A STRUCTURE-BASED VISCOSITY MODEL AND DATABASE FOR MULTICOMPONENT OXIDE MELTS
Guixuan Wu1, Sören Seebold1, Elena Zayhenskikh1, Klaus Hack2, and Michael Müller1
1Institute of Energy and Climate Research (IEK–2), Forschungszentrum Jülich GmbH, Wilhelm- Johnen-Straße, 52425 Jülich, Germany.
2GTT Technologies, Kaiserstraße 100, 52134 Herzogenrath, Germany.
Keywords: Molten slag, Thermodynamic modelling, Viscosity, Structure, Model Abstract
Based on the thermodynamic associate species model, a new viscosity model and database are currently being developed for the fully liquid system SiO2–Al2O3–CaO–MgO–Na2O–K2O–FeO–
Fe2O3–P2O5 and its subsystems in the Newtonian range. The modified Arrhenius model, employed in this database, is a structure-based viscosity model, in which the associate species link the viscosity to the internal structure of melts. Both the temperature- and composition- induced structural changes of melts are then described by a set of monomeric associate species in combination with the critical clusters induced by self- and inter-polymerization. The viscosity, therefore, is well described over the whole range of compositions and a broad range of temperatures by using only one set of model parameters, which have clear physico-chemical meaning. Moreover, the model is self-consistent, meaning the extension of viscosities from lower order systems to higher order systems is valid, and vice versa.
Introduction
The knowledge of viscosity as a function of temperature and composition has important practical applications to a variety of industrial processes, such as soldering, fusion casting of ceramic materials, metallurgical processes, glass production, and coal combustion and gasification. By way of example, slag viscosity is identified as a significant process variable for metallurgical smelting and refining and also plays a key role in determining the optimum continuous casting conditions [1, 2]. In the field of glass, glass viscosity is regarded as an important parameter for all stages of the manufacturing process. Successful melting and fining require viscosities below 20 Pa·s, whereas other processes such as blowing, pressing, drawing, and rolling require viscosities in the range from 102 to 106 Pa·s [3]. Accurate viscosity control improves the performance and efficiency of these processes, and for this purpose, viscosity-temperature- composition relationships are needed. Extensive viscosity measurements have been made, but it is not practicable to provide data for all possible combinations of temperatures and compositions encountered in practice. Moreover, measuring viscosity is not only time consuming but also expensive. These factors result in an obvious demand for viscosity models, by which viscosities are rapidly calculated from temperature and composition. Due to complicated nonlinear relationships between viscosity and composition, many early viscosity models such as the Lakatos model [4], Urbain model [5] and Riboud model [6] are only valid in a limited range of compositions. A new model is therefore developed to predict the viscosity of molten slags over
Advances in Molten Slags, Fluxes, and Salts: Proceedings of The 10th International Conference on Molten Slags, Fluxes and Salts (MOLTEN16) Edited by: Ramana G. Reddy, Pinakin Chaubal, P. Chris Pistorius, and Uday Pal TMS (The Minerals, Metals & Materials Society), 2016
the whole range of compositions. The methodology is outlined using the system SiO2–Al2O3– CaO–Na2O.
The viscosity model
The viscosity as a function of temperature and composition essentially depends on the structure.
A comprehensive description of structural dependence is therefore required for the development of a new viscosity model. The idea of the current model is to link the viscosity to the associate species distribution obtained from the thermodynamic modified associate species model [7], which has been successfully applied to describe the Gibbs energy of oxide melts. Due to the common structural basis of viscosity and Gibbs energy [8], the associate species in turn are assumed to be the common structural units. To effectively describe the structural dependence of viscosity, each associate species is transformed to a monomeric associate species (i.e. a discrete structural unit expressed with the formula that contains only one reference atom, marked in bold, in Table I). The monomeric associate species are defined as basic structural units, some of which can produce larger structural units (i.e. clusters) through self- and inter-polymerization, which cause a higher degree of roughness at atomic scale. Such increased roughness leads to an increased viscosity. This is because the nature of melt viscosity is an internal fluid friction. It is noted that the self- and inter-polymerization are taken into account only for silica and silicon- aluminium based ternary associate species such as CaSi2Al2O8. Such ternary associate species contain the Al3+-based quasi-tetrahedra, which can further self-polymerize and inter-polymerize with the silica tetrahedra. Only when the size of a resulting cluster reaches a critical value [9], does the viscosity increase significantly. For the self-polymerization, in the current model only two critical silica clusters are employed, while for each ternary associate species involved only one critical cluster is employed. For the inter-polymerization, only one critical cluster is incorporated into the model.
