1 CFD Open Series Revision 1.65 1a

Multiphase Flow Ideen Sadrehaghighi, Ph.D.

Multiphase Flow

Multicomponent Flow

Chemical Reaction

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ANNAPOLIS, MD

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Contents 1

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Introduction .................................................................................................................................. 7

1.2 Multiphase Flow ..................................................................................................................................................... 7 1.2.1 Solids Phase ......................................................................................................................... 8 1.2.2 Liquids Phase ....................................................................................................................... 8 1.2.3 Gases Phase ......................................................................................................................... 8 1.2.4 Phase Rule ........................................................................................................................... 8 1.2.4.1 Pure Substances (one component) ................................................................................ 9 1.3 Multicomponent Flow .......................................................................................................................................... 9 1.4 Multiscale Flow .................................................................................................................................................... 10

Multi /Phase/Component/Scale/ Flows ........................................................................... 12

2.1 Equations of Multiphase Flow ....................................................................................................................... 12 2.2 Multiphase Coupling .......................................................................................................................................... 13 2.3 Examples of Multiphase Flow ........................................................................................................................ 13 2.4 Modeling Approach Defined Based on Interface Physics ................................................................... 14 2.4.1 VOF Model ......................................................................................................................... 16 2.4.2 Eulerian Multiphase Model ............................................................................................... 16 2.4.2.1 Equations of Eulerian 2-Phase Model .......................................................................... 16 2.4.2.2 Simplification Applied to 2 Phase Flows ...................................................................... 18 2.4.2.3 2-Phase Flow Instability Mechanisms .......................................................................... 18 2.4.2.4 3-Phase Flow................................................................................................................. 18 2.4.2.5 Poly-Dispersed Flow ..................................................................................................... 18 2.4.2.6 Inhomogeneous Multiphase Flow ................................................................................ 18 2.4.2.7 Homogeneous Multiphase Flow ................................................................................... 19 2.4.2.8 Multi-Component Multiphase Flow ............................................................................. 19 2.4.2.9 Volume of Fraction ....................................................................................................... 19 2.4.2.10 Free Surface Flow .................................................................................................... 19 2.4.2.11 Surface Tension ....................................................................................................... 19 2.5.3 Mixture Model ..................................................................................................................... 19 2.4.3 Dispersed Phase Model (DPM) .......................................................................................... 19 2.4.4 Porous bed Model ............................................................................................................. 21 2.5 Guidelines for Selecting a Multiphase Model ........................................................................................... 21 2.6 Volume Averaging Formulation .................................................................................................................... 22 2.7 Constitutive Relations ....................................................................................................................................... 23 2.8 Some Thought in Multiphase CFD for Industrial Processes .............................................................. 25 2.9 Multicomponent Flow ....................................................................................................................................... 27 2.9.1 Integral and Differential Balances on Chemical Species ................................................... 30 2.9.1.1 Molar Basis ................................................................................................................... 30 2.9.1.2 Mass Basis..................................................................................................................... 31 2.9.2 Diffusion Fluxes.................................................................................................................. 32 2.9.3 Fick's Law ........................................................................................................................... 33 2.9.4 Species' Balances for Systems Obeying Fick's Law ............................................................ 34 2.10 Multiscale Modeling ..................................................................................................................................... 34 2.10.1 Traditional Approaches to Modeling ................................................................................. 35 2.10.2 Multiscale Modeling .......................................................................................................... 36

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2.10.3 Sequential Multiscale Modeling ........................................................................................ 36 2.10.4 Concurrent Multiscale Modeling ....................................................................................... 37 2.10.5 Two types of Multiscale Problems..................................................................................... 37 2.10.6 Modeling Approach defined based on Length Scale ......................................................... 37 2.10.6.1 Micro Approach (Fluid–Micro, Particle-Micro) ....................................................... 38 2.10.6.2 Meso Approach (Fluid–Meso, Particle-Meso) ......................................................... 38 2.10.6.3 Macro Approach (Fluid–Macro, Particle-Macro) .................................................... 38 2.10.6.4 Macro‐Micro Approach (Fluid–Macro, Particle-Micro)........................................... 38 2.10.6.5 Meso‐Micro Approach (Fluid–Meso, Particle-Micro) ............................................. 39 2.10.7 Block-Spectral Method of Solution.................................................................................... 39

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3.1 3.2

Case Studies for Composite Fluid ........................................................................................ 41

Case Study 1 - Liquid-Particle Suspension ............................................................................................... 41 Case Study 2 - Two Fluid Flow....................................................................................................................... 41 3.2.1 Mixture Viscosity ............................................................................................................... 41 3.2.2 Drag force .......................................................................................................................... 41 3.3 Case Study 3 - Unsteady MHD 2-Phase Flow of Fluid-Particle Suspension between Two Concentric Cylinders ...................................................................................................................................................... 43 3.3.1 Statement of Problem ....................................................................................................... 43 3.3.1.1 Literature Survey and Background ............................................................................... 43 3.3.2 Mathematical Formulation ................................................................................................ 44 3.3.3 Analytical Approach ........................................................................................................... 46 3.3.4 Comparison with Numerical .............................................................................................. 47 3.4 Case Study 4 - Simulation of Compressible 3-Phase Flows in an Oil Reservoir ........................ 48 3.4.1 Problem Statement ........................................................................................................... 48 3.4.2 Mathematical Modeling .................................................................................................... 49 3.4.3 Temporal and Spatial Discretization Methods .................................................................. 50 3.4.4 Results and Discussion....................................................................................................... 50 3.5 Case Study 5 - Effects of Mass Transfer & Mixture of Non-Ideality on Multiphase Flow ...... 51 3.5.1 Problem Statement ........................................................................................................... 51 3.5.2 Mathematical Model ......................................................................................................... 52 3.5.2.1 Bulk Species Transport ................................................................................................. 52 3.5.2.2 Interphase mass transfer.............................................................................................. 53 3.5.3 Simulation Procedure ........................................................................................................ 53 3.5.4 Results and Discussion....................................................................................................... 54 3.5.5 Concluding Remarks .......................................................................................................... 55 3.6 Case Study 6 - Numerical Study of Turbulent Two-Phase Coquette Flow................................... 55 3.6.1 Motivation and Literature Survey ..................................................................................... 55 3.6.2 Objectives .......................................................................................................................... 56 3.6.3 Problem Statement ........................................................................................................... 56 3.6.4 Governing Equations and Numerical Method ................................................................... 57 3.6.5 Initial and Boundary Conditions ........................................................................................ 58 3.6.6 Grid Resolution and Time Step Requirement .................................................................... 58 3.6.6.1 Turbulent length scale .................................................................................................. 58 3.6.6.2 Interface length scale ................................................................................................... 58 3.6.7 Results ............................................................................................................................... 59 3.6.8 Influence of the water depth............................................................................................. 61 3.6.9 Conclusions ........................................................................................................................ 61

