Hybrid Particle Laden Flow Modelling

Inhaltsverzeichnis

1 Introduction and motivation
1.1 Numerical simulation of particle-laden flow
1.1.1 Numerical modelling of dilute flows
1.1.2 Numerical modelling of dense flows
1.1.3 Numerical modelling of intermediate dilute/dense particle-laden flows
1.2 Aim of this thesis
1.3 Organization of this thesis

2 Eulerian granular phase modelling
2.1 Continuity equation
2.2 Momentum balance
2.3 Granular temperature
2.4 Radial distribution function
2.5 Drag coefficient and interphase momentum exchange
2.5.1 Wen and Yu
2.5.2 Gidaspow
2.5.3 Huilin and Gidaspow
2.6 Solids Stresses
2.6.1 Kinetic and collisional stresses
2.6.2 Frictional stresses
2.7 Turbulence modelling
2.8 Boundary Conditions
2.8.1 Johnson and Jackson
2.8.2 Li and Benyahia
2.8.3 Jenkins and Louge
2.8.4 Schneiderbauer et. al

3 Lagrangian discrete phase modelling
3.1 Force balance and torque balance
3.2 Forces on a particle
3.2.1 Drag force
3.2.2 Particle rotation and Magnus force
3.2.3 Saffman force
3.2.4 Additional forces
3.3 Torque
3.4 Turbulent fluctuations
3.5 Particle wall collisions
3.5.1 Restitution coefficient model
3.5.2 Rough wall-particle collisions

4 The hybrid model EUgran+Poly
4.1 Motivation and overview
4.2 Coupling and exchange forces
4.3 Coupling forces on the Eulerian granular phase
4.3.1 Magnus force
4.3.2 Particle-wall interaction
4.3.3 Modified drag law for poly-dispersity
4.4 Coupling forces on the Lagrangian tracer particles
4.4.1 Collisional particle-solid force
4.4.2 Granular pressure force
4.4.3 Collisional torque
4.5 Simulation sequence and implementation

5 Agglomeration
5.1 Simple models
5.1.1 Agglomerated filling
5.1.2 Linear agglomeration
5.2 Particle population balance equation
5.2.1 Assumptions
5.2.2 Collision rates
5.2.2.1 Kinematic collision rate
5.2.2.2 Brownian collision rate
5.2.2.3 Turbulent collision rate
5.2.2.4 Comparison of collision rates
5.2.3 Effective collision rate
5.2.4 Sticking probability
5.3 Bus stop model
5.3.1 Implementation
5.4 Volume population balance model

6 Validation by lab-scale experiments
6.1 Dilute poly-dispersed flow in a duct
6.1.1 Boundary conditions and simulation setup
6.1.2 Results and discussion
6.2 Mono-dispersed flow in a medium laden duct
6.2.1 Boundary conditions and simulation set up
6.2.2 Results and discussion
6.3 Agglomeration of poly-dispersed particulate flow in a vertical pipe

7 Application to cyclone separation
7.1 Hybrid Model
7.1.1 Boundary conditions and simulation setup
7.1.2 Results and discussion
7.1.3 Results and discussion for separation of limestone material
7.1.4 Discussion of computational efficiency
7.2 Agglomeration

8 Conclusions and Outlook

A Restitution coefficients are no constants

B Computation of Lagrangian particle-wall collision

C UDF Structure of hybrid model

D Cyclone dimensions based on Muschelknautz theory

E Nomenclature

F Curriculum Vitae

List of Figures

List of Tables

Bibliography

Abbreviations

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Eidesstattliche Erklärung

Ich erkläre an Eides statt, dass ich die vorliegende Dissertation selbstständig und ohne fremde Hilfe verfasst, andere als die angegebenen Quellen und Hilfsmittel nicht benutzt bzw. die wörtlich oder sinngemäß entnommenen Stellen als solche kenntlich gemacht habe. Die vorliegende Dissertation ist mit dem elektronisch übermittelten Textdoku- ment identisch.

Graz, am 29. Jänner 2013

David Schellander

Acknowledgements

Completing my PhD degree was probably one of the most challenging activities of my life. It was a great time to spend several years in the Christian-Doppler Laboratory on Particulate Flow Modelling and the Department on Particulate Flow Modelling at Johannes Kepler University of Linz, and its members will always remain dear to me.

Special thanks go to

- Privatdozent Dipl.-Ing. Dr. Stefan Pirker,
- Dipl.-Ing. Mag. Dr. Simon Schneiderbauer,
- Ulrich Voss,
- the team of the CD-Laboratory on Particulate Flow Modelling,
- my wife Ulrike Schellander,
- my family,
- my uncle Karl-Heinz Schellander(+).

Furthermore, the author want to thank Evan Smuts (Department of Chemical Engineering, University of Cape Town, Africa),who thoroughly reviewed this work.

This work was funded by the Christian-Doppler Research Association, the Austrian Federal Ministry of Economy, Family and Youth, the Austrian National Foundation for Research, Technology and Development.

