−1]hd mass transfer coefficient in the diffusive stage [m s−1]hr mass transfer coefficient in the reactive stage [m s−1]H settling time [s]Ii, j intensity of diffraction line i for crystalline

specie j[counts s−1]

j mass flux relative to mass-average velocity [kg m−2 s−1]J gas volume flux [m3 m−2]k reaction rate constant [s−1]Ki, j constant for diffraction line i for crystalline

specie jK1 constant in equation 3.14 [s−1]K2 constant in equation 3.14 [K]K3 constant in equation 3.14 [K]Ke X-ray diffraction constant [-]l liquid phasem mass [kg]M molar weight [kg mol−1]n mass flux [kg m−2 s−1]nro reaction order [-]r∗ reaction rate [s−1]r radius [m]ro radius of a sand grain and the surrounding melt

phase shell[m]

rs sand grain radius [m]rt radius of a sand grain, the surrounding melt

phase shell and the surrounding soda ash shell[m]

R universal gas constant [J K−1 mole−1]Re Reynolds number [-]RIR Reference Intensity Ratio [-]s solid phaseSc Schmidt number [-]Sh Sherwood number [-]


Sh′ Sherwood number corrected for moving bound-ary

[-]

t time [s]te duration time of the transient stage [s]T temperature [K]v velocity [m s−1]vr radial velocity [m s−1]V volume [m3]V molar volume [m3 mol−1]w weight fraction [-]x weight fraction [-]

Greek symbols

β heating rate [K s−1]δ melt phase thickness [m]φ dimensionless concentration (see equation

3.34)[-]

ρ density [kg m−3]σ uncertainty [-]θ diffraction angle [o]τ Fourier number [-]ϑ fraction [-]ν stoichiometric reaction coefficientµ mass absorption coefficient [-]ξ degree of conversion [-]


b bulkc culletdl large diameterds small diameterGB Ginstling-Brounsteini interfaceL1 liquidus line SiO2L2 liquidus line Na2O ·2SiO2L3 liquidus line Na2O ·SiO2m meltn numbernm numerical model0 initialr radials sand grainsa soda asht total

126 Bibliography

3.8 Bibliography[1] P. Hrma. Complexities of batch melting. In: Proc. of the 1st International Conference on

Advances in the Fusion of Glass, pages 10.1–10.18, Alfred University, Alfred, New York,June 14-17, 1988.

[2] L. Bodalbhai and P. Hrma. The dissolution of silica grains in isothermally heated batchesof sodium carbonate and silica sand. Glass Technology, 27(2):72–78, 1986.

[3] J.R. Frade and M. Cable. Reexamination of the basic theoretical model for the kinetics ofsolid-state reactions. J. Am. Ceram. Soc., 75(7):1949–1957, 1992.

[4] H. Salmang and H. Scholze. Die physikalischen und chemischen Grundlagen der Keramik.Springer-Verlag, Berlin/Heidelberg, Germany, 5 edition, 1968.

[5] R.E. Carter. Kinetic model for solid-state reactions. J. Chem. Phys., 34(6):2010–2015,1961.

[6] M. Muhlbauer and L. Nemec. Dissolution of glass sand. Am. Ceram. Soc. Bull.,64(11):1471–1475, 1985.

[7] R.G.C. Beerkens, H.P.H. Muijsenberg, and Heijden van der T. Modelling of sand graindissolution in industrial glass melting tanks. Glastech. Ber. Glass Sci. Technol., 67(7):179–188, 1994.

[8] P. Hrma. Reaction between sodium carbonate and silica sand at 874oC<T<1022oC.J.Am.Ceram.Soc., 68(6):337–341, 1985.

[9] P. Hrma. A kinetic equation for interaction between grain material and liquid with appli-cation to glass melting. Silikaty, 24(1):7–16, 1980.

[10] P. Hrma, J. Barton, and T.L. Tolt. Interaction between solid, liquid and gas during glassbatch melting. J. Non-Cryst. Solids, 84:370–380, 1986.

[11] R.B. Bird, W.E. Stewart, and E.N. Lightfoot. Transport phenomena. John Wiley & Sons,Inc., New York, 1960.

[12] D.W. Ready and A.R. Cooper. Molecular diffusion with a moving boundary and sphericalsymmetry. Chem. Engng. Sci., 21:917–922, 1966.

[13] R.V. Harrington, J.R. Hutchins, and J.D. Sherman. The kinetics and mechanisms of subliq-uidus alkali carbonate-silica reactions. Advances in Glass Technology, 8-14 July, 1962.USA Washington DC.

[14] M. Cable and D Martlew. Effective binary diffusivities for the dissolution of silica in meltsof the sodium carbonate - silica system. Glass Technology, 26(5):212–217, 1985.

[15] K. Kautz and Stromburg G. Untersuchungen der Vorgange beim Einschmelzen vonGlasgemengen im Gradientofen. Glastech. Ber., 42(7):309–317, 1969.

Bibliography 127

[16] C. Kroger. Gemengereaktionen und Glasschmelze. Glastechn. Ber., 25(10):307–324,1952.

[17] C. Kroger. Die ternaren und quaternaren Systeme Alkalioxyd-CaO-SiO2, Gleichgewichte, Reaktionsgeschwindigkeiten und ihre Beziehung zumGlasschmelzprozess, Teil I. Glastechn. Ber., 15(9):335–346, 1937.



[20] C. Kroger. Die ternaren und quaternaren Systeme Alkalioxyd-CaO-SiO2, Gleichgewichte, Reaktionsgeschwindigkeiten und ihre Beziehung zumGlasschmelzprozess, Teil II. Glastechn. Ber., 22(12):248–261, 1949.

[21] C. Kroger. Die ternaren und quaternaren Systeme Alkalioxyd-CaO-SiO2, Gleichgewichte, Reaktionsgeschwindigkeiten und ihre Beziehung zumGlasschmelzprozess, Teil II. Glastechn. Ber., 22(15):331–338, 1949.

[22] C. Kroger and G. Ziegler. Uber die Geschwindigkeiten der zur Glasschmelze fuhren-den Reaktionen. II. Die Umsetzung von Natriumdisilikat mit Soda und von Quarz mitKalkstein. Glastechn. Ber., 26(11):346–353, 1953.

[23] C. Kroger and G. Ziegler. Uber die Geschwindigkeiten der zur Glasschmelze fuhrendenReaktionen. III. Reaktionsgeschwindigkeiten im quaternaren System Na2O-CaO-SiO2-CO2. Glastechn. Ber., 27(6):199–212, 1954.

[24] C. Kroger and F. Marwan. Uber die Geschwindigkeiten der zur Glasschmelze fuhrendenReaktionen. IV. Die Druckabhangigkeit der Umsetzgeschwindigkeiten im quaternarenSystem Na2O-CaO-SiO2- CO2. Glastechn. Ber., 28(2):51–57, 1955.

[25] C. Kroger and F. Marwan. Uber die Geschwindigkeiten der zur Glasschmelze fuhrendenReaktionen. VI. Der Einfluß von Zusatzen auf die Reaktionsgeschwindigkeit eines Soda-Kalkstein-Quarz-Grundgemengen. Glastechn. Ber., 29(7):275–289, 1956.

[26] C. Kroger. Uber die Geschwindigkeiten, den Mechanismus und die Phasenneubilding beiden unter Schmelzbilding ablaufenden Festkorperreaktionen. Glastechn. Ber., 30(2):42–52, 1957.

[27] E.M. Levin, C.R. Robbins, and H.F. McMurdie. Phase Diagrams for Ceramics. TheAmerican Ceramic Society, 1964.

[28] M.E. Savard and R.F. Speyer. Effect of particle size on the fusion of soda-lime-silicateglass containing NaCl. J. Am. Ceram. Soc., 76(3):671–677, 1993.

[29] L. Stoch and S. Kraishan. Interface phenomena accompanying the early stages of glassbatch reactions - a model study. Glastech. Ber. Glass. Sci. Technol., 70(10):298–305, 1997.

128 Bibliography

[30] R. Rottenbacher and H.E. Engelke. Reaktionsvorgange zwischen SiO2 und einerNatriumsilikatschmelze, Teil 1. Auflosung von SiO2 in der Schmelze. Glastechn. Ber.,49(11):257–263, 1976.

[31] A. Shimizu and Y. Hao. Influence of particle contact on the estimation of powder reactionkinetics of binary mixtures. J. Am. Ceram. Soc., 80(3):557–568, 1997.

[32] H.S. Carslaw and J.C. Jaeger. Conduction of heat in solids. Clarendon Press, Oxford,U.K., 2nd edition, 1959.

[33] L. Shartsis, S. Spinner, and W. Capps. Density, expansivity, and viscosity of molten alkalisilicates. J. Am. Ceram. Soc., 35(6):155–160, 1952.

[34] J. Hermans. Personal communication.

[35] Manual FactSage 5.1. Centre for Research in Computational Thermochemistry, EcolePolytechnique (Universite de Montreal), Montreal, Quebec, Canada.

[36] I.C. Madsen, N.V.Y. Scarlett, L.M.D. Cranswick, and T. Lwin. Outcomes of the interna-tional union of crystallography commision on powder diffraction round robin on quantita-tive phase analysis: samples 1a to 1h. J. Appl. Cryst., 34:409–426, 2001.

[37] R. Jenkins and R.L. Snyder. Introduction to X-ray powder diffractometry. John Wiley &Sons, Inc., 605 Third Avenue, New York, NY, USA, 1996.

[38] H.W. Coleman and W.G. Steele. Experimentation and uncertainty analysis for engineers.John Wiley & Sons, Inc., 2nd edition, 1999.

Chapter 4

Heat conductivity of glass forming batches

4.1 IntroductionAs mentioned in chapter 1, a glass forming batch can be regarded as a mixture of a condensedphase, which encloses the solid batch particles and the formed melt phases, and a gas phase.The Lagrangian description of the energy equation of the two-phase glass batch is given by

∂∂t

(

ρ cp T)

mn = −∇ ·(



∂t,

(4.1)in which

(

ρ cp T)

mn represents the mean value of the enthalpy of the glass batch, ρg and cp,gare the density and heat capacity of the gas phase, Tmn is the mean temperature of the glassbatch, t is the time, εp is the porosity of the glass batch, vg,z is the vertical velocity of the gasphase relative to the condensed phase1, λeff is the effective heat conductivity of the glass batch,−→q r,eff is the effective radiative heat flux through the glass batch, and Hchem is the temperaturedependent energy per unit volume of the glass batch required for batch reactions.

The heat transport in the interior of the glass batch is determined by a combination of threemodes of heat transfer:

• convective heat transport by the ascending gas phase (characterized by the first right-hand-side term in equation 4.1),

• conductive heat transport by mutual contact between solid particles and between solidparticles and liquid phases (characterized by the second right-hand-side term in equation4.1),

• radiative heat transport through the transparent phases (characterized by the third right-hand-side term in equation 4.1).

Since (experimental) discrimination between these modes of heat transfer in glass batches iscomplex, in general the heat penetration in a glass batch is described by an effective heat con-ductivity λeff, which encloses the contributions of the three modes of heat transfer and theenergy required for the chemical batch reactions. In the past, several attempts have been made


129

130 Chapter 4. Heat conductivity of glass forming batches

to measure and describe the thermal behavior of glass batches during heating. In general, thethermal behavior of glass batches is studied by measuring the temperature at different positionsin the glass batch as function of time during heating of the glass batches [2–6]2. However,neither of these studies provided values for the temperature dependent heat conductivity of theglass batches. Jack and Jacquest [2], Faber et al. [5] and Conradt et al. [6] calculated a (temper-ature dependent) thermal diffusivity3 of glass batches derived from the measured temperaturesin the glass batches during heating. The calculated temperature dependent thermal diffusivityfor different batches showed that at temperatures above 1173 K, a sudden increase in the ther-mal diffusivity of the glass batches is observed. This increase in thermal diffusivity is attributedto the formation of primary formed melt phases, which are transparent for radiative energy andtherefore enhance the heating of glass batches [6]. At the onset temperature for primary meltphase formation, radiative heat transport (or photon conductivity) through the glass batch pre-dominates over conductive heat transport (or phonon conductivity) in the glass batch. Conradtet al. [6] also observed that the onset for enhanced thermal diffusion in a glass batch is notonly dependent on temperature, but also on the position in the glass batch. The position depen-dent onset temperature for enhanced thermal diffusion was explained by local differences in theamount of entrapped batch gases in the melting glass batch, which decrease the mean free pathfor radiative energy in the glass batch. This indicates that the thermal diffusivity (and therewiththe heat conductivity) is dependent on local conditions in the glass batch and therefore theseproperties are time- and temperature.

A quantitative study to the temperature dependent heat conductivity of glass batches wasperformed by Kroger and Eligehausen [8]. The heat conductivity of glass batch componentsand complete glass batches was measured with a so-called static ’hollow cylinder technique’.With this technique, two cylinders with different diameters are placed in each other. The spacebetween both cylinders is filled with glass batch (components). During heating of this system,both the temperature difference over the glass batch shell and the heat flow at the outer boundaryof the glass batch shell are measured. The heat conductivity λ of the glass batch is calculatedby

λ =Q

Tr2 −Tr1

ln(d1/d2)

2π L, (4.2)

in which Q is the heat flow through the glass batch, Tr1 and Tr2 are the temperatures at theboundaries of the glass batch, d1 and d2 are the diameters of both cylinders, and L is the lengthof the cylinders over which heat is transferred. In this experimental set-up, the thickness of theglass batch was approximately 17 mm. It was observed that in the temperature range at whichno melt phases are generated, which is approximately below 1073 K, the temperature dependentheat conductivity can mostly be described by a first order temperature dependency.

Because the presence and the amount of generated melt phases in a glass batch, which en-hances the diffusion of heat through the glass batch, is dependent on the kinetics of melt phaseforming and dissolution reactions, the effective heat conductivity of glass batches is both time-and temperature dependent. Therefore, a drawback of the use of a static technique to determinethe effective heat conductivity of glass batches is that time dependent effects (caused by the

2Next to the determination of the heat conductivity of glass batches by measuring the temperature at differentpositions in the glass batch as function of time, Krzoska et al. [7] developed an alternative method with which thelocal thermal diffusivity of melting glass batches can be determined using a pulsing heat source.

3The thermal diffusivity is defined as(

λρcp

)

4.1. Introduction 131

time dependent melting of glass batches) on the heat conductivity of glass batches are not takeninto account.

The purpose of this chapter is to develop a transient method, with which the temperaturedependent heat conductivity of glass batches can be determined quantitatively. This transientmethod encloses both the development of an experimental set-up and a mathematical methodwith which the temperature dependent heat conductivity of glass batches can be estimated fromthe experiments performed in the experimental set-up. Similar to most experimental studies,in this study it is chosen to study the heat conductivity of glass batches by measuring the tem-peratures at different positions in a glass batch as function of temperature. Figure 4.1 showsan example of the measured temperatures at different positions in a silica sand batch which ismeasured in this study. Throughout this section, the heating experiments of mixtures of glassbatch components is denoted as ’heat penetration experiments’.

0 1000 2000 3000 4000 5000 6000 7000 8000 9000

400

600

800

1000

1200

1400

1600

4

3

2

1

PSfrag replacements Time [s]

Tem

pera

ture

[K]

Figure 4.1: Measured temperatures at different positions from the hot boundary of a silica sand batch.

In section 4.2, a description is given of the phonon and photon conductivity of individualspecies and of mixtures of species. In section 4.3, the most suitable mathematical method forestimating the effective heat conductivity of glass batches from heat penetration experiments isstudied. The developed experimental set-up for performing these heat penetration experimentsis discussed in section 4.4. Section 4.5 describes the estimation of the temperature dependentheat conductivity of solid particle mixtures up to a temperature of maximum 1400 K and themodelling of the temperature dependent heat conductivity of these mixtures. Section 4.6 pro-vides a summary of the results obtained in this section.


4.2 Phonon and photon conductivityThe effective heat conductivity of a mixture of solid particles surrounded by a static gas phaseis determined by both phonon and photon conduction through the gas-solid mixture. In sections4.2.1 and 4.2.2, the intrinsic phonon and photon conductive properties of both crystalline andamorphous solid species is discussed. Sections 4.2.3 and 4.2.4 describe the phonon and photonconductive properties of a mixture of solid particles surrounded by a static gas phase.

4.2.1 Intrinsic phonon conduction of solid speciesPhonon conduction in a solid specie is the transfer of vibrational energy of atoms via latticewaves, i.e. phonons, through the solid specie. The phonon conductivity λphonon, as describedby Kingery [9], is given by

λphonon =13

∫

ωc(ω) v(ω) l(ω) dω, (4.3)

in which c is the heat capacity of the solid, expressed in J kg−1 m−3, v is the phonon velocityin the solid4, l is the mean free path of the phonons, and ω is the frequency of the phonons.According to Kingery et al. [9], the heat capacity of a solid increases with temperature up to amaximum limiting value expressed in J mole−1 K−1 equal to (3 N R), in which N is the numberof atoms per molecule and R is the universal gas constant. The mean free path of the phononsis limited by the scattering of phonons at lattice imperfections. Because in general the numberof lattice imperfections increases with temperature, the phonon conductivity decreases withtemperature and becomes independent on the frequency ω. As an example, Slifka et al. [10]measured the phonon conductivity of polycrystalline MgO over the temperature range from400 K up to 1300 K. In this temperature range, the phonon conductivity decreases from 30 Wm−1 K−1 at 400 K down to 8 W m−1 K−1 at 1290 K. According to figure 4.2, the phononconductivity is linear dependent on the reciprocal value of the absolute temperature.

At high temperatures, the mean free path of the phonons in crystalline solids approachesa value similar to the lattice spacing of the crystalline solid. Now, the mean free path of thephonon is independent on the phonon frequency. For a constant mean free path of the phononsand writing the phonon velocity as the ratio of the velocity of light in vacuum, vvacuum, and therefractive index of the solid n, the temperature dependency of the phonon conductivity is givenby

λphonon =vvacuum l

3 nc, (4.4)

in which c is the temperature dependent heat capacity of the solid. Because at high tempera-tures, the heat capacity of a solid is constant, also the phonon conductivity attains a constantvalue. Figure 4.3 shows the measured phonon conductivity of crystalline silica, which is givenby Eligehausen and Kroger [11], as function of the reciprocal absolute temperature. The lin-ear dependency of the phonon conductivity with the reciprocal absolute temperature in the lowtemperature range clearly diminishes at higher temperatures.

Because in amorphous species, such as glass and glass melts, no short-range structural or-dering of atoms is present, the mean free path for phonons in amorphous species is limited. The

4The phonon velocity in the solid is given by the ratio of the velocity of light and the refractive index of thesolid, which is only slightly dependent on the wavelength ω of the phonon.

4.2. Phonon and photon conductivity 133

0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 5

10

15

20

25

30

PSfrag replacements


Phon

onco

nduc

tivity

[Wm

−1

K−

1 ]

T=1290 K

T=400 K

Figure 4.2: Phonon conductivity of polycrystalline MgO as function of the reciprocal absolute temper-ature [10]. The temperature dependency of the phonon conductivity is given by λphonon =−1.67+1.269 ·104 T−1.

temperature dependency of the phonon conduction in amorphous species is mainly given bythe temperature dependency of the specific heat of the amorphous specie, which increases withtemperature. Mann et al. [12] measured the intrinsic phonon conductivity of float glass culletup to about 780 K, which is given by

λint,phonon,float cullet = 0.97+6.24 ·10−4 T, (4.5)

in which T is expressed in K.