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3.7 Case Study 7 - Slug Flow in Horizontal Air and Water Pipe Flow ................................................... 62 3.7.1 Motivation and Statement of Problem ............................................................................. 62 3.7.2 Slug Flow and Slug Formation in Pipe................................................................................ 63 3.7.3 Baker Chart ........................................................................................................................ 63 3.7.4 Problem Formulation......................................................................................................... 64 3.7.4.1 Boundary Condition ...................................................................................................... 65 3.7.5 Volume of Fluid (VOF)........................................................................................................ 66 3.7.6 Results and Discussion....................................................................................................... 67 3.7.6.1 Slug Initiation ................................................................................................................ 67 3.7.6.2 Slug Length ................................................................................................................... 67 3.7.6.3 Slug Volume Fraction .................................................................................................... 68

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Chemical Reaction .................................................................................................................... 70 4.1 Overview of Chemical Reaction Engineering........................................................................................... 70 4.1.1 Variables Affecting the Rate of Reaction ........................................................................... 70 4.1.2 Definition of Reaction Rate ............................................................................................... 71 4.1.3 Speed of Chemical Reactions............................................................................................. 71 4.1.4 Classification of Reactions ................................................................................................. 72 4.1.5 The Common Types of Chemical Reactions....................................................................... 72 4.1.5.1 Combination Chemical Reactions ................................................................................. 72 4.1.5.2 Decomposition Chemical Reactions ............................................................................. 73 4.1.5.3 Single Displacement Chemical Reactions ..................................................................... 73 4.1.5.4 Double Displacement Chemical Reactions ................................................................... 73 4.1.5.5 Precipitation Reactions................................................................................................. 73 4.1.5.6 Neutralization Reactions .............................................................................................. 74 4.1.5.7 Combustion Chemical Reactions .................................................................................. 74 4.1.5.8 Redox Chemical Reactions............................................................................................ 74 4.1.5.9 Organic Reaction .......................................................................................................... 74 4.1.6 Chemkin ............................................................................................................................. 74 4.2 CFD Applied To Chemical Reaction Engineering ................................................................................... 75 4.2.1 Reactor Design and CFD .................................................................................................... 75 4.2.2 Gas-Phase Reacting Flow Models ...................................................................................... 76 4.2.3 Liquid-Phase Reactions ...................................................................................................... 76 4.3 Basic Equations of Chemically Reacting Flows in CFD ........................................................................ 77 4.3.1 Flow and Reaction Interactions ......................................................................................... 78 4.3.2 Governing Equations ......................................................................................................... 78

List of Tables Table 1 Single and Multi-Phase flow vs Single and Multi-Component ........................................................ 14 Table 2 Modeling available for Multi-Phase flows............................................................................................... 15 Table 3 Suggested Multiphase Models for some common processes (Courtesy of Andresen et al.) ....................................................................................................................................................................................................... 21 Table 4 Comparison of numerical velocity values (Riemann Sum vs. Finite Difference).................... 48 Table 5 Slug length at different air-water velocities .......................................................................................... 68

List of Figures Figure 1

Example of Multi-Phase flow ........................................................................................................................ 7

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Figure 2 Carbon Dioxide (CO2) pressure-temperature phase diagram showing the triple point and critical point of CO2 ................................................................................................................................................................ 9 Figure 3 Multi-gas separated by a wall .................................................................................................................... 10 Figure 4 Theories and methods for different temporal and spatial scales................................................ 10 Figure 5 Description of Multiphase flow ................................................................................................................. 12 Figure 6 Coupling in multiphase flows ..................................................................................................................... 13 Figure 7 Shock wave generation in a gaseous medium due to a high pressure and supersonic jet flow. The image of shock wave is captured using synchrotron x-radiography. ......................................... 14 Figure 8 Solid and Fluid in 2-Phase flow and Transport analysis between them .................................. 15 Figure 9 Schematic guide for the selection of Multiphase Models................................................................ 20 Figure 10 Average Volume V and three phases α, β, γ ....................................................................................... 22 Figure 11 Binary System of Gases .............................................................................................................................. 27 Figure 12 Volumetric Flux ............................................................................................................................................. 28 Figure 13 Volume Swept ................................................................................................................................................ 29 Figure 14 Divergence Theorem applied to chemical species.......................................................................... 30 Figure 15 Illustration of the multi-physics hierarchy ........................................................................................ 36 Figure 16 Modeling Scales in Fluid-Particle Systems ......................................................................................... 39 Figure 17 Contour plots for particle volume fraction ........................................................................................ 42 Figure 18 Sketch of the problem ................................................................................................................................. 42 Figure 19 Schematic diagram of the problem....................................................................................................... 45 Figure 20 Sketch of the reservoir with the four injection wells at the corners and the production well in the center ................................................................................................................................................................... 50 Figure 21 Cumulative flow in the production well for a production day ................................................... 51 Figure 22 Contour of gas volume fraction at different time levels ............................................................... 55 Figure 23 Schematic illustration of the flow geometry ..................................................................................... 57 Figure 24 Snapshots of the air-water interface at different times................................................................ 59 Figure 25 Turbulent statistics: time- and stream wise-averaged velocity field...................................... 60 Figure 26 Turbulent statistics for two-phase Couette flow ............................................................................. 61 Figure 27 Air-water interface at fully developed state...................................................................................... 62 Figure 28 Hydrodynamic slug formation (Courtesy of Z. I. Al-Hashimy et al.) ......................................... 63 Figure 29 Baker chart where (.) Operating conditions of water–air two-phase flow ......................... 64 Figure 30 Boundary condition for water-air slug flow through a pipe....................................................... 65 Figure 31 Slug initiation of the air-water slug flow ............................................................................................ 67 Figure 32 Slug length calculation of air-water slug flow .................................................................................. 68 Figure 33 Cross section of the fluid domain for the extraction of volume fraction for Case 3 .......... 69 Figure 34 Typical Chemical Process .......................................................................................................................... 70 Figure 35 Moles of A disappearing Rate of reactions ........................................................................................ 72 Figure 36 Essential Aspects of Chemical Reactor Design ................................................................................. 75 Figure 37 Physical reactor configurations essential in predicting liquid .................................................. 77