Table I. Associate species and structural units employed for the system SiO2–Al2O3–CaO
Compounds Associate species Structural units
SiO2 Si2O4 SiO2
Al2O3 Al2O3 AlO1.5
CaO Ca2O2 CaO
Al6Si2O13 1
4·Al6Si2O13 Al3SiO6.5
CaSiO3 CaSiO3 CaSiO3
Ca2SiO4 2
3·Ca2SiO4 Ca2SiO4
CaAl2O4 2
3·CaAl2O4 Ca0.5AlO2
CaSi2Al2O8 2
5·CaSi2Al2O8 Ca0.5SiAlO4
Each resulting structural unit has a relative contribution to viscosity, which can be described by the Arrhenius model. Consequently, the total viscosity of melts can be calculated by a linear combination of such relative contributions, given in Eq. (1).
lnη= ln ηideal+ ln ηexcess
= (∑iXi∙lnηi) + (lnηself−pol.+ lnηinter−pol.) (1)
where: lnηi= Ai+ Bi⁄T,
lnηself−pol.=∑j(A(SiO2)nj+ B(SiO2)nj⁄T)∙(XSiOnj 2) +∑k(A(Si−Al)k+ B(Si−Al)k⁄T)∙(X(Si−Al)nk k),
lnηinter−pol.=∑m(A(Si−Al)m+ B(Si−Al)m⁄T)∙(X(Si−Al)m∙XSiOnm2), j = 1, 2.
k = 1, 2, 3 … m = 1, 2, 3 …
(Si−Al)m and (Si−Al)k: the silicon-aluminium based ternary associate species.
ηideal and ηexcess are the ideal viscosity and the excess viscosity respectively; Xi is the mole fraction of the monomeric associate species i; ηi is the viscosity contribution from the monomeric associate species i; ηself−pol. is the excess viscosity resulting from the critical self- polymerizations; ηinter−pol. is the excess viscosity resulting from the critical inter- polymerizations; nj, nk, and nm are the integer coefficients that relate to a particular degree of polymerization; Ai and Bi are the temperature- and composition-independent constants, respectively, for the ideal viscosity; A(SiO2)nj, A(Si−Al)k, and A(Si−Al)m are the temperature- independent constants for the excess viscosity; B(SiO2)nj, B(Si−Al)k, and B(Si−Al)m are the composition-independent constants for the excess viscosity; T is the absolute temperature; XSiOnj 2, X(Si−Al)nk k, and (X(Si−Al)m∙XSiOnm2) are the weighting factors indicating relative contribution of each associate species involved to the excess viscosity. The weighting factor is derived from the mole fraction of the critical clusters. In view of chemical equilibrium, for example, there is a dynamic equilibrium between the monomeric associate species SiO2 and the critical silica cluster (SiO2)nj, as shown in Eq. (2).
njSiO2= (SiO2)nj (2)
The mole fraction of the critical silica cluster, therefore, can be calculated by the mole fraction of the monomeric associate species SiO2, as shown in Eq. (3).
X(SiO2)nj= Knj∙XSiOnj 2 (3)
where: X(SiO2)nj is the mole fraction of the critical silica cluster (SiO2)nj; Knj is an equilibrium constant for a particular degree of polymerization. Here, the mole fraction in place of activity is used to calculate the mole fraction of the critical silica cluster. To simplify the equation to estimate the excess viscosity, the equilibrium constant Knj is implicitly incorporated into the model parameters A(SiO2)nj and B(SiO2)nj. It should be noted that a possible dependence of Knj on temperature is ignored.
Results and discussion
Pure oxides
Figure 1 shows the calculated viscosities for pure oxides SiO2 and Al2O3 are in good agreement with the experimental data [10-15] and the viscosity decreases with increasing temperature. The temperature-induced structural change for the pure oxides Al2O3, CaO, and MgO is described using only the monomeric associate species. In contrast, the temperature-induced structural change for the SiO2 is described by the coexisting monomeric associate species SiO2 and two common critical silica clusters (SiO2)6 and (SiO2)109, which are obtained by fitting experimental data for both pure molten silica and the relevant SiO2-based binary systems.
Figure 1. Model performance for pure oxides (a) SiO2 and (b) Al2O3
It is noted that the viscosity of SiO2 is much greater than that of Al2O3. In a pure silica melt, the basic structural units (silica tetrahedra) can interconnect with each other and thereby produce various large structural units such as chain structures, ring structures, and network structures, which can cause a significant increase in viscosity. A pure alumina melt does not have such structural features.