4.2.2 Intrinsic photon conduction of solid speciesIn addition to vibrational energy transfer in solids, energy is also transferred by electromagneticradiation, which is called photon conduction. According to Kingery [9], the intrinsic photonconductivity λint,photon is given by

λint,photon =163

σ n2 T 3 lr, (4.6)

in which σ is the Stefan-Boltzmann constant, n is the refractive index, T is the temperature inK, and lr is the mean free path of photons in the solid. Single defect free crystals are, to a


0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 3

4

5

6

7

8

9

PSfrag replacements


Phon

onco

nduc

tivity

[Wm

−1

K−

1 ]

T=1373 K

T=273 K

Figure 4.3: Intrinsic phonon conductivity of crystalline silica as function of the reciprocal absolute tem-perature given by Eligehausen and Kroger [11].

certain extent, transparent in the electromagnetic spectrum for wavelengths between 1 µm and4 µm. Both at wavelengths shorter than 1 µm and longer than 4 µm, the crystals are opaquefor radiative energy. In the UV-range of the electromagnetic spectrum (at wavelengths largerthan 4 µm), the transmission of radiative energy is lost by electron excitation, whereas in theIR-range, strong absorption is present due to atomic vibrational phenomena. In the visibleregion of the electromagnetic spectrum, the mean free path for a single alumina crystal in thetemperature range from 500 K up to 2000 K increases from 1 cm up to 10 cm [9]. However, incase pores are present in polycrystalline alumina, the photon conduction is strongly limited dueto the scattering of photons at the surface of the pores. The transmission of photons throughpolycrystalline alumina, containing 1 volume percent of pores with a diameter of 2 µm, equalsabout 1 % of the transmission of photons through a single alumina crystal [9]. Therefore, ingeneral polycrystalline solids can be regarded as opaque for photon conduction.

For translucent species, such as glass and glass melts, the photon conductivity λphoton isgiven by

λint,photon,translucent =163

σ n2 T 3

a, (4.7)

in which a is the wavelength dependent absorption coefficient, which is the reciprocal of themean free path of the photons. The absorption coefficient of a glass (melt) is mainly dependent


on the absorption of radiation by electronic transitions of transition elements such as Fe, Cr.Each valency state of these polyvalent ions, such as Fe2+ and Fe3+, absorbs radiation of a typicalown wavelength (e.g. Fe2+ around 1050 nm and Fe3+ around 380 nm) and gives a temperaturedependent absorption spectrum. The valency state of the polyvalent ions in the glass (melt) iscoupled to the oxygen activity in the glass (melt). Therefore, the photon conductivity of a glass(melt) is dependent on both the total amount of transition elements and the oxygen activity inthe glass (melt).

4.2.3 Phonon conduction of mixtures of solid speciesIn the solid-state regime5, a glass batch is regarded as a mixture of a solid phase and a gas phase.For the description of the heat conductivity of gas-solid mixtures, different geometrical modelsare proposed in literature [13, 14]. These geometrical models predict the heat conductivity ofthe gas-solid mixture, assuming a static gas phase, as a function of

• the volume fraction of the gas phase,

• the heat conductivity of the gas phase,

• the heat conductivity of the solid phase(s), and

• the structural arrangement of the two phases.

In general, the geometrical models describe the heat conductivity through a gas-solid mixture6

by a heat flow through a three dimensional arrangement of thermal resistances. For a solidparticle mixture, each individual thermal resistance characterizes the resistance towards heattransport through either a solid particle or through the gas phase between the solid particles.The heat flow Q through each thermal resistance is described by the ratio of the temperaturedifference over the resistance, T2 −T1, and value of the heat resistance RT according to

Q =T2 −T1

RT. (4.8)

The heat resistance RT is defined by

RT =L

λ A, (4.9)

in which L is a characteristic length scale for heat transport through a solid particle or a staticgas phase volume, A is a characteristic cross-section of the thermal resistance perpendicular onthe heat flow direction, and λ is the heat conductivity of the solid particle or gas phase. Thetwo extreme arrangements of the thermal resistances are shown in figures 4.5(a) and 4.5(b).Figure 4.5(a) shows the arrangement of the thermal resistances in case the heat flow in the solidparticle mixture is regarded as a parallel process of heat transfer through the solid and the gasphase. Figure 4.5(b) shows the arrangement of the thermal resistances in case the heat flow inthe solid particle mixture is regarded as a serial process of heat transfer through the solid and

5The solid-state regime is the temperature regime during which no melt phases are present. A typical value forthe onset temperature for melt phase formation, which finishes the solid-state regime, for a soda-lime-silica glassequals approximately 1073 K.

6Further throughout this thesis, a gas-solid mixture is denoted as a solid particle mixture.

136 Chapter 4. Heat conductivity of glass forming batchesPSfrag replacements

T1 T2

Q Resistance RT

L

A

Figure 4.4: Schematic representation of the heat flow through a thermal resistance in which Q is the heatflow, T2 −T1 is the temperature difference over the thermal resistances, RT is the thermal re-sistance, L is a characteristic length scale for heat transport through a solid particle or a staticgas phase volume, A is a characteristic cross-section of the thermal resistance perpendicularon the heat flow direction

the gas phase. The mathematical formulation describing the heat transport through these twoextreme arrangements is denoted as the Parallel Network Model (PNM) or the Serial Networkmodel (SNM).

PSfrag replacementsSolid phase

Gas phase

T1 T2

Q

RTg

RTs

(a) Parallel arrangement of thermal resis-tances

PSfrag replacementsSolid phase

Gas phase

T1 T2

Q RTg RTs

(b) Serial arrangement of thermal resis-tances

Figure 4.5: Models of the arrangements of thermal resistances in a solid particle mixture, in which Q isthe heat flow, T2 −T1 is the temperature difference over the thermal resistances, RTg is thethermal resistance of the gas phase, and RTs is the thermal resistance of the solid phase.

The effective heat conductivity of the packed bed derived from both models is given by

λPNM = εp λg +(1− εp) λs, (4.10)

andλSNM =

1εpλg

+1−εp

λs

, (4.11)

in which λPNM is the heat conductivity according to the parallel network model, λSNM is theheat conductivity according to the serial network model, λs is the heat conductivity of the solidphase, λg is the heat conductivity of the gas phase, and εp is the volume fraction of the gasphase in the solid particle mixture.

According to Kingery [9], a mathematical formulation of the heat conductivity of the solidparticle mixture based on a more realistic structure of a solid particle mixture, which is shownin figures 4.6(a) and 4.6(b), is proposed. Figure 4.6(a) shows the structure of a solid particlemixture in case the major phase is a continuous phase in which the minor phase is dispersed.Figure 4.6(b) shows the structure of a solid particle mixture in case the minor phase is a con-tinuous phase (gas) in which the major phase (solid particles) is dispersed. The mathematicalformulations describing the heat transport process through these two structures is denoted as theContinuous Phase Network model (CPNM) and the Dispersed Phase Network model (DPNM).


(a) Major phase is continuous phase (b) Major phase is dispersed phase

Figure 4.6: Schematic structural arrangements for solid particle mixtures.

Kingery applied the so-called Eucken-model for the description of the heat conductivity ofthese continuous-dispersed models. The heat conductivity of a solid particle mixture containingspherical particles is given by this model is

λEucken = λc

1+2 εp

(

1− λcλd

)

\(

2 λcλd

+1)

1− εp

(

1− λcλd

)

\(

λcλd

+1) , (4.12)

in which λc is the heat conductivity of the continuous phase, and λd is the heat conductivity ofthe dispersed phase.

Resuming, the effective phonon conductivity of a mixture of solid particles surrounded bya gas phase is dependent on the intrinsic properties of the solid and gas phase species andon the three dimensional arrangement of the solid particle mixture. The identification of themost suitable structural model for the description of the phonon conductivity of solid particlemixtures composed of industrial grade raw material components of glass batches is consideredin section 4.5.1.

4.2.4 Photon conduction in mixtures of solid speciesThe photon conduction in mixtures of solid species is determined by the mean free path forphotons in the mixture, which is dependent on:

• the absorption and transmission of photons in and through the solid particles and thesurrounding gas phase, and

• the reflection of photons at the surface of solid particles and pores in the mixture.

Most of the industrial raw material components of glass batches, such as silica sand and lime-stone, are opaque for radiative energy transfer and will therefore not transmit radiation. In casecullet, which is (partly) transparent for radiation, is used as alternative raw material component


during glass batch melting, the mean free path for photons in the solid particle may increaseresulting in a higher value for the photon conductivity of the solid particle mixture.

According to Kingery [9], the reflecting properties of solid particles is characterized by aso-called scattering coefficient. The degree of photon scattering at the surface of particles isdependent on

• the ratio of the particle diameter to the wavelength of the incident photons,

• the relative refractive index, which is defined as the ratio of the refractive index of theparticles and the refractive index of the medium surrounding the particles.

The degree of photon scattering increases with the relative refractive index. At a particle sizeless than the wavelength of the incident photons, the degree of scattering increases up to amaximum value, which is attained at a particle size equal to the wavelength of the incidentphotons. At a particle size larger than the wavelength of the incident photons, the degree ofscattering becomes proportional to the reciprocal particle size.

According to Holman [15], the photon conductivity of a pore is given by

λphoton,pore = 4 dp n2gσ εg T 3, (4.13)

in which dp is the diameter of the pore, ng is the refractive index of the gas phase present inthe pore, and εg is the emissivity of the radiating gas. The radiative conductivity of the poreincreases with pore size, emissivity and temperature. Therefore, especially small pores in aglass batch act as barriers for radiative heat transfer.

Resuming, the photon conductivity of a glass batch is dependent on

• the intrinsic photon conductivity of, especially, the glass cullet and the radiative gas in thepores,

• the dimensions of the glass cullet and the pores,

• the surface roughness of the glass cullet, and

• the volume fraction of the glass cullet and the pores per unit volume of the glass batch.

In general, the contribution of the intrinsic photon conductivity of the glass cullet to the heatconductivity of a glass batches decreases with decreasing particle size, increased surface rough-ness of the glass cullet and with decreasing volume fraction of the glass cullet in the glassbatch. Faber et al. [5] remarked that the beneficial effect of glass cullet on the heating rate ofglass batches is apparent for cullet fractions above 50 %.

The heat conductivity of a glass batch containing glass cullet can hardly be predicted fromthe intrinsic photon conductivity of the glass cullet because of the complexity concerning dif-ferent batch components, cullet sizes, gas pore sizes, etc, . The photon conductivity of a glassbatch containing glass cullet is mainly dependent on the presence of scattering sources for pho-ton conduction, such as small bubbles and very fine glass powder and glass cullet with a roughsurface. The fusion of glass cullet, the disappearance of small bubbles in the glass batch and theformation of new melt phases will determine the actual photon conductivity of the glass batch.This indicates that knowledge of the reactive melting of the raw materials of glass batches willnot be sufficient to predict the photon conductivity of the glass batch.

4.3. Estimation of the heat conductivity from heat penetration experiments 139

4.3 Estimation of the heat conductivity from heat penetra-tion experiments

In this section, different mathematical methods are studied to estimate the temperature depen-dent effective heat conductivity of a solid particle mixture from heat penetration experiments.From the accuracy of the estimated effective heat conductivity, which is derived from a fictiveheat penetration experiment using a known value for the heat conductivity to reconstruct theheat penetration in the solid particle mixture, the most accurate mathematical method is iden-tified.

After a general introduction in section 4.3.1, section 4.3.2 describes the accuracy of estimat-ing the effective heat conductivity from heat penetration experiments in case a-priori informa-tion of λ = f (T ) is missing. The influence of a-priori information of λ = f (T ) on the accuracyof the estimated heat conductivity is studied in section 4.3.3. In this section, three differentmathematical methods are described, viz. a differential approach, an integral approach, and anumerical-experimental approach.

4.3.1 IntroductionDifferent mathematical methods can be used for the determination of the temperature dependentheat conductivity of a solid particle mixture7 from heat penetration experiments. In this section,the most accurate mathematical method is identified from the calculated accuracy of the heatconductivity of a solid particle mixture derived from fictive heat penetration experiments witha known value for the heat conductivity of the solid particle mixture. To identify the mostaccurate mathematical method, the following steps are performed:

1. Simulation of the heat penetration in a solid particle mixture

2. Registration of the temperature as function of time at predefined positions in the solidparticle mixture.

3. Estimation of the known temperature dependent heat conductivity based on the fictivetemperature measurements.

The fictive heat penetration experiment is obtained by numerical simulation of the heat penetra-tion process in a solid particle mixture, for which a schematic representation is given in figure4.7. The indicated locations in the solid particle mixture denote the positions of the infinitesmall fictive thermocouples.

The numerical simulations are performed with the finite element code SEPRAN [16] bysolving

ρ cp∂T∂t

=∂∂x

(

λ∂T∂x

)

, (4.14)

which describes the one-dimensional heat penetration in a non-reacting solid particle mixture.At both boundaries of the solid particle mixture, the temperature is prescribed, i.e. at x=0 and

7Throughout this section is referred to a solid particle mixture instead of a glass batch. The reason for this is 1)the mathematical methods can be applied to solid particle mixtures in general and not only to glass batches, and 2)for the evaluation of the most accurate mathematical method, the heat penetration in a non-reacting solid particlemixture is simulated.


PSfrag replacements

Tx3

Tx2

Tx1

Tx4

x3=30 mm

x2=20 mm

x1=10 mm

x4=40 mm

x = 0

x = dspm Tb

Tb

Center line

Packed bed

Figure 4.7: Schematic representation of the geometry of the solid particle mixture for which the heatpenetration is simulated. The indicated locations in the solid particle mixture denote thepositions of fictive thermocouples, from which the temperature as function of time are shownin figure 4.8.

x=dspm, the temperature equals Tb. The initial solid particle mixture temperature is T0. Thevalues for the solid particle mixture properties ρ, cp and λ, the thickness of the solid particlemixture dspm and the initial and boundary temperatures T0 and Tb, for the heat penetration sim-ulations are listed in table 4.1.Figure 4.8 shows the calculated temperatures as function of time at the fictive thermocouplepositions. For the evaluation of the different mathematical methods to estimate the heat conduc-tivity from these heat penetration curves, it is assumed that the solid particle mixture propertiesρ and cp, the initial and boundary temperatures, the temperatures at the indicated positions inthe solid particle mixture and the indicated positions self are known a-priori and are exact. Thetemperatures at the predefined positions in the solid particle mixture are stored each 5 s. Fromthese stored temperatures, the heat conductivity of the solid particle mixture is estimated. First,in section 4.3.2, the temperature dependent heat conductivity λ is estimated using a differentialmethod and in case a-priori information on λ = f (T ) is missing. In section 4.3.3, the estimationof the heat conductivity is performed in case beforehand is known that the heat conductivity ofthe solid particle mixture is given by λ = cλa + cλb T . For this estimation, three mathematicalmethods are used, i.e. a differential method, an integral method and a numerical-experimentaltechnique.

The deviation between the estimated and the true value of cλa and cλb are an indication forthe accuracy of the mathematical method, with which the estimation is performed. Next to theestimation of cλa and cλb in case the other model parameters ρ and cp, the temperatures and


Table 4.1: Physical properties of the solid particle mixture, thickness of the solid particle mixture andboundary and initial temperatures used for simulating the heat penetration in the solid particlemixture as function of time. These simulations are used for the evaluation of the accuracy ofdifferent mathematical methods for the estimation of the temperature dependent heat conduc-tivity.

Bulk density ρ 1500 [kg m−3]Heat capacity cp 1200 [J kg−1 K−1]Heat conductivity λ 0.2+3.0 ·10−4 ·T [W m−1 K−1]Thickness dspm 0.10 [m]Initial temperature T0 298 [K]Boundary temperature Tb 1673 [K]

the positioning of the fictive thermocouples are exact, also the effect of uncertainties8 in thesemodel parameters on the estimated value for cλa and cλb is studied. The mathematical methodwhich shows the lowest uncertainty in the temperature dependent heat conductivity is chosen tobe the most accurate method and will be used further throughout this chapter.

4.3.2 Parameter estimation without a-priori knowledge of λ = f (T )

In case a-priori information of the temperature dependency of the heat conductivity of the solidparticle mixture is missing, a first estimate of the temperature dependency of the heat conduc-tivity can be derived by neglecting the second right-hand-side term in

ρ cp∂T∂t

= λ(

∂2T∂x2

)

+∂λ∂x

∂T∂x

. (4.15)

Figure 4.9 shows a schematic representation of the geometry of the solid particle mixture. Incase the spatial distribution of thermocouples is uniform, i.e. ∆x1 = ∆x2 = ∆x, the heat conduc-tivity at position i and temperature T can be calculated from three spatial distributed tempera-tures according to

λρ cp

=∆Ti

∆t/

(

Ti+1 +Ti−1 −2Ti

∆x2

)

, (4.16)

in which Ti is the temperature at position i in the solid particle mixture, Ti−1 and Ti+1 are thetemperatures at the positions at −∆x and ∆x distance from xi, and ∆Ti is the temperature differ-ence at position xi in the solid particle mixture after a time step of ∆t [5]. The denominator inequation 4.16 is derived by assuming that the spatial temperature distribution between the threethermocouples is described by a 2nd order polynomial (see Appendix C).

Plotting the heat conductivity calculated with equation 4.16 versus temperature provides thetemperature dependent heat conductivity. Figure 4.10 shows both the true and the estimatedheat conductivity of the solid particle mixture as function of temperature in case ∆x=10 mmand for xi=20 mm, xi=30 mm and xi=40 mm.

8According to Coleman and Steele [17], the uncertainty in a parameter value is defined as the standard deviationof the parameter value.


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1200

1400

1600

PSfrag replacements

Time [s]

Tem

pera

ture

[K]

x=10 mm

x=20 mm

x=30 mm

x=40 mm

Figure 4.8: Calculated temperature as function of time at different positions in a solid particle mixturesimulated with SEPRAN [16]. The solid particle mixture properties, the solid particle mix-ture thickness and the initial and boundary conditions are listed in table 4.1

It is observed that large deviations in the temperature dependent heat conductivity are ob-tained in case a-priori knowledge of the temperature dependency of the heat conductivity ismissing. Combining equations 4.15 and 4.16 show that the accuracy of the estimated heat con-ductivity at position i is dependent on:

• the accuracy of approximating the second order derivative of the temperature to positiongiven by the denominator in equation 4.16,

• the accuracy of approximating the heating rate by the first order derivative of the temper-ature to position, and

• the value for the neglected term from equation 4.15, which equals cλb

(

∂T∂x

)2in case

λ = cλa + cλb T .

Because(

∂2T∂x2

)

= lim∆x→0

(

Ti+1+Ti−1−2Ti∆x2

)

, it is expected that the approximation of the secondorder derivative of the temperature to position by the denominator in equation 4.16 improvesin case the spacing of the fictive thermocouples (∆x) decreases. Figure 4.11 shows the effectof the spacing of the fictive thermocouples on the estimated value for the heat conductivity. Asexpected, a closer spacing of the thermocouples results in a smaller difference between the true


PSfrag replacements

Ti+1

Ti

Ti−1

xi+1

xi

xi−1

x = 0

x = dspm

∆x1

∆x2

Packed bed

Figure 4.9: Schematic representation of the geometry of the solid particle mixture through which theheat penetration is simulated.

and estimated heat conductivity.In case the approximation of the heating rate by a first order derivative of the temperature

to time affects the estimation of the heat conductivity, a higher sampling frequency of the mea-sured temperatures is expected to result in a better estimation of the heat conductivity. However,additional simulations did not show a significant change in the estimated heat conductivity withincreasing sampling frequency, which indicates that the approximation of the heating rate doesnot have a significant effect on the estimated heat conductivity in case the sampling frequencyis less than 0.2 s−1.