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1 Introduction Multiphase Flow The term Multi-phase flow is used to refer to any fluid flow consisting of more than one phase or component. For brevity and because they are covered in other texts, we exclude those circumstances in which the components are well mixed above the molecular level. Consequently, the flows considered here have some level of phase or component separation at a scale well above the molecular level. This still leaves an enormous spectrum of different multiphase flows. One could classify them according to the state of the different phases or components and therefore refer to gas/solids flows, or liquid/solids flows or gas/particle flows or bubbly flows and so on. Many texts exist that limit their attention in this way2. Consequently, the flows considered here have some level of phase separation at a scale well above the molecular level. Some treatises are defined in terms of a specific type of fluid flow and deal with low Reynolds number suspension flows, dusty gas dynamics and so on. Others focus attention on a specific application such as slurry flows, cavitation flows, aerosols, debris flows, fluidized beds and etc. Again there are many such texts and here, we attempt to identify the basic fluid mechanical phenomena and to illustrate those phenomena with examples from a broad range of applications and types of flow (see Figure 1). Virtually every processing technology must deal with multiphase flow, from activating pumps and turbines to electro photographic processes. Clearly the ability to predict the fluid flow behavior of these processes is central to the efficiency and Figure 1 Example of Multi-Phase flow effectiveness of those processes. For example, the effective flow of toner is a major factor in the quality and speed of electro-photographic printers. Multi-Phase flows are also a pervasive feature of our environment whether one considers rain, snow, fog, avalanches, mud slides, sediment transport, debris flows, and countless other natural phenomena. Very critical biological and medical flows are also multiphase, from blood flow to the bends to lithotripsy to laser surgery cavitation and so on. No single list can adequately illustrate the diversity; consequently any attempt at a comprehensive treatment of multiphase flows is flawed unless it focuses on common phenomenological themes and avoids the temptation to deviate into lists of observations. Be aware that there are situations in which the Multi-Phase homogeneous and Multi-Component cases are overlap and hard to distinguished. The difference is the Multi-Component model assumes they mix into a single phase, which can be represented by a bulk density, viscosity etc. and the components are mixed on a microscopic scale. On the other hand, Multi-Phase homogeneous means you have multiple phases (e.g., gas and liquid) and they are separated on a resolvable scale. Some commercial software (i.e., Ansys CFX®) simulating it as multiphase homogenous flow it solves for volume fractions, after that calculates density (mean density?) and then solves momentum equations. If it is multicomponent than it solves for mass fractions, after that calculates mean density and momentum and so on in the end. 2

Christopher E. Brennen, “Fundamentals of Multiphase Flows”, Cambridge University Press 2005.

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Two general topologies of multiphase flow can be usefully identified at the outset, namely Disperse flows and Separated flows. By disperse flows we mean those consisting of finite particles, drops or bubbles (the disperse phase) distributed in a connected volume of the continuous phase. On the other hand separated flows consist of two or more continuous streams of different fluids separated by interfaces. In multiphase flows, solid phases are denoted by the subscript S, liquid phases by the subscript L and gas phases by the subscript G. Some of the main characteristics of these three types of phases are as follows: Solids Phase In a multiphase flow, the solid phase is in the form of lumps or particles which are carried along in the flow. The characteristics of the movement of the solid are strongly dependent on the size of the individual elements and on the motions of the associated fluids. Very small particles follow the fluid motions, whereas larger particles are less responsive. Liquids Phase In a multiphase flow containing a liquid phase, the liquid can be the continuous phase containing dispersed elements of solids (particles), gases (bubbles) or other liquids (drops). The liquid phase can also be discontinuous, as in the form of drops suspended in a gas phase or in another liquid phase. Another important property of liquid phases relates to wettability. When a liquid phase is in contact with a solid phase (such as a channel wall) and is adjacent to another phase which is also in contact with the wall, there exists at the wall a triple interface, and the angle subtended at this interface by the liquid-gas and liquid-solid interface is known as the Contact Angle. Gases Phase As a fluid, a gas has the same properties as a liquid in its response to forces. However, it has the important additional property of being (in comparison to liquids and solids) highly compressible. Notwithstanding this property, many multiphase flows containing gases can be treated as essentially incompressible, particularly if the pressure is reasonably high and the Mach Number, with respect to the gas phase, is low (e.g., < 0.2). Phase Rule Gibbs's phase rule3 was proposed by Josiah Willard Gibbs in his landmark paper titled On the Equilibrium of Heterogeneous Substances. The rule applies to non-reactive multi-component heterogeneous systems in thermodynamic equilibrium and is given by the equality

F CP2 where F number of degrees of freedom Eq. 1.1

C number of components P number of phases

The number of degrees of freedom is the number of independent intensive variables, i.e. the largest number of thermodynamic parameters such as temperature or pressure that can be varied simultaneously and arbitrarily without affecting one another. An example of one-component system is a system involving one pure chemical, while two-component systems, such as mixtures of water and ethanol, have two chemically independent components, and so on. 3

Gibbs, J. W., Scientific Papers (Dover, New York, 1961).

