The effect of composition, size distribution and porosity
ter verkrijging van de graad van doctor aan de Technische Universiteit Delft,
op gezag van de Rector Magnificus prof. ir. K.C.A.M. Luyben, voorzitter van het College voor Promoties,
in het openbaar te verdedigen op maandag 23 januari 2012 om 10.00 uur
Gregory Andrew GEORGALLI
Magister Ingenieurswese in Natuurwetenskappe Stellenbosch University, Zuid-Afrika geboren te Krugersdorp, Zuid-Afrika
Prof. D.Eng M.A. Reuter (PhD) Prof. Dr. R. Boom
Rector Magnificus voorzitter
Prof. D.Eng M.A. Reuter (PhD) University of Melbourne/ Outotec Ltd., Australia, promotor
Prof. Dr. R. Boom Technische Universiteit Delft, promotor Prof. K. Heiskanen Helsinki University of Technology, Finland Prof.Dr.Ir. J. Sietsma Technische Universiteit Delft
Prof.Dr.Ir. M.P.C. Weijnen Technische Universiteit Delft Prof.Dr. B.J. Thijsse (reserve) Technische Universiteit Delft
Dr. Y. Xiao Technische Universiteit Delft
Dr. W. Hüsslage Tata Steel
Copyright ©2011, by G.A. Georgalli
All rights reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior written permission of the author.
The work presented in this thesis formed part of the VG2 (Vergroening van Gas: The Greening of Gas) project which was funded by the EET (Economy, Ecology and Technology) program. EET is a joint programme of three Dutch Ministries: the Ministry of Economic Affairs, the Ministry of Education, Culture and Science, and the Ministry of Housing, Spatial Planning and the Environment.
The badger that sings is the badger that knows W. Austin
Isn’t it the most beautiful negative landscape? Just see on the left that pile of ashes they call a dune here, the great dyke on the left, the livid beach at our feet and, in front of us, the sea looking like a weak lye-solution with the vast sky reflecting the colourless waters. A flabby hell, indeed! Everything horizontal, no relief; space is colourless and life dead.
Many people contributed to the completion of this thesis. I would like to thank prof. Markus Reuter and prof. Jacques Eksteen for giving me the opportunity to work towards completing a PhD degree at the TU Delft. I also valued the support and guidance of prof. Markus Reuter, prof. Rob Boom and Dr Yanping Xiao throughout the process and Dr Renate van der Weijden during the initial stages. I would also like to thank Mr Wim Kuilman who greatly assisted me with the large amount of Delphi programming that this project involved.
I would also like to thank prof. Guven Akdogan and prof. Jacques Eksteen of the Process Engineering department of Stellenbosch University for allowing me to continue with my PhD research while working there.
For the financial assistance offered for the travel involved with the defence of this thesis I would like to thank Anglo American Platinum, and in particular Mr Bertus de Villiers, Mr July Ndlovu and Mrs Paula Oberholzer for their assistance in this regard.
To my fellow PhD candidates at Mijnbouw, Jim, Lenka, Christa, Olga and Erwin, I enjoyed sharing the great culinary feasts that were served up in the canteen together with you. I reserve special thanks for Steven Lans and the late Pascal Visser, both of whom showed great hospitality and friendship to me when I arrived in Delft, and helped me through some very hard times. I will also not forget the assistance of Joke Baan and Louise Eksteen on my arrival in Delft.
I would like to thank all of my friends that I shared time with in the Netherlands, I may inadvertently forget someone so I shall not name them all, but you guys know who you are. Your continued friendship and the memories of the time we spent together are the most valuable thing that I got out of my time in the Netherlands. I nonetheless want to include a special mention for Emile, Colossus, Claudia and Giacomo. To my friends in London who supplied me with a home away from home, Kevin, Ed, Eli (and your mom and gran), Chris and Barnyard, julle is piele.
None of this would have been possible were it not for my family. I dedicate this thesis to my mother and my father, who instilled in me the resilience that allowed me to complete this project.
Finally to Alet, who has put up with me for 16 years (and counting) and is probably the only philosophy major who understands particle packing (after a fashion). This PhD project affected your life as much as it did mine. I love you and I will never forget the sacrifices you made to come and join me in the Netherlands. This is as much yours as it is mine. Jy is nou amptelik “Mev. Dokter”.
1 Introduction 1
1.1 Background . . . 1
1.2 Effect of transient liquid phases on metallurgical processes . . . 2
1.2.1 Physical metallurgical processes . . . 3
1.2.2 Extractive pyro-metallurgical processes . . . 3
1.3 Aim and innovation of the thesis . . . 4
1.4 Layout of the thesis . . . 7
1.5 Publications from this thesis . . . 9
2 Theory and behaviour of transient liquid phases 11 2.1 Transient liquid phases . . . 12
2.1.1 Mechanism of formation and solidification of a transient liquid phase 14 2.2 Corbin’s diffusion-based model of the isothermal solidification kinetics of a transient liquid phase . . . 15
2.2.1 Iterative solution of isothermal solidification model . . . 19
2.3 Copper-nickel . . . 21
2.3.1 Phase equilibria . . . 21
2.3.2 Diffusivity . . . 22
2.3.3 Density of copper liquid . . . 24
2.4 Mechanism of melt propagation within the pellet . . . 24
2.5 Conclusion . . . 28
3 Coordination number model 31 3.1 Introduction . . . 32
3.2 Model equation for estimating the co-ordination number . . . 32
3.3 Experimental . . . 34
3.4 Image analysis . . . 37
3.5 Results . . . 39
3.5.1 Validation of image analysis output . . . 40
3.5.2 Size distributions . . . 44
3.5.3 Coordination number model . . . 44
4 Particle packing algorithm 51 4.1 Introduction . . . 52 4.2 Simulated annealing . . . 53 4.3 Objective function . . . 54 4.3.1 Coordination number . . . 55 4.3.2 Overlap volume . . . 55
184.108.40.206 Combining the coordination number and overlap volume functions to form the objective function . . . 58
4.4 Algorithm structure . . . 59
4.5 Concluding remarks . . . 64
5 Determining the compositional probability density function 69 5.1 Introduction . . . 70
5.2 Numerical procedure for determining the compositional probability density function . . . 71
5.3 Determining the solidification profile using the compositional probability density function . . . 75
5.4 Concluding remarks . . . 75
6 Transient liquid phase sintering experiments 79 6.1 Introduction . . . 80
6.2 Materials used . . . 81
6.3 Tube furnace setup . . . 82
6.3.1 Temperature profile of furnace . . . 83
6.3.2 Validation of furnace temperature . . . 83
6.3.3 Gas delivery system . . . 85
6.4 Sample preparation . . . 86
6.5 Experimental procedure . . . 87
6.6 Experiments conducted . . . 87
6.7 Analytical techniques and procedures . . . 89
7 Modelling and experimental results 93 7.1 Introduction . . . 94
7.2 Model results . . . 95
7.2.1 Particle packing algorithm . . . 95
220.127.116.11 Inputs to the particle packing algorithm . . . 95
18.104.22.168 Discussion of packing solutions . . . 97
7.2.2 Compositional probability density function . . . 99
7.2.3 Predicted solidification profiles . . . 102
7.3 Drop-quench experimental results . . . 106
7.3.1 Results and discussion . . . 106
7.3.2 Qualitative validation of the model . . . 111
7.4 Application for the prediction of transient liquid phase formation in pyro-metallurgical applications . . . 116
7.5 Conclusions . . . 119
8 Conclusions and recommendations 121
A Analysis of packing outputs 125
A.1 Particle packing algorithm parameters and minimisation profiles . . . 125 A.2 Analysis of the coordination number and overlap volume properties of the
simulated pellets . . . 129 A.3 Graphical reconstructions of simulated pellets . . . 137 A.4 Comparison of outputs from identical packing simulations . . . 143
B Determination of melt infiltration mechanism 145
C Copper maps 149 C.1 Experimental set A . . . 150 C.2 Experimental set B . . . 154 C.3 Experimental set C . . . 160 Bibliography 166 Summary 173 Sammenvating 176 List of abbreviations 180
Packed particulate assemblies are used as feed material to numerous processes in both physical and extractive metallurgy. In physical metallurgy supported packed beds of par-ticles are used as the starting point in the manufacture of various sintered metallic or ceramic materials. In extractive metallurgy pellets comprised of ore and flux materials bound together into an unsupported structure are used as the value bearing feed to numerous pyro-metallurgical processes. For the sake of brevity packed particulate as-semblies, be they supported or unsupported, will hence forth be referred to as pellets.