Danksagung

In den Jahren in denen ich an meiner Dissertation gearbeitet habe ist viel passiert, gutes und schlechtes, fröhliches und trauriges und all dies hat mich zu dem Punkt gebracht wo ich heute stehe. Viele Menschen haben mich auf dem Weg begleitet, manche die gesamte Zeit, manche nur Teile davon. Einige wurden zu echten Freundinnen und Freunden, hiermit möchte ich euch allen dafür danken.

Besonders hervorheben möchte ich

- Privatdozent Dipl.-Ing. Dr. Stefan Pirker,
- Dipl.-Ing. Mag. Dr. Simon Schneiderbauer,
- Ulrich Voss für die fachlichen Diskussionen und gute Zusammenarbeit,
- das gesamte Team des CD-Labors für partikuläre Strömungen an der Johannes Kepler Universität,
- die Katholische Hochschuljugend, das Forum Sankt Severin und die Redaktion der QUART,
- meine gesamte Familie,
- meine Ehefrau Ulrike Schellander,
- meinen Onkel Karl-Heinz Schellander(+), dem ich besonders danke für die schönen und diskussionsreichen Stunden, oft bis spät in die Nacht.

Einen speziellen Dank an Evan Smuts (Department of Chemical Engineering, University of Cape Town, Africa) für sein Feedback und das Korrekturlesen dieser Arbeit.

Kurzfassung

In dieser Dissertation wurde das numerische hybrid Modell EUgran+ [Pirker et al., 2010] modifiziert und erweitert. Das EUgran+ Modell ist ein Euler-Euler granular Mod- ell mit zusätzlichen Informationen vom diskreten Euler-Lagrange Modell. Die Haupt- modifikationen betreffen die Implementierung von I) einem neuem Widerstandsmodell für poly-disperse Partikelverteilungen II) eine neue Wandbehandlung für Partikel in der soliden Phase des Euler-Euler Modells und III) eine neue Implementierung der Populationsbilanzgleichung innerhalb der Euler-Lagrangen Modellierung. Mit diesen Modifikationen ist das EUgran+ Modell zum EUgran+Poly Modell erweitert geworden, welches verwendet werden kann um poly-disperse Partikelströmungen im Bereich der pneumatischen Förderung und Separation am Computer zu simulieren. In dieser Dok- torarbeit werden 3 spezifische Anwendungen mit unterschiedlichen Massenbeladungen simuliert um das EUgran+Poly Modell zu validieren: I) pneumatische Förderung von poly-disperse verteilten Partikeln in einem rechteckigen Kanal mit 90 ◦ Krümmer bei geringer Massenbeladung mit L = 0 . 00206; II) eine pneumatische Förderung von mono- dispersen Partikeln durch einen Doppellooping zur Validierung der Partikelsträhnenbil- dung bei Beladung L = 1 . 5; III) eine Fallröhre zur Untersuchung der Implementierung der Populationsbilanzgleichung im Agglomerationsfall. Weiters, um die Anwendbarkeit des Modells zu zeigen, wurde ein Industriezyklon in Experimentalgröße simuliert und die Ergebnisse mit Messungen verglichen. Die Validierungen und Anwendung zeigen, dass vor allem die Verwendung eines poly-dispersen Widerstandsmodell für die Partikel sehr wichtig und einen großen Einfluss auf die Simulationsergebnisse hat. In dieser Ar- beit wurde gezeigt, dass im Bereich der partikulären Strömungen der hybride Ansatz gut geeignet ist für die Simulation von industriellen Anwendungen.

The numerical hybrid model EUgran+ [Pirker et al., 2010], which is an Eulerian- Eulerian granular phase model extended with models from the Eulerian-Lagrangian model for dense rapid particulate flows, is modified to account for poly-dispersed par- ticle diameter distributions. These modifications include the implementation of I) a new poly-dispersed drag law and of II) new particle boundary conditions distinguishing between sliding and non-sliding particle-wall collisions and III) a new implementation of the population balance equation in the agglomeration model using the Eulerian- Lagrangian approach, referred to as Bus-stop model. Further, the applicability of the EUgran+ model is extended to cover dilute to dense poly-disperse particulate flows. Furthermore, this provides an improvement in the numerical simulation of dust sepa- ration and the formation of particle strands in industrial scale cyclones. In this PHD thesis, the EUgran+Poly model is validated at 3 specific cases with different mass load- ings: I) poly-dispersed particle conveying in a square pipe with a 90 ◦ bend at low mass loading (L = 0 . 00206); II) a particle conveying case in a rectangular pipe with a double-loop at high mass loading (L = 1 . 5); III) in a vertical pipe the implementation of the agglomeration model is validated. To show the applicability of the presented models a simulation of an industrial cyclone in experimental scale is presented. The validation and application shows that considering a poly-disperse Eulerian-Eulerian granular phase improves the accordance of the simulation results with measurements significantly. Finally, the hybrid model is a good compromise for a computational ef- ficient simulation of particulate transport and separation with different mass loading regimes.

Introduction and motivation

Particle-laden flows are present all over the world. They can be found in our environ- ment, in life science and in industrial processes. Sandstorms in the desert, wind blowing up leaves and dust, sediment transport in river beds are only a few examples of environ- mental flows. In life science blood flow can be considered as particle-laden flow, with the blood cells representing the particles. Moreover, many industrial processes involve particle-laden flows including the conveying of particles in transport pipes, the separa- tion of dust in cyclones and filters or heating and reaction of particles in countercurrent gas flows.