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1.2.4.1 Pure Substances (one component) For pure substances C = 1 so that F = 3 − P. In a single phase (P = 1) condition of a pure component system, two variables (F = 2), such as temperature and pressure, can be chosen independently to be any pair of values consistent with the phase. However, if the temperature and pressure combination ranges to a point where the pure component undergoes a separation into two phases (P = 2), F decreases from 2 to 1. When the system enters the two-phase region, it becomes no longer possible to independently control temperature and pressure. If the pressure is increased by compression, (see Figure 2) some of the gas condenses and the temperature goes up for CO2. If the temperature is decreased by cooling, some of the gas condenses, decreasing the pressure. Throughout both processes, the temperature and pressure stay in the relationship shown by this boundary curve unless one phase is entirely consumed by evaporation or condensation, or unless the critical point is reached. As long as there are two phases, Figure 2 Carbon Dioxide (CO2) pressure-temperature there is only one degree of freedom, which phase diagram showing the triple point and critical point corresponds to the position along the of CO2 phase boundary curve. The critical point is the black dot at the end of the liquid–gas boundary. As this point is approached, the liquid and gas phases become progressively more similar until, at the critical point, there is no longer a separation into two phases. Above the critical point and away from the phase boundary curve, F = 2 and the temperature and pressure can be controlled independently. Hence there is only one phase, and it has the physical properties of a dense gas, but is also referred to as a supercritical fluid. Of the other two-boundary curves, one is the solid–liquid boundary or melting point curve which indicates the conditions for equilibrium between these two phases, and the other at lower temperature and pressure is the solid–gas boundary. Even for a pure substance, it is possible that three phases, such as solid, liquid and vapor, can exist together in equilibrium (P = 3). If there is only one component, there are no degrees of freedom (F = 0) when there are three phases. Therefore, in a single-component system, this three-phase mixture can only exist at a single temperature and pressure, which is known as a triple point. In the diagram for CO2 (see Figure 2), the triple point is the point at which the solid, liquid and gas phases come together, at 5.2 bar and 217 K. It is also possible for other sets of phases to form a triple point, for example in the water system there is a triple point where ice I, ice III and liquid can coexist4.

Multicomponent Flow The multi-component model assumes they mix into a single phase, which can be represented by a bulk density, viscosity etc. The components are mixed on a microscopic scale. The multi-component flow (species transport) refers to flow that the components are mixed at molecular level and can be

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From Wikipedia, the free encyclopedia.

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characterized by a single velocity and temperature field for all species5. A simple example of such a multicomponent system is a binary (two component) solution consisting of a solute in an excess of chemically different solvent. (see Figure 3).

Figure 3

Multi-gas separated by a wall

Multiscale Flow Multiscale modeling refers to a style of modeling in which multiple models at different scales are used simultaneously to describe a system. The different models usually focus on different scales of resolution. They sometimes originate from physical laws of different nature, for example, one from continuum mechanics and one from molecular dynamics. In this case, one speaks of multi-physics modeling even though the terminology might not be fully accurate. The need for multiscale modeling comes usually from the fact that the available macroscale models are not accurate enough, and the microscale models are not efficient enough and/or offer too much information. By combining both viewpoints, one hopes to arrive at a reasonable compromise between accuracy and efficiency. The subject of multiscale modeling consists of three closely related components: multiscale analysis, multiscale models and multiscale algorithms. Multiscale analysis tools allow us to understand the relation between models at different scales of resolutions. Multiscale models allow us to formulate models that couple together models at different scales. Multiscale algorithms allow us to use multiscale ideas to design computational algorithms.

Figure 4

Theories and methods for different temporal and spatial scales

Associate Professor Britt M. Halvorsen Amaranath S. Kumara, ”Computational Fluid Dynamics (CFD) and Multiphase Flow Modelling”, Telemark University College, Porsgrunn, Norway. 5

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Figure 4 summarizes theories and typical numerical methods for different temporal and spatial scales. When the continuum assumption breaks down, the fluid has to be described by atomistic point of view, such as the molecular dynamics as a microscopic method or statistical rules for molecular groups, i.e. kinetic theories, as the mesoscopic methods for a larger scale. If the characteristic length is smaller than 1 nm (1 Nano-meter is 1×10−9 m) or the characteristic time is shorter than 1 fs (1 femtosecond is equal to 10−15 seconds), the quantum effect may be not negligible for the concerned system and the quantum mechanics has to be brought in to describe the transport as a result. In fact modeling from a smaller scale may lead to a more accurate description of the problem, but will bring much more computational cost as well. Therefore we may have to find an appropriate tradeoff for our concerned fluid behaviors in engineering6.

6 Shiyi

Chen, Moran Wang, Zhenhua Xia, “Multiscale fluid mechanics and modeling”, Procedia IUTAM 10 ( 2014).

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2 Multi /Phase/Component/Scale/ Flows Up to this point we were dealing with single phase flows. To get matters complicated, we now concern ourselves with multi-phase flows which exist in many industrial applications such as Oil & Gas, Power Generation, Biomedical, Automotive, Chemical Processing and Aerospace among others. Multiphase flows refer to flows of several fluids in the domain of interest. In general, we associate fluid phases with gases, liquids or solids and as such some simple examples of multi-phase flows are: air bubbles rising in a glass of water, sand particles carried by wind, rain drops in air. In fact, the definition of ‘phase’ can be generalized and applied to other fluid characteristics such as size and shape, density, temperature, etc. With this broader definition, multiple phases can be used to represent the entire size distribution of particles in several size groups or ‘phases’ of a multi-phase model. In fluid mechanics, multiphase flow is simultaneous flow of (a) materials with different states or phases (i.e. gas, liquid or solid), or (b) materials with different chemical properties but in the same state or phase (i.e. liquid-liquid systems such as oil droplets in water). Generally, a multiphase fluid is composed of two or more distinct phases which themselves may be fluids, gases or Figure 5 Description of Multiphase flow solids, and has the characteristic properties of a fluid. Within the discipline of multi-phase flow dynamics the present status is quite different from that of the single phase flows. The theoretical background of the single phase flows is well established and apparently the only outstanding practical problem that still remains unsolved is turbulence, or perhaps more generally, problems associated with flow stability. Generally, a phase is a class of matter, with a definable boundary and a particular dynamic response to the surrounding flow and potential field. Phases are generally identified by solid, liquid or gaseous states of matter but can also refer to other forms: Materials with different chemical properties but in the same state or phase (i.e. liquid-liquid, such as, oil-water). The fluid system is defined by a primary and multiple secondary phases (See Figure 5). There may be several secondary phase denoting particles with different sizes.

Equations of Multiphase Flow While it is rather straightforward to derive the equations of the conservation of mass, momentum and energy for an arbitrary mixture, no general counterpart of the Navier-Stokes equation for multiphase flows have been found. Using a proper averaging procedure it is however quite possible to derive a set of equations of multiphase flow which in principle correctly describes the dynamics of any multiphase system and is subject only to very general assumptions7. A direct consequence of the complexity and diversity of these flows is that the discipline of multiphase fluid dynamics is and may long remain a prominently experimental branch of fluid mechanics. Preliminary small scale model testing followed by a trial and error stage with the full scale system is still the only conceivable solution for many practical engineering problems involving multiphase flows. Inferring the necessary constitutive relations from measured data and verifying the final results are of vital importance also within those approaches for which theoretical modeling and subsequent numerical solution is considered feasible. 7

Multiphase flow Dynamics, Theory and Numeric.