Definition 1.1 A pellet is a collection of particles that are packed
to-gether either in a supporting structure (i.e. packed bed) or bound to-gether without external support to form a self supporting structure.
The behaviour or pellets in different metallurgical processes can be predicted both thermodynamically and kinetically, but these predictions are more often than not based on the assumption that the particles of different composition in the pellet are well mixed. In the manufacture of sintered materials such as ceramics or composites, the particulate feed materials used have generally been pre-processed to a high degree, both physically and chemically, with the sintering process in mind. They tend to have well controlled and narrow size distributions, and contain low levels of impurities. As the aim in these
processes is to create homogeneous sintered materials in the most efficient manner pos-sible, every effort is made to ensure homogeneity within the green pellet prior to sintering. These pellets can therefore be seen as well mixed, or homogeneous, and their kinetic be-haviour during sintering modelled accordingly.
The same cannot be said for pellets which are used as value bearing feed to extrac-tive pyro-metallurgical processes. These pellets are generally made up of ore and flux materials with various mineralogical composition. These materials are usually only sub-ject to physical preparation, i.e. size reduction and physical separation, and tend to have a broader particle size distribution. Modelling thermodynamic behaviour in these pellets is therefore more complicated due to the internal compositional distribution which needs to be accounted for. This compositional distribution is a function of the quantity and size distribution of each constituent within the pellet, which must be accounted for in order to properly describe the effect of heterogeneity within the pellet on its thermodynamic behaviour.
In this thesis a specific phenomenon of high temperature pellet processing, the for-mation and behaviour of a transient liquid phase (TLP), will be investigated. The focus of the work will be on deriving a method for predicting the probability of formation of a TLP in extractive metallurgical processes, which has never been done in the past. Such a method would significantly reduce the amount of high cost, time intensive experimental work necessary for pellet feed optimisation to these processes. Validation of this method will, however, be done by modelling the kinetics of TLP solidification in the copper-nickel system, which is an important system for physical metallurgists and one of the systems which has received the most attention with regards to TLP behaviour.
Effect of transient liquid phases on metallurgical
Depending on the application, the formation of a TLP can either be advantageous or disadvantageous. In general, one can say that for physical metallurgical processes it is advantageous while for extractive metallurgical processes it is disadvantageous. The focus of the research done on TLPs also differs for different metallurgical processes.
Physical metallurgical processes
In physical metallurgy, particularly the creation of alloys through sintering, the formation of a TLP has numerous advantages. This process, called Transient Liquid Phase Sintering (TLPS) offers a number of advantages over solid-state sintering including rapid densifi-cation, lower sintering temperatures, use of less expensive elemental powders vs alloyed powders, and reduced microstructural coarsening [1, 2]. TLPS also has advantages over conventional liquid-phase sintering in that the liquid disappears from the microstructure prior to cooling. This leads to a more homogeneous and higher strength compact . The removal of the transient liquid also has the advantage of forming a new solid alloy from an initial solid/liquid two-phase mixture, which exhibits an increased melting point after initial processing . This behaviour is being exploited in the development of TLPS solder and brazing compounds [2, 4–8].
The main disadvantage of TLPS is that it is quite sensitive to processing conditions such as the heating rate, particle size distribution as well as the solute/base metal initial powder distribution. This can lead to variable liquid formation and sintered densities . The formation of a TLP in the processes is never in question, but the amount and distribution of the TLP within the pellet can vary (depending on the processing conditions listed above) which will effect the solidification time and the properties of the product.
Extractive pyro-metallurgical processes
For extractive pyro-metallurgical processes, specifically in those processes where pellets are used as feedstock and gas-solid reactions are present, the formation of a TLP (“local melt” or “non-equilibrium melt” as it is referred to by pyro-metallurgists) is generally disad-vantageous. This is because the liquid can retard gas diffusion into the pellet by lowering the pellet porosity, thereby slowing the desired reactions and significantly diminishing the strength of the pellet, causing the bed to compress and reducing the overall porosity of the packed bed of pellets [9, 10].
In pyro-metallurgy the interest lies in predicting the probability of TLP formation and not in the subsequent behaviour, as once a TLP has formed the damage has essentially been done. The probability of formation of a TLP is dependant on the compositional distribution within the pellet, which is a function of the pellet porosity and the respective amounts and size distribution of the various constituents of the pellet.
A good example of this is the iron blast furnace. Coke, while important for the reduc-tion of iron oxides in the blast furnace, is also responsible for maintaining bed porosity in
the blast furnace burden due to the fact that it remains solid at extremely high tempera-tures. In fact pyro-metallurgists have been manipulating blast furnace pellets or sinter to try and retard melt formation (and associated reduction retardation and pellet softening) as much as possible for many years. Barnaba  showed that grain size of flux addi-tion to sinter (i.e. olivine and dolomite) to a certain degree influenced both softening and melting behaviour. The softening temperature was shifted to slightly higher temperatures (by± 30°C) in case of fluxes of greater fineness. A more extravagant approach to pellet manipulation is the so-called two layered structure [9, 12, 13]. In this approach different chemistries are used for the central and peripheral regions of the pellet to deliberately manipulate melt formation and thus postpone reduction retardation and softening. Thus the bulk behaviour of the bed in the blast furnace is greatly affected by the pellet prop-erties, that are in-turn affected by the particles from which the pellet is made up of, as is illustrated in Figure 1.1.
While the coal feed to a blast furnace can be replaced by cleaner (with regard to greenhouse gases) reductants such as hydrogen or natural gas, the coke cannot be completely replaced within the framework of the blast furnace operation due to the im-portant physical support role it plays in the bed. Coke is more expensive than coal and therefore it is more feasible to replace it with a cleaner reductant source, but this can only be done if pellet strength can be increased.
The blast furnace is a significant producer of CO2. Gielen  reported that in 1999
420 Mt of coke and coal were used in blast furnaces  which amounted to 1.1 Gt of CO2
emissions. Global CO2emissions in 1999 (excluding deforestation and land use change)
amounted to 23.6 Gt of CO2. Blast furnaces were therefore responsible for 4.6% of
the total global CO2emissions. More recent estimates of CO2emissions from the steel
industry reported by Birat  are in the region of 4 to 7 % of global anthropogenic CO2
emissions. In a traditional Integrated Steel Mill roughly 69% of the CO2 emissions are
derived from the blast furnace , making it the source of roughly 3-5% of global CO2
emissions. Any decrease in coke consumption in the blast furnace process will therefore have a significant effect on global CO2emissions.