The effect of the temperature dependency of the solid particle mixture on the accuracy ofthe estimated heat conductivity, which is given by the parameter cλb in λ = cλa + cλb T , isshown in figure 4.12. This figure shows the heat conductivity of the solid particle mixture es-timated from a fictive heat penetration experiment for which the true heat conductivity is givenby λ = 0.2 + 1 ·10−3 T instead of λ = 0.2 + 3 ·10−4 T . It is observed that for this strongertemperature dependent heat conductivity, non-realistic values for the heat conductivity are ob-tained. From figures 4.10, 4.11, and 4.12, it is concluded that in case a-priori knowledge ofthe temperature dependency of the heat conductivity is missing, non-realistic values form thetrue heat conductivity are obtained. The confidence of the estimated heat conductivity by thismethod depends on the solid particle mixture properties (especially the heat conductivity), thetemperature gradient at the position of thermocouple for which the heat conductivity is esti-mated and the spacing of the three thermocouples, which are required for the estimation of theheat conductivity.

4.3.3 Parameter estimation with a-priori knowledge of λ = f (T )

This section describes the estimation of the temperature dependent heat conductivity from heatpenetration experiments in case beforehand is known that λ = cλa + cλb T . The parameter esti-mation is performed by a differential method, an integral method and a numerical- experimentalmethod.

The differential method


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Temperature [K]

Hea

tcon

duct

ivity

[Wm

−1

K−

1 ]

True heat conductivity

A

B

C

Figure 4.10: True (dotted line) and estimated (solid lines) heat conductivity of a solid particle mixture asfunction of temperature. A, B and C indicate the estimated heat conductivity for a fictive heatpenetration experiment in case a-priori information of λ = f (T ) is missing for xi=20 mm,xi=30 mm and xi=40 mm, respectively. The spacing between the fictive thermocouples ∆x is10 mm. The properties of the solid particle mixture and the initial and boundary conditions,with which the fictive heat penetration experiment is simulated, are listed in table 4.1.

In case the temperature dependency of the heat conductivity is known beforehand and is de-scribed by λ = cλa + cλb T , equation 4.14 can be written as

ρ cp∂T∂t

= cλa

(

∂2T∂x2

)

+ cλb

[

(

∂T∂x

)2

+T(

∂2T∂x2

)

]

. (4.17)

Describing the spatial dependency of the temperature with a 2nd order polynomial, the bestestimates for the parameters cλa and cλb can be obtained from a least squares approach. Equation4.17 is now rewritten to

k1,i = cλa k2,i + cλb k3,i (4.18)

in which i refers to the i-th time period, k1,i is the left-hand-side term of equation 4.17 at t = ti,k2,i is the first right-hand-side term of equation 4.17 at t = ti, and k3,i is the second right-hand-side term of equation 4.17 at t = ti. The least squares approach requires a criterion qls, whichquantifies the difference between the measured and the simulated temperatures. The criterion


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PSfrag replacements

Temperature [K]

Hea

tcon

duct

ivity

[Wm

−1

K−

1 ]


A

B

Figure 4.11: True (dotted line) and estimated (solid lines) heat conductivity of a solid particle mixture asfunction of temperature. A and B indicate the estimated heat conductivity for a fictive heatpenetration experiment when a-priori information of λ = f (T ) is missing for xi=30 mm andin case the spacing between the fictive thermocouples are ∆x=10 mm and ∆x=6 mm, respec-tively. The spacing between the fictive thermocouples ∆x is 10 mm. The properties of thepacked bed and the initial and boundary conditions, with which the fictive heat penetrationexperiment is simulated, are listed in table 4.1.

qls describing the squared difference of the measured and simulated temperatures over the wholetime domain in which the simulations and the measurements are performed is given by

qls =n

∑i=1

(

k1,i − cλa k2,i − cλb k3,i)2

, (4.19)

in which cλa and cλb are the estimates for cλa and cλb and n is the number of time-periods atwhich the temperature data are collected from the fictive heat penetration experiment. Esti-mates for the parameters cλa and cλb providing the smallest difference between measured andcalculated temperatures are obtained by minimization of the criterion qls with respect to the un-known estimates cλa and cλb . This results in a set of two linear algebraic equations from which


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-10

0

10

20

30

40

50

PSfrag replacements

Temperature [K]

Hea

tcon

duct

ivity

[Wm

−1

K−

1 ]


A

Figure 4.12: True (dotted line) and estimated (solid lines) heat conductivity of a solid particle mixtureas function of temperature. A indicates the estimated heat conductivity for a fictive heatpenetration experiment in case a-priori information of λ = f (T ) is missing for xi=30 mm.The spacing between the fictive thermocouples ∆x is 10 mm. The properties of the packedbed ρ and cp and the initial and boundary conditions, with which the fictive heat penetrationexperiment is simulated, are listed in table 4.1. In contrast to table 4.1, the heat conductivityis given by λ=0.2 + 1 ·10−3 T.

estimates of the parameters cλa and cλb can be derived.

∂qls

∂cλa

= −2n

∑i=1

k2,i(

k1,i − cλa k2,i − cλb k3,i)

= 0 ⇒n

∑i=1

k1, j k2,i − cλa

n

∑i=1

k22,i − cλb

n

∑i=1

k3,i k2,i = 0

(4.20)∂q

∂cλb

= −2n

∑i=1

k3,i(

k1,i − cλa k2,i − cλb k3,i)

= 0 ⇒n

∑i=1

k1,i k3,i − cλa

n

∑i=1

k2,i k3,i − cλb

n

∑i=1

k23,i = 0

(4.21)This results in the following equations for the estimates for the parameters cλa and cλb , withK1 = ∑n

i=1 k1,i k2,i, K2 = ∑ni=1 k2

2,i, K3 = ∑ni=1 k3,i k2,i, K4 = ∑n

i=1 k1,i k3,i, K5 = K3 and K6 =

∑ni=1 k2

3,i.

cλa =K4 − K1K6

K3

K3 − K2K6K3

(4.22)


cλb =K1 − cλaK2

K3(4.23)

The estimated heat conductivity of the solid particle mixture as function of temperature in case∆x=10 mm and for xi=20 mm, xi=30 mm and xi=40 mm is shown in figure 4.13.

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0.9

1.0

1.1

PSfrag replacements

Temperature [K]

Hea

tcon

duct

ivity

[Wm

−1

K−

1 ]


A

B

C

Figure 4.13: True (dotted line) and estimated (solid lines) heat conductivity of the solid particle mixtureas function of temperature using the differential method with a-priori knowledge of λ =f (T ). A, B and C indicate the estimated heat conductivity at xi=20 mm, xi=30 mm andxi=40 mm, respectively. The spacing between the fictive thermocouples is ∆x is 10 mm.The properties of the solid particle mixture and the initial and boundary conditions, withwhich the fictive heat penetration experiment is simulated, are listed in table 4.1. The trueheat conductivity is given by λtrue=0.2 + 3.0 ·10−4 T. The estimated heat conductivity forthe different positions in the solid particle mixture are λA=0.279 + 3.93 ·10−4 T, λB=0.267+ 4.41 ·10−4 T and λC=0.228 + 6.79 ·10−4 T.

Although a-priori information of the temperature dependency of the heat conductivity is takeninto account, the minimum deviation between the true and estimated heat conductivity equals34 %.

The integral methodThe integral form of equation 4.14 is given by

ρcp

∫ x+

x−

(

∂T∂t

)

∂x =∫ x+

x−

(

∂∂x

(

λ∂T∂x

))

∂x, (4.24)


in which the spatial dependency of the temperature and the heating rate are described by a 2nd

order polynomial given by T (x) = m1x2 + m2x + m3 and ∂T/∂t = n1x2 + n2x + n3. The valuesfor n1, n2, n3, m1, m2 and m3 at a certain time t and position x are determined by the methoddescribed in Appendix C. The integration domain around position i is given by x−, which isdefined by 0.5 (xi−1 + xi), and x+, which is defined by 0.5 (xi + xi+1). Integration of equation4.24 results in

ρ cp

[n1

3x3 +

n2

2x2 +n3x

]x+

x−=(

cλa + cλb T)

[2m1x+m2]x+x− . (4.25)

Figure 4.14 shows both the estimated and the true heat conductivity of the solid particle mixtureas function of temperature for xi=20 mm and ∆x=10 mm calculated with equation 4.25. It isobserved that the difference between the true and the estimated heat conductivity is only verysmall, which indicates that the integral method provides an accurate estimation of the heatconductivity in case the model parameters ρ and cp, the temperatures and the positioning of thefictive thermocouples are known exactly beforehand.

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0.40

0.45

0.50

0.55

0.60

0.65

PSfrag replacements

Temperature [K]

Hea

tcon

duct

ivity

[Wm

−1

K−

1 ]

1

2

Figure 4.14: True and estimated heat conductivity of the solid particle mixture as function of temper-ature for xi=40 mm and ∆x=10 mm with a-priori information on λ = f (T ) according tothe integral method. Line 1 indicates the true thermal heat conductivity, which is given byλ = 0.20 + 3.0 ·10−4 W m−1 K−2. Line 2 indicates the estimated heat conductivity, whichis given by λ = 0.18+3.2 ·10−4 W m−1 K−1.

Next, the sensitivity of the estimated temperature dependent heat conductivity on errors in


the exactly known solid particle mixture properties, the measured temperatures and the positionsof the thermocouples is studied. According to Coleman and Steele [17], the uncertainty9 in aparameter Y , which is depending on n parameters X , viz. Y = Y (X1, ..,Xi, ..,Xn), is given by

UY =

[

(

∂Y∂X1

UX1

)2

+ ...+

(

∂Y∂Xi

UXi

)2

+ ...+

(

∂Y∂Xn

UXn

)2]0.5

, (4.26)

in which UXi is the uncertainty in parameter Xi. The effect of random errors on the estimatedtemperature dependent heat conductivity is studied in order to identify

• the effect of uncertainties in the model parameters10 on the uncertainty in the estimatedheat conductivity, and

• the model parameter, which has the major contribution to the uncertainty in the estimatedheat conductivity.

The uncertainty in the estimated heat conductivity at position i in the solid particle mixture isgiven by

Uλ = f(

Uρ,Ucp,UTi−1,UTi,UTi+1,Uxi−1,Uxi,Uxi+1,UT0,UTb

)

. (4.27)

To identify the major source for the uncertainty in the estimated heat conductivity, the percentualcontribution of the uncertainty of each parameter (ρ, cp, Ti−1, Ti, Ti+1, xi−1, xi, xi+1, T0 and Tb)to the squared total uncertainty of the estimated heat conductivity, which is given by equation4.28, are derived by additional simulations with the numerical model.

UPCi =

(

∂Y∂Xi

)2U2

Xi

U2Y

100%. (4.28)

During these additional simulation studies, the effect of small changes on the a-priori knownmodel parameters on the estimated value for the heat conductivity is determined. The smallchange of the parameter values is defined by the uncertainty of the specific model parameter.The contribution of the uncertainty of parameter i to the squared total uncertainty of the heatconductivity is given by the so-called Uncertainty Percentage Contribution of a parameter i [17](see equation 4.28).

Figure 4.15 shows the true and the estimated heat conductivity of the solid particle mixtureas function of temperature in case ∆x=10 mm for xi=20 mm and xi=40 mm. The uncertainty inthe model parameters which are used to calculate the uncertainty in the estimated heat conduc-tivity are Uρ=5 %, Ucp=5 %, UT =5 K and Ux=1 mm. The lines A indicate the confidence limitsin case xi=20 mm, whereas the lines B indicate the confidence limits in case xi=40 mm.

It is observed that the accuracy of the heat conductivity estimated from the temperature atxi=20 mm is higher (i.e. the confidence limits of the estimated heat conductivity are smaller)than for xi=40 mm. This indicates that the uncertainty in the estimated temperature dependentheat conductivity increases with decreasing temperature gradients in the solid particle mixture.

9The uncertainty of a measured parameter is described by the standard deviation of the measured parameter incase the spread of measured values are normal distributed around the mean value.

10With model parameters is ment the packed bed properties ρ, cp and the measured temperatures and the mea-sured positions.


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Temperature [K]

Hea

tcon

duct

ivity

[Wm

−1

K−

1 ]

Estimated heat conductivity

A

A

B

B

Figure 4.15: True and the estimated heat conductivity of the packed bed as function of temperature incase ∆x=10 mm for xi=20 mm and xi=40 mm. The estimation is performed with a-prioriinformation of λ = f (T ) according to the integral method. Line 1 indicates the true thermalheat conductivity, which is given by λ = 0.20+3.0 ·10−4 W m−1 K−2. The lines A indicatethe confidence limits in case xi=20 mm, whereas the lines B indicate the confidence limitsin case xi=40 mm.

An increased accuracy of the heat conductivity is obtained in case the heat conductivity is esti-mated at a position in the solid particle mixture where large temperature gradients are present.

Figure 4.16 shows the contribution of the parameter (ρ, cp, Ti−1, Ti, Ti+1, xi−1, xi and xi+1) tothe squared total uncertainty of the heat conductivity as function of temperature. It is observedthat the major source for the inaccuracy in the estimated heat conductivity is the positioning ofthe thermocouples. This indicates that for improving the accuracy of the estimated heat con-ductivity, the uncertainty in the positioning of the thermocouples should be increased. In thefollowing section, the third mathematical technique, i.e. the numerical-experimental techniqueis discussed.

Numerical-experimental techniqueFor the estimation of the effective heat conductivity of a solid particle mixture from heat pene-tration measurements, also a numerical-experimental technique [18] was used. This techniqueconsists of three parts, i.e. an experiment, a mathematical simulation of the experiment and an


400 600 800 1000 1200 1400 1600 0

10

20

30

40

50

60

70

PSfrag replacements

Temperature [K]

UPC

[%]

A

B

C

D−H

Figure 4.16: The contribution of the parameter (ρ, cp, Ti−1, Ti, Ti+1, xi−1, xi and xi+1) to the squared totaluncertainty of the heat conductivity, indicated by the Uncertainty Percentage Contributionas function of temperature. The lines A, B and C indicate the UPC values for the parametersxi, xi−1 and xi+1, respectively. The lines D−H indicate the UPC values for ρ, cp, Ti−1, Ti,and Ti+1, respectively.

estimation algorithm (see figure 4.17).During the experiment, the temperatures at different positions x in the solid particle mixture

are measured as function of time, with xT =[

x1, ..,x j, ..,xnrT]

, j is the thermocouple num-ber and nrT is the total number of thermocouples in the solid particle mixture. The mea-sured temperatures in the solid particle mixture at time ti are stored in the column mi, withmT

i =[

Ti,1, ..,Ti, j, ..,Ti,nrT]

. During the numerical simulation of the experiment, the tempera-tures at the same positions x in the solid particle mixture as in the experiment are calculated bysolving the one-dimensional thermal energy equation

ρ cp∂T∂t

=∂∂x

(

λ∂T∂x

)

. (4.29)

For the simulation of the heat penetration in the solid particle mixture, the finite element packageSEPRAN [16] is used. The thermal heat equation is solved in one-dimension, because in theexperimental set-up, which is discussed in section 4.4, the horizontal temperature gradient inthe solid particle mixture can be neglected with respect to the vertical temperature gradient in


PSfrag replacements

Experiment

Numerical simulation of experiment

Estimation algorithm

u

θ0

mi

yi

εi

θ Hi

−+

Figure 4.17: Scheme of the numerical-experimental technique according to Op den Camp et al. [18],in which u contains the input data for the (numerical simulation of the) heat penetrationprocess, mi contains the measured temperatures at time i in the packed bed, y

icontains the

calculated temperatures at time i, εi contains the differences of the measured and calculatedtemperatures at time i in the solid particle mixture, H

iis the sensitivity matrix at time i, θ

contains the values for the estimates of the a-priori unknown parameters and θ0 contains theinitial estimates for the unknown parameters.

the solid particle mixture. The calculated temperatures at the different positions in the solidparticle mixture at time ti are stored in the column yi, with yT

i =[

yi,1, ..,yi, j, ..,yi,nrT]

. The fixedinput data for the numerical simulation of the experiment, which are stored in the column u,enclose:

• the thickness dspm of the solid particle mixture,

• the initial temperature of the solid particle mixture T0,

• the time-dependent temperatures at the boundaries of the solid particle mixture Tb1(t) andTb2(t),

• the a-priori known or estimated values for the density ρ and the heat capacity cp of thesolid particle mixture, and

• the positions xi of the thermocouples in the solid particle mixture.

Next to these fixed input parameters, the numerical simulation of the heat penetration processalso requires information of the model parameters to be estimated. In the current study, first es-timates for the model parameters describing the temperature dependent heat conductivity givenby, for this case, λ = cλa + cλb T are required. The estimates for these model parameters are

stored in the column θ, with θT=[

cλa, cλb

]

. The initial estimates of these model parameters is


stored in the column θ0.Because the initial estimates for the model parameters cλa and cλb will deviate from the true

values of the model parameters (θ0 6= θtrue), the calculated temperatures at time ti (yi) differfrom the measured temperatures at time ti (mi). The deviation column of temperatures at time tiis given by

εi = mi − yi. (4.30)

To obtain a more accurate simulation of the measured heat penetration in the solid particle mix-ture with respect to the estimation based on the initial estimates θ0, improved estimates for themodel parameters are required. The improved estimates are determined by the estimation algo-rithm, which minimizes the deviation matrix εi with respect to the unknown model parametersθ by a least squares approach. For this minimization procedure, a weighted square criterion qlsis defined, which is given by

qls =n

∑i=1

[

(

mi − yi

)TW

i

(

mi − yi

)

]

. (4.31)

The matrix Wi

is a time dependent positive weighting matrix, with which differences in relia-bility of the different measured temperatures can be accounted for. According to Hendriks [19],the matrix W

ihas to be chosen such that temperatures in the solid particle mixture, which are

measured with high accuracy have a higher contribution in the least square criterion qls com-pared to temperatures which are measured with a lower accuracy. Because the thermocouples,which are used in the experimental set-up to measure the temperature as function of positionin the solid particle mixture, are made from the same thermocouple wire, it is assumed that thevalues indicated by the thermocouples have an identical accuracy. Now, the matrix W

iequals

the unity matrix.The temperatures at time t = ti in the solid particle mixture can be determined by

yi = hi(θ), (4.32)

in which hi(θ) characterizes the numerical calculation of the temperatures at the different posi-tions in the solid particle mixture at time t = ti. In this specific case, the column θ consists ofthe estimates for the model parameters cλa and cλb , which describe the temperature dependenteffective heat conductivity of the solid particle mixture according to λ = cλa + cλb T . The cal-culated temperatures in the solid particle mixture at time t = ti depend in a non-linear way uponthe model parameters cλa and cλb . Therefore, the operation hi(θ) is non-linear in the modelparameters and an iterative procedure is required to estimate θ.

The new estimates for the model parameters during the iterative estimation procedure arederived by minimization of the criterion qls with respect to the unknown model parameters θ,which results in (see Appendix B):

θk+1 = θk +n

∑i=1

(

HTi H i

)−1 n

∑i=1

HTi

(

mi −hi(θk))

. (4.33)

in which

H i =∂hi(θ)

∂θ, (4.34)


and θk and θk+1 are the estimates of the model parameters after iteration k and k +1. The iter-ative estimation procedure is repeated until ‖ θk+1 − θk ‖ has become smaller than a predefinedvalue ε0 > 0.

In the following, the accuracy of the numerical-experimental technique is studied by esti-mating the model parameters from a fictive heat penetration experiment. The measured temper-atures from the fictive heat penetration experiment are derived by numerical simulation of theheat penetration process with fixed input parameters.