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Multiphase Coupling Phase coupling, in terms of momentum, energy, and mass, is a basic concept in the description of any multiphase flow. The coupling can occur through exchange of momentum, energy, and mass among phases as shown in Figure 8. In principle, fluid‐particulate properties can be described by position, velocity, size, temperature, and species concentration of fluid and/or particle. While the phenomenological description of multiphase flow can be applied to classify flow characteristics, it also can be used to determine appropriate numerical formulations. In various modes of coupling, depending on the contribution of phases and phenomena, different coupling schemes can be adapted. This may allow independent treatment of phases or simultaneous integration of momentum, heat, and mass exchanges between phases. In general, modeling complexity increases as more effects associated with time and length scales are included in the simulation. In general, coupling depends on particle size, relative velocity, volume fraction. Three ways that coupling could be presents as shown in Figure 6:

One Way

Two Way

Four Way

1. One-way coupling: Sufficiently dilute such that fluid feels no effect from presence of Figure 6 Coupling in multiphase flows particles. Particles move in dynamic response to fluid motion. 2. Two-way coupling: Enough particles are present such that momentum exchange between dispersed and carrier phase interfaces alters dynamics of the carrier phase. 3. Four-way coupling: Flow is dense enough that dispersed phase collisions are significant momentum exchange mechanism8.

Examples of Multiphase Flow While the modeling and numerical simulation of multiphase and multicomponent flows poses far greater challenges than that of single-phase and single-component flows, their accessibility in nature is numerous. Rain and snow, and a vaporing tea pot is among prime example of multiphase flow. Others include, Spray drying, Pollution control, Pneumatic transport, Slurry transport, Fluidized beds, Spray forming, Plasma spray coating, Abrasive water jet cutting, Pulverized coal fired furnaces, Solid propellant rockets, Fire suppression and controls9. These challenges are due to interfaces between phases and large or discontinuous property variations across interfaces between phases and/or components. High-pressure and supersonic multiphase and multicomponent jet flow is one of the most challenging problems in multiphase flow due to the complexity of the dynamics of the jet. For example, the presence of cavitation and gas entrapment inside the nozzle orifice can greatly affect 8 9

Ken Kiger, “Multiphase Turbulent Flow”, UMCP presentation. Grétar Tryggvason, “Multiphase Flow Modeling”, spring 2010.

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the development and formation of the external jet. Another case involves high-pressure fuel spray has never been recognized as supersonic under typical fuel injection conditions10-11. Recently, MacPhee et al12 have used a synchrotron x-radiography and a fast x-ray detector to record the time evolution of the transient fuel sprays from a high-pressure injector. In their experiment, the propagation of the spray-induced shock waves in a gaseous medium were captured and the complex nature of the spray Figure 7 Shock wave generation in a gaseous medium due hydrodynamics were revealed. They to a high pressure and supersonic jet flow. The image of have found out that under injection shock wave is captured using synchrotron x-radiography. conditions similar to those in operating engines, the fuel jets can exceed supersonic speeds and result an oblique shock wave in the gaseous medium, see Figure 7. However, the effect of this shock wave to the atomization of the fuel and the combustion processes is currently not known. There are 4 distinctive new attitude which cover the following flow regimes13 in Table 1.

Single - Phase

Multi - Phase

Table 1

Single - Component

Multi - Component

Water flow Pure Nitrogen flow Steam bubble in H20 Freon-Freon Vapor flow Ice Slurry flow

Air flow H20 + Oil blends Coal particles in air Sand particle in H20

Single and Multi-Phase flow vs Single and Multi-Component

Modeling Approach Defined Based on Interface Physics

Genuine models for multiphase flows have been developed mainly following two different approaches. Within the Eulerian Approach all phases are treated formally as fluids which obey normal one phase equations of motion in the unobservable ’mesoscopic’ level (e.g., in the size scale of suspended particles) with appropriate boundary conditions specified at phase boundaries. The macroscopic flow equations are derived from these mesoscopic equations using an averaging T. Nakahira, M. Komori, K. Nishida, and K. Tsujimura. Shock W aves, K. Takayama, Ed., 2:1271–1276, 1992 H.H. Shi, K. Takayama, and O Onodera. JSME Intl. J. Ser. B, 37:509, 1994. 12 A.G. MacPhee, M.W. Tate, C.F. Powell, Y. Yue, M.J. Renzi, A. Ercan, S. Narayanan, E. Fontes, J. Walther, J. Schaller, S.M. Gruner, and J. Wang, ”X-ray imaging of shock waves generated by high pressure fuel spray”, Science, 295:1261–1263, 2002. 13 Randy S. Lagumbay, “Modeling and Simulation of Multiphase/Multicomponent Flows”, A thesis submitted to the Faculty of the Graduate School of the University of Colorado in partial fulfillment of the requirements for the degree of Doctor of Philosophy Department of Mechanical Engineering -2006. 10 11

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procedure of some kind. This averaging procedure can be carried out in several alternative ways such as time averaging, volume averaging and ensemble averaging. Various combinations of these basic methods can also been considered. Irrespective of the method used, the averaging procedure leads to equations of the same generic form, namely the form of the original physical equations with a few extra terms. These extra terms include the interactions (change of mass, momentum etc.) at

Interface

Figure 8

Solid and Fluid in 2-Phase flow and Transport analysis between them

phase boundaries and terms analogous to the ordinary Reynolds stresses in the turbulent single phase flow equations. Each averaging procedure may however provide a slightly different view in the physical interpretation of these additional terms and, consequently, may suggest different approach for solving the closure problem that is invariably associated with the solution of these equations. The manner, in which the various possible interaction mechanisms are naturally divided between these additional terms, may also depend on the averaging procedure being used. The advantage of the Eulerian method is its generality means that in principle it can be applied to any multiphase system, irrespective of the number and nature of the phases. A drawback of the straightforward Eulerian approach is that it often leads to a very complicated set of flow equations and closure relations. In some cases, however, it is possible to use a simplified formulation of the full Eulerian approach, namely Mixture Model (or Algebraic Slip Model). The mixture model may be applicable, e.g., for a relatively homogeneous suspension of one or more species of dispersed phase that closely follow the Approaches