Aim and innovation of the thesis
The phenomenon of TLP formation within pellets in extractive pyro-metallurgical pro-cesses is well known (albeit by a different name), and methods for avoiding or minimising the occurrence have been investigated, at least on a laboratory scale. However no
quan-Figure 1.1: From the blast furnace to pellets to particles
titative modelling technique has as yet been developed to predict TLP formation in these processes.
This thesis aims to rectify this by creating a novel simulation technique that can be applied to these pellets to predict the probability of TLP formation and its subsequent be-haviour as a function of the size distribution of the reagents and the overall pellet porosity. The advantage of using such a method for pyro-metallurgical processes is that it could be used as a feed-forward simulation method for pellet design to avoid TLP formation. Generally, in pyro-metallurgical processes, designing pellets or predicting their behaviour within a process is done using time consuming experimental methods [9, 11–13] which need to be repeated every time a new ore or flux is used in the pellets. In the global economy of today, metal producers are using ores from all over the planet, with deals for new ores being made based primarily on financial considerations. To an extent the metal-lurgists are expected to make the best out of what they get. While a simulation technique can never totally replace experimental work, it can significantly reduce it.
While pyro-metallurgical operations have the most to gain from such work, the mod-elling and accompanying experimental work in this thesis will be done using the copper-nickel system, which is important in physical metallurgy. There are numerous reasons why the Cu-Ni system was chosen (see Section 6.1), suffice to say that this system proves ideal to use in a proof-of-principle exercise for the concept proposed in this thesis. The modelling approach adopted in this thesis revolves around the creation of a virtual
pellet, which simulates the packing arrangement of particles in a real pellet as realistically as possible. This was achieved by writing a custom particle packing algorithm (PPA) [18, 19]. The packing of the particles in the PPA is controlled by the coordination number (CN) of the multi-sized particles, which is determined using an experimentally validated CN model which calculates the CN of the particles in the pellet based on their size distribution and the overall pellet porosity. The validation of the chosen CN model was done using a CT scanner, the first time this technology has been used in such an application. Custom image processing software was written in order to analyse the CT scanner output so that the measured CN of the spheres could be compared to the model predictions.
Once a virtual pellet is created using the PPA, the next step is to use this to deter-mine the probability of formation and/or the subsequent behaviour of the TLP within the pellet. For this to be achieved the compositional distribution within the pellet needs to be determined both prior to and after the formation of the TLP. This was done by creating a Monte Carlo search algorithm, which is in essence a numerical form of sampling theory. The output from this algorithm is a compositional probability density function (PDF) of the pellet. The advantage of this approach over conventional sampling theory is that the compositional PDF can be created for pellets both prior to and after the formation of the TLP, i.e. the algorithm is able to determine the effect of melt percolation through the pellet matrix on the compositional distribution within the pellet. This is made possible through a simplified approach to the capillary flow which the melt will be subject to within the pellet .
It is these compositional PDFs of the pellet at various stages of the TLP process which are the true contribution of this thesis. Validation of these distributions does, however, require additional modelling as well as experimental work. It was decided that validation of the compositional PDF of the pellet after the formation and percolation of the TLP would be attempted. While validating this PDF is somewhat more complicated than that of the green pellet, the structure of the green pellet is inherent in the pellet containing the TLP and as such if this distribution can be validated it will automatically validate that of the green pellet.
The method chosen to perform the validation was to compare the solidification profiles of experimental Cu-Ni pellets sintered in a tube furnace to a model generated solidifica-tion profile of pellets with the same properties (composisolidifica-tion, size distribusolidifica-tion, porosity) created by the PPA. The model generated solidification profile is constructed by using the compositional PDF of the pellet just after TLP formation, together with a kinetic model for the isothermal solidification of a TLP in the Cu-Ni system [3, 21]. Pellets with different
copper and nickel powder size distributions and overall compositions will be sintered for varying amounts of time and the disappearance of the TLP during isothermal solidifica-tion monitored using SEM and microprobe analysis.
Thus the modelling philosophy explained above seeks to quantify what powder met-allurgists have known for decades, that pellet porosity and particle size distribution effect both the formation and behaviour of TLPs.
Layout of the thesis
All of the TLP theory relevant to this thesis will be summarised here. The formation of a TLP, the manner in which it percolates through the solid pellet structure and its isother-mal solidification kinetics will be described. The relevant thermodynamic, diffusional and physical properties of the chemical system used in the TLPS experiments conducted in this thesis are also given in this chapter.
The CN model used in the PPA is described mathematically and the experimental valida-tion of the model shown. This validavalida-tion was done using packed beds with different size distributions made up of glass beads. The packed beds were then put through a CT scan-ner and the output images were processed using custom written image analysis software in order to determine the CN of the glass spheres, which could then be compared to the model predicted CN.
A full description of the custom written PPA is supplied in this chapter. The PPA uses the Monte Carlo method, with the objective function minimisation done using the simulated annealing minimisation algorithm. The objective function and algorithm structure, along with all techniques used to speed up the operation of the algorithm, are described in detail.
The Monte Carlo algorithm used to generate the compositional PDF of the virtual pellet (both before and after TLP formation) is described in this chapter. The technique used to account for the effect of the melt percolation through the solid matrix of the pellet on the PDF is shown. Finally, the manner in which the compositional PDF is combined with a kinetic model for the isothermal solidification of a TLP to determine the solidification profile of the pellet is explained.
In this chapter the drop-quench TLPS experiments conducted on the Cu-Ni system are described. The reasoning behind the selection of the drop-quench experimental method over other methods will be defended, and the selection of the Cu-Ni system discussed. The equipment and the reagents used in the experiments will be fully described. The analytical techniques and the methodology used to search for evidence of a TLP in the samples are explained.
The results of both the modelling exercise and experimental work will be shown, com-pared and discussed. For the model results this includes an analysis of the performance of the PPA and the effect of different pellet compositions and particle size distributions on the compositional PDF and predicted solidification time. The SEM and microprobe analyses done on the experimental pellets will be used as an indication of the solidifi-cation time of the TLP, which will then be compared to the model results. A qualitative comparison of the model results with other work done on the solidification kinetics of a TLP in the copper-nickel system will also be completed. The model will also be applied to predict the probability of formation of a TLP in the FeO-CaO system, and the results compared to experimental data from the literature.
Publications from this thesis
• G.A. Georgalli and M.A. Reuter, Modelling the co-ordination number of a packed bed of spheres with distributed sizes using a CT scanner. Minerals Engineering, vol. 19, p. 246-255, 2006.
• G.A. Georgalli and M.A. Reuter, A particle packing algorithm for pellet design with a predetermined size distribution, Powder Technology, vol. 173, pp. 189-199, 2007. • G.A. Georgalli and M.A. Reuter, A particle packing algorithm for packed beds with
size distribution, Granular Matter, vol. 10, pp. 257-262, 2008.
• G.A. Georgalli and M.A. Reuter, Modelling the coordination number of a multi-component packed bed of spheres with size distribution using a CT scanner. In: Pyrometallurgy 05, 14-15 March 2005, Cape Town, South Africa. (CD-Rom). • G.A. Georgalli and M.A. Reuter, A particle packing algorithm for packed beds with
size distribution, CHoPS-05: The 5th International Conference for Conveying and Handling of Particulate Solids, 27-31 August 2005, Sorrento, Italy (poster presen-tation).