Also, in cement industry, particle-laden flows are dominant. At the beginning of the cement production process, lumpy stones are milled to small particles. The resulting particle diameters are so small that the powder is prone to particle cohesion and subse- quentially agglomeration. Next, this limestone powder is introduced into the top of the preheating tower (Fig. 1.1). The preheating tower resembles a system of interconnected cyclones, which should guarantee that the downward stream of particles is heated up by the countercurrent hot exhaust gas efficiently. At the same time any particle loss by the exhaust gas should be avoided. Finally, at bottom of the preheating tower the conditioned particles enter the calciner.

Figure 1.1: Picture of a preheater tower (www.polysius.de)

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Understanding the physical behavior of this particle-laden flow is crucial for the overall process efficiency. The actual realization of the downward particle flow has a direct impact on particle losses and offgas temperature. Obviously, this depends on the specific design and geometry of the components of the preheating tower.

Commonly, three main investigation methods for the behavior of particles in a gas flow are reported in literature: analytic or empirical considerations, experiments and numerical simulations.

A lot of empirical correlations are available for conveying and separation of dust in simplified geometries. In the case of straight conveying lines the movement of solid bulk material can be estimated by simple force balances (e. g. Muschelknautz [2010]). In cyclones the separation of dust particles can be pictured by either trajectory models or by separation surface models. A review of these methods can be found in Hoffmann and Stein [2008]. Generally all these methods are restricted to simplified geometries and boundary conditions. For example, in case of cyclone separation, it has been reported that the overall separation efficiency depends on the geometry and orientation of the upstream conveying line. If the upstream duct is curved shortly before the cyclone entrance the overall separation efficiency will be affected [Abrahamson et al., 2002, Pirker and Kahrimanovic, 2007]. None of the available empirical correlations are able to account for this dependency. Nevertheless, the system of interlinked cyclones in an industrial preheating tower consists of a lot of complicated geometries which can not be modelled by standard correlations. Therefore, in our case, analytical considerations can only be used for highlighting the underlying physical phenomena.

Much knowledge of the behavior of conveying and separation of powder was gained by experiments. In principle, experiments in an industrial cement production plant could provide valuable insights into the particle flow behavior of the real process. Nev- ertheless, in the case of the preheating tower, any experiments must be conducted in a very harsh environment. The tower is high, the gas is hot and the platforms are not weather proof. Another crucial point is that experiments should not impair the continuous cement production. On-site experiments are very time consuming, nearly al- ways expensive and a logistical challenge. Therefore, experiments are commonly done at laboratory-scale. In this case a big challenge is that the physical flow regimes of the lab-scale experiments agree with the behavior of the real industrial process. Nev- ertheless, in case of the preheating tower this requirement can hardly be met. For instance, if the gas flow velocity is adjusted such that the conveying conditions are the same as in the industrial plant, the resulting centrifugal forces on the particles inside a down-scaled cyclone are orders of magnitude larger than in the corresponding industrial cyclone. Consequently, experimental findings have to be evaluated carefully prior to drawing conclusions for industrial plants. Furthermore, even lab-scale experiments are always expensive and time consuming. Therefore, experimental methods are on the one hand limited in their predictability, and on the other hand they are not flexible enough for short term design issues.

Numerical simulations are pooled under the catchphrase Computational Fluid Dynam- ics (CFD). They provide flexible, fast and cheap methods for predicting the behavior of particle-laden flows. The method is based on approximation for solving the well estab- lished Navier-Stokes flow equations. In principle this method can be applied to arbitrary geometries. CFD software packages are available as open source (e. g. OpenFOAM ®) and as commercial products (e. g. ANSYS FLUENT). Common numerical simulation approaches are organized into Lagrangian and Eulerian methods. The Lagrangian methods compute the movement of individual particles and their interaction with the surrounding fluid. In contrast to that, Eulerian methods assume that the multitude of particles behaves as an artificial particulate phase that interacts with the fluid phase. Using well established models guarantees that simulation results are consistent. In the case of the preheating tower, dense and dilute particle-laden regions are to be expected. Especially in case of dense laden flows numerical models are still under investigations and simulation results have to be taken with care. Therefore, an in depth knowledge of the simulated process and a careful interpretation of the simulation results is im- portant. Consequently, CFD simulations have to be checked for plausibility and they should always be validated prior to application.