Accuracy of the numerical-experimental technique in case no uncertainty in the inputparameters is presentThe accuracy of the numerical-experimental technique is studied by estimating the model pa-rameters from a fictive heat penetration experiment. The fictive heat penetration experiment isobtained by numerical simulation of the heat penetration in a solid particle mixture with theinput data as described in table 4.1. The initial estimated values for the parameters cλa and cλb ,which are stored in θ0, are 0.1 W m−1 K−1 and 1.0 ·10−4 W m−1 K−2, respectively. Figure4.18 shows the calculated temperature at 5 mm from the boundary of the solid particle mixtureas function of time.

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Time [min.]

Tem

pera

ture

[K]

1

2

Figure 4.18: Calculated temperature in the solid particle mixture at 5 mm from the hot side of the solidparticle mixture as function of time. The black dots indicate the experimental data, fromwhich the unknown material parameters cλa and cλb are estimated.

Because only two model parameters have to be estimated, only two measured temperatures


are required for estimation of these unknown model parameters. Figure 4.19 shows the esti-mation for the model parameters cλa and cλb as function of the iteration number in case thetemperature values 1 and 2 in figure 4.18 are used for the estimation procedure. It is observedthat for both model parameters only five iterations are required to obtain the true value of thesemodel parameters.

0 1 2 3 4 5 6 0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

0.45

-5.0x10 -5

0.0

5.0x10 -5

1.0x10 -4

1.5x10 -4

2.0x10 -4

2.5x10 -4

3.0x10 -4

3.5x10 -4

PSfrag replacements

Iteration number

Mod

elpa

ram

eter

c λa

[Wm

−1

K−

1 ]

Mod

elpa

ram

eter

c λb

[Wm

−1

K−

2 ]

Figure 4.19: Estimations of model parameter cλa and cλb as function of the iteration number. The truevalues for cλa and cλb equal 0.2 W m−1 K−1 and 3 ·10−4 W m−1 K−2, respectively. Theinitial estimates for cλa and cλb equal 0.1 W m−1 K−1 and 1 ·10−4 W m−1 K−2, respectively.

Accuracy of the numerical-experimental technique in case uncertainties in the input pa-rameters are presentSimilar to the integral method, the accuracy of the numerical-experimental technique in caseuncertainties in the input parameters u are present is calculated by equation 4.26. The uncer-tainties in the model parameters are Uρ=5 %, Ucp=5 %, UT =5 K and Ux=1 mm. Figure 4.20shows

• The estimated heat conductivity at xi=20 mm in the packed bed (thick solid line), whichequals the true heat conductivity of the solid particle mixture, estimated with the numerical-experimental method.

• The estimated heat conductivity at xi=20 mm in the packed bed (thick dotted line) esti-mated with the integral method.

• The confidence limits of the estimated heat conductivity at xi=20 mm in the solid particlemixture (thin solid lines) estimated with the numerical-experimental method.


• The confidence limits of the estimated heat conductivity at xi=20 mm in the solid particlemixture (thin dotted lines) estimated with the integral method.

It is observed that the accuracy of the estimated heat conductivity derived from the numerical-experimental technique is higher compared to the accuracy of the estimated heat conductivityderived from the integral method. Further throughout this section, this mathematical techniqueis used to estimate the heat conductivity of solid particle mixtures.

400 600 800 1000 1200 1400 1600

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1.1

1.2

PSfrag replacements Temperature [K]

Hea

tcon

duct

ivity

[Wm

−1

K−

2 ]

Figure 4.20: The estimated heat conductivity at xi=20 mm in the solid particle mixture estimated withboth the numerical-experimental method (thick solid line) and the integral method (thick)dotted line. Also the confidence limits of the estimated heat conductivity at xi=20 mm inthe solid particle mixture derived from the numerical-experimental method (thin solid lines)and the integral method (thin dotted lines) are indicated.

4.4 Experimental set-up for measuring the heat penetrationin solid particle mixtures

Figure 4.21 shows a cross-sectional view of the experimental set-up, which was developed formeasuring the temperatures as function of time at different positions in a mixture of solid parti-cles. In the experimental set-up, the mixture of solid particles is heated in vertical direction byconductive and radiative heat transfer from a radiative SiC-plate, which is in direct contact with

4.4. Experimental set-up for measuring the heat penetration in solid particle mixtures 157

the solid particle mixture. The experimental set-up consists of 3 sections, i.e. a heating section,a sample section and a top section.

PSfrag replacements

Thermocouple joints

1873 Kinsulation bricks

1873 Kinsulation bricks

Hea

ting

sect

ion

Sam

ple

sect

ion

Top

sect

ion

Thermocouple wires

Furnace lid

Ceramic board

SiC plate

250 mm

H

T

Heating elements5.4 kW each

Cer

amic

bloc

ksfo

rtig

hten

ing

ther

moc

oupl

ew

ires

z

r

600 mm

Figure 4.21: Cross-sectional view of the experimental set-up, with which the temperature as function oftime at different positions in a mixture of solid particles is measured.

The heating section, which is a box with a square surface area of 600 mm x 600 mm with aheight of 300 mm, contains 1873 K insulation bricks. In the insulation bricks, a square apertureis made of 250 mm x 250 mm and a depth of 150 mm. In this aperture, 6 SiC elements with amaximum power of 5.4 kW each are positioned. The aperture is closed on top by a SiC plate. Inthe aperture, a Pt/Pt-10% Rh (type S) thermocouple is positioned. To avoid overheating of theSiC elements, the maximum temperature allowed temperature in this section is fixed at 1673 K.On top of the SiC-plate, a second SiC-plate is positioned, which separates the heating sectionand the sample section. This additional SiC-plate is present to avoid damage of the SiC heatingelements if the SiC-plate which is in direct contact with the solid particle mixture breaks. Theheating elements in the heating section were controlled by a control unit based on the value ofthe thermocouple, which is positioned between the two SiC-plates.

On top of the heating section, the sample section is located. The sample section is insulatedby ceramic board, with an aperture of 250 mm x 250 mm. The height of the sample section is150 mm. The aperture is surrounded by highly insulating ceramic board of 170 mm thickness.The width of the aperture and the thickness and properties of the ceramic insulation board arechosen in such a way that the heat penetration in the mixture of solid particles can be regardedas one dimensional (vertical) heat transport. Later in this section, heat transport in only vertical


direction in the solid particle mixture is proven.The sides of the aperture in the sample section is covered at the inside by a thin layer of in-

sulation bricks through which 0.35 mm Pt/Pt-10% Rh (type S) thermocouple wires are guided.The joints of the thermocouples wires are positioned at defined vertical distances from theSiC-plate at the center of the aperture. The positioning of the thermocouple wires is fixed bytightening the wires in ceramic blocks located on top of the sample section. Five thermocouplewire joints are positioned at the center of the sample section at about 10 mm, 20 mm, 30 mm,40 mm and 50 mm from the SiC-plate. A sixth thermocouple joint is positioned at a distanceof 30 mm from the SiC-plate and at a horizontal distance of about 10 mm from the center ofthe sample section. This sixth thermocouple is used for comparing the heat flow in horizontaldirection relative to the heat flow in vertical direction in the solid particle mixture.

The thermocouple wires, which are connected with compensation wires, are connected toa data acquisition system. This data acquisition system monitors and records all thermocouplereadings from the experimental set-up. On top of the sample section a furnace lid, which iscomposed of insulation material, is located.

Experimental procedureFirst, the bulk density of the mixture of solid particles is determined by measuring the weightof a defined volume of the solid particle mixture. After positioning the thermocouple wires andmeasuring their vertical position inside the sample section, the sample section is slowly filledwith the raw material batch at room temperature via the walls of the sample section to avoidvertical displacement of the thermocouple wires. After filling the sample section up to a heightof approximately 8 cm, the furnace lid is put on top of the sample section. The set point of thethermocouple, which is located between the two SiC-plates, is set to the desired final tempera-ture.

The thermocouples in the sample section were prepared from the same wire. The uncer-tainty in the thermocouple values is estimated to be ± 5 K over the whole temperature range.As mentioned in the previous section, the uncertainty of the thermocouple readings is not themajor source for the uncertainty in the estimated values for the heat conductivity of the mix-ture of solid particles. The measured uncertainty level of the positioning of the thermocouplejoints equals 1 mm. It is assumed that the uncertainty of the positioning of the thermocouplejoints is independent on temperature. However, especially at high temperatures, at which thethermocouple wire is slightly expanded, and in case large volumes of batch gases are releasedduring heating, the positioning of the thermocouples may fluctuate slightly due to the ascensionof batch gases.

The heat conductivity of the mixture of solid particles is estimated by applying the numerical-experimental approach described in the previous section. The temperatures indicated by thethermocouples, which are positioned at about 10 mm and 50 mm from the SiC-plate, serve astransient boundary condition for the numerical simulation of the heat penetration experiment.The value for the temperature dependent heat capacity of the mixture of solid particles is de-rived from thermodynamic tables [20].

Evaluation of the radial temperature gradientTo evaluate whether it is allowed to consider the heat flow in the experimental set-up as one-dimensional, the horizontal and vertical temperature gradient at about 30 mm from the SiC-plateis calculated from the temperatures in a silica sand batch with a bulk density of 1487 kg m−3.

4.4. Experimental set-up for measuring the heat penetration in solid particle mixtures 159

The vertical and horizontal temperature gradients, expressed in K per mm, are approximated by(

∂T∂x

)

vert,x=30mm=

Tx=40mm,z=0mm −Tx=20mm,z=0mm

20, (4.35)

and(

∂T∂x

)

hor,x=30mm=

Tx=30mm,z=0mm −Tx=30mm,z=10mm

10. (4.36)

Figure 4.22 shows both the horizontal and the vertical temperature gradient as function of thetemperature at the center of the sample section, at 30 mm from the SiC-plate.

400 600 800 1000 1200 -20

-18

-16

-14

-12

-10

-8

-6

-4

-2

0

2

PSfrag replacements Temperature [K]

Tem

pera

ture

grad

ient

[Km

m−

1 ]

Figure 4.22: Horizontal (dotted line) and vertical (solid line) temperature gradient as function of themeasured temperature at the center of a silica sand batch at 30 mm from the SiC-plate.

Figure 4.22 clearly shows that the horizontal temperature gradient is negligible compared to thevertical temperature gradient, which indicates that the heat transport process in the solid particlemixture can merely be regarded as a one-dimensional process.

Accuracy and reproducibilityThe accuracy and reproducibility of the estimation of the thermal heat conductivity of a silicasand batch from the measured heat penetration in the silica sand batch is performed by esti-mating the heat conductivity from two subsequent heat penetration experiments with the same


silica sand type. Figure 4.23 shows both the calculated heat conductivity of the silica sandbatches and the 95 % confidence limits of the calculated heat conductivity values in case theuncertainty in the positioning of the thermocouples equals ±1 mm. The uncertainty in the ef-

300 400 500 600 700 800 900 1000 1100 1200 1300 1400 0.10

0.15

0.20

0.25

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0.35

0.40

0.45

0.50

0.55

0.60

0.65

0.70

PSfrag replacements

Temperature [K]

Hea

tcon

duct

ivity

[Wm

−1

K−

1 ]

1

2

Figure 4.23: Heat conductivity of a silica sand batch estimated with the numerical-experimental tech-nique from two heat penetration experiments with the same silica sand type.

fective heat conductivity of the silica sand batch equals approximately 18 % at 1373 K. It isseen that the heat conductivity estimated from the two subsequent heat penetration experimentslies well within their 95 % confidence limits. It was also shown that two sequential performedheat penetration measurements provided almost similar estimates for the heat conductivity ofthe silica sand batch.

4.5 Experimental determination of heat conductivity in a par-ticle bed

In this section, the heat conductivity of solid particle mixtures, in the solid-state regime is stud-ied. The identification of the most suitable structural model describing the heat conductivity ofsolid particle mixtures composed of industrial grade raw material components of glass batches,is discussed and shown for a silica sand batch in section 4.5.1. In section 4.5.2, it is investi-gated whether the identified structural model is also able to describe the heat conductivity ofmulti-component mixtures.

4.5. Experimental determination of heat conductivity in a particle bed 161

4.5.1 Heat conductivity of a silica sand batchThe most suitable network model describing heat conductivity of solid particle mixtures is iden-tified by comparison of the predicted heat conductivity of a mixture of silica sand and air for thedifferent structural models described in section 4.2.3 with the estimated heat conductivity fromheat penetration experiments in well defined silica sand batches. The bulk density of the silicasand batch, in which the heat penetration was measured, equals 1487 kg m−3. The porosity ofthe silica sand batch is given by

εp = 1− ρb

ρSiO2

, (4.37)

in which ρb is the bulk density of the silica sand batch, and the ρSiO2 is the true density of silica.The true density of silica crystals equals 2650 kg m−3, which results in a value for the porosityof the silica sand batch equal to 0.44. The temperature dependent heat conductivity of air isgiven by Wakao and Kaguei [13] and is given by

λair = 1.20 ·10−2 +5.50 ·10−5 T, (4.38)

in which T is the temperature in K. The intrinsic temperature dependent heat conductivity ofcrystalline silica is fitted from figure 4.3 and equals

λsilica = 2.81+8.60 ·102 T−1 +2.11 ·105 T−2, (4.39)

in which T is the temperature in K. Figure 4.24 shows the calculated heat conductivity of thesilica sand batch for the different network models. Next to these calculated values for theheat conductivity, also the heat conductivity estimated from the heat penetration experiment isshown.

From figure 4.24 it can be seen that neither of the four heat conductivity models predicts atemperature dependent heat conductivity for the silica sand batch equal to the heat conductivityestimated from the heat penetration experiment. Although the SN- and the DPN-model seemto provide a reasonable prediction of the temperature dependent heat conductivity of the silicasand batch, the difference between the measured and predicted heat conductivity of the silicasand batch11 equals 43 % and 66 % at 1273 K for the DPN- and the SNM-model, respectively.It appears that the heat flow through the silica sand batch is described by an intermediate formof the four heat conductivity models presented in section 4.2.3. It is also observed that, similarto the DPNM- and the SNM-model, the heat conductivity of the silica sand batch increaseswith increasing temperature. Because the intrinsic heat conductivity of silica decreases withtemperature and the heat conductivity of air increases with temperature, it appears that theconduction of heat in the silica sand bed is predominated by the heat conductivity of air.

Figure 4.25 shows the schematic representation of the intermediate form of parallel andserial arranged thermal resistances, which will be used to describe the heat flow through thesilica sand batch.

The heat conductivity of the silica sand batch is now given by

λ =1

ψλPNM

+ 1−ψλSNM

, (4.40)

11The difference of the measured and predicted heat conductivity is defined as λest−λpredλest

, in which λest is the heatconductivity, which is estimated from the heat penetration experiments and λpred is the heat conductivity, which ispredicted by either of the two models.


200 400 600 800 1000 1200 1400 0

1

2

3

4

5

PSfrag replacements

Temperature [K]

Hea

tcon

duct

ivity

[Wm

−1

K−

1 ]

PNM

SNM

CPNM

DPNM

Figure 4.24: The calculated heat conductivity of a particle bed composed of silica sand grains and airaccording to the PN-, CPN-, DPN-, and the SN-model is indicated by solid lines. The dottedline indicates the heat conductivity of the particle bed estimated from a heat penetrationexperiment.

in which ψ is the parameter, which distinguishes between the contribution of the parallel and theserial arrangement of thermal resistances to the overall heat flow through the silica sand batch.For the silica sand batch out of the heat penetration experiments, the experimental derived valuefor ψ equals 0.29. This value is calculated from equations 4.10, 4.11 and 4.40 using the intrinsicvalue of the heat conductivity of silica and air given by equations 4.38 and 4.39, respectively.

This indicates that the major part of the silica sand batch can be regarded as a serial arrange-ment of solid particles and enclosures of gas phase. Now, the heat conductivity of the silicasand batch is not only determined by the relative amount of silica sand and air per unit volume,which is characterized by the porosity, but also by the mutual contact between the silica sandparticles. The parameter ψ can be regarded as a measure for the heat transfer area between thesilica sand particles in the bed. The value for ψ depends on both the porosity and the shape ofthe silica particles. A low porous bed will have a higher contact area between the silica parti-cles compared to a high porous bed. This results in a higher value for ψ. Prediction of the heatconductivity of a particle bed requires knowledge of the porosity εp, which can easily be mea-sured, and ψ, which is not known a-priori and can not easily be determined. In the following,the accuracy of the prediction of the heat conductivity of a silica sand batch as function of the


PSfrag replacements

T1 T2

Q

RTg

RTg

RTs

RTs

serialconnection

(1-ψ)

parallelconnection

ψ

Figure 4.25: Structural arrangement of thermal resistances.

porosity assuming a constant value for ψ is evaluated based on heat conductivity data providedby Eligehausen [11]. Eligehausen measured the heat conductivity of a silica sand batch for twodifferent bulk densities, i.e. 1505 kg m−3 and 1605 kg m−3. The value for the shape factorψ for the silica sand batch with a bulk density of 1505 kg m−3 was calculated using equations4.10, 4.11 and 4.40 and equals 0.24. Figure 4.26 shows the predicted, using equation 4.40,versus the measured heat conductivity of the silica sand batch with a bulk density of 1605 kgm−3 assuming that the shape factor equals 0.24. The largest difference between the predictedand measured heat conductivity is observed in the low temperature range and equals 17 % at273 K. In the temperature range from 673 K up to 1273 K, the difference between the predictedand measured heat conductivity is less than 2 %.

The measured temperature dependent heat conductivity of a silica sand batch with a bulkdensity of 1605 kg m−3 is given by

λsand/air,ρ=1605,measured = 0.235+3.07 ·10−4 T, (4.41)

while the predicted temperature dependent heat conductivity of this silica sand batch is givenby

λsand/air,ρ=1605,predicted = 0.173+3.77 ·10−4 T. (4.42)

The effect of the difference in the measured and the predicted temperature dependent heat con-ductivity of the silica sand batch with the bulk density of 1605 kg m−3 (see figure 4.26) on theheating of this batch is evaluated by simulating the heat penetration in a silica sand batch with athickness of 10 cm, a heat capacity of 1200 J kg−1 K−1, a bulk density of 1605 kg m−3, an ini-tial temperature of 293 K, and a constant boundary temperature at both sides of the silica sandbatch layer of 1673 K. Figure 4.27 shows the temperature at the center of the silica sand batchas function of time for both measured and predicted temperature dependent heat conductivity.The maximum temperature difference over the total temperature range is 27 K after 60 minutesof heating up.

Resuming, in this section it is shown that the heat conductivity of a silica sand batch asfunction of bulk density can be predicted with a reasonable accuracy in case the shape fac-tor ψ is determined from a heat penetration experiment in a silica sand batch with a knownporosity. However, it is remarked that for dense batches, the parameter ψ is likely to increase.Also in case a solid particle bed is remained a long time at a certain high temperature, the heatconductivity may become time dependent due to sintering of the solid particles.


0.0 0.2 0.4 0.6 0.8 1.0 0.0

0.2

0.4

0.6

0.8

1.0

PSfrag replacements

Measured heat conductivity [W m−1 K−1]

Pred

icte

dhe

atco

nduc

tivity

[Wm

−1

K−

1 ]

T ↑

Figure 4.26: Predicted versus measured [11] heat conductivity of a silica sand batch with a bulk densityof 1605 kg m−3.

4.5.2 Heat conductivity for multi-component mixturesBased on the previous section, the prediction of the heat conductivity of a bed of a multi-component mixture of solid particles in a gas phase, requires knowledge of the following prop-erties

• the intrinsic heat conductivity of the pure solid components,

• the intrinsic heat conductivity of the gas phase,

• the porosity of the multi-component mixture, and

• the shape of the overall mixture of the individual mixture components.