• G.A. Georgalli, M.A. Reuter, Y. Xiao and R. Boom, Solidification behaviour of a transient liquid phase (TLP) in pellets. Minerals Processing 2007, Cape Town, South Africa.
Theory and behaviour of
transient liquid phases
A more thorough description of the mech-anism of TLP formation and solidification is the starting point of this chapter. Re-search into TLPS by physical metallurgists has resulted in a model which can be used to describe the kinetics of solidification of a TLP [3, 21]. This model and its numeri-cal solution will be described in this chap-ter along with all of the relevant thermo-dynamic, diffusional and physical proper-ties for the Cu-Ni system used in the TLPS experiments conducted in this thesis. Ex-tending the work done by physical met-allurgists to systems where the assump-tion of homogeneity becomes unrealistic,
which is the major contribution to this field of this thesis, requires a method to quantify the distribution of the TLP within the pellet prior to the onset of solidification. This was accomplished by adopting a simplified thermodynamic approach to the capillary action experienced by melt within a pellet structure , also explained here.
Figure 2.1: Two binary-phase diagrams showing conditions where a TLP may form, either by the melting of the low temperature constituent or by eutectic formation. The overall composition is where the dashed lines intersect and is in the single-phase region.
Transient liquid phases
Definition 2.1 A Transient Liquid Phase (TLP) is a liquid phase which forms due to
pro-cessing conditions either by the melting of the low temperature constituent or by eutectic formation in systems whose overall composition and temperature are below the solidus line on the phase diagram.
A TLP, as defined in Definition 2.1, is formed in binary systems either by the melting of a low melting point constituent or by the formation of a eutectic . From Figure 2.1 it is obvious that the overall composition (Xo) is in the single phase region at the
operating temperature (To). If the system were to be analysed considering only bulk
thermodynamics, the existence of a liquid phase would be deemed impossible. The formation of a TLP, as well as the amount formed and the time that it survives, are a function of operating conditions such as the heating rate, particle size and diffusivity .
For extractive metallurgy applications this concept is best explained if one considers the iron blast furnace. Blast furnace pellets, contain (among other components) “FeO” and CaO. The nature of the binary “FeO”-CaO phase diagram is a contentious issue within the metallurgical world. The issue is whether this phase diagram is of a eutectic or eutectoid nature, with some favouring the eutectic [10, 23] and others the eutectoid
[24–26]. Determining the nature of this system is beyond the scope of this thesis, and in order to explain possible TLP formation in blast furnace pellets the eutectic diagram, as shown in Figure 2.2 with a eutectic temperature of 1103°C, will be used.
Figure 2.2: “FeO”-CaO binary eutectic 
The “FeO”-CaO systems is a definite candidate for the formation of TLP and was evaluated by Bakker  by doing experiments on the iron rich side of the phase diagram (5 wt% CaO) at a temperature of 1120°C, just above the eutectic temperature of 1103°C. Bakker experimented with both homogeneous and heterogeneous conditions by manip-ulating the particle size of the CaO in the mixtures. For the homogeneous conditions the FeO and CaO particles were of the same size, while for the heterogeneous conditions the CaO particle size was much larger than that of the FeO.
For the heterogeneous conditions Bakker found evidence of melt formation, which was referred to as non-equilibrium or local melt, even though the equilibrium products at the experimental conditions are solids (calciowustite for “FeO”-CaO and wustite+fayalite for “FeO”-SiO2) when considering the phase diagrams. The melt formed was no doubt
a TLP called by a different name. Although iron blast furnace pellets contain other com-ponents besides wustite, silica and lime, the formation of a TLP in the pellets during the blast furnace operation is considered plausible.
Figure 2.3: TLPS mechanism 
Mechanism of formation and solidification of a transient liquid
Figure 2.3 shows the different stages that have been identified for the formation and behaviour of a TLP [3, 21] formed during the sintering of a pellet composed of pure particles of two species, A and B, who form an ideal binary solution as depicted in Figure 2.1. The low melting point species will hence forth be referred to as component A or the solute. The other species will be referred to as the component B or the base phase.
During the first, or heating stage, solid-state interdiffusion between the atoms of parti-cles A and B occurs. The extent to which this interdiffusion occurs is strongly dependant on the heating rate of the pellet. If the heating rate during this stage is sufficiently high, then solid-state interdiffusion can be considered to be negligible. Stage two begins once the temperature reaches the melting point of pure A. At this point all remaining pure A in the pellet will melt. As stage 2 progresses and the temperature increases to the op-erating temperature, the A rich regions in the pellet will also melt. If, once the opop-erating temperature has been reached, the melt is not as yet saturated with component B, then dissolution of B in the melt will occur.
Stage 3 begins once the operating temperature has been reached and the melt is saturated with component B. During this stage isothermal solidification occurs by diffusion of the melt into the base phase. Once all of the TLP has solidified stage 4 begins, during which the pellet homogenises to eliminate compositional gradients.
Corbin’s diffusion-based model of the isothermal
solidification kinetics of a transient liquid phaseBased on the mechanism described in the proceeding section, Corbin  derived a kinetic model to predict the solidification time of a TLP. While the general form of Corbin’s model is always the same, some modifications are required in the derivation depending on the binary system being considered and the experimental conditions, specifically the heating rate of the sample.
The model of Corbin  has been applied to both binary eutectic (Pb-Sn [2, 7]) and binary isomorphous (Cu-Ni ) systems. The solidification times predicted by the model were compared to results obtained from differential scanning calorimetry (DSC) and de-spite the simplicity of the model, the results compared favourably. The models simplicity, which makes it so attractive to use, is achieved by assuming that the sample undergoing TLPS is homogeneous and that the transient liquid is evenly distributed throughout its volume. These conditions were approached in the experimental validation of the model [2, 3, 7] by using quite high fractions of solute with particle sizes smaller than that of the base phase powders . Thus there will initially be a large volume of TLP in the sample which is widely and evenly distributed throughout the sample.
This, however, would seem to exclude the use of Corbin’s model for predicting solid-ification times in more heterogeneous systems. This is overcome in this thesis by the generation of the compositional PDF (described in Chapter 5) of the pellet at the stage just prior to the onset of solidification, as was alluded to in Chapter 1. Thus rather than assuming one overall composition for the pellet, the approach in this thesis is to consider the pellet to be made up of independant zones with different compositions. The solidifica-tion time of each zone can then be calculated using Corbin’s model. Each zone will also have a probability of existence which is read from the compositional PDF. By summing the product of the solidification time and probability of existence for all of the zones in a pellet the overall solidification time of a TLP in a heterogeneous pellet can be calculated. In the mechanism of formation and solidification of a TLP, as discussed in Section
2.1.1, 4 stages were identified; 1. Solid-state interdiffusion
2. Melting, dissolution, capillary flow, densification 3. Isothermal solidification
4. Completion of isothermal solidification and beginning of homogenisation
In the drop-quench TLPS experimental work carried out for this thesis (Chapter 6), the samples were introduced into the pre-heated furnace at the desired operating tempera-tureToand as such solid-state interdiffusion can be considered negligible. The melting
of the solute phase at this temperature is assumed to be instantaneous and as such the melting time is not considered. The dissolution time of the base phase into the original liquid phase formed due to the melting of the pure low melting point solute component is also considered to be negligible. Thust=0 is taken as the point where the solute phase has a composition equal to the liquidus composition for the alloy system at the temper-ature under investigation. Only the time taken for the isothermal solidification (which begins once the solute phase has dissolved enough base phase to be at the liquidus composition for the temperature evaluated) is determined using the model presented in this section.