1.1 Numerical simulation of particle-laden flow

In numerical simulation of particle-laden flows, the most important challenge is that different flow regimes are governed by completely different physics. Commonly particle- laden flows are classified in dilute, medium and dense flow regimes [Dartevelle, 2003]. Therefore, the decision which regime is present is judged by the volume fraction

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The volume fraction is described by the ratio of particle/solid volume V s in a specific volume V cell. The ratio has an upper threshold which is given in a packed bed of mono-dispersed spherical particles by α max,p = 0 . 64 [Lun et al., 1984]. Hence, there is always a minimum of 36 % fluid inside the specific volume. In numerical simulation the volume fraction can be evaluated for each computational cell. Following Elghobashi

Figure 1.2: Sketch of particle-laden flow regimes

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[1994] and the literature cited therein, Figure 1.2 shows the different possible particle- laden flow regimes. Dilute particulate flows are characterized by a volume fraction α s of particles lower than 10 − [6]. In this regime only a coupling from fluid to the particles is important (one way coupling). Particle trajectories can be calculated independently. Therefore, just the kinetics of particles are taken into account. In medium-density particle flows, 10 − [6] < α s < 10 − [3], the influence of the particles on the surrounding fluid should be considered (two way coupling). Additionally, the increasing number of particle-particle collisions must be taken into account for α s > 10 − [3] (four way coupling). With increasing volume fraction the influence of collisions on the behavior of the flow increases. In dense particle flows, α s > 0 . 5 the friction between particles becomes important for overall flow behavior. Following this classification of particle-laden flows,

Figure 1.3: Sketch of single particle that is followed by time.

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methods for dilute and dense particle-laden flows have been developed. In case of the preheating tower, nearly all particle-laden flow regimes can occur within the process. Hence, a numerical simulation model which can handle dilute to dense particle flow regimes is needed.

1.1.1 Numerical modelling of dilute flows

As defined previously, in dilute gas solid flows the particle movement depends only on the flow. Hence, the influence of particles on the gas flow and inter-particle collisions are small and can be neglected. Every single particle can be considered independently by Newton’s second law, given by

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where m p denotes the mass of the particle, up the particle velocity, t the time and ∑F p the sum of forces acting on the particle. With this force balance, the motion of each particle inside the flow can be traced in time (Figure 1.3). Nevertheless, with increasing density of particles (α s > 10 − [6]) the method has to be extended in order to account for the influence of the particles on the gas flow. This approach is named Particle-Source-In Cell model [Crowe et al., 1977] and provides exchange terms from particles to the fluid phase. In doing so, it regards the discrete particle phase as a source of mass, momentum and energy to the fluid phase. As a next step, particle-particle interactions should be included to the simulation model. This is especially important if the particles under consideration are prone to agglomeration. There are two possibili- ties to achieve this. The collisions can be included either by a stochastic consideration [Oesterlé, 1993] or by deterministic methods [Cundall and Strack, 1979]. Stochastic methods compute the collisions of a particle with virtual collision partners based on the probability for a collision. The stochastic method was developed by Oesterlé [1993] and extended and validated by Sommerfeld [1996], Frank [2002] and Hussmann et al. [2007]. A brief introduction and comparison of the models can be found in Rao and Nott [2008]. Deterministic models resolve each collision of a particle by searching for real collision partners in the near region of the particle. Simulation methods using the deterministic collision detection are named Discrete Element Methods (DEM). For example, this method is used in the open source code LIGGGHTS [Kloss, 2011]. This DEM code has been coupled to OpenFOAM to account for the interstitial gas flows [Go- niva et al., 2012]. This coupling is distributed at www.cfdem.com. An advantage of the discrete phase model is that every impact force on a particle is resolved. Additionally, it is straight forward to include particle rotation (Magnus force) and to model particle reflection at walls. However, in the case of DEM the computational efforts rise by the square of the number of particles. Hence, the approach is limited by computational re- sources, even new computers can only handle up to few million particles. Finally, with respect to the preheating tower, the DEM method will fail due to an excess number of particles inside the computational domain. Even in case of the stochastic approach there are still to many particles involved. Therefore, this case of dense particle-laden flow can not be simulated with a pure Lagrangian modelling approach.

1.1.2 Numerical modelling of dense flows

As mentioned before, dense particle-laden flows depend on the gas flow, particle-particle collisions and for flows near the packing bed volume fraction, the friction between par- ticles. Obviously, this kind of flow contain an immense number of particles. In many cases it seems as they act like an independent phase. Hence, a concept named par- ticular phase was developed and introduced [Ishii, 1975]. This method considers the particles as a continuous medium. Hence, the discrete particle properties are replaced by quantities of velocity, density and volume fraction for the particle phase. These quantities are assumed to be smooth functions of position and time. Therefore, a mass and force balance can be built for the particle medium inside a fixed control volume [Agrawal et al., 2001] (Figure 1.4). With this approach a multi-phase Navier-Stokes equation model can be taken to model granular flows [Gidaspow, 1994]. Thus, specific correlations for granular temperature and granular stresses have been deduced from ki- netic theory. Furthermore, a momentum exchange term between the fluid and granular phases has to be included. An introduction to this model, including a discussion of its sub-models, can be found in Gidaspow [1994], Brilliantov, N. V. and Pöschel, T. [2004] and Rao and Nott [2008]. The advantage of this model is that for each phase just a set of balance equations has to be solved. In case of a two-phase gas and solid flow the computational effort is rather low. This continuum model is a good choice for particle- laden flows which are dominated by inter-particle collisions. Negative aspects of this model are that the Magnus force and realistic particle-wall interactions are currently not included. Furthermore, poly-dispersed flows can only be handled by computing a continuum particulate phase for each diameter. This would provide the chance to include the famous population balance equation of Smoluchowski [1917] for the compu-

Figure 1.4: Sketch of granular flow through fixed control volume

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tation of agglomeration, but as a consequence, a new set of balance equations must be solved for each particle diameter. This increases the computational effort dramatically. In the case of the preheating tower example, the continuous phase method seems to be a good choice. The model counts for many of the physical effects in the convey- ing and separation of dense particle-laden flows. But, there are still some questions remaining regarding the Magnus force, the particle-wall treatment and poly-dispersed particle-laden flow handling.