Model

Definitions

Eulerian

Volume of Fluid Model (VOF)

Eulerian

Eulerian Model

Eulerian

Mixture Model

Lagrangian

Dispersed Phase Model (DPM)

Direct method of predicting interface shape between immiscible phases Model resulting from averaging of VOF model applicable to dispersed flows Simplification of Euler model; applicable when inertia of dispersed phase is small Lagrangian particle/bubble/droplet tracking

Table 2

Modeling available for Multi-Phase flows

Flow Regions Stratified Flow Dispersed Flow Dispersed Flow Dilute Flow

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motion of the continuous carrier fluid. For such a system the mixture model includes the continuity equation and the momentum equations for the mixture, and the continuity equations for each dispersed phase. The slip velocities between the continuous phase and the dispersed phases are inferred from approximate algebraic balance equations. This reduces the computational effort considerably, especially when several dispersed phases are considered. Another common approach is the so called Lagrangian Method which is mainly restricted to particulate suspensions. Within that approach only the fluid phase is treated as continuous while the motion of the discontinuous particulate phase is obtained by integrating the equation of motion of individual particles along their trajectories. Table 2 represents the different modeling approaches. VOF Model The VOF Model is designed to track the position of the interface between two or more immiscible fluids. Tracking is accomplished by solution of phase continuity equation resulting volume fraction abrupt change points out the interface location. A mixture fluid momentum equation is solved using mixture material properties. Thus the mixture fluid material properties experience jump across the interface. Turbulence and energy equations are also solved for mixture fluid. Surface tension and wall adhesion effects can be taken into account. VOF is an Eulerian fixed-grid technique. Eulerian Multiphase Model Eulerian Multiphase Model is a result of averaging of NS equations over the volume including arbitrary particles + continuous phase. The result is a set of conservation equations for each phase (continuous phase + N particle “media”). Both phases coexist simultaneously: conservation equations for each phase contain single-phase terms (pressure gradient, thermal conduction etc.). Interfacial terms express interfacial momentum (drag), heat and mass exchange. These are nonlinearly proportional to degree of mechanical (velocity difference between phases), thermal (temperature difference). Hence equations are harder to converge. Within the Eulerian-Eulerian Model, certain inter phase transfer terms used in momentum, heat, and other interphase transfer models, can be modeled using either the Particle Model, the Mixture Model or the Free Surface Model. In particular, the calculation of the interfacial area density, used for all inhomogeneous transfer models for a given fluid pair, is calculated according to one of these models. The available options depend on the morphology of each phase of the pair (for example, continuous, dispersed, etc.), and as settings in (homogeneous options, free surface model option). In the Eulerian Multiphase Model, the phases are treated as interpenetrating continua coexisting in the flow domain. Equations for conservation of mass, momentum and energy are solved for each phase. The share of the flow domain occupied by each phase is given by its volume fraction and each phase has its own velocity, temperature and physical properties. Interactions between phases due to differences in velocity and temperature are taken into account via the inter-phase transfer terms in the transport equations. Eulerian multi-phase modelling provides a general framework for all types of multi-phase flows; both dispersed (e.g. bubble, droplet, and particle flows) and stratified (e.g. freesurface flows) flows can be modelled. Comparing the Eulerian multi-phase with the Lagrangian twophase method, the former has the advantage of being computationally more efficient in situations where the phases are widely dispersed and/or when the dispersed phase volume fraction is high. For free-surface flows, similar advantages could be found in the Eulerian model relative to the previously developed approach. However, the free surface calculated will be less sharp in comparison with the VOF method. 2.4.2.1 Equations of Eulerian 2-Phase Model While it is rather straightforward to derive the equations of the conservation of mass, momentum and energy for an arbitrary mixture, no general counterpart of the Navier-Stokes equation for multiphase flows have been found. Using a proper averaging procedure it is however quite possible

17

to derive a set of Equations of Multi-Phase Flow which in principle correctly describes the dynamics of any multiphase system and is subject only to very general assumptions14. A direct consequence of the complexity and diversity of these flows is that the discipline of multiphase fluid dynamics is and may long remain a prominently experimental branch of fluid mechanics. Preliminary small scale model testing followed by a trial and error stage with the full scale system is still the only conceivable solution for many practical engineering problems involving multiphase flows. Inferring the necessary constitutive relations from measured data and verifying the final results are of vital importance also within those approaches for which theoretical modeling and subsequent numerical solution is considered feasible. The six-equations, one pressure model is currently implemented, supplemented with the High Reynolds number k-ε model for turbulence. A set of distinct mass, momentum and energy conservation equations is solved for each phase, and the phases are coupled via momentum and heat transfer terms. The pressure is assumed to be the same in each phase. Sub-models are provided to describe the interphase exchange terms and close the equations. In the following, we first present the fundamental equations for generic phase k (where k could be either the continuous or dispersed phase), before going on to present the models and sub-models which are implemented to obtain closure. The fundamental equations for the Eulerian two-phase model are: N - phase jk m kj ) (α k ρ k ) .(α k ρ k u k ) Ι k (m t j 1 j k

where

α k volume fraction , I k the rate of mass transfer to the phase k from the other phases N - phase

per unit total volume & since mass as a whole must be conserved

I k 1

ρ k density , u k mean phase velocity and

Dispersed

α

k k Continous

k

0

1

(α k ρ k u k ) .(α k ρ k u k u k ) - k p k k g .[ k ( k kt )] M k (Eint ) k t where k and kt molecular and turbulent stress respectively p pressure to be assumed to be equal for both phases M k inter - phase momentum trasfer per unit volume (Fint ) k internal forces g gravity vector Eq. 2.1

The inter-phase momentum transfer represents the sum of all the forces the phases exert on one another and satisfies Mc = - Md. The internal forces represent forces within a phase. In the current form, they are limited to particle-particle interaction forces in the dispersed phase. The derivation

14

Multiphase flow Dynamics, Theory and Numeric.