Thus, for a two component system made up of low melting solute particles (con-stituentA, whose melting point is below To) and base particles (constituentB, whose
melting point is aboveTo) the mass balance can be expressed using the following
XO= XAWA+ XBWB (2.1)
WhereXOis the bulk solute composition,XAWAandXBWBare the solute contents
within theA and B constituents, WAandWBare the mass fractions of each phase, and
XAandXBare the compositions of solute in each phase at timet. Now let us define WA0
as the original mass fraction of the solute liquid att=0. As was mentioned, the samples in this study were introduced directly into the pre-heated furnace and as such no solid-state interdiffusion could occur, thus the mass of initial melt formed prior to isothermal solidification is determined as follows:
Figure 2.4: Ideal binary phase diagram depictingXBf
Rearranging Eq. 2.1 and dividing byWA0gives:
=XO− XBWB XAWA0
(2.3) WhereWA/WA0 is then the normalised fraction of liquid remaining in the system at
a given time. Att=0 there is no solute dissolved in the base phase and as such XB=0.
Thus, from Eq. 2.1:
XO= XAWA0 (2.4)
Substituting this into Eq. 2.3 gives: WA
=XO− XBWB XO
(2.5) Thus from Eq. 2.5 the fraction of the TLP remaining in the system at a given time can be determined from knowledge of the solute content within the base phase (XBWB), as
long as the bulk composition for the system is known.
Crank  developed mathematical expressions and graphical solutions for solid-state diffusion of solute into a spherical particle. Of particular interest is his solution for the fractional solute uptake (Mt/Mf) of a sphere:
Mt Mf = 1−π6 ∞ X n=1 1 n2 exp−Da ·n 2π2t a (2.6)
WhereDais apparent diffusivity in the base metal particle,a is the base metal particle
radius, t is the time, Mtis the solute content at timet and Mf is the solute content at
infinity. Using the notation developed from Eq. 2.1 through 2.5 an alternate expression can be developed forMt/Mf:
(2.7) WhereXBf andWBf are the solute concentration and mass fraction of the base
phase at infinity respectively. Here infinity corresponds to the point where the base phase is completely saturated with solute which would be the solidus composition, as shown in Figure 2.4 for an ideal binary system (such as Cu-Ni, the system used in this study). This would also correspond to the complete solidification of the TLP and as suchWBf = 1.
Substituting these values into Eq. 2.7 and rearranging results in the following expression forXBWB: XBWB = Mt Mf XS (2.8)
Substituting this into Eq. 2.5 results in: WA WA0 = 1 − M t Mf X S XO (2.9) As was mentioned, the Crank solution assumes complete saturation of the base phase with solute. This situation will, however, never occur during TLPS as for TLPS to occur the overall composition of the pellet must be below the solidus line, as explained in Section 2.1. However, Corbin  states that the insertion ofMt/Mf into the mass
balance equation allows cases where incomplete saturation occurs (XO < XS) to be
considered through theXS/XO term in Eq. 2.9. As such, Eq. 2.9 is valid for all values
ofXO ≤ XS, which is the entire range available for TLPS.
From Eq. 2.6 it is clear that the particle radius of the base phasea plays an important part in the Crank solution. During the isothermal solidification of the TLP, when the solute phase is diffusing into the base phase, the base phase particle will grow as more and more solute diffuses into it, this needs to be accounted for. Corbin  accounted for this in the following manner. At any time, the ratio between the current volume of the solid base phase particles is:
= WB/ρ WB0/ρ0
100 101 102 103 104 105 1.25 1.30 1.35 1.40 1.45 1.50 n Sum
Figure 2.5: Plot of the value of the summation term in Eq. 2.7 vs. the end value ofn
For the Cu-Ni system evaluated in this study, it can be assumed that the density of the base phase is not greatly affected by the solute diffusing into it, as the densities of the two components are extremely close to one another. Thus:
aB= aB0 W B WB0 1/3 = aB0 1 − WA 1 − WA0 1/3 (2.11) WhereWA0andWAcan be determined using Eqs. 2.2 and 2.9 respectively.
Iterative solution of isothermal solidification model
In solving the equations derived in Section 2.2 for determining the isothermal solidification time Corbin  used an iterative procedure. This procedure, for which the flowsheet is shown in Figure 2.6, calculates the liquid fraction present at increments of isothermal solidification time in the following manner.
Firstly, Mt/Mf is calculated using Eq. 2.7. For this equation it is clear that the
summation term should be summed from n=1 to infinity. Obviously this is impractical and a lower end value ofn is required. This was determined by seeing at what value the summation term converged and the results are shown in Figure 2.5. It is clear from this plot that the summation term converges above an end value ofn=100. Thus for the
START i=0, t=0, WA0=XO/CL WA>0 T F i=i+1, t=i, Sum=0 n = 1 n = n + 1 n <= 1000 T F Sum=Sum+... Mt/Mf=1-Sum Calculate WA[i] Calculate aB[i+1] STOP
algorithmic solution an end value ofn of 1000 was chosen.
Subsequent to this calculation,WAis then calculated using the following form of Eq.
2.9: WA[i]= WA0 1 − Mt Mf XS XO (2.12) The particle radius of the base phase,aBmust then be updated for the next iteration
which represents an incremental increase int, this is done using Eq. 2.11 in the following form: aB[i] = aB[i−1] 1 − WA[i] 1 − WA[i−1] 1/3 (2.13) The value oft is incremented in this manner until complete solidification is achieved, which corresponds to the point whereWA[i]=0.
For the TLPS drop-quench experiments work conducted in this thesis (Chapter 6) the copper-nickel system was selected. The reasoning behind this selection are discussed in Section 6.1. This thermodynamic, diffusional and physical properties of this system are well quantified in the literature and the relevant properties will be summarised in this section.
The copper-nickel system is a binary isomorphous system which contains a solid-solid miscibility gap at lower temperatures (see Figure 2.7). The system has received much attention over the years with some disagreement regarding the miscibility gap as well as the wideness of the two phase solid-liquid region.
Initial work done on the two phase solid-liquid region (the region of interest in this thesis), which was summarised by Hansen and Anderko , under-estimated the solidus temperature by roughly 35 °C. This was corrected by Feest and Doherty  as well as Bastow and Kirkwood , who attributed this to undercooling.
More recently the focus has been on the lower temperature region and its wide mis-cibility gap. The most recent re-evaluation of the system was done by an Mey  where
LIQUID FCC FCC1 + FCC2 Ni (mass fraction) T (C ) 0 .2 .4 .6 .8 1 0 300 600 900 1200 1500
Figure 2.7: Cu-Ni phase diagram. Drawn using the SGTE database in FactSage
the influence of magnetic effects on the thermodynamic properties was included. The phase diagram shown in Figure 2.7 was drawn with FactSage 5.5 using the SGTE solution models. While it is difficult to determine exactly which data SGTE used for their solution models, comparing the phase diagram shown in Figure 2.7 to the calculated phase diagram by an Mey  (shown by Xiong et al. ) resulted in an almost perfect match.