1.1.3 Numerical modelling of intermediate dilute/dense particle-laden flows

The transport of powder, as it is observed in the preheating tower, contains a huge number of small particles which are arranged in dilute and dense particle-laden flow situations. While in some regions strand formations are observed, in other regions a dispersed flow is present. This calls for an approach that considers regime changes in particle-laden flows. To achieve this, a combination of dilute and dense flow models can be used (Figure 1.5). Two methods are conceivable, an embedded method [Zwinger, 2000, Pirker and Kahrimanovic, 2007] and a joint domain method [Pirker et al., 2010]. In the embedded method the computational domain is divided into two distinct re- gions. In dilute laden regions it computes the particulate flow with a numerical model for dilute flows. In all other regions a numerical model for dense flow is used. For this approach the volume fraction α s in the geometry must be estimated a priori. In contrast to this, the joint domain model computes both dilute and dense particle flow approaches in the whole domain and weights the impact of each method according to the actual volume fraction. Remembering the preheating tower, the positions of dilute or dense flow regimes are not known a priori. Therefore, a joint method should be taken for the simulation of this industrial application. For reasons of simplicity, this joint domain combination will be abbreviated as hybrid model throughout this the- sis. These hybrid models are currently under development and intended for modelling

Figure 1.5: Sketch of particle-laden flow regimes; depicting the applicability of (I) La- grangian particle tracing, (II) Eulerian particulate phase model and (III) hybrid particle-laden flow modelling industrial applications containing particle-laden flows, where different flow regimes are present.

1.2 Aim of this thesis

This thesis aims at further developing a joint domain hybrid model for picturing particle-laden flows with locally and time dependent changing flow regimes. Therefore, special emphasis should be given to poly-dispersity, particle-wall boundary conditions and particle rotation. Furthermore, a concept for incorporating the effect of agglomer- ation should be suggested as an extension of the developed hybrid model. Finally, this hybrid model will be applied to particle-laden flows in the cement industry. In industri- ally used numerical simulation codes the numerical stability is important. A modular concept should be used in the design and programming of the method. Then, there should be the possibility to change individual sub-models in a simple way. Finally, the model should be open for further extensions in future, e. g. including the consideration of heat transfer and chemical reactions.

1.3 Organization of this thesis

This thesis is organized in three main steps: model development, model validation and model application (Figure 1.6). Sticking to the work plan this thesis starts with model development, which consists of four chapters. In Chapter 2 the numerical modelling of dense particle-laden flows with the Eulerian granular phase model is presented. Later on in Chapter 3 a numerical model for single particle trajectory calculation - the Euler- Lagrangian discrete phase model - is shown. In Chapter 4 a hybrid model for poly- dispersed granular flows, entitled EUgran+Poly, is presented. Furthermore, in Chapter 5, a concept study on agglomeration modelling is presented including a new imple- mentation method of the population balance equation by Smoluchowski [1917]. Later on in Chapter 6 the hybrid and agglomeration models are tested by three validation examples. First, based on the work of Mohanarangam et al. [2007], the hybrid model is validated for the poly-dispersed, dilute particle flow case in a duct bend. Secondly, based on the work of Pirker et al. [2010], the hybrid model is validated for the mono- dispersed, dense particle-laden flow case in a curved rectangular duct. The plausibility of the agglomeration model is demonstrated by a poly-dispersed particle-laden down- pipe. In this example the agglomeration of particles during sedimentation is simulated. The presented implementation of the agglomeration model is compared to the standard implementation of the population balance equation by Smoluchowski [1917]. Finally, in Chapter 7 the hybrid model is applied to a real cyclone geometry. The cyclone, designed with theory of Muschelknautz et al. [1994] and Hoffmann and Stein [2008] was created during a diploma thesis at the Institute’s experimental facility [Krainz, 2007]. Simulations with the EUgran+Poly model are compared to measurements and analytical results of the cyclone. Additionally, an industrial cyclone at experimental scale is simulated and compared to measured results. In this application the impact of the new developed sub-models in the hybrid model can be highlighted. The thesis is closed by conclusions and an outlook. In the appendix, additional information such as: restitution coefficients, the modelling of the cyclone with Muschelknautz’s theory, notations and curriculum vitae are given.

Figure 1.6: Organization of this thesis

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In faith there is enough light for those who want to believe and enough shadows to blind those who don’t.