18

for energy equation and interphase momentum transfer can be applied through CD-adapco® methodology manual. 2.4.2.2 Simplification Applied to 2 Phase Flows Certain statement can be made for simplification of the equation with regard to the case investigated. For example in gas-particle two-phase flows, when the concentration of the dispersed phase is low, certain assumptions may be made which simplify considerably the equations to solve. The gas and particle flows are then linked only via the interaction terms. One may therefore uncouple the full system of equations into two subsystems: one for the gas phase, whose homogeneous part reduces to the Euler equations; and a second system for the particle motion, whose homogeneous part is a degenerate hyperbolic system. The equations governing the gas phase flow may be solved using a high-resolution scheme, while the equations describing the motion of the dispersed phase are treated by a donor-cell method using the solution of a particular Riemann problem. Coupling is then achieved via the right-hand-side terms15. 2.4.2.3 2-Phase Flow Instability Mechanisms The objective here is to review the main kinds of instabilities occurring in two-phase flows. It complements previous reviews, putting all two-phase flow instabilities in the same context and updating the information including coherently the data accumulated in recent years. In the first section, a description of the main mechanisms involved in the occurrence of two-phase flow instabilities is made. In order to get a clear picture of the phenomena taking place in two-phase flow systems it is necessary to introduce some common terms used in this field. The first distinction should be made between microscopic and macroscopic instabilities. The term microscopic instabilities is used for the phenomena occurring locally at the liquid–gas interface; for example, the Helmholtz and Taylor instabilities, bubble collapse, etc. The treatment of this kind of instabilities is out of the scope of this work. On the other hand, the macroscopic instabilities involve the entire twophase flow system. In this review, the main focus is kept on macroscopic phenomena. The most popular classification, introduced in [Bouré]16, divides two-phase flow instabilities in static and dynamic. In the first case, the threshold of the unstable behavior can be predicted from the steadystate conservation laws. On the other hand, to describe the behavior of dynamic instabilities it is necessary to take into account different dynamic effects, such as the propagation time, the inertia, compressibility, etc. In addition, the term compound instability is normally used when several of the basic mechanisms, described later, interact with each other17. 2.4.2.4 3-Phase Flow It is possible to have more than one dispersed phase in a continuous phase. For example, certain regimes of water-oil-gas flow in an oil pipeline may involve both oil droplets and gas bubbles immersed in a continuous water phase. 2.4.2.5 Poly-Dispersed Flow The above dispersed flow examples assume a single mean particle diameter for the dispersed phases. Poly-dispersed flows involve dispersed phases of different mean diameters. 2.4.2.6 Inhomogeneous Multiphase Flow Inhomogeneous multiphase flow refers to the case where separate velocity fields and other relevant R. Saurel, A. Forestier, D. Veyret And J. C. Loraud, “A Finite Volume Scheme For Two-Phase Compressible Flows”, International Journal For Numerical Methods In Fluids, Vol. 18, 803-819 (1994). 16 J. Bouré, A. Bergles, L. Tong, Review of two-phase flow instabilities, Nucl. Eng. Des. 25 (1973) 165–192. 17 Leonardo Carlos Ruspini, Christian Pablo Marcel, Alejandro Clausse, “Two-phase flow instabilities: A review”, International Journal of Heat and Mass Transfer 71 (2014) 521–548. 15

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fields exist for each fluid. The pressure field is shared by all fluids. The fluids interact via interphase transfer terms. The Particle and Mixture Models are both inhomogeneous multiphase models. 2.4.2.7 Homogeneous Multiphase Flow Homogeneous multiphase flow is a limiting case of Eulerian-Eulerian multiphase flow where all fluids share the same velocity fields, as well as other relevant fields such as temperature, turbulence, etc. The pressure field is also shared by all fluids. 2.4.2.8 Multi-Component Multiphase Flow It is possible to combine the notions of multicomponent and multiphase flows. In this case, more than one fluid is present, and each such fluid may be a mixture of chemical species mixed at molecular length scales. An example is air bubbles in water in which ozone gas is dissolved in both the gaseous and liquid phases. In this case, mass transfer of common species may occur by diffusion across the phase interface. 2.4.2.9 Volume of Fraction Multi-Phase modeling employs the notion of interpenetrating continua. Although phases are mixed at length scales much larger than molecular, they are also assumed to be mixed at length scales smaller than you want to resolve. Thus, each phase is assumed to be present in principle in each control volume, and assigned a volume fraction equal to the fraction of the control volume occupied by that phase. 2.4.2.10 Free Surface Flow Free Surface flow refers to a multiphase situation where the fluids (commonly water and air) are separated by a distinct resolvable interface. 2.4.2.11 Surface Tension Surface tension is a force that exists at a free surface interface which acts to minimize the surface area of the interface. It gives rise to effects such as a pressure discontinuity at the interface and capillary effects at adhesive walls. 2.5.3 Mixture Model Mixture Model is a simplified Eulerian approach for modeling n-phase flows. The simplification is based on the assumption that the Stokes number is small (particle and primary fluid velocity is nearly equal in both magnitude and direction). Solves the mixture momentum equation (for mass-averaged mixture velocity) and prescribes relative velocities to describe the dispersed phases. Interphase exchange terms depend on relative (slip) velocities which are algebraically determined based on the assumption that St is defined here as a volume average instead of a time average as the usual Reynolds stress. It also contains momentum fluxes that arise both from the turbulent fluctuations of the mesoscopic flow and from the fluctuations of the velocity of phase due to the presence of other phases. Consequently, it does not necessarily vanish even if the mesoscopic flow is laminar. The so called transfer integrals, Ϻα, Гα are defined as

Γα

1 ρα (uα u A ).nˆ αdA V Aα

Mα

1 ραuα (uα u A ).nˆ αdA V Aα

Eq. 2.5

nˆ α unit outward normal vector of phase α Constitutive Relations Eqn. (5.4) are, in principle, exact equations for the averaged quantities. So far, they do not contain much information about the dynamics of the particular system to be described. That information must be provided by a set of system dependent constitutive relations which specify the material properties of each phase, the interactions between different phases and the (pseudo)turbulent stresses of each phase in the presence of other phases. These relations finally render the set of equations in a closed form where solution is feasible. At this point we do not attempt to elaborate in detail the possible strategies for attaining the constitutive relations in specific cases, but simply state the basic principles that should be followed in inferring such relations. The unknown terms that appear in the averaged equations (5.4) such as the transfer integrals and stress terms that still contain macroscopic quantities, should be replaced by new terms. Typically, constitutive relations are given in a form where these new terms include free parameters which are supposed to be determined experimentally. In some cases constitutive laws can readily be derived from the properties of the mixture, or from the properties of the pure phase. For example, the incompressibility of the pure phase α implies the constitutive relation ρα = constant. Similarly, the equation of state Pα = Cρα, where C = constant for the pure phase, implies Pα = Cρα. In most cases, however, the constitutive relations must be either extracted from experiments, derived analytically under suitable simplifying assumptions, or postulated.