However, data on the intrinsic heat conductivity of the pure solid components, which are usedin glass batches, are scarce. Also the a-priori estimation of the net effect of the shapes of thedifferent mixture components on the heat conductivity of the multi-component mixture is noteasy. Therefore, it is decided to characterize the heat conductivity of a solid component byan apparent heat conductivity, which contains both the intrinsic heat conductivity of the puresolid component and the contribution of the shape factor ψ. To determine the apparent heatconductivity of a solid component i, a heat penetration experiment with a particle mixture ofsolid component i is required.

The value for the apparent heat conductivity of silica sand, estimated from a heat penetrationexperiment in a silica sand batch, is determined both regarding the silica sand batch as a serial


0 20 40 60 80 100 120 140 160 180

400

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800

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1200

1400

1600

-15

-10

-5

0

5

10

15

20

25

30

PSfrag replacements

Time [min]

Tem

pera

ture

[K]

Tem

pera

ture

diff

eren

ce[K

]

Figure 4.27: Calculated temperature as function of time at the center of a silica sand batch with a thick-ness of 10 cm. The heat capacity and the density of the silica sand batch equal 1200 J kg−1

K−1 and 1605 kg m−3, respectively. The initial temperature and the constant boundary tem-perature at both sides of the silica sand batch layer equal 293 K and 1673 K, respectively.For the solid line, λ equals 0.173 + 3.77 ·10−4 T , whereas for the dotted line, λ equals0.235+3.07 ·10−4 T .

and as a parallel arrangement of thermal resistances. In case of a serial arrangement of thermalresistances, negative values for the apparent heat conductivity of silica sand were obtained.Therefore, the apparent heat conductivity of a solid phase i, estimated from a heat penetrationexperiment in a batch containing solid phase i as the only solid phase, is given by a parallelarrangement according to

λi,app =λ− εp λg

1− εp. (4.43)

Assuming that the net apparent heat conductivity of a mixture of solid components is given by

λs,app =n

∑i=1

xi λi,app, (4.44)

in which n is the number of solid species in the mixture, xi is the weight fraction of solid phasei, and λi,app is the apparent heat conductivity of solid phase i, the heat conductivity of a multi-component mixture is estimated by

λ = εp λg +(1− εp)n

∑i=1

xi λi,app. (4.45)


The porosity of the multi-component mixture is given by

εp = 1−ρbulk

[

n

∑i=1

xi ρtrue,i

]−1

, (4.46)

in which ρtrue,i is the true intrinsic density of component i.The applicability of equation 4.45 describing the heat conductivity of multi-component par-

ticle mixtures, is evaluated and checked for a three component mixture containing industrialgrade silica sand, soda ash and limestone. The values for the effective heat conductivity ofthese three components are listed in table 4.2. The values for both silica sand and limestonehave been determined with a heat penetration experiment, whereas for the data for soda ash isreferred to Eligehausen [11].

Table 4.2: effective heat conductivity of silica sand, soda ash and limestone in case the effective heatconductivity is given by λ = cλa +cλb T The regression coefficients of this equation are deter-mined with using equation 4.43.

Specie cλa cλb

[W m−1 K−1] [W m−1 K−2]

Silica sand 0.18 5.40 ·10−4

Soda ash 0.01 6.46 ·10−5

Limestone 0.36 2.84 ·10−4

Figure 4.28 shows the measured temperatures as function of time at different positions in athree component particle bed containing silica sand, soda ash and limestone. From these heatpenetration curves, the heat conductivity of the three-component mixture is estimated using thenumerical-experimental technique.

Figure 4.29 shows the predicted versus the measured effective heat conductivity of the three-component mixture. It appears that for the three-component mixture, with a bulk density of1394 kg m−3, the predicted effective heat conductivity deviates less than 2 % from the mea-sured effective heat conductivity. This indicates that accurate values for the heat conductivityof glass batches in the solid-state regime can be predicted in case both the batch compositionand the porosity of the glass batch are known. However, it is remarked that this validation ex-periment is performed only once. It is recommended to perform a similar experiment in case another type of batch is used.

4.5.3 Heat conductivity of particle beds containing culletIn contrast to opaque industrial glass batch components, such as silica sand, dolomite and lime-stone, it is expected that a glass batch containing cullet is (partly) transparent for radiative heattransfer. In the following part, the contribution of radiative heat transfer to the net effective heatconductivity of two solid partly transparent particle mixtures is discussed.


0 50 100 150 200 250 300 350 400 450

300

400

500

600

700

800

900

1000

PSfrag replacements

Time [min]

Tem

pera

ture

[K]

x=0 mmx=10 mmx=20 mmx=30 mmx=40 mm

Figure 4.28: Measured temperature as function of time at different positions in a batch layer containingsilica sand, soda ash and limestone.

0.15 0.20 0.25 0.30 0.35 0.40 0.15

0.20

0.25

0.30

0.35

0.40

PSfrag replacementsMeasured λapp [W m−1 K−1]

Pred

icte

dλ a

pp[W

m−

1K−

1 ]

Figure 4.29: Predicted versus measured effective heat conductivity for a glass batch composed of silicasand, soda ash and limestone.


One mixture is composed of float glass cullet12 (48.6 wt.%) and silica sand (51.4 wt.%)and a second mixture, contains only float glass cullet. According to section 4.2.2, the pho-ton conductivity is proportional to the absolute temperature to the power of 4. The effectiveheat conductivity of the two mixtures is estimated with the numerical-experimental methoddescribed in section 4.3.3. The temperature dependent effective heat conductivity is fitted byλ = cλa + cλb T + cλc T 3, in which cλa + cλb T represents the conduction of heat and cλc T 3

represents the radiation of heat. The percentual contribution of radiative heat transfer to the neteffective heat conductivity of the mixtures, κ, is in this study estimated by

κ =cλc T 3

cλa + cλb T + cλc T 3 100%. (4.47)

Figures 4.30 and 4.31 show the estimated temperature dependent effective heat conductivity ofthe two mixtures and the contribution κ of radiative heat transfer to the net effective heat con-ductivity. From figure 4.30 it is observed that the contribution of radiative heat transfer in themixture containing both silica sand and float glass cullet in the temperature range up to 1173 Kis less than 14 %. This indicates that the major mode of heat transfer in this mixture (of a typicalbatch layer thickness) is phonon conduction. Photon conduction (or radiative heat transfer) is ofminor importance, which is likely because the amount of non-radiative conducting silica sand issufficient to block radiative heat transport through the thick layer of the solid particle mixture.

300 400 500 600 700 800 900 1000 1100 1200 0.26

0.28

0.30

0.32

0.34

0.36

0.38

0.40

0.42

0.44

0.46

0.48

0

2

4

6

8

10

12

14

PSfrag replacements

Temperature [K]

Eff

ectiv

ehe

atco

nduc

tivity

[Wm

−1

K−

1 ]

κ[%

]

Figure 4.30: Temperature dependent effective heat conductivity and the contribution κ of radiative heattransfer to the net effective heat conductivity for a bed composed of float glass cullet (48.3wt.% ) and silica sand (51.7 wt.% ).

Figure 4.31 shows that for the 100 % float glass cullet mixture a remarkable higher effective

12The particle size of the float glass cullet ranges from 4 mm up to 8 mm


heat conductivity is estimated compared to the mixture containing both float glass cullet andsilica sand. Also the contribution of radiative heat transfer to the net effective heat conductivityof the 100 % float glass cullet mixture is higher. For this mixture, radiative heat transfer con-tributes to a large extent to the effective heat conductivity of the mixture.

300 400 500 600 700 800 900 1000 1100 1200 0.0

0.5

1.0

1.5

2.0

2.5

0

10

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30

40

50

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90

100

PSfrag replacements

Temperature [K]

Eff

ectiv

ehe

atco

nduc

tivity

[Wm

−1

K−

1 ]

κ[%

]

Figure 4.31: Temperature dependent effective heat conductivity and the contribution κ of radiative heattransfer to the net effective heat conductivity for a 100 % float glass cullet bed.

The effect of the higher effective heat conductivity from mixtures with transparent culletgrains can easily be seen from figures 4.32 and 4.33, which show the temperatures as functionof time at different positions in both solid particle mixtures.

Resuming, these experiments indicate that radiative heat transfer in glass batches increaseby the use of cullet as raw material. However, at least below a weight fraction of cullet in aglass batch of 50 %, the contribution of radiative heat transfer to the total heat transfer in theglass batch is not or hardly noticeable, which agrees with the observation mentioned by Faber etal. [5]. The beneficial effect of glass cullet on the heating rate of glass batches is only apparentfor cullet fractions above 50 wt.%. As mentioned in section 4.2.4, it is expected that the photonconduction in glass batches is not only dependent on the cullet fraction in the glass batch, butbecause of photon scattering, also on the particle size and the surface roughness of the cul-let. With the determination of the contribution κ of radiative heat transfer to the net effectiveheat conductivity, the effect of these cullet properties on the effective heat conductivity can bedetermined in future studies according to the here presented method.


0 10 20 30 40 50 60 70 80 90 100

300

400

500

600

700

800

900

1000

1100

PSfrag replacements

Time [min]

Tem

pera

ture

[K] x=0 mm

x=10 mm

x=20 mmx=30 mm

x=40 mm

Figure 4.32: Measured temperature as function of time at different positions in a 100 % float glass culletbatch. The cullet particle size varies between 4 mm and 8 mm.

0 10 20 30 40 50 60 70 80 90 100 110 120 130 140

300

400

500

600

700

800

900

1000

1100

PSfrag replacements

Time [min]

Tem

pera

ture

[K]

x=0 mm

x=10 mm

x=20 mm

x=30 mm

x=40 mm

Figure 4.33: Measured temperature as function of time at different positions in a mixture containingsilica sand (51.7 wt.%) and float glass cullet (48.3 wt.%). The cullet particle size variesbetween 4 mm and 8 mm.


4.6 Concluding remarksThe purpose of this section was three-fold, viz:

1. Identification of the most accurate mathematical method to estimate the temperature de-pendent heat conductivity in powder layers from heat penetration experiments.

2. The development of an experimental set-up to measure the temperatures at different po-sitions in a solid particle mixture as function of time.

3. Modelling of the temperature dependent heat conductivity of solid particle mixtures basedon the estimated heat conductivity of single type particle mixtures.

The most accurate mathematical method to estimate the temperature dependent heat conduc-tivity of solid particle mixtures from heat penetration experiments is identified by comparisonof the accuracy of the estimated heat conductivity derived from fictive heat penetration experi-ments with different mathematical models. The most accurate mathematical method appearedto be the numerical-experimental method during which the heat conductivity is estimated fromheat penetration experiments by iteratively adapting the relation of the heat conductivity withtemperature in a numerical model simulating the heat penetration in a solid particle layer in afictive experiment.

With this mathematical method, the heat conductivity can be estimated using only one ther-mocouple value in contrast to the other discussed techniques for which at least three temperaturevalues are required. The numerical-experimental method appeared to be less sensitive for in-accuracies in the other a-priori known or estimated material properties compared to the otherdescribed mathematical methods. A sensitivity analysis for the numerical-experimental methodshowed that the accuracy of the positioning of the thermocouple(s) in the solid particle mixtureis the major source for the uncertainty in the estimated heat conductivity.

Next, structural models for the description of heat flow through a three dimensional ar-rangement of thermal resistances simulating the heat transport in a solid particle mixture areexamined. It is shown that the effective heat conductivity of mixtures of solid particles canbe predicted based on the (apparent) heat conductivity of the individual solid particle mixturecomponents and the particle bed porosity. Finally, a start was made with the determination ofthe effect of glass cullet in the batch on the effective heat conductivity of solid particle mixtures.It was shown by estimation of the contribution κ of radiative heat transfer to the net effectiveheat conductivity, that the effect of cullet on the radiative heat conductivity is only observed incase the weight fraction of cullet in the particle bed exceeds at least 50 %.



a absorption coefficient [m−1]A cross section [m2]c heat capacity [J m−3 K−1]cλ,a constant in the temperature dependent heat con-

ductivity given by λ = cλ,a + cλ,b T + cλ,c T 3[W m−1 K−1]

cλ,b constant in the temperature dependent heat con-ductivity given by λ = cλ,a + cλ,b T + cλ,c T 3

[W m−1 K−2]

cλ,c constant in the temperature dependent heat con-ductivity given by λ = cλ,a + cλ,b T + cλ,c T 3

[W m−1 K−4]

cp heat capacity [J kg−1 K−1]d diameter [m]dp pore diameter [m]H i sensitivity matrix at time tiHchem chemical energy demand [J mol−1]i indicatorl mean free path [m]L length [m]mi column containing measured temperatures at

time ti[K]

n refractive index [-]qls least square criterionq heat flux [W m−2]Q heat flow [W]R universal gas constant [J K−1 mole−1]RT heat resistance [K W−1]T temperature [K]t time [s]U uncertaintyUPC uncertainty percentage coefficient [%]v velocity [m s−1]yi column containing calculated temperatures at

time ti[K]

Greek symbols

εp porosity [-]ε emissivity [-]εi column containing the differences in measured

and calculated temperatures at time ti[K]

κ contribution of radiative heat transfer to the neteffective heat conductivity [%]

λ heat conductivity [W m−1 K−1]ω frequency [s−1]


ψ packed bed parameterρ density [kg m−3]σ Stefan-Boltzmann constant [W m−2 K−4]θ column containing the values for the parameters

to be estimated cλ,a and cλ,b


app apparentb boundaryc continuous phaseCPNM continuous phase network modeld dispersed phaseDPNM dispersed phase network modeleff effectiveg gas phaseint intrinsicmn meanPNM parallel network model0 initialr radiatives solid phaseSNM serial network modelspm solid particle mixturez vertical directionˆ estimates

174 Bibliography



[2] H.R.S. Jack and J.A.T. Jacquest. Heat transfer in glass batch materials. Symposium sur lafusion du verre, Bruxelles, pages 339–360, 1958.

[3] M. Daniels. Einschmelzverhalten von Glasgemengen. Glastechn. Ber., 46(3):40–46, 1973.

[4] P. Costa. Untersuchung des Einschmelzverhaltens von pelletiertem Gemenge zurGlasherstellung. Glastech. Ber., 50(1):10–18, 1977.

[5] A.J. Faber, R.G.C. Beerkens, and H. de Waal. Thermal behaviour of glass batch on batchheating. Glastech. Ber. Glass Sci. Technol., 7(7):177–185, 1992.


[7] J. Krzoska, N. Samkham, and R. Conradt. Bestimmung der lokalenTemperaturleitfahigkeit in aufschmelzendem Gemenge. In 73. Glastechnische Tagung,pages 214–217, Halle, Germany, 1999.

[8] C. Kroger and H. Eligehausen. Uber das Warmeleitvermogen des einschmelzendenGlasgemenges. Glastechn. Ber., 32(9):362–373, 1959.

[9] W.D. Kingery, H.K. Bowen, and D.R. Uhlmann. Introduction to ceramics. John Wiley &Sons, 2nd edition, 1976.

[10] A.J. Slifka, B.J. Filla, and J.M. Phelps. Thermal conductivity of magnesium oxide fromabsolute, steady-state measurements. J. Res. Natl. Inst. Stand. Technol., 103:357–363,1998.

[11] H. Eligehausen. Uber die Warmeleitfahigkeit von Glasern und einschmelzenden Glasge-mengen. PhD thesis, Rheinisch-Westfalischen Technischen Hochschule Aachen, 1957.

[12] D. Mann, R.E. Field, and R. Viskanta. Determination of specific heat and true thermal con-ductivity of glass from dynamic temperature data. Warme- und Stoffubertragung, 27:225–231, 1992.


[14] P. Zehner and E.U. Schlunder. Warmeleitfahigkeit von Schuttungen bei massigenTemperaturen. Chemie Ing. Techn., 42(14):933–941, 1970.

[15] J.P. Holman. Heat Transfer. McGraw-Hill Book Company, 5 edition, 1981.

Bibliography 175

[16] A. Segal. SEPRAN users manual, standard problems and programming guide. Ingenieurs-bureau SEPRA, Leidschendam, The Netherlands, 1993.

[17] H.W. Coleman and W.G. Steele. Experimentation and uncertainty analysis for engineers.John Wiley & Sons, Inc., 2nd edition, 1999.

[18] O.M.G.C. Op den Camp, C.W.J. Oomens, F.E. Veldpaus, and J.D. Janssen. An efficient al-gorithm to estimate material parameters of biphase mixtures. Int. J. Numer. Meth. Engng.,45:1315–1331, 1999.

[19] M.A.N. Hendriks. Identification of the mechanical behaviour of solid materials. PhDthesis, Eindhoven University of Technology, Eindhoven, The Netherlands, 1991.


176 Bibliography

Chapter 5

Complete simulation model for the heatingof glass forming batches

5.1 Energy conservation equation and temperature depen-dent glass batch properties

The simulation of the heating of glass batches requires the solution of the energy conservationequation of the glass batch together with appropriate boundary conditions describing the heattransport to the top and boundary of the glass batch. According to section 1.4, the Lagrangiandescription of the energy equation of the two-phase glass batch is given by

∂∂t

(

ρ cp T)

mn = −∇ ·(



∂t,

(5.1)in which

(

ρ cp T)

mn represents the mean value of the enthalpy of the glass batch, ρg and cp,gare the density and heat capacity of the gas phase, Tmn is the mean temperature of the glassbatch, t is the time, εp is the porosity of the glass batch, vg,z is the vertical velocity of the gasphase relative to the condensed phase1, λeff is the effective heat conductivity of the glass batch,−→q r,eff is the effective radiative heat flux through the glass batch, and Hchem is the temperaturedependent energy per unit volume of the glass batch required for batch reactions. For a completedescription of the heating of glass batches, values for the parameters ρm, cp,m, εp, ρg, cp,g, vg,z,λeff, −→q r,eff, and Hchem are required. The temperature dependent average heat capacity of areacting glass batch is given by

cp,mn =nb

∑i=1

wi cp,i, (5.2)

in which nb is the number of (intermediate) glass batch components (including the gas species),wi is the time and temperature dependent weight fraction of component i and cp,i is the temper-ature dependent heat capacity of component i. The heat capacity of the individual glass batchcomponents can be derived from thermodynamic tables [2].

In chapter 2 of this thesis, the energy required for chemical reactions was studied. Accord-ing to Madivate [3], the chemical reaction energy is mainly dependent on the energy required


177

178 Chapter 5. Complete simulation model for the heating of glass forming batches

for calcination reactions. It was shown that for a float glass batch, the CO2-release from thecomplete batch can be described by three (almost) independent calcination reactions, i.e. thethermal calcination of dolomite and limestone and the reactive calcination of soda ash. For theestimation of the time and temperature dependent chemical energy demand Hchem of the floatglass batch, this allows the determination of the kinetics of the individual calcination reactions.Combination of the measured calcination kinetics of the individual decomposition reactionswith the calcination enthalpies, which were derived from thermodynamic tables [2], resulted inan expression for the chemical energy required for complete calcination of the float glass batchas function of time, temperature and partial CO2-pressure:

∂Hchem

∂t=

nc

∑i=1

(

Hr,i w0i ρc

Mi

∂ξi

∂t

)

, (5.3)

in which nc is the total number of carbonates in the glass batch, Hr,i is the enthalpy requiredfor the calcination of carbonate i expressed in J mole−1, w0

i is the initial weight fraction ofcarbonate i in the condensed phase, ρc is the density of the condensed phase, Mi is the molarmass of carbonate i, ξi is the time and temperature dependent degree of conversion2 of carbonatei, and t is the time. The conversion rate of carbonate i is given by

∂ξi

∂t= Ai e

−Ea,iR T (1−ξi) , (5.4)

in which A is a pre-exponential factor and Ea is an apparent reaction activation energy. Thereaction energy and the reaction kinetic parameters for the three calcination reactions occurringduring heating of the float glass batch are listed in table 5.1.