Blaise Pascal (1623-1662)

Eulerian granular phase modelling

As mentioned in the introduction, the total number of particles involved in most practi- cally relevant particulate flows is extremely large. Hence, it may be impractical to solve the equations of motion for each particle [Agrawal et al., 2001, Schneiderbauer et al., 2012a]. It is therefore common to average the equations of motion of the individual particles, as for example in the Eulerian-Eulerian granular phase model. The model describes the flow of solid particles as a fluidized medium [Ishii, 1975, Agrawal et al., 2001]. The quantities describing the flow, e. g. velocity, density and volume fraction, are assumed to be smooth functions of position and time [Rao and Nott, 2008]. The field of application of the Eulerian granular phase model is shown in Figure 2.1. With this approach multi-phase Navier Stokes equations (NSE) are modified to account for the granular flow [Gidaspow, 1994]. This approach requires the definition of granular equivalents of temperature, pressure and stresses. For granular phases, a granular tem- perature (Section 2.3), a solids pressure and a solid shear stress tensor (Section 2.6) are included. For example, the solids shear stresses consist of a kinetic and a collisional con- tribution and a frictional part. Each of them contains a viscosity, which is the kinetic viscosity divided into a shear and bulk term. An overview of the Eulerian-Eulerian granular phase model and the models used therein is given in Figure 2.2.

Eulerian granular phase modelling 11

Figure 2.1: Range of application of Eulerian-Eulerian granular phase model.

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2.1 Continuity equation

In a closed system, energy and mass is conserved. Hence, in standard fluid flows [Crowe et al., 1998] - without considering chemical reactions - a mass balance, referred to continuity equation, can be defined. In case of compressible flows for phase q it is written as

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where α q denotes the volume fraction, ρ q the density and uq the velocity. The continuity equation shows that mass entering a region increases the density. In case of incompressible flows, e. g. ρ = const., the equation can be simplified to

2.2 Momentum balance

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The fluid momentum balance for multi-phase flows is nearly the same as in case of a single phase flow. It is extended by a force exchange term β (ug − us) between the particle and the gas phase and the volume fraction α g of the phase. The momentum balance is given by

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Figure 2.2: Schematic of Eulerian granular phase model, without considering any boundary conditions. The grayed ellipse describes the gas or granular phase. Inside each ellipse the computed variables are shown. Each phase model is described by a set of balancing equations which are represented by the grayed rectangles. Further models, represented by white ellipses, provide information for the balancing equations.

where Tg denotes the shear stress tensor for the gas phase (see also 2.6) and is given by,

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where μ g denotes the gas viscosity and Dg the rate of strain tensor, which is given for phase q as

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Furthermore, the momentum balance for a solid phase s is given by [Ding and Gidaspow, 1990, Schneiderbauer et al., 2012a],

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where Skcs denotesthesolidsstresstensorarisingfromthekineticandcollisionalcontributions, Sfrs thestresstensorfromfrictionalcontributionsandfs,add additionalforces. With respect to the formulation of the hybrid particle model, only one particulate phase in the Eulerian granular phase model is used in this work.

2.3 Granular temperature

Corresponding to the thermal fluctuations of gas molecules a granular temperature Θ, which corresponds to the fluctuations of the particle velocities, is introduced for the granular phase. This granular temperature is defined by

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where D denotes the number of dimensions, u the velocity and 〈 u 〉 the average granular velocity [Goldhirsch, 2008]. The balance equation for the granular temperature is given by

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The first term − Skc s: ∇ us denotes the generation of pseudo-thermal energy (PTE). The second term ∇ · q represents the diffusion of pseudo-thermal energy q, which is given by [Schneiderbauer et al., 2012a]

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where e p denotes the particle-particle restitution coefficient. g 0 denotes the radial distribution function to include the maximum packing limit (Chapter 2.4), L C denotes the characteristic length scale of the actual physical system [Hrenya and Sinclair, 1997] and l s is the particle mean free path given by

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Eulerian granular phase modelling 14

Hence, the term l s recognizes the presence of boundaries [Schneiderbauer et al., 2012a].

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D Θ is the dissipation of PTE due to particle-particle collisions and is given by [Lun et al., 1984]

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γ Θ describes the dissipation of PTE by viscous damping [Gidaspow et al., 1992, Gidaspow, 1994], given by

γ Θ = 3 β sΘ , (2.15)

where β s denotes the interphase momentum exchange (e. g. Chapter 2.5). φ s describes the PTE production by gas-particle slip between the solid phase and the gas phase [Koch, 1990]

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In dense systems the balance equation for granular temperature can be simplified to an equilibrium algebraic equation by neglecting convection and diffusion terms [van Wachem et al., 2001]. This simplified algebraic equation is used as the standard method in the commercial code FLUENT/ANSYS. Furthermore, a maximum threshold should be included to avoid discontinuities and instabilities in the granular temperature field.

2.4 Radial distribution function

The radial distribution function g 0 is a correction factor to incorporate the maximum packing limit α s,max into the collision integral in the Boltzmann equation (e. g. Bril- liantov, N. V. and Pöschel, T. [2004] and Schneiderbauer et al. [2012a]). At positions with low volume fraction α s = 0, the radial distribution function tends to g 0 = 1 and has no impact on the collision integral. In the dense regime, where α s approaches α s,max, the radial distribution function tends to infinity (g 0 → ∞), because the number of collisions tends to infinity. The maximum volume fraction α s,max for an assembly of identical spheres is about 0 . 74 (e. g. named the Kepler problem) and defines the theoretical limit for a packing of mono-dispersed spheres. A detailed discussion of the hexagonal packing of spheres can be found in Conway and Sloane [1993]. In real world applications the theoretical value will never be reached and the maximum packing limit is about α s = 0 . 64. This is named the random closest packing for a mono-dispersed particle packing. The radial distribution function g 0 is described with the average par- ticle surface to surface distance s and the distance of particle center of gravity. It is