The equilibrium data (such as the solidus or liquidus compositions) used in the isothermal solidification model (see Section 2.2) are obtained using FactSage 5.5 in con-junction with the SGTE database.
Model prediction of the isothermal solidification time for TLPS in the Cu-Ni system has already been done by Turriff and Corbin . While this was done for homogeneous systems, this makes no difference to the diffusion properties of the system when handling a heterogeneous system in the manner proposed in this thesis. Thus an exhaustive review of diffusion in the adopted experimental system is unnecessary and beyond the
scope of this thesis.
In fact, the diffusion which needs to be accounted for in the current study is somewhat simpler than that which was necessary in the work done by Turriff and Corbin . The reason for this is that the experimental procedure in this study is different to that adopted by Turriff and Corbin. In their study, DSC was used to validate their model predictions. As samples cannot be introduced into the DSC apparatus at the final TLPS temperature, Turriff and Corbin needed to account for solid state interdiffusion which occurs in this system during the heat up phase prior to the melting of the solute component in the sample. In the TLPS experimental work conducted in this study samples are lifted directly into the hot-zone of a tube furnace at the TLPS temperature and as such no solid state interdiffusion occurs.
Thus, only the diffusion of the solute phase into the base phase need be considered in this study. Turriff and Corbin handled this by using an apparent diffusivity,Da(see Eq.
2.6), which includes the contributions of lattice and grain boundary diffusion: Da = Dl 1 + k δ · Db d · Dl (2.14) WhereDl and Db are the lattice and grain boundary diffusivity of the solute (Cu)
in the base phase (Ni) respectively,δ is the grain boundary width for which Turriff and Corbin used a value of 0.5 nm which they obtained from Porter and Earterling ,d is the average base metal grain size andk is a geometric constant between 0.5 and 1.5 depending on the grain geometry . Turriff and Corbin found thatk values greater than 1.5 gave better model predictions in their study which they said indicates that there is a greater contribution from the grain boundary diffusion than expected. Nonetheless they used a value of 1.5 in their calculations. The uncertainty over the contribution of grain boundary diffusion [3, 35] means that differentk values will be investigated in this thesis. The effect of temperature onDawas taken into account by Turriff and Corbin using an
Arrhenius expression, and the values ofDoandQ for the lattice  and grain boundary
diffusion . Dl = 5.7 × 10−5exp −258300 8.314 · T (2.15) Db = 1.1 × 10−4exp −124700 8.314· T (2.16)
As the nickel powder used in this study was from the same manufacturer as that used by Turriff and Corbin, it was decided to adopt the same values for the grain boundary width and base metal grain size. While Turriff and Corbin did not give the values of the nickel powders grain size, they did state theirDavalues for experiments conducted at 1140 and
1200 °C, the same experimental temperature as used in this study. Back calculations of d is thus possible through re-arrangement of Eq. 2.14, combined with Eqs. 2.15 and 2.16 and substitution of same values ofk and δ used by Turriff and Corbin .
Density of copper liquid
As can be seen from the phase diagram given in Figure 2.7, copper is the low melting (solute phase) in the Cu-Ni system. The density of copper liquid as a function of tem-perature is required when generating the compositional PDF of the experimental pellets using the numerical method put forward in Chapter 5. Brillo and Ergy  measured the density of copper liquid at temperatures above the melting point using electromagnetic levitation. They derived the following equation for calculating the density of liquid copper:
ρ(T ) = 7900 − 0.765(T − 1083) (2.17)
Mechanism of melt propagation within the pellet
The construction of the compositional PDF of the pellet prior to the onset of isothermal solidification is key to the application of Corbin’s model to heterogeneous pellets. The manner in which the molten TLP percolates through the pellet microstructure must there-fore be accounted for, which is no easy task. Research using CFD to model flow through packed beds of spheres has been done by Guardo et al.  and Nijemeisland . Their applications were somewhat different, and somewhat less complex than that encountered in this thesis in that they were attempting to model continuous flow through structured packed beds made up of mono-sized spheres; however some of the pitfalls they encoun-tered are common to this work. The main problem encounencoun-tered by both Guardo et al. and Nijemeisland involved meshing the porous volume and particle surfaces, which proved to be impossible due to the angles generated at the sphere-sphere contact points. They overcame this problem by either separating the spheres slightly or introducing an overlap between the spheres. While it would be relatively easy to re-construct a virtual pellet
0 10 20 30 40 50 60 70 80 0 2 4 6 8 10 12 14 16 18
Contact angle (Degrees)
Critical liquid to solid ratio
f 0=0.4 f 0=0.7 spreading no spreading
Figure 2.8: Liquid phase spreading as a function of the contact angleθ for powder com-pacts with different packing factorsf0
created using the PPA in CFD meshing software, the actual meshing of the system would prove impossible and the solution applied by Guardo et al. and Nijemeisland would not be viable.
In any case, such a complex solution to the problem of melt percolation through the pellet structure is against the spirit of this thesis. As such a thermodynamic approach to the melt distribution, as proposed by Zheng and Lim , was adopted. Zheng and Lim determined maps of the spreading liquid as a function of the packing factor of the solid (base phase) particles and the contact angle of the liquid on the solid. They assumed that the solute particle, once melted, are spherical liquid droplets and that the base phase particles are rigid spheres. They then determined the critical radius ratio of the solute particles to the base phase particles. If the liquid to solid particle radius ratio fell below this critical ratio then the melt was defined as not spreading, i.e. the melt would not percolate through the base phase matrix. If, on the other hand, the critical ratio was exceeded then the melt would spread through the base phase matrix. An example of their map curve is shown in Figure 2.8, with different curves for different packing factors (f0). If the liquid to solid ratio is below the line for the relevant packing factor then the
liquid will not spread, and vice versa.
While the model of Zheng and Lim  is well suited to the application in this thesis, it cannot be used as is due to the fact that it is derived for powder compacts where the
Figure 2.9: Graphical representation of the two liquid spreading mechanisms, black=liquid (solute), grey=solid (base), white=porous. (a) A molten solute particle em-bedded in a matrix of base particles. (b) Liquid generated by the molten solute particle spreading through the base phase matrix.
solute and base particles, while different in diameter from one another, are mono-sized. This is not the case in this thesis, and as such the model needs to be adapted accordingly. This adaptation is made easier by the fact that a virtual pellet exists from the PPA output, and as such a numerical solution of the problem can be obtained.