Table 5.1: Reaction enthalpy and reaction kinetic parameters for the calcination reactions occurring dur-ing heating of the float glass batch.

Reaction Hr,i Ai Ea,i

[kJ mol−1] [s−1] [kJ mol−1]

Thermal decomposition of dolomite 129 2.58 ·1022 513Thermal decomposition of limestone 172 1.80 ·107 191Reactive decomposition of soda ash 118 4.72 ·1021 500

Below 1200 K, the effective heat conductivity of solid particle mixtures such as glass batchesis in the order of 0.1-1.0 W m−1 K−1 [4]. Above 1200 K, the heat transfer process in glassbatches is relatively fast compared to temperatures below 1200 K, because the effective heatconductivity of melting glass batches increases steeply above 1200 K [4–6]. Because of the loweffective heat conductivity below 1200 K, the residence time of a glass batch in this temperaturerange is relatively long. To predict the heating process of a glass batch accurately, detailed dataon the low temperature effective heat conductivity of glass batches are required. In chapter 4,the temperature dependent effective heat conductivity of solid particle mixtures containing glassbatch components such as silica sand, soda ash and limestone is determined up to about 1200

2The degree of conversion of a carbonate is defined as the ratio of the weight loss with respect to the totalweight loss when the carbonate has completely been dissociated.

5.1. Energy conservation equation and temperature dependent glass batch properties 179

K, quantitatively. It was shown that the temperature dependent heat conductivity of the solidparticle mixture can be estimated from

λeff = εp λg +(1− εp)ns

∑i=1

wi λi,app. (5.5)

in which λg is the true heat conductivity of the gas phase in the solid particle mixture, λi,app isthe apparent heat conductivity of solid component i, ns is the number of solid particles in thesolid particle mixture and εp is the porosity of the solid particle mixture which is given by

εp = 1−ρbulk

[

ns

∑i=1

wi ρtrue,i

]−1

, (5.6)

in which ρtrue,i is the true intrinsic density of component i. The true temperature dependentheat conductivity values of different gas phases are given by Wakao and Kaguei [7]. The in thisstudy determined apparent heat conductivity of silica sand, soda ash and limestone is given byλapp = cλ,a + cλ,b T , with cλ,a and cλ,b listed in table 5.2.

Table 5.2: Apparent heat conductivity of silica sand, soda ash and limestone in case the apparent heatconductivity is given by λapp = cλ,a + cλ,b T .

Specie cλ,a cλ,b

[W m−1 K−1] [W m−1 K−2]

Silica sand 0.18 5.40 ·10−4

Soda ash 0.01 6.46 ·10−5

Limestone 0.36 2.84 ·10−4

In this study, the effect of float glass cullet on the effective radiative heat flux in solid particlemixtures was studied. The effective radiative heat flux was estimated by

−→q r,eff = λr,eff ·∇Tm, (5.7)

in which λr,eff is the effective radiative heat conductivity, which is described by λr,eff = cλ,c T 3.The contribution of radiative heat transfer to the net total effective heat conductivity in solidparticle mixtures containing cullet appeared to be dependent on the cullet fraction in the solidparticle bed. Below a cullet fraction of 50 wt.%, the effective radiative heat conductivity wasonly about maximum 13 % at 1200 K. A more extensive study to the effect of cullet fraction andcullet size on the heat penetration in solid particle mixtures is required to model the contributionof the effective radiative heat transfer coefficient to the net effective heat conductivity in solidparticle mixtures as function of these cullet parameters.

In the solid particle mixture, the position dependent vertical gas velocity can be calculatedfrom a local gas balance describing the incoming gas flux from below and the generation of gasdue to the calcination reactions. The position dependent vertical gas velocity in the solid particlemixture containing dolomite, silica sand and soda ash due to the calcination of the carbonatescan be calculated from

vg,z|z=z =

(

R Tg

p εp As

)

z=z

[

φm,g|z=z−dz

Mg+

nc

∑i=1

(

As dz w0i ρc

Mi

∂ξi

∂t

)

]

, (5.8)


in which vg,z|z=z is the vertical gas velocity at position z = z, R is the universal gas constant, pis the pressure in Pa, As is the cross section of the solid particle mixture volume perpendicularon the gas flow, φm,g|z=z−dz is the mass flow of the gas phase in kg s −1 at position z = z−dz,Mg is the average molar weight of the gas phase and nc is the total number of carbonates in theglass batch.

In this study, no quantitative description of the complete melting of glass batches is given.To simulate this process, additional information is required concerning the change in meandensity cp,m and porosity εp of the glass batch. The change in the porosity directly affects theeffective phonon conductivity as is shown by equation 5.5.

5.2 Heat transfer towards a batch blanket

The heat transfer towards the boundaries of a batch blanket in an industrial furnace is depen-dent on both (free and forced) convective and radiative heat transport. In glass tank simulationmodels, no heat transfer coefficient at the bottom of the batch blanket is required, becausethe heat transfer between the glass melt underneath the batch blanket and the batch blanketis calculated by coupling the differential equations describing the energy conservation of bothsections.

In the following section, a description is given of the heat transfer process to the top layerof the batch blanket and the major mode of heat transfer is identified.

5.2.1 Forced convective heat transfer

Approximating the batch blanket as a flat plate, the forced convective heat transfer from thehot combustion gases towards the relative cold batch blanket can be estimated by standardempirically found heat transfer relations for gas flow over a flat plate. These empirical heattransfer relations are based on heat transfer over a thermal boundary layer, which is build upwhen a free streaming gas flow meets the tip of the flat plate as is indicated in figure 5.1. Inthe case that a gas flow with uniform velocity reaches the tip of the flat plate, the gas flow inthe vicinity of the surface of the flat plate is retarded due to frictional forces. In case of no-slipconditions at the surface of the flat plate, the gas velocity equals zero at the flat plate surface.As a consequence of viscous forces, the gas velocity parallel to the flat plate decreases in thedirection of the plate and a gaseous boundary layer with a velocity gradient is build up. Becauseof differences in gas flow parallel to the flat plate, also stronger temperature gradients are buildup resulting in a thermal boundary layer. The thickness of the thermal boundary layer increaseswith increasing distance from the leading edge3 of the flat plate for laminar flow. Near theleading edge of the flat plate the flow in the boundary layer is laminar. At a certain distancefrom the leading edge, the flow in the boundary layer may become turbulent. The flow type inthe boundary layer, which is either laminar or turbulent, is dependent on the gas flow velocity,the surface structure of the plate, the properties of the gas phase and the distance from theleading edge. In general, for a smooth plate, the transition from the laminar flow regime intothe turbulent flow regime is characterized by a Reynolds number, calculated with equation 5.9,equal to 5.0 ·105 [8].

3The tip of the flat plate, which is the first position where the gas flow hits the flat plate

5.2. Heat transfer towards a batch blanket 181

PSfrag replacements

Leading edgeGas flow

δ

Flat plate

Laminar regime Turbulentregime

Transitionregime

v=0

v=v∞v∞, T∞

T=Ts

T=T∞

x

Figure 5.1: Schematic view of boundary layer in which v∞ is the free stream gas velocity, T∞ is the freestream temperature, Ts is the surface temperature of the flat plate, x is the flow direction ofthe gas stream and δ is the boundary layer thickness for velocity which is dependent on x.

The Reynolds number at distance x from the leading edge is given by

Rex =ρ v∞ x

µ, (5.9)

in which ρ is the density of the gas phase, v∞ is the free stream gas phase velocity and µ is thedynamic viscosity of the gas phase. Although the critical Reynolds number for the transitionfrom laminar to turbulent flow regime is about 5.0 ·105, the practical transition value is stronglydependent on the roughness of the surface layer and on the turbulence level of the free streamgas flow. With a high degree of surface roughness and with large turbulences present in the freestream, the Reynolds number for transition may shift down to values of 2.0 ·105 [8].

The heat transfer coefficient by forced convection from the hot combustion gases to thebatch blanket surface at position x, hfo,x, can be described by the local Nusselt number given by

Nux =hfo,x x

λg, (5.10)

in which λg is the heat conductivity of the gas phase in the boundary layer. The heat flux perunit batch area at position x is now described by

qfo,x = hfo,x (T∞ −Ts) , (5.11)

in which Ts is the temperature of the surface of the batch blanket and T∞ is the free streamtemperature of the gas. Characteristic values for the over length L averaged Nusselt numberNuL describing the forced convective heat transfer towards the batch blanket are given by

NuL,lam = 0.664 Pr1/3 Re1/2 (5.12)

for the laminar flow regime [9], and

NuL,turb = 0.664 Pr1/3 Re1/2 +0.0359[

Re0.8L −Re0.8

x,crit]

Pr1/3 (5.13)

for the turbulent flow regime [9], respectively. The Prandtl number Pr is described by

Pr =µ cp

λ, (5.14)


in which cp is the heat capacity of the gas phase in the thermal boundary layer.To determine the forced convective heat transfer coefficient from the hot combustion gases

to the batch blanket in an industrial furnace, typical values for the free stream gas velocity andtemperature and the gas phase composition have to be known. However, the assumptions thathave been made for deriving the Nusselt numbers are a uniform gas flow (v∞) and a constanttemperature for the batch blanket surface (Ts), which are not full-filled in practice. To estimatethe forced convective heat transfer coefficient, characteristic values for the free gas stream ve-locity and temperature have to be estimated.

For a known gas phase composition, the gas phase properties can be calculated at the knowngas phase temperature. Because there is a temperature difference over the thermal boundarylayer, the gas phase properties will change with the distance to the batch blanket. To take thisvariation in gas phase properties into account, the gas phase properties are evaluated at theso-called film temperature Tf , which is the mean value of the surface temperature Ts and thefree-stream temperature T∞. Table 5.3 lists a typical gas phase composition for an oxy-firedglass melting furnace.

Table 5.3: Gas phase composition

Gas phase component Molar fraction

CO2 0.33H2O 0.58N2 0.07O2 0.02

The density of the gas phase is calculated assuming ideal gas behavior in which p is the pres-sure, M is the mean molar mass of the gas phase which equals 27.6 g mol−1 for this gas phaseand T is the temperature expressed in K.

ρ =p MR T

. (5.15)

The dynamic viscosity of a gas phase mixture, µmix, is calculated by the semi-empirical formulaof Wilke given in [10] and described by

µmix =ng

∑i=1

wi µi

∑nj=1 w j Φi j

, (5.16)

in which ng is the number of gas phase components, µi is the viscosity of the pure gas componenti, and wi and w j are the molar fractions of the components i and j in the gas phase. The constantΦi j is given by

Φi j =1√8

(

1+Mi

M j

)−1/2[

1+

(

µi

µ j

)1/2(M j

Mi

)1/4]2

, (5.17)

in which Mi and M j are the molar weights of components i and j in the gas phase mixture [10].The viscosity of a pure gas phase component is estimated by

µi = 2.6693 ·10−5√

Mi Tσ2

i Ωi, (5.18)


in which σi is the Lennard-Jones collision diameter of gas species i, T is the temperature and Ωiis the collision integral. The collision integral Ωi is dependent on the dimensionless temperatureκi T/εi. The ratio κi/εi is specific for each gas phase component and is tabulated for differentgas phase species in [10].

Similar to the description of the viscosity of the gas phase mixture, the thermal heat con-ductivity of a gas phase mixture, λmix, is given by

λmix =n

∑i=1

wi λi

∑nj=1 w j Φi j

. (5.19)

The thermal heat conductivity of a pure gas phase component is given by the Chapman-Enskogformula given by

λi = 1.9891 ·10−4

√

T/Mi

σ2i Ωi

, (5.20)

in which λi is the thermal heat conductivity of the pure gas phase component i. The generalequation describing the temperature dependency of the heat capacity of a gas phase componentis given by [11]

cp,i = cp,A,i + cp,B,i T + cp,C,i T 2 + cp,D,i T 3 + cp,E,i T 4, (5.21)

in which cp,A, cp,B, cp,C, cp,D and cp,E are coefficients which are specific for each gas phasecomponent and can be derived from thermodynamic tables [11] and

cp,mean =n

∑i=1

wi cp,i. (5.22)

Figure 5.2 shows the calculated Reynolds number of the gas flow over a flat plate as a functionof the average film temperature in case of a free-stream gas velocity of 10 m s−1 and a typicalplate length of 5 m. These values are representative for situations of batch blankets in glassfurnaces.

It is observed that for the current case, the Reynolds number is lower than the critical Reynoldsnumber at which the transition to the turbulent flow regime takes place for smooth plates. Inthis flow regime, the heat transfer coefficient is in the order of 3.5 W m−2 K−2. However, asmentioned above, the critical Reynolds number for the transition from the laminar flow regimeinto the turbulent flow regime for a rough batch surface and gas flows with turbulence eddiesmay be lower than the 5.0 ·105. Assuming that the critical Reynolds number equals 1.0 ·105,the forced convective heat transfer coefficient, which is calculated with equation 5.13, variesbetween 7 - 10 W m−2 K−2.

5.2.2 Free convective heat transfer coefficientAccording to Holman [9], the average free convective heat transfer coefficient for the case of ahorizontal flat plate with a gas phase above the flat surface can be derived from

Nu = C (Gr Pr)m , (5.23)


1100 1200 1300 1400 1500 1600 1700

2.0x10 5

2.5x10 5

3.0x10 5

3.5x10 5

4.0x10 5

4.5x10 5

5.0x10 5

3.50

3.55

3.60

3.65

3.70

3.75

PSfrag replacements

Film temperature Tf [K]

Rey

nold

snu

mbe

r[-]

Ave

rage

h c,f

o[W

m−

2K−

1 ]

Figure 5.2: Reynolds number (solid line) and average forced convective heat transfer coefficient for lam-inar flow (dotted line) as function of the film temperature.

in which C and m are constants which are both dependent on the geometry of the system andGr is the Grashof number. The Grashof number is defined by

Grx =g β (Ts −T∞) x3 ρ2

µ2 (5.24)

in which g is the gravitational acceleration and β is the volumetric expansion coefficient of thegas phase mixture. For the composition given in table 5.3, the Rayleigh number (GrPr) rangesfrom 2.4 ·109 down to 1.1 ·109 in the temperature range from 1073 K up to 1673 K. For theseRayleigh numbers, the values for C and m are 0.15 and 0.33, respectively [9]. This results in aheat transfer coefficient of approximately 1.5 W m−2 K−2.

5.2.3 Radiative heat transfer coefficientAccording to Holman [9], the radiative heat transfer coefficient hr between two parallel plateswith surface areas of A1 and A2 is given by

hr =σ(

T 21 +T 2

2)

(T1 +T2)

1ε1−1+ 1

F + A1A2

(

1ε2−1) , (5.25)

in which σ is the Stefan-Boltzmann constant, T1 and T2 are the temperatures of the plates, ε1and ε2 are the emissivities of the plates and F is the fraction of energy which is emitted fromplate 1 and reaching plate 2. To estimate the order of magnitude of the radiative heat transfercoefficient from the hot superstructure of a glass furnace (or a flame) towards the top layer of


400 600 800 1000 1200 1400 1600 0

50

100

150

200

250

300

350

400

450

500

PSfrag replacements

Surface temperature batch blanket [K]

Rad

iativ

ehe

attr

ansf

erco

effic

ient

[Wm

−2 K

−1 ]

εsup = 0.6

εsup = 0.5

εsup = 0.4

εsup = 0.3

εsup = 0.2

Figure 5.3: Radiative heat transfer coefficient as a function of the surface temperature of the batch blan-ket for different emissivity levels of the superstructure in case that the superstructure temper-ature equals 1773 K and the emissivity of the batch blanket equals 0.7.

the batch blanket, it can be assumed that the surface area of the superstructure perpendicular tothe surface of the superstructure and the batch blanket are similar and that all energy leaving thesuperstructure reaches the surface of the batch blanket. The radiative heat transfer coefficientfor this situation is now given by

hr =σ(

T 2sup +T 2

b

)

(Tsup +Tb)1

εsup+ 1

εb−1

, (5.26)

in which Tsu and Tb are the temperatures of the superstructure (or flame) and the surface of thebatch blanket. The emissivities of the surfaces are indicated by εsup and εb, respectively.

Figure 5.3 shows the radiative heat transfer coefficient hr as function of the surface tem-perature of the batch blanket for different emissivity levels of the superstructure in case of atemperature of the crown of 1773 K. According to Ungan et al., a typical value for εb is 0.7 [1].

From this figure it is observed that, for this idealized case, the radiative heat transfer coeffi-cient is about two orders of magnitude larger than the convective heat transfer coefficient. Thisindicates that heat transfer by radiation is the main mode for heat transfer towards the batchblanket.



A pre-exponential factor [s−1]As heat transfer area per unit volume [m−1]cp heat capacity [J kg−1 K−1]cp,A heat capacity constant [J kg−1 K−1]cp,B heat capacity constant [J kg−1 K−2]cp,C heat capacity constant [J kg−1 K−3]cp,D heat capacity constant [J kg−1 K−4]cp,E heat capacity constant [J kg−1 K−5]cλ,a constant in the temperature dependent heat con-

ductivity given by λ = cλ,a + cλ,b T + cλ,c T 3[W m−1 K−1]

cλ,b constant in the temperature dependent heat con-ductivity given by λ = cλ,a + cλ,b T + cλ,c T 3

[W m−1 K−2]

cλ,c constant in the temperature dependent heat con-ductivity given by λ = cλ,a + cλ,b T + cλ,c T 3

[W m−1 K−4]

C geometry dependent constants in Nu =C (Gr Pr)m

Ea (apparent) reaction activation energy [J mol−1]F Fraction of radiative energy emitted from a sur-

face, which reaches an opposite surface[-]

Gr Grashof number [-]h heat transfer coefficient [W m−2 K−1]Hchem chemical energy demand [J m−3]Hr reaction enthalpy [J mol−1]i, j indicatorsm geometry dependent constants in Nu =

C (Gr Pr)m

M molar mass [kg mol−1]M mean molar mass [kg mol−1]nb number of (intermediately formed) glass batch

components[-]

nc number of different types of carbonates in aglass batch

[-]

ng number of different types of gases [-]ns number of different types of solid particles in a

glass batch[-]

Nu Nusselt number [-]NuL over length L averaged Nusselt number [-]p pressure [Pa]Pr Prandtl number [-]q heat flux [W m−2]Re Reynolds number [-]R universal gas constant [J K−1 mole−1]


t time [s]T temperature [K]T∞ free stream temperature [K]v velocity [m s−1]v∞ free stream velocity [m s−1]w weight fraction [-]x horizontal position [-]

Greek symbols

β volumetric expansion coefficient [m−3]ε emissivity [-]εp porosity [-]ε/κ Lennard-Jones potential parameter [K]λ heat conductivity [W m−1 K−1]µ dynamic viscosity [kg m−1 s−1]Ω collision integral [-]φm mass flow rate [kg s−1]Φi j constant in equation 5.17ρ density [kg m−3]ρbulk bulk density [kg m−3]ρtrue true density [kg m−3]σ Lennard-Jones collision diameter [m]ξ degree of conversion [-]

Subscripts

app apparentb batchc condensed phasecrit criticaleff effectivefo forcedg gas phaselam laminarmn meanmix mixturer radiatives surfacesu superstructureturb turbulentz vertical direction

188 Bibliography





[4] C. Kroger and H. Eligehausen. Uber das Warmeleitvermogen des einschmelzendenGlasgemenges. Glastechn. Ber., 32(9):362–373, 1959.

[5] A.J. Faber, R.G.C. Beerkens, and H. de Waal. Thermal behaviour of glass batch on batchheating. Glastech. Ber. Glass Sci. Technol., 7(7):177–185, 1992.



[8] W.S. Janna. Engineering heat transfer. PWS Publishers, Hong Kong, 1988.