Eulerian granular phase modelling 15

Figure 2.3: Simple model of particle packing [Sinclair and Jackson, 1989, Gidaspow and Huilin, 1998] given by

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To derive an expression for g 0 a model for the packing of the particles has to be assumed (e. g. Figure 2.3). In Figure 2.3 the particle packing assumption for the model derived by Sinclair and Jackson [1989] and Gidaspow and Huilin [1998] is shown. It shows that the particles are arranged on a cubic grid. With this assumption g 0 can be described with α s and α s,max. The ratio of d s and l in volumetric consideration can be used to calculate the volume fraction given by

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where α s,max = π / 6, which leads to (

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α s,max and l is calculated to

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Recognizing that s = l − d s yields ((

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Eulerian granular phase modelling 16

Inserting this into equation (2.17) gives [Sinclair and Jackson, 1989]

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Table 2.1: Radial distribution functions

Model Equation

Carnahan and Starling [1969] g 0 =

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Sinclair and Jackson [1989] g 0 = 1 −

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Gidaspow and Huilin [1998]

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Iddir and Arastoopour [2005] g 0 =

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However, the assumption that particles will arrange on a cubic grid may be inappro- priate. The measured data of Gidaspow and Huilin [1998] show that equation (2.22) underestimates g 0 in regions with low α s. It is clear that with other assumptions on the particle packing or direct measurements of g 0 different mathematical descriptions for g 0 are found. Common models for the radial distribution function are listed in Table 2.1 and compared to measurements and computational simulations [van Wachem et al., 2001] in Figure 2.4. At the limits α s = 0 and α s = α s,max most of the models approach the correct value for g 0. At α s = 0 the model of Gidaspow approaches g 0 = 3 / 5, whereas the correct value should be one. The model of Carnahan and Starlin gives for α s → α s,max, g 0 ≪ ∞. In the intermediate region of α s, all radial distribution functions differ from measurements and an enhancement seems to be possible [Schellander et al., 2011]. This difference can be explained by the behavior of the particles, which do not tend to create a grid array, but rather form heterogeneous structures. Recognizing this, higher values for g 0 at low α s as given in the measurements are expected. Later in this thesis this phenomena will be considered again in case of particle strand formation in cyclones. However, we propose to use a piecewise defined function for g 0. This function

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Figure 2.4: Comparison of the different models for the radial distribution function and measurements

is fitted to the experiment data from Gidaspow and Huilin [1998], the computational data from Alder and Wainwright [1960] and model for the radial distribution function presented by Gidaspow and Huilin [1998]. The function is defined as

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In this equation the derivative of the resulting function is not completely smooth. Nev- ertheless, test-simulations proved that this does not impair the simulation stability. Additionally, it is assumed that the maximum packing limit is α s,max = 0 . 67. Consid- ering poly-dispersity of the granular material, which increases the maximum packing limit this assumption can be justified. If a model based on common radial distribu- tion models should be used, a mix between Carnahan and Starling [1969] and Iddir and Arastoopour [2005] is suggested [Schneiderbauer et al., 2012a, Schneiderbauer and Pirker, 2012b],

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For the simulation of fluidized beds and particle bin discharge, equation (2.24) proves to apply well. For the case of the preheating tower, where particle strand building is expected, the new piecewise defined function is suggested.

2.5 Drag coefficient and interphase momentum exchange

Physically, the fluid phase exerts several forces on the particle, e. g. drag force, Magnus force, Saffman force. The most dominant force is the drag on the particle. Hence, only the drag is considered for a coupling between the phases. Many drag laws have been proposed in the literature. In general, the interphase momentum exchange by drag is written as

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where β s is described by several models. Note that, the following models for the interphase momentum exchange are based on the assumption of homogenously distributed particles with the same diameter d s. Three commonly used drag laws for the simulation of particle-laden flows are the models of Wen and Yu [1966], Gidaspow et al. [1992] and Huilin et al. [2003]. In dense regime, e. g. for a flow through a fixed packed bed of particles, the model of Ergun [1952] is commonly used.

2.5.1 Wen and Yu

The model of Wen and Yu [1966] is valid for particle-laden flows up to volume fractions of α s = 0 . 6, where β s is given by

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For dense particulate flows Gidaspow et al. [1992] suggested a combination of the Wen and Yu correlation with the Ergun equation [Ergun, 1952]. For α s < 0 . 2 β is set to equation (2.26) and for α s ≥ 0 . 2 the Ergun equation is used

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One problem of this model is the discontinuity at the switching position of the two combined models. For the case of the preheating tower, where the complete range of α s is present, a model like this should be used to get correct results for packed bed regions.