The assumptions and methodology used by Zheng and Lim are nonetheless retained. They state that in a particulate assembly where you have a solute and base phase, once the solute phase melts (and in the absence of particle re-arrangement) it will behave in one of two ways, as shown in Figure 2.9. In Figure 2.9(a) the melt does not spread through the base phase matrix, and thus retains it original spherical shape. For this scenario the liquid-solid interfacial area is negligible and as such the free energy of the system is expressed by Zheng and Lim as:
Ga= 4πr2lγlv+ n0 4 3πR 3 SC− 4 3πr 3 l 4πr2sγsv (2.18)
whererlandrsare the radii of the liquid sphere and the base phase particle
respec-tively. The liquid-vapour and solid-vapour interfacial tensions are represented byγlvand
γsvrespectively.RSC is the radius of the sample cell encompassing the particles andn0
is the number of base phase particles per unit volume. This expression is inconvenient for this study, due to the fixed size ofrl. Dependency onrlandrsin Eq. 2.18 is however
easily removed by using the surface area of the liquid (Sl) and solid (Ss) phases, both of
Ga = Slγlv+ Ssγsv (2.19)
Figure 2.9(b) shows the scenario where the molten solute phase does spread through the base phase matrix through capillary action. Here, for simplicity, Zheng and Lim as-sume that the volume of the original solute particle is equal to the total volume of the void space in between the base phase particles and that the total volume of the sample cell remains unchanged after spreading. Here the solid-vapour interfacial area is negligible and so the free energy of the system, when removing the dependency onrlas before, is
Gb = (SSC + Sp) γlv+ Ssγsl (2.20)
whereSpis the surface area of the pore in the centre of the sample cell andγslis the
solid-liquid interfacial tension. Combining Eqs. 2.19 and 2.20 for spreading conditions Gb− Ga< 0:
SSCγlv+ Spγlv+ Ssγsl− Slγlv− Ssγsv< 0 (2.21)
in Eq. 2.21,Sp(surface area of white region in Figure 2.9(b)) is equal toSl(surface
area of black region in Figure 2.9(a)), rearranging thus results in: SSC
(2.22) Young’s equation relates the interfacial energies using the contact angleθ, cos θ = (γsv− γsl)/γlv. Eq. 2.22 thus becomes:
< cos θ (2.23)
This equation was used to draw up the curve in Figure 2.10, which is similar to that shown in Figure 2.8 with the dependency on rl removed. This type of curve can now
be used to determine the dominant spreading mechanism for the pellets in this thesis. Once the spreading mechanism is determined, the algorithm which generates the com-positional PDF of the pellets (Chapter 5) can be constructed accordingly.
The spreading mechanism is determined by creating numerous virtual samples cells (SCs) of random position and volume within the pellet space, and then calculatingSSC
0 10 20 30 40 50 60 70 80 0 1 2 3 4 5
Contact angle (Degrees)
Critical surface area ratio
Figure 2.10: Liquid phase spreading as a function of the contact angleθ with the depen-dency ofrlremoved
is shown that for each packing configuration investigated in this thesis more than 99% of the SC’s generated display liquid spreading characteristics.
In this chapter the relevant thermodynamic, kinetic and physical properties of the Cu-Ni system have been summarised. The mathematical description of these properties given in this chapter allow one to determine the quantity, rate of solidification and spreading of a TLP formed in a Cu-Ni particulate assembly, so long as a compositional distribution of the green pellet is available. The construction of such a distribution, as a function of the particle size distribution and pellet porosity, is the focus of the following three chapters.
Da Apparent diffusivity in the base metal particle [m2/s].
Db Grain boundary diffusivity [m2/s].
Dl Lattice diffusivity of solute in base metal [m2/s].
Ga Free energy of the particulate assembly with no spreading [J].
Gb Free energy of the particulate assembly with spreading [J].
Mf Solute content att=∞ [-].
Mt Solute content at timet [-].
RSC Radius of the sample cell [m].
Sl Surface area of the molten phase [m2].
Ss Surface area of the solid phase particles [m2].
SSC Surface area of the sample cell [m2].
T Temperature [°K].
Ta Melting point of pure A [°C].
To Operating temperature [°C].
WA Mass fraction of the solute (liquid) phase [-].
WB Mass fraction of the base (solid) phase [-].
WA0 Initial mass fraction of solute [-].
WBf Mass fraction of the base phase att=∞ [-].
XA Solute content in constituent A [-].
XB Solute content in constituent B [-].
XL Liquidus composition expressed as solute mass fraction [-].
XO Bulk solute content [-].
XS Solidus composition [-].
XBf Solute concentration in base phase att=∞ [-].
a Base metal particle radius [m]. d Average base metal grain size [m].
k Geometric constant between 0.5 and 1.5 depending on the grain geome-try [-].
n0 Number of base phase particles per unit volume.
rl Radius of molten solute particles [m].
rs Radius of the base phase particles [m].
t Time [s].
δ Grain boundary width [m].
γlv Liquid-vapour interfacial tension [J/m2].
γsl Solid-liquid interfacial tension [J/m2].
γsv Solid-vapour interfacial tension [J/m2].
ρ Density [kg/m3].
θ Contact angle [◦
Coordination number model
The CN model is crucial in controlling the structure of the virtual pellets created us-ing the PPA. The model, which will be fully described in this chapter, was formulated by Suzuki and Oshima [41–43]. Consid-ering how crucial the CN model is to the work in this thesis a rigorous validation was required, something which could not be achieved with previously used meth-ods for experimental CN measurement. A novel method using a CT scanner and custom written image analysis software was created. The validation was done on numerous cylindrical packed beds with
different size distributions of spherical glass beads. This non-destructive and repeatable approach proved very successful and the validity of Suzuki and Oshima’s model was confirmed .
In the past much work done on the construction of empirical models for the estimation of the average coordination number (CN) of packed beds made up of mono-sized spheres as a function of the overall porosity of the bed [45–52]. Multi-component mixtures of spheres have however received less attention, with much of the work being qualitative in nature . Suzuki and Oshima [41–43] derived a semi-empirical model for the esti-mation of the CN of spheres with size distribution as a function of porosity. The model was evaluated by the authors using a particle packing algorithm, as they considered this to be more accurate than the physical techniques available at the time (liquid bridging technique). The model was, however, never evaluated experimentally. As the CN model is to be a cornerstone of the PPA, it is necessary that there is a high level of confidence in its performance, which is not obtained by validation with a particle packing algorithm.
For this reason it was decided that the model derived by Suzuki and Oshima [41–43] should be evaluated experimentally using a CT scanner, which has not been done in the past. This technology, along with image analysis techniques allows for a large number of samples to be evaluated accurately and quickly in a non destructive and reproducible manner. Glass spheres of 5, 6, 7, 8 and 9mm were used to construct randomly packed cylindrical beds with different size distributions which were than evaluated using the CT scanner.
Model equation for estimating the co-ordination
Suzuki and Oshima proposed the following model for the estimation of the mean CN of spheres in a multi component packed bed . The mean CN,Nk, is defined as the
mean CN for all of the spheres within size intervalk:
Nk = o
Sa(l)· Nk,l (3.1)
whereo is the number of size intervals. Sa(l)is the fractional area of all the particles
in size intervall, which is calculated from the fractional volume Sv(l)and diameterDp(l)
Sa(l)= ml X i=1 Sv(i)/Dp(i) m X i=1 Sv(i)/Dp(i) = ml· D 2 p(l) m X i=1 Dp(i)2 (3.2)
wherem is the total number of particles and ml is the number of particles in size
intervall. Nk,lin Eq. 3.1 is the CN of a particle in size intervalk in direct contact with a
particle in size intervall.
Nk,l= 2αk D p(k) Dp(l) + 1 1 + Dp(k) Dp(l) − D p(k) Dp(l) D p(k) Dp(l) + 2 1/2 (3.3)
It must be noted that the value ofNk,lmust always be greater or equal to 2, as this is
the lowest CN for which a particle can be supported in a packed bed. The constantαkin
Eq. 3.3 is calculated using the mean CN ( eNk) of a one component packed bed consisting
only of particles from size intervalk.