[9] J.P. Holman. Heat Transfer. McGraw-Hill Book Company, 5th edition, 1981.

[10] R.B. Bird, W.E. Stewart, and E.N. Lightfoot. Transport phenomena. John Wiley & Sons,Inc., New York, 1960.

[11] Gasunie. Physical properties of natural gases. N.V. Nederlandse Gasunie, 1980.

Appendix A

Calcination of a TV-panel glass batch

A.1 Description of the calcination mechanism of a TV-panelglass forming batch

As discussed in section 2.1, the chemical energy demand of glass batches is to a large extentdependent on the energy required for decomposition reactions, such as calcination reactions.For the estimation of the time and temperature dependent chemical energy demand of a glassbatch, the kinetics of the different decomposition reactions are required. For the float glassbatch, which was discussed in section 2.4, the calcination of dolomite, limestone and soda ashoccurred (almost) independently from each other. This allows the determination of the kineticsof the individual calcination reactions to describe the calcination behavior of the complete floatglass batch. Combination of the measured kinetics of the individual calcination reactions withthe calcination enthalpies, which were derived from thermodynamic tables [1], resulted in anexpression for the chemical energy required for complete calcination of the float glass batch asfunction of time, temperature and partial CO2-pressure.

Kramer [2] and Kawachi et al. [3] measured the release of batch and fining gases as functionof temperature for a TV-panel glass batch. It was observed that the release of CO2 occurs ina broad temperature range from 700 K up to 1500 K with a maximum CO2 release between1100 K and 1200 K. To determine the chemical energy required for complete calcination of aTV-panel glass batch as function of time, temperature and partial CO2-pressure, similar to thefloat glass batch, the kinetics of the different calcination reactions occurring during heating ofthe TV-panel glass batch need to be determined. In this study, the calcination behavior of a TV-panel glass batch, with as major constituents silica sand, soda ash, potash (K2CO3), potassiumnitrate (KNO3), nepheline (Na2O ·Al2O3 ·SiO2), strontium carbonate (SrCO3), barium carbon-ate (BaCO3), and zircon silicate (ZrO2 ·SiO2), is investigated.

In section A.2, it is shown by thermogravimetric analysis that, in contrast to the decom-position of the earth-alkaline carbonates MgCO3 and CaCO3 in the float glass batch, the de-composition of the earth-alkaline carbonates SrCO3 and BaCO3 occurs via reactive calcinationwith primary formed melt phases. In the float glass batch, the primary formed melt phases aremainly generated by the reactive calcination of soda ash with silica sand (see section 2.5.3). Inthe TV-panel glass batch, the primary melt phases are (at least partly) formed by a combinationof reactive calcination of both potash and soda ash with silica sand. Thermogravimetric analy-

189

190 Appendix A. Calcination of a TV-panel glass batch

sis of a mixture of potash and silica sand1 shows a similar calcination behavior as mixtures ofsoda ash and silica sand. The measured onset temperature for reactive calcination of potash ina mixture with silica sand in a 1 bar CO2-atmosphere is about 970 K, which equals the onsettemperature for reactive soda ash calcination with silica sand.

As mentioned above, the reactive calcination of the earth-alkaline carbonates SrCO3 andBaCO3 in the TV-panel glass batch is dependent on the presence of melt phases. After theprimary melt phases have been formed, the amount of melt phase in the TV-panel glass batchincreases due to the (reactive) dissolution of silica sand, nepheline, strontium carbonate, bariumcarbonate (BaCO3), and zircon silicate. Now, the reactive calcination of the earth-alkaline car-bonates SrCO3 and BaCO3 is not only dependent on the primary formed melt phases, but alsoon the formation of the melt phases generating by the other batch reactions. The dependencyof the reactive calcination of SrCO3 and BaCO3 on different batch reactions does not allow theeasy determination of the kinetics of the individual calcination reactions in the TV-panel glassbatch. Therefore, in contrast to the float glass batch, an expression for the chemical energyrequired for complete calcination of the TV-panel glass batch as function of time, temperatureand partial CO2-pressure cannot easily be determined.

Although detailed modelling of the kinetics of the different calcination reactions in the TV-panel glass batch is complex, the calcination of the complete TV-panel glass batch can beapproximated by identification of the temperature regimes in which the different calcinationreactions occur. Based on the results of the thermogravimetric analysis, both soda ash andpotash decompose under a 1 bar CO2-atmosphere in the temperature range from 970 K up toabout 1120 K. The onset temperature for the reactive calcination of the earth-alkaline carbon-ates SrCO3 and BaCO3 equals the onset temperature for reactive calcination of potash and sodaash. The decomposition of these carbonates may last up to about 1250 K. This maximum tem-perature for reactive calcination of the earth-alkaline carbonates is derived from phase analysison quenched TV-panel batch samples, which were ramp heated with 10 K min−1 in an ambientatmosphere in the temperature range from 920 K up to 1720 K (see figure A.1 and [4]).

From figure A.1 it is also observed that at 920 K, all TV-panel batch components, exceptKNO3 and K2CO3, are still present in the heat-treated batch. The reason for the absence ofcrystalline KNO3 is the low melting temperature of KNO3 (607 K). This indicates that in theTV-panel glass batch, already at 607 K energy is consumed, which is used for the KNO3 melt-ing. Although, based on the thermogravimetric analysis, it is expected that K2CO3 is stillpresent at 900 K, it is not identified with X-ray diffraction. The cause for not identifying crys-talline K2CO3 has not been studied in detail, but may be due to low diffraction peak intensitiesof K2CO3 and peak overlap of the diffraction lines of K2CO3 with other batch components.

Because the detailed analysis of the calcination rate of the complete TV-panel batch is time-consuming, the chemical energy demand of the TV-panel batch is not easy to determine. Beforethe kinetics of the individual calcination reactions in the TV-panel glass batch is studied in de-tail, it should be known how important a detailed description of the chemical energy demand isfor an accurate prediction of the heating process of a glass batch.

A first estimation of the chemical energy required for complete calcination of the TV-panelglass batch can be obtained by combination of assumed degree of conversion of the differentcarbonates in their specific calcination temperature range with the enthalpy of these calcinationreactions. Because melt phases are formed during the calcination of K2CO3, Na2CO3, SrCO3,

1These thermogravimetric analysis are not reported in this thesis.

A.1. Description of the calcination mechanism of a TV-panel glass forming batch 191

900 1000 1100 1200 1300 1400 1500 1600 1700

PSfrag replacements

Temperature [K]

KNO3

K2CO3

Na2CO3

SrCO3

BaCO3

Na2O ·Al2O3 ·SiO2

ZrO2 ·SiO2

SiO2(q)Na2O ·SiO2

2SrO ·SiO2

Na2O ·2SrO ·2SiO2

Na2O ·2BaO ·2SiO2

SiO2(cr)

Figure A.1: Identified crystalline species during heating of a complete TV-panel glass batch contain-ing silica sand, soda ash, potash, potassium nitrate, strontium carbonate, barium carbonate,nepheline, zircon and some minor constituents. The TV-panel batch is ramp heated with 10K min−1 in an ambient atmosphere. Phase identification is performed between 1020 K and1720 K with temperature steps of 50 K. SiO2(q) and SiO2(cr) indicate SiO2 in the quartz-and cristobalite modification, respectively.

and BaCO3, the determination of the enthalpy of the calcination reactions requires the knowl-edge of the thermodynamic properties of a glass melt containing SiO2, K2O, Na2O, SrO andBaO. To estimate the thermodynamic properties of this glass melt, a thermodynamic model asdiscussed by Shakhmatkin et al. [5] and Conradt [6] is used. Therefore, for a description of thechemical energy demand of the TV-panel glass batch, first the thermodynamics of the TV-panelglass needs to be determined.

In the next section, a description is given of the calcination behavior of the TV-panel glassbatch carbonates SrCO3 and BaCO3. The conditions for either thermal or reactive calcinationof SrCO3- and BaCO3-grains during heating is studied by

• thermogravimetric analysis of (particle mixtures containing) SrCO3-grains heated in dif-ferent atmospheres, and

• phase analysis on quenched samples of heat-treated glass batch mixtures containing SrCO3or BaCO3.


A.2 Calcination of SrCO3- and BaCO3-grainsThis section describes the results of an experimental study into the calcination behavior of bothSrCO3- and BaCO3-grains.

Figure A.2 shows the standard Gibbs free energy for the thermal decomposition of strontiumcarbonate and barium carbonate as function of temperature.

1200 1300 1400 1500 1600 -8.0x10 4

-6.0x10 4

-4.0x10 4

-2.0x10 4

0.0

2.0x10 4

4.0x10 4

6.0x10 4

8.0x10 4

1.0x10 5

1.2x10 5

PSfrag replacements

BaCO3(s) ⇐⇒ BaO(s)+CO2(g)SrCO3(s) ⇐⇒ SrO(s)+CO2(g)

Temperature [K]

Stan

dard

Gib

bsfr

eeen

ergy

ofre

actio

n[J

mol

−1 ]

10−4

10−3

10−2

10−1

10−0.3

100

100.7

101

102

Figure A.2: Standard Gibbs free energy of spontaneous thermal decomposition of SrCO3 and BaCO3.The dashed lines represent isobars for the partial CO2-pressure in bar above the carbonates.

The calcination temperature for strontium carbonate and barium carbonate in a 1 bar CO2-atmosphere is 1510 K and 1646 K, respectively. Figure A.3 shows the degree of SrCO3-calcination measured with thermogravimetric analysis for the following four cases:

• individual SrCO3-grains in a N2-atmosphere,

• individual SrCO3-grains in a 1 bar CO2-atmosphere,

• a mixture of SrCO3- and sand grains in a 1 bar CO2-atmosphere, and

• a mixture of SrCO3-, soda ash- and sand grains in a 1 bar CO2-atmosphere.

A.2. Calcination of SrCO3- and BaCO3-grains 193

From figure A.3, it is observed that the measured onset temperature for thermal calcination ofSrCO3 in a 1 bar CO2-atmosphere equals 1540 K (curve B), which is 30 K higher than thecalcination temperature of 1510 K in a 1 bar CO2-atmosphere which is determined by thermo-dynamic calculation (see figure A.2). In figure A.3, the impact of the partial CO2-pressure onthe thermal calcination of SrCO3 is clearly shown by the shift of the temperature dependentdegree of SrCO3 calcination with about 300 K to lower temperatures when changing from a 1bar CO2-atmosphere (curve B) to a N2-atmosphere (curve A). This indicates that the calcinationof individual SrCO3 grains is (at least partly) governed by thermodynamic driving forces.

900 1000 1100 1200 1300 1400 1500 1600 0.0

0.2

0.4

0.6

0.8

1.0

PSfrag replacements

Temperature [K]

Deg

ree

ofSr

CO

3ca

lcin

atio

n[-

]

A BCD

Figure A.3: Measured degree of the SrCO3 calcination in A) a N2-atmosphere, B) a 1 bar CO2-atmosphere, C) when mixed with sand in a 1 bar CO2-atmosphere and D) when mixed withsoda ash and sand in a 1 bar CO2-atmosphere.

According to thermodynamics, a mixture of SrCO3 and SiO2 may form crystalline strontiumsilicates via a solid-state calcination reaction according to

2 SrCO3(s)+SiO2(s) 2SrO.SiO2(s)+2 CO2(g), (A.1)

andSrCO3(s)+SiO2(s) SrO.SiO2(s)+CO2(g). (A.2)

The calcination temperatures of these mixtures in a 1 bar CO2-atmosphere, which are calculatedfrom thermodynamic tables [1], are 668 K and 825 K, respectively. This means that reactivesolid-state calcination of SrCO3 in a 1 bar CO2-atmosphere is expected to start at temperaturesabove 668 K. However, the measured onset temperature for SrCO3-calcination in a 1 bar CO2-atmosphere when mixed with silica sand is approximately 1470 K (curve C in figure A.3). Thisindicates that, similar to a mixture of soda ash and silica sand, the formation of a crystalline


silicate by a solid-state calcination reaction of SrCO3 and silica does hardly contribute to thecalcination of the carbonate. This conclusion is supported by the observation that phase anal-ysis on quenched samples of a heat-treated mixture of silica sand and SrCO3 did not show thepresence of crystalline strontium silicates at temperatures up to 1200 K.

During thermogravimetric analysis of a mixture of soda ash, silica sand and SrCO3 in a 1bar CO2-atmosphere, a sharp increase in weight loss for the mixture is observed at about 970K (curve D in figure A.3), which is similar to the calcination onset temperature for a mixtureof soda ash and silica in a 1 bar CO2-atmosphere (see section 2.5.3). Complete calcination2 ofthe three-component mixture in a 1 bar CO2-atmosphere is observed at about 1220 K, which isapproximately 330 K lower than the thermal calcination temperature of individual SrCO3 grainsin a 1 bar CO2-atmosphere. This indicates that, similar to reactive dissolution of silica sand (seesection 2.5.3), the decomposition of SrCO3 is enhanced in the presence of a silicate melt phase.

Figure A.4 shows the identified crystalline phases as function of temperature in quenchedsamples of heat-treated mixtures of silica sand, soda ash and strontium carbonate. These glassbatch mixtures were ramp heated with 10 K min−1 in an ambient atmosphere. Similar to amixture of soda ash and silica sand, the first formed intermediate crystalline phase is sodiummetasilicate starting at about 1073 K. The SrCO3 grains in the mixture react with this primaryformed melt phase forming a sodium strontium silicate melt phase. At about 1123 K, the pres-ence of crystalline 2SrO ·SiO2 is observed in the partly reacted three-component mixture. Thedissolution of SrO in the primary melt phase leads to an increase of the SiO2 solubility in themelt phase. Therefore, the reactive dissolution of SrCO3 is likely to affect the dissolution rateof sand grains.

The determination of the residual amount of silica during ramp heating of a glass batch mix-ture containing silica sand, soda ash and SrCO3 with 10 K min−1 in an ambient atmosphere upto 1273 K shows that, for a given particle size of the silica sand and soda ash grains, a smallerparticle size of the SrCO3 grains resulted in a lower residual amount of undissolved silica inthe heat-treated glass batch mixture. In case of a particle size of the SrCO3 grains between 250µm and 500 µm, the percentage of the initial amount of silica sand in the ternary glass batch,which is still present as crystalline silica equals 60 % at 1273 K. For a SrCO3 grain size lessthan 63 µm this percentage equals 50 %. The amount of silica sand, which is dissolved in thethree components glass melt, increases with decreasing particle size of the strontium carbonategrains, which indicates that decreasing the particle size of the SrCO3 grains results in an en-hanced reactive calcination of strontium carbonate.

Figure A.5 shows the identified crystalline phases as function of temperature in quenchedsamples of heat-treated mixtures of silica sand, soda ash and barium carbonate. These glassbatch mixtures were ramp heated with 10 K min−1 in an ambient atmosphere. Similar to mix-tures of soda ash and silica sand and a three-component mixture of silica sand, soda ash andstrontium carbonate, the first formed intermediate crystalline phase is sodium metasilicate start-ing at about 1073 K. At about 1123 K, both crystalline 2BaO ·SiO2 and Na2O ·2BaO ·2SiO2 areobserved. The crystalline Na2O ·2BaO ·2SiO2 was also observed by Krol and Janssen [7] dur-ing heating of a mixture containing silica sand, soda ash and barium carbonate.

2Calcination of both Na2CO3 and SrCO3 to Na2O and SrO.

A.2. Calcination of SrCO3- and BaCO3-grains 195

900 950 1000 1050 1100 1150 1200 1250 1300 1350 1400

PSfrag replacements

Temperature [K]

Na2CO3

SrCO3

SiO2(q)

Na2O ·SiO2

2SrO ·SiO2

Figure A.4: Identified crystalline species in quenched samples of a glass batch containing silica sand,soda ash and strontium carbonate as function of temperature. The glass batch is ramp heatedwith 10 K min−1 in an ambient atmosphere. Phase identification was performed up to 1350K with temperature steps of 50 K. SiO2(q) indicates SiO2 in the quartz modification.

900 950 1000 1050 1100 1150 1200 1250 1300 1350 1400

Temperature [˚C]


Na2CO3

BaCO3

SiO2(q)

Na2O ·SiO2

2BaO ·SiO2

Na2O ·2BaO ·2SiO2

Figure A.5: Identified crystalline species in quenched samples of a glass batch containing silica sand,soda ash and barium carbonate as function of temperature. The glass batch is ramp heatedwith 10 K min−1 in an ambient atmosphere. Phase identification was performed up to 1350K with temperature steps of 50 K.SiO2(q) indicates SiO2 in the quartz modification.

196 Bibliography

A.3 Bibliography[1] O. Knacke, O. Kubaschewski, and K. Hesselmann. Thermochemical Properties of Inor-

ganic Substances. Springer-Verlag Berlin, Heidelberg, Germany, 2nd edition, 1991.

[2] F. Kramer. Gasprofilmessungen zur Bestimmung der Gasabgabe beim Glasschmelzprozeß.Glastechn. Ber., 53(7):177–188, 1980.

[3] S. Kawachi, M. Kato, and Y. Kawase. Evaluation of reaction rate of refining agents.Glastech. Ber. Glass Sci. Technol., 72(6):182–187, 1999.

[4] R.S.A.H. Huijben. Modeling the melt behavior of tv-panel batch. Afstudeerverslag, Eind-hoven University of Technology, Eindhoven, The Netherlands, 2001.

[5] B.A. Shakhmatkin, N.M. Vedishcheva, and C.A. Wright. Thermodynamic properties: Areliable instrument for predicting glass properties. In Proc. Int. Congr. Glass, volume 1,pages 52–60, Edinburgh, Scotland, 1-6 July 2001.

[6] R. Conradt. A simplified procedure to estimate thermodynamic activities in multicompo-nent oxide melts. Molten Salt Forum, 5-6:155–162, 1998.

[7] D.M. Krol and R.K. Janssen. Raman study of the chemical reactions during the melting ofa 15Na2CO3 ·18BaCO3 ·75SiO2 batch. J. Physique, 43(12), 1982.

Appendix B

Estimation of model parameters by a leastsquares approach

B.1 Description of the least squares approach for parameterestimation

To estimate the model parameters stored in the column θ, a criterion of the measured and cal-culated temperatures in a solid particle mixture, that is defined by

qls =n

∑i=1

(

yi −mi

)T (yi −mi

)

, (B.1)

is minimized. Herein i is the time indicator, n is the total number of points of time at whichtemperatures in the solid particle mixture are measured, mi is the column containing the mea-sured temperatures at different positions in the solid particle mixture at t = ti and yi is a columncontaining the calculated temperatures at these positions in the solid particle mixture at t = ti.Given an estimate θ of the model parameters θ, temperatures in the solid particle mixture attime ti can be determined by

yi = h(θ, ti) = hi(θ), (B.2)

in which hi characterizes the numerical calculation of the temperatures at the different positionsin the solid particle mixture at t = ti. In this specific case, the column θ consists of the modelparameters cλa and cλb , which describe the temperature dependent effective heat conductivityof the solid particle mixture according to λ = cλa + cλb T .

The estimates for the model parameters, θ, are derived by minimization of the criterion qlswith respect to the unknown model parameters θ. This requires the derivative of q with respectto each component of θ to be equal to 0. This derivative is written as

∂q∂θ

= limδθ→0

1δθ

n

∑i=1

[

(

hi(θ+δθ)−mi)T (

hi(θ+δθ)−mi)

−(

hi(θ)−mi)T (

hi(θ)−mi)

]

.