2.5.3 Huilin and Gidaspow

To get a smooth switching between the drag given by the Ergun equation and the Wen and Yu model, Huilin et al. [2003] have rewritten the model of Gidaspow as follows

where ψ is defined as

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2.6 Solids Stresses

The stress tensor for the granular phase consists of two parts. On one hand it contains the stresses arising from kinetic and collisional contributions and on the other hand the frictional stresses. These stresses can be calculated independently and the resulting solid stress is assumed to be the summation of the kinetic, collisional and frictional stresses. The closure of the system with modelling solid stresses is done with the model of Hrenya and Sinclair [1997]. It is analogous to kinetic theory of gases, based on a pseudo-thermal energy balance of the velocity fluctuations [Schneiderbauer and Pirker, 2012a].

2.6.1 Kinetic and collisional stresses

The stress tensor, arising from kinetic and collisional contributions for the solid-phase is given by

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where p kcs denotesthesolidpressureand λ sc thebulkviscosity.devDs denotesthe deviatoric part of Ds, which is defined as

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The shear viscosity arises from translational kinetic motion and the collisional particleparticle interactions. The solids viscosity μ kcs inagranularmediumisingeneraldecomposed into kinetic viscosity and collisional viscosity [Bakker, [2008]].

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It is calculated with a modified version of Hrenya and Sinclair [1997] and Agrawal et al.

[2001] that accounts for the influence of boundaries, and is given by Schneiderbauer et al. [2012a] with

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which is proportional to the particle collisions frequency (f ∝

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The bulk viscosity accounts for the resistance of the particle ensemble to compression and expansion. It is given by

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In the Eulerian granular model, a thermodynamic collisional pressure for the solid phase is defined such that it is equivalent to the gas pressure. It is described by the model of Hrenya and Sinclair [1997] and Lun et al. [1984]. It consists of a kinetic term and a term due to particle collisions and is given by

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2.6.2 Frictional stresses

In the frictional regime (volume fraction α s > 0 . 5) the particle-particle collisions are no longer instantaneous as assumed by kinetic theory. In this regime there are long sliding frictional contacts which does not apply in the kinetic theory. Therefore, the frictional part of the solids shear viscosity does not depend on the amount of pseudo-thermal energy [Srivastava and Sundaresan, 2003b, Schneiderbauer et al., 2012a]. It is common to model the yield stress, at which the granular material begins to flow, by Coulomb’s law [Schaeffer, 1987, van Wachem et al., 2001]

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where τ frs denotesthefrictionalshearstress, p sr thefrictionalpressureand μ i =sin φ i is the coefficient of internal friction with φ i the angle of the internal friction. The frictional stresses are usually written in non-Newtonian form [Schneiderbauer et al., 2012a, Schaeffer, 1987]. The rigid-plastic rheological model for the frictional stresses is given by [Srivastava and Sundaresan, 2003b]

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where devDs denotes the deviatoric part of Ds (2.32). By assuming Coulomb friction, Schaeffer [1987] derived the frictional viscosity from first principles to

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with ∥ devDs ∥ = Ds : Ds / 2. Note that μ frs divergesbynumericalcomputation.Vari- ous formulations have been proposed in literature for the frictional pressure. For exam- ple Johnson and Jackson [[1987]], Johnson et al. [[1990]] and Jackson [[2000]] suggested

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where F, r and p are material constants. For example, F = 0 . 05, r = 2 and p = 5 are typically used for glass beads. α s,min denotes the value of volume fraction at which frictional interaction occurs, which is typically around α s,min = 0 . 5. Also Syamlal [1987] proposed a simple empirical function for the frictional pressure,

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where A = 10[25] and n = 10 are material constants and common values used for the simulation of frictional pressure in a mixture of glass beads. The two presented func- tions for frictional pressure are monotonically increasing with increasing α s [Srivastava and Sundaresan, 2003a]. Schneiderbauer et al. [2012a] and Schneiderbauer and Pirker

Eulerian granular phase modelling 22

[2012b] suggested a model that recognizes the shear rate dependent rheology and dilata- tion in the frictional regime [da Cruz et al., 2005, Chialvo et al., 2012]. The frictional pressure reads

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where b = 0 . 2 [Forterre and Pouliquen, 2008] for mono-dispersed glass beads. The

frictional viscosity is given by

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obtained by Jop et al. [2006], where μ ci, μ it and I 0 areconstants. Typical values for the constants, for mono-dispersed glass beads are I 0 = 0 . 279, μ sti = tan (20 . 9 ◦) and μ c = tan (32 . 76 ◦) [Forterre and Pouliquen, 2008].

2.7 Turbulence modelling

In the preheating tower, the gas flow has a Reynolds number of more than Re = 10[5]. Furthermore, the gas flow is turbulent. One problem due to computational limitations, is that the grid size must be in a range where it is not possible to resolve all turbulent scales. Hence, the turbulence of the gas phase is commonly represented by turbulence models to account for the influence of unresolved eddies in the mean flow. Turbulence models that are commonly used in industrial cases are the Reynolds Averaged Navier- Stokes (RANS) models (for example k ϵ, RSM) because they include the influence of small vortices. In literature there are two basic approaches to extend single phase RANS to multiphase flows. One approach is the mixture approach and the second one is the dispersed approach. In the mixture approach a dilute to medium density particulate flow is assumed and for the the dispersed approach a dilute particulate flow is assumed. Furthermore, in dense particulate flow regimes, the gas flow through the particles is assumed to be laminar.

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