4 = 0.067 eNk (3.4)
There has been much work done to construct empirical models which can estimate the CN of spherical particles in a one component packed bed. Suzuki et al.  proposed the following: e Nk= 2.812(1 − eǫk )−1/3 B Dp(k) 2" 1 + B Dp(k) 2# (3.5)
where the constantB is calculated using eǫk:
Dp(k) = 7.318 × 10 −2
+ 2.193eǫk− 3.357eǫ2k+ 3.194eǫ3k (3.6)
whereeǫk is the porosity of a packed bed comprising only of uniformly sized spheres
in size intervalk. This value is obtained experimentally as will be explained in Section 3.3. The contributions of other authors are summarised in Table 3.1:
Table 3.1: Correlations for average CN in a packed bed of mono-sized spheres
Equation Constraints Ref.
e Nk= 3.1/ǫ None  e Nk= 2 · e2.4(1−ǫ) None  ǫ = 1.072 − 0.1193 eNk+ 0.00431 eNk2 None  e Nk= 22.47 − 39.39ǫ ǫ ≤ 0.5  e Nk= 1.61ǫ−1.48 ǫ ≤ 0.82  e Nk= 26.49 − 10.73/(1 − ǫ) ǫ < 0.595  e Nk= 20.7(1 − ǫ) − 4.35 0.3 ≤ ǫ ≤ 0.53 
To test the models described in the previous section experimentally, packed beds were made up of glass spheres with five different diameters, namely 5, 6, 7, 8, and 9 mm. The glass spheres were housed in two cylindrical clear PVC sample holders (see Figure 3.1), which were identical in every aspect except the internal diameter. Two diameters were used in order to evaluate whether there was any noticeable effect on the results as a consequence of length to diameter ratio of the packed bed. The sample holders consisted of a plastic tube 140 mm in length, with the internal diameters of the two holders being 40 and 62 mm. The holders had a removable plastic lid which was fastened onto the plastic tube using wing nut bolts. The removable lid had a threaded hole in the centre through which a steel piston rod was screwed. The steel piston rod had a plastic disk with a diameter equal to that of the internal diameter of the tube connected to its end. This was used to compress the sample so that there was no movement of the spheres during scanning. Each sample was compressed as much as possible, until no further movement of the piston rod was possible. The volume of the sample space in which the spheres were packed could then also be easily measured, so that the porosity of the packed bed could be accurately calculated. In order to measure the value ofeǫk (as
mentioned in Section 3.2), the same procedure as described above was used, except that the sample in the holder was comprised of only mono-sized spheres. The mass of mono-sized spheres introduced into the holder for this measurement was the same as the mass of spheres used for the samples with size distribution being investigated. The porosity calculated for each holder is then the value ofeǫkfor spheres in size intervalk for
said holder. These values are shown in Table 3.2.
The samples were made up using three different types of size distribution (based on volume), namely log-uniform, Rosin-Rammler and Andreasen (Gaudin-Schuhmann)
Figure 3.1: Sample holder used for CT scans, the two diameters are for the small and large sample holders respectively
with three separate samples each made up for the Rosin-Rammler and log-uniform, and four samples for the Andreasen. The size distributions were made up by first setting the relevant parameters for each correlation, and then calculating the required mass of each distinct sphere size needed in order to obtain the required distribution.
The log-uniform cumulative distribution function appears linear when plot on semi-logarithmic axes, where the logarithm of particle size is taken:
D = A · log Dp Dp50 + 50 (3.7)
whereA is the slope of the straight line. The Rosin-Rammler distribution function is
D = 1 − exp − Dp De n (3.8) wheren is the distribution constant and Deis the size atD=0.632.
Table 3.2: Values ofeǫkper sample holder for spheres in each size interval
Size class eǫkper sample holder
(mm) D=40 mm D=62 mm 4.5-5.5 0.399 0.398 5.5-6.5 0.411 0.405 6.5-7.5 0.421 0.418 7.5-8.5 0.434 0.424 8.5-9.5 0.442 0.430
In the Andreasen distribution the maximum and minimum particle sizes are set as constant. This results in a distribution where the fraction of each particle size varies with the expansion of the distribution. This is contrary to the log-uniform and Rosin-Rammler distributions where the minimum and maximum particle sizes vary with the expansion of the distribution. The cumulative Andreasen distribution is defined by the following equation: D = 100 D p Dpmax q (3.9) whereq is the Fuller constant.
Each sample was subject to three scans, two in the large and one in the small di-ameter sample holder. This was done in order to ascertain the reproducibility of the results, and as mentioned previously, to see if there was any effect on the results due to the differing length to diameter ratio of the packed bed. Prior to the scans, the spheres were poured into the sample holder, the removable plastic lid was then bolted tight and the sample shaken vigorously to ensure that the spheres were well mixed. The steel piston rod was then screwed tighter pushing the plastic disk down onto the sample. Sub-sequently the holder was tapped repeatedly and the piston rod screwed tighter until no more movement was possible. The length of the sample space was then measured, after which the sample holder was placed horizontally onto the moving table of the CT scanner and the scan completed.
The experiments were carried out using a Siemens Volume Zoom CT scanner. The scan were done with a slice width of 1 mm using a tube voltage of 80 kV and a tube current of 218 mA. Once the scans were completed the spheres in each sample were counted by hand so that the results obtained from the image analysis could be verified.
Figure 3.2: Flowchart of algorithm used for image processing of CT scanner images
Once the scans were completed, the image data obtained were processed using the Image Processing toolbox of Matlab 6.5. A flowchart of the algorithm used to analyse the image data is shown in Figure 3.2.
Starting with the first image of the scan in which the spheres were visible, the global image threshold was calculated using Otsu’s method . This was done to convert the image to binary black and white (see Figure 3.3(a)). Once the thresholding was com-pleted the image was then eroded (see Figure 3.3(b)). This was done using a structuring element object, which in this case was a disk. After erosion the objects were identified and each object was dilated back to its original size. This scheme of erosion and dilation was done to ensure the correct identification of each separate object within an image, and thus prevent a situation where two spheres are identified as one object due to con-tact between them. Subsequently the properties of each object (centroid coordinates and equivalent diameter) were calculated.
These properties were used to identify the corresponding object (if any) in the next image. This was done by calculating the two dimensional Euclidean distance between
(a) Before erosion (b) After erosion
Figure 3.3: Black and white binary images before and after erosion
the centroids of all objects in consecutive images. If this distance was less than 0.4 mm, the objects were assumed to be part of one sphere. The scheme of erosion and dilation ensured that two spheres sitting directly over each other in the vertical plane (i.e. their x and y centroids were the same) were not identified as one sphere. This is because any cross sectional slice of the sphere which was below± 3.5 mm in diameter was eroded away, and thus not identified as an object and therefore not dilated back to its original size. This ensured that there was always at least one image in between spheres which had no object corresponding to either sphere.
Once all of the images in the scan were subject to these calculations, a numbered array with an entry for each identified sphere was created. Each entry in this array con-tained a sub-array in which thex and y coordinates and the equivalent diameter of all of the objects identified as belonging to this sphere were stored. The centroidx and y coor-dinates for each sphere were calculated by taking the arithmetic average of those values for each entry in the sub-array. These coordinates, along with the equivalent diameter, were used to compute the equation of the sphere and solve for thez centroid coordinate as well as the diameter. This was done using unconstrained non-linear optimisation . In order to compute the sphere equation, at least two rows of data were required in each sub-array.
These calculations resulted in an array containing thex, y and z centroid coordinates as well as the diameter of each sphere. Subsequently this was used to calculate the CN