(B.3)The calculated temperatures in the solid particle mixture depend in a non linear way uponthe model parameters cλa and cλb . Therefore, the operation hi(θ) is non linear in the modelparameters. When δθ is chosen to be sufficiently small, the model output hi(θ + δθ) given in

197

198 Appendix B. Estimation of model parameters by a least squares approach

equation B.3 can be approximated by

hi(θ+δθ) = hi(θ)+H i δθ, (B.4)

when second or higher order terms of δθ are assumed to be negligible.In case m is the number of positions at which for each point of time a temperature measure-

ment is available in the solid particle mixture, the matrix H i is given by

H i =

∂h1δcλa

∂h1δcλb

......

∂hmδcλa

∂hmδcλb

t=ti

. (B.5)

For the solid particle mixture, as described in this thesis, the derivatives of the components ofh with respect to each of the model parameters can easily be determined in a numerical way.For this purpose, the model parameters are slightly changed and the impact on the result of h iscomputed.

Setting the derivative ∂q∂θ in equation B.3 to zero results in

limδθ→0

1δθ

n

∑i=1

[

hTi (θ+δθ) hi(θ+δθ)−2 mT

i hi(θ+δθ)−hTi (θ) hi(θ)+2 mT

i hi(θ)]

= 0.

(B.6)Combining equation B.4 with equation B.6 results in

limδθ→0

1δθ

n

∑i=1

[

2 hTi (θ) H i δθ+

(

H i δθ)T

H i δθ−2 mTi H i δθ

]

= 0. (B.7)

Assuming that δθ is chosen to be sufficiently small and neglecting the second and higher orderterms of δθ yields

n

∑i=1

[

(

2 hTi (θ) −2 mT

i)

H i

]

= 0. (B.8)

Since the operation hi(θ) is non-linear in the model parameters θ, an iterative procedure is re-quired to estimate θ. Suppose that a previous estimate θk of the model parameters θ is available,such that equation B.8 is not fully satisfied:

hi(θk) 6= mi; for all i. (B.9)

As θk is the best available estimate for the model parameters, it may be assumed that a new andbetter estimate θk+1 of θ will not deviate much from the previous estimate θk plus a correctionterm ∆θ according to

θk+1 = θk +∆θ. (B.10)

The estimation algorithm is aimed at finding the correction term ∆θ to improve the estimateof the model parameters. For this reason, equation B.10 is substituted in equation B.8, whichresults in:

n

∑i=1

[(

hTi (θk) +

(

H i ∆θ)T

−mTi

)

H i

]

= 0 (B.11)

B.2. Bibliography 199

It must be denoted that H i is determined based on the previous estimate θk.According to Kreyszig [1], the transpose of a product equals the product of the transposed

factors, taken in reverse order, i.e. (A B)T = BT AT . Now, the change in estimation of the modelparameters can be calculated from rewriting equation B.11 to:

∆θ =n

∑i=1

(

HTi H i

)−1 n

∑i=1

HTi

(

mi −hi(θk))

. (B.12)

Once the correction ∆θ has been computed from equation B.12, the estimate of underlineθ canbe updated according to equation B.10. The new estimate of the model parameters is conse-quently written as

θk+1 = θk +n

∑i=1

(

HTi H i

)−1 n

∑i=1

HTi

(

mi −hi(θk))

. (B.13)

Because the calculated temperatures in the solid particle mixture depend nonlinearly on themodel parameters cλa and cλb , this iterative estimation procedure must be repeated until ‖ ∆θ ‖has become smaller than a predefined value ε0 > 0.

B.2 Bibliography[1] E. Kreyszig. Advanced engineering mathematics. John Wiley and Sons, Inc., 6 edition,

1988.

200 Bibliography

Appendix C

Derivation of 2nd order derivative ofposition dependent temperature

Estimating the 2nd order spatial derivative of temperature requires at least a 2nd order spatialdependency of temperature, viz. T = ax2 + bx + c. The determination of the 2nd order spatialderivative of temperature, given by 2a, requires at each time i information of at least threetemperatures at three different positions j distributed in the solid particle mixture:

Tj−1,i = ai x2j−1 +bi x j−1 + ci (C.1)

Tj,i = ai x2j +bi x j + ci (C.2)

Tj+1,i = ai x2j+1 +bi x j+1 + ci (C.3)

From equation C.1 an equation for the coefficient ci is derived:

ci = Tj−1,i −ai x2j−1 −bi x j−1 (C.4)

Combining equation C.4 with equations C.2 and C.3 results in equations C.5 and C.6.

Tj −Tj−1 = ai(

x2j − x2

j−1)

+bi(

x j − x j−1)

(C.5)

Tj+1 −Tj−1 = ai(

x2j+1 − x2

j−1)

+bi(

x j+1 − x j−1)

(C.6)

From equation C.5 an equation for the coefficient bi is derived.

bi =

(

Tj −Tj−1)

−ai

(

x2j − x2

j−1

)

(

x j − x j−1) (C.7)

Combining equations C.6 and C.7 results in an expression for the coefficient ai.

ai =

(

Tj+1 −Tj−1)

− x j+1−x j−1x j−x j−1

(

Tj −Tj−1)

(

x2j+1 − x2

j−1

)

− x j+1−x j−1x j−x j−1

(

x2j − x2

j−1

) (C.8)

In case that(

x j+1 − x j−1)

=(

x j − x j−1)

= ∆x than equation C.8 simplifies to

ai =Tj+1 −2Tj +Tj−1

2∆x2 . (C.9)

201

202 Appendix C. Derivation of 2nd order derivative of position dependent temperature

The 2nd order spatial derivative of temperature is now given by

∂2T∂x2 = 2

Tj+1 −2Tj +Tj−1

2∆x2 . (C.10)

Samenvatting

De kwaliteit van de glassmelt die geproduceerd wordt in continu bedreven glasovens wordtvoor een belangrijk deel bepaald door het temperatuur-tijd traject dat vers ingesmolten glasdoorloopt in de smeltwan van een glasoven. Tijdens het transport van het vers ingesmolten glasdoor de glasoven dient de verblijftijd en de temperatuur van de glassmelt voldoende te zijn voorhet oplossen van nog ongesmolten materiaal, het verwijderen van de in de glassmelt opgelostegassen en het homogeniseren van de gevormde glassmelt. Berekeningen met glasovensimu-latiemodellen1 tonen aan dat de dikte, lengte, reactiviteit en thermische eigenschappen van degemengdeken, bestaande uit glasvormend gemeng2, in een industriele glasoven een grote in-vloed hebben op de glassmeltstroming en de temperatuurverdeling in de glassmelt in de geheleoven. Hiermee beınvloedt de gemengdeken indirect de kwaliteit van het geproduceerde glas.Om glasovensimulatiemodellen te kunnen gebruiken als hulpmiddel bij het begrijpen, beheersenen optimaliseren van de glassmeltkwaliteit en bij het verbeteren van glasovenontwerp, zijn gede-tailleerde mathematische modellen nodig die het thermisch en chemisch gedrag van glasgemengbeschrijven.

Tijdens het opwarmen van glasgemeng treedt een verscheidenheid aan chemische reac-ties op, zoals ontwateringsreacties, kristalovergangen, vaste-stof reacties, ontledingsreacties,smeltvormende reacties en oplosreacties. De opwarming van het uit drie fasen bestaande (rea-gerende) glasgemeng wordt bepaald door warmte-overdracht van zowel de verbrandingsruimteboven de gemengdeken als van de hete smelt onder de gemengdeken naar de randen van degemengdeken en door het warmtetransport in het inwendige van de gemengdeken. Vanwegede complexiteit van zowel het smelt- als het opwarmproces van een gemengdeken en vanwegehet gebrek aan voldoende sensoren en nauwkeurige analysetechnieken om de voortgang van hetopwarm- en smeltproces van glasgemeng te kunnen meten, is tot op heden geen kwantitatievebeschrijving van het opwarmen en reageren van glasgemeng bekend.

De doelstelling van dit proefschrift is het kwantitatief beschrijven van het opwarmprocesvan een glasgemeng en de omzettingssnelheid van het glasgemeng in de glassmelt. Het warmte-transport in een gemengdeken kan worden beschreven door:

∂(ρm cp,m Tm)

∂t=−∇ · (ε ρg cp,g vg,z Tg)−∇ · (λeff∇Tm)+∇ ·−→q r,eff +(1− ε)

∂Hchem

∂t, (C.11)

Het linkerlid van deze energievergelijking beschrijft de lokale ophoping van warmte in degemengdeken. De eerste, tweede en derde term in het rechterlid van de energievergelijkingbeschrijven het transport van energie door de gemengdeken via convectief, diffusief en stra-lingswarmtetransport. De laatste term in de energievergelijking beschrijft de energie die nodig

1Glasovensimulatiemodellen beschrijven onder andere het insmelten van glasgemeng, de stroming van de glass-melt en andere thermische processen tijdens het industrieel glassmelten.

2Het glasvormend gemeng wordt in het algemeen glasgemeng genoemd.

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204 Samenvatting

is voor de chemische reacties die optreden tijdens het opwarmen van het glasgemeng. Vooreen gedetailleerde beschrijving van het opwarmen en smelten van een glasgemeng is kwan-titatieve kennis nodig van de temperatuurafhankelijke eigenschappen van het glasgemeng. Indeze studie is de nadruk gelegd op het bepalen van de temperatuurafhankelijke chemische ener-giebehoefte en warmtegeleidbaarheid van glasgemeng. Voor het beschrijven van de omzettingvan glasgemeng tot een glassmelt is het oplossen van zandkorrels in het reagerende glasgemengonderzocht.

In dit promotiewerk zijn een drietal activiteiten te onderscheiden:

1. De identificatie van de belangrijkste gemengreacties voor een vlakglasgemeng en eenTV-schermengemeng.

2. Het ontwikkelen van zowel experimentele als mathematische methodieken waarmee desnelheid van gemengreacties en de warmtegeleidbaarheid van een glasgemeng kwanti-tatief bepaald kunnen worden.

3. Het bepalen en modelleren van de temperatuurafhankelijke chemische energiebehoefte,de warmtegeleidbaarheid en de reactiekinetiek voor het vlakglasgemeng, waarbij onder-meer gebruik gemaakt wordt van de in deze studie ontwikkelde technieken.

Voor een vlakglasgemeng bestaande uit zand, soda en dolomiet, kan de calcinering van hetcomplete gemeng beschreven worden door de thermische ontleding van dolomiet en kalk ende reactieve ontleding van soda met zand. De reactieve ontleding van soda met zand bepaaltde vorming van primaire smeltfasen beginnend bij circa 1070 K. De vorming van deze smelt-fasen zorgt voor een versneld oplossen van het zand. In tegenstelling tot het vlakglasgemengkan de calcinering van een typisch TV-schermengemeng niet per afzonderlijke carbonaat in hetgemeng beschreven worden.

De chemische energiebehoefte van een glasgemeng wordt grotendeels bepaald door de energiebenodigd voor het calcineren van de carbonaten die aanwezig zijn in het glasgemeng. Daaromis voor het bepalen van de chemische energiebehoefte van vlakglasgemeng de kinetiek vande thermische ontleding van dolomiet en kalk en de kinetiek van de reactieve calcinering vansoda in deze studie bepaald. Combinatie van de gemeten calcineringskinetiek met de reactie-enthalpien van betreffende calcineringsreacties heeft geresulteerd in een beschrijving van dechemische energiebehoefte van vlakglasgemeng als functie van tijd, temperatuur en partieleCO2-druk in het gemeng.

Het warmtetransport in het inwendige van een gemengdeken wordt bepaald door drie ther-mische processen, te weten conductieve, convectieve en stralingswarmtegeleiding. De bijdragevan deze afzonderlijke thermische processen op het totale warmtetransport in de gemengdekenis bestudeerd met behulp van een speciaal ontwikkelde experimentele opstelling waarmee opverschillende plaatsen en als funktie van tijd de temperaturen in een opwarmend glasgemenggemten kan worden. Om uit deze gemeten temperaturen de temperatuurafhankelijke warmte-geleidbaarheid te bepalen is een numeriek-experimentele techniek gebruikt.

Met deze numeriek-experimentele techniek is de temperatuurafhankelijke warmtegeleid-baarheid van individuele glasgemengcomponenten en van mengsels van glasgemengcompo-nenten bestudeerd. Het is aangetoond dat de warmtestroom door deze mengsels beschreven kan

Samenvatting 205

worden als een warmtestroom door een combinatie van serieel en parallel geschakelde thermi-sche weerstanden.

De voorspelling van de effectieve warmtegeleidbaarheid van mengsels van glasgemengcom-ponenten op basis van gemengsamenstelling en de porositeit van het glasgemeng is mogelijkdoor het glasgemeng te beschouwen als een parallele schakeling van thermische weerstanden.Hierbij worden, in deze studie bepaalde, schijnbare warmtegeleidbaarheidswaarden van deglasgemengcomponenten gebruikt in plaats van de intrinsieke waarden voor de warmtegelei-dbaarheid. Het verschil in de gemeten en de voorspelde warmtegeleidbaarheid van een drie-component mengsel met deze aanpak bedraagt ca. 0.01 W m−1 K−1 bij een waarde voor dewarmtegeleidbaarheid varierend tussen de 0.15 en 0.40 W m−1 K−1.

De invloed van de aanwezigheid van scherven in het glasgemeng op de netto effectievewarmtegeleidbaarheid van glasgemeng in het temperatuurgebied tot circa 1200 K is bestudeerdmet de numeriek-experimentele techniek. De berekening van de bijdrage van stralingswarmte-transport op de netto effectieve warmtegeleidbaarheid van glasgemeng toont aan dat de ver-wachte toename in de warmtegeleidbaarheid bij schervenhoudend glasgemeng (met een scherf-grootte varierend van 4 tot 8 mm) alleen waargenomen wordt indien de glasscherffractie in hetglasgemeng groter is dan tenminste 50 %.

Het opwarmen van een glasgemeng wordt niet alleen bepaald door het warmtetransport inhet inwendige van een glasgemeng, maar tevens door de warmteoverdracht vanuit de verbran-dingsruimte naar de toplaag van het glasgemeng. Het is aangetoond dat warmteoverdracht naarde gemengdeken toe voornamelijk bepaald wordt door stralingswarmteoverdracht en in minderemate door warmteoverdracht via vrije en gedwongen convectieve stroming van hete gassen overde gemengdeken.

In het algemeen wordt de omzettingsgraad van een glasgemeng gerelateerd aan de omzettings-graad van zandkorrels in het reagerende glasgemeng. Voor de modelmatige beschrijving vandit stofoverdrachtsgelimiteerde proces dienen processen op microschaal, zoals bijvoorbeeld hetbevochtigen van zandkorrels door laag visceuze smeltfasen, beschreven te worden samen metcomplexe tijdsafhankelijke randvoorwaarden. Voor het gebruik van glasovensimulatiemodellenals hulpmiddel in de glasindustrie is het gewenst om een eenvoudige uitdrukking te hebbenvoor het tijd- en temperatuurafhankelijk oplossen van zandkorrels in plaats van het toepassenvan complexe oplosmodellen voor beschrijvingen op microschaal. Een voorwaarde voor hetgebruik van deze eenvoudige uitdrukking is dat deze uitdrukking voor het beschrijven van hetoplossen van zandkorrels in een reagerend gemeng het praktisch waargenomen oplosgedraggoed beschrijft en dat de afhankelijkheid met korrelgrootte, opwarmsnelheid en scherffractiegoed weergegeven wordt. Het is algemeen gebruikelijk om benaderende theoretische modellente gebruiken voor het beschrijven van complexe processen zoals het oplossen van zand tijdensgemengsmelten.

Het meest geschikte benaderende theoretische model dat de snelheid van een driedimen-sionaal diffusie gelimiteerd proces, zoals het oplossen van zandkorrels in een reagerend glas-gemeng beschrijft, is het Ginstling-Brounstein model (GB-model). De toepasbaarheid van hetGB-model is bestudeerd door simulatieresultaten van het GB-model te vergelijken met resul-taten verkregen met een gedetailleerder numeriek model. Het blijkt dat het onder bepaalde om-standigheden mogelijk is om met een aangepast GB-model het oplossen van zandkorrels in eenreagerend glasgemeng te voorspellen als functie van tijd en temperatuur. Dit geldt voornamelijkvoor grote zandkorrels die oplossen in relatief dikke smeltlagen. Voor dunne smeltlagen blijkt

206 Samenvatting

de voorspelling van het GB-model niet accuraat te zijn.Het gebruik van het aangepaste GB-model is experimenteel gevalideerd door het restgehalte

zand te meten in een reagerend vlakglasgemeng. Het restgehalte zand in gedeeltelijk inge-smolten vlakglasgemeng is bepaald via kwantitatieve fase analyse met Rontgendiffractie. Voorhet vlakglasgemeng zijn de GB-model parameters bepaald als functie van zandkorrelgrootte,opwarmsnelheid en schervenfractie.

Dankwoord

Het promotieonderzoek ’Thermal and chemical behavior of glass forming batches’ heeft plaats-gevonden in het kader van ontwikkeling van het TNO Glass Tank Simulation Model. Dit onder-zoek werd gefinancierd door het Nationaal Comite van de Nederlandse Glasindustrie (NCNG),Glaverbel, LG.Philips Displays, PPG Industries, Schott Glas, Thomson Multimedia en Visteon.Bij deze wil ik mijn dank uitspreken voor de financiele ondersteuning van dit promotieonder-zoek. TNO bedank ik voor de mogelijkheid die mij geboden is om mij via dit promotiewerkzowel op technisch als op persoonlijk vlak verder te kunnen ontwikkelen.

Prof. Ruud Beerkens wil ik bedanken voor zijn initiatie van dit promotiewerk, zijn goedeadviezen en zijn grote betrokkenheid. Ich mochte Herrn Prof. Conradt fur die Ubernahme desKorreferats danken, wodurch ich mich sehr geehrt fuhle.

Mijn collega’s en afstudeerders bij TNO en het promovendi/post-doc groepje van de leer-stoel Glastechnologie wil ik bedanken voor hun steun en bijdrage bij zowel het experimenteelonderzoek als het modelleerwerk. In het bijzonder ben ik Olaf zeer dankbaar voor zijn altijdluisterende oor, zijn goede adviezen en voor zijn geestelijke steun tijdens mijn promotiewerk-zaamheden.

Familie en vrienden wil ik bedanken voor hun steun en voor hun begrip voor het feit datik er de afgelopen tijd vaak niet was. Arjen wil ik bedanken voor het delen van alles wat eenpromotie raakt en ook wat daarbuiten van belang is. De vele biertjes zullen niet snel vergetenzijn!

Tot slot en in het bijzonder wil ik Roelie bedanken voor alle steun die zij mij gegeven heefttijdens mijn promotiewerkzaamheden bij TNO en vooral gedurende de lange tijd dat ik thuismijn proefschrift aan het schrijven was. Zeker ook door haar grote doorzettingsvermogen is hetpromotiewerk tot een goed einde gekomen. De tijd van genieten breekt nu aan!

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Curriculum Vitae

Oscar Verheijen werd op 3 augustus 1970 geboren te Arnhem. Hij behaalde in 1988 het VWO-diploma aan het Nederrijn College te Arnhem. In 1988 begon hij met de studie ChemischeTechnologie aan de Hogeschool Eindhoven, alwaar hij in 1992 afstudeerde. Van 1992 tot1995 volgde hij de verkorte opleiding Scheikundige Technologie aan de Technische Univer-siteit Eindhoven. In mei 1995 trad hij in dienst bij TNO TPD te Eindhoven als wetenschappelijkmedewerker bij de afdeling Glastechnologie. In de periode van juli 1997 tot en met december2001 verrichte hij een promotie-onderzoek bij TNO TPD. Sinds januari 2002 is hij werkzaamals projectleider bij de afdeling Glastechnologie van TNO TPD met als voornaamste werkge-bieden smelttechnologie en thermochemie.

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thermal and chemical behavior of glass forming batches

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