Binary systems
The viscosity of molten silica decreases significantly when a small amount of network modifiers such as Al2O3 and Na2O is added into a pure silica melt, as shown in Figure 2. This is the so- called lubricant effect reported by Avramov et al. [16], in which the network modifiers play the role of lubricants allowing silica clusters to glide more easily past each other. This challenging viscosity behavior can be described by the current model. For the description of the composition- induced structural change in the system SiO2–Al2O3, the binary associate species Al6Si2O13 is used in addition to the monomeric associate species for the pure oxides and two common critical silica clusters (SiO2)6 and (SiO2)109. Then, the calculated viscosities at temperatures from 1800
oC to 2000 oC sufficiently agree with the experimental data [11, 12, 14, 17]. In contrast, the structural change in the system SiO2–Na2O is more complex and three binary associate species Na4SiO4, Na2SiO3, and Na2Si2O5 are therefore employed. The experimental data [18-22] are
consequently well reproduced with the current model. In addition to the lubricant effect, another dramatic drop in viscosity (the so called weak lubricant effect due to the possible ring structures [23]) occurs in somewhere in the middle (see Figure 2(b)), which is also well predicted.
Figure 2. Model performance for the binary systems (a) SiO2–Al2O3 and (b) SiO2-Na2O Moreover, the viscosity extension from the binary system SiO2–Al2O3 and SiO2–Na2O to the pure SiO2 is valid. In contrast, the viscosity values extrapolated from different SiO2-based binary systems to the pure SiO2 are different in the model of Zhang and Jahanshahi [24]. This indicates the current model is self-consistent.
Multicomponent systems
The viscosity of the system SiO2–CaO–Na2O is extrapolated from the corresponding lower order systems SiO2–CaO and SiO2–Na2O, where the ternary associate species Na2Si6Ca3O16 is employed. As shown in Figure 3(a), the calculated viscosities agree well with the experimental data [25-27]. For a constant SiO2 content of 0.75 mole fraction, the viscosity increases when replacing Na2O with CaO at temperatures from 1100 oC to 1300 oC. This indicates that Na2O has a greater ability to decrease the viscosity than CaO in the system SiO2–CaO–Na2O. When the model is extended to the system SiO2–Al2O3–CaO, the ternary associate species CaSi2Al2O8 is introduced to describe the Al3+-induced structural change. The monomeric associate species Al2O3, Al6Si2O13, and CaAl2O4 for the pure oxides and binary systems are also included. The associate species Al2O3 and Al6Si2O13 behave as network modifiers, whereas the associate species CaAl2O4 and CaSi2Al2O8 behave as network formers, which are employed to describe the charge compensation effect. Figure 3(b) shows that the viscosity maximum for the system SiO2– Al2O3–CaO is well described by the current model for a constant SiO2 content of 0.5 mole fraction at temperatures from 1600 oC to 2000 oC. Using the current model, the calculated viscosities are in good agreement with experimental data [11, 17]. It is noted that the position of the viscosity maximum is slightly shifted towards the Al2O3-rich side with increasing temperature. The viscosity maximum tends to be less pronounced due to lower stability of the Al3+-based quasi-tetrahedral structures. Moreover, increased separation of the viscosity isotherms occurs when approaching the fully charge-compensated composition. Similarly, the
ternary associate species NaSiAlO4, and NaSi3AlO8 are employed to describe the Al3+-induced structural change in the system SiO2–Al2O3–Na2O, the experimental data [28-32] are then reproduced well with the current model, as shown in Figure 3(c).
Figure 3. Model performance for the systems (a) SiO2–CaO–Na2O, (b) SiO2–Al2O3–CaO, (c) SiO2–Al2O3–Na2O and (d) SiO2–Al2O3-CaO–Na2O
When the model is extended to the system SiO2–Al2O3–CaO–Na2O, no additional associate species and model parameters are employed. The calculated viscosities still agree well with the experimental data [33-36], as shown in Figure 3(d). It is noted that the viscosities of the three slags follow the sequence: slag 1 > slag 2 > slag 3, which is consistent with the sequence of the sum of the SiO2 and Al2O3 contents although the amount of the charge compensators for Al3+
might be not enough for slag 1. Due to the charge compensation effect, Al2O3 behaves as a network former and the effective concentration of SiO2 is increased, both of which lead to an increase in viscosity.
Conclusion
A new structure-based model, designated as the modified Arrhenius model, has been developed to describe the viscosity of multicomponent oxide melts. In the new model, the structural dependence of the viscosity is successfully described by using the associate species distribution and the critical clusters derived from the associate species. The new model provides a reliable prediction over the whole range of compositions and a broad range of temperatures using only one set of model parameters, all having clear physico-chemical meaning. The challenging viscosity behaviors such as the lubricant effect and charge compensation effect are well captured.
Moreover, the model is self-consistent. The model parameters have been assessed for the system SiO2–Al2O3–CaO–MgO–Na2O–K2O–FeO–Fe2O3 and further relevant oxides such as P2O5 are currently being assessed and incorporated.
Acknowledgments
The work described in this article has been performed in the framework of the HotVeGas Project supported by Bundesministerium für Wirtschaft und Technologie (FKZ 0327773) and the HVIGasTech Project supported by the Helmholtz Association of German Research Centers (VH–VI–429).
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