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Including a given physical mechanism in the model by imposing proper constitutive relations is not always straightforward even if adequate experimental and theoretical information is available. In particular, making specific assumptions concerning one of the unknown quantities may induce the constraints on other terms. For example, the transfer integrals Гα and Mα contain the effect of exchange of mass and momentum between the phases. According to Eqn. (5.4), the quantity Гα gives the rate of mass transfer per unit volume through the phase boundary Aα into phase α from the other phases. In a reactive mixture, where phase α is changed into phase, the mass transfer term Гα might be given in terms of the experimental rate of the chemical reaction α → γ, correlated to the volume fractions ϕα and ϕγ, and to the temperature of the mixture T. Similarly, the quantity Mα gives the rate of momentum transfer per unit volume into phase α through the phase boundary Aα. The second integral on the right side of Eqn. (5.4) contains the transfer of momentum carried by the mass exchanged between phases. It is obvious that this part of the momentum transfer integral Mα must be consistently correlated with the mass transfer integral Гα. Similarly, the first integral on the right side of Eqn. (5.4) contains the change of momentum of phase α due to stresses imposed on the phase boundary by the other phases. Physically, this term contains forces such as buoyancy which may be correlated to average pressures and gradients of volume fractions, and viscous drag which might be correlated to volume fractions and average velocity differences. For instance in a liquid-particle suspension, the average stress inside solid particles depends on the hydrodynamic forces acting on the surface of the particles. The choice of e.g., drag force correlation between fluid and particles should therefore influence the choice of the stress correlation for the particulate phase. While this particular problem can be solved exactly for some idealized cases, there seems to be no general solution available. Perhaps the most intricate term which phase boundary Aα. The second integral on the right side of Eqn. (5.4) contains the transfer of momentum carried by the mass exchanged between phases. It is obvious that this part of the momentum transfer integral Mα must be consistently correlated with the mass transfer integral Гα. Similarly, the first integral on the right side of Eqn. (5.4) contains the change of momentum of phase α due to stresses imposed on the phase boundary by the other phases. Physically, this term contains forces such as buoyancy which may be correlated to average pressures and gradients of volume fractions, and viscous drag which might be correlated to volume fractions and average velocity differences. For instance in a liquid-particle suspension, the average stress inside solid particles depends on the hydrodynamic forces acting on the surface of the particles. The choice of, e.g., drag force correlation between fluid and particles should therefore influence the choice of the stress correlation for the particulate phase. While this particular problem can be solved exactly for some idealized cases, there seems to be no general solution available. Perhaps the most intricate term this is to be correlated to the averaged quantities through constitutive relations is the tensor given by Eqn. (5.4). It contains the momentum transfer inside phase α which arises from the genuine turbulence of phase α and from the velocity fluctuations due to presence of other phases, and which are present also in the case that the flow is laminar in the mesoscopic scale. Moreover, the truly turbulent fluctuations of phase α may be substantially modulated by the other phases. Bearing in mind the intricacies that are encountered in modeling turbulence in single phase flows, it is evident that inferring realistic constitutive relations for tensor remains as a considerable challenge. It may, however, be attempted, e.g., for fluid-particle suspensions by generalizing the corresponding models for single phase flows, such as turbulence energy dissipation models, large-eddy simulations or direct

25

numerical simulations.

Some Thought in Multiphase CFD for Industrial Processes CFD is a rapidly evolving discipline oriented on developing computational tools for solving problems related to transport processes: fluid mechanics, heat and mass transfer, reactive flow, and multiphase flow, [Eskin and Derksen]24 . In narrow terms CFD is the numerical solution of the mass, momentum, and energy conservation equations with properly defined boundary conditions. Those equations may be supplemented with (Newtonian or non-Newtonian) constitutive equations and equations of state for compressible fluids. In broader terms CFD also involves modelling (parameterization) of phenomena at length and time scales that are too small to be fully resolved computationally; the three most prominent examples being turbulence, multiple phases flows, and reactive flows. In strongly turbulent flows, the spectrum of length and time scales is simply too wide to be completely resolved in a single computation. Models for small-scale turbulence are used to alleviate the computational burden and make simulations of large-scale industrial turbulent flows possible. Multiphase flows usually take the form of a continuous phase that carries one or more dispersed phases. The solid particles, or droplets, or gas bubbles that constitute the dispersed phases are often too small to be fully resolved; their impact on the macroscopic flow patterns needs to be modelled. A similar multiscale issue relates to chemically reacting flow where mixing at the micro (molecular)-scale defines the rate of chemical reactions. The most important issue in predictive modelling of chemical industrial processes is how to deal with their multiphase character. Process equipment (chemical reactors, burners, mixers, crystallizers, hydro and pneumatic conveying pipe lines, fluidized beds, flotation cells) usually operates with multiple phases, modelling of which is much more complicated than that of a single phase flow. In dependence on the phases composing the flow system, the geometry of the flow domain and the process conditions (flow rates, agitation speeds), an abundance of flow regimes and flow phenomena can be distinguished. Resolving and predicting these in a numerical simulation is a clear and grand challenge. Key in virtually any simulation effort is to distinguish between the relevant and irrelevant physics and model what is relevant. Though, general mathematical descriptions of multiphase processes are known, it is practically impossible to solve all the conservation equations numerically without simplifications. There are the two major groups of approaches, which are currently used in engineering and science: 1. Methods based on a simplified model representation of some processes involved in a certain multiphase system. 2. Methods of direct numerical simulation. The simplified model representation focuses on the macroscale processes and global flow patterns, and uses simplifying assumptions and models to represent micro-scale effects. Direct simulations aim at fully resolving the micro-scale including the behavior (motion, deformation, breakup, coalescence, aggregation) of individual dispersed phase particles (solids, droplets, bubbles). Given the high resolution at the micro-scale, direct simulations are only able to simulate small volumes and

24 Dmitry Eskin and Jos Derksen, “Introduction To A

Series Of Featured Article-Multiphase Computational Fluid Dynamics For Industrial Processes”, The Canadian Journal Of Chemical Engineering, Volume 89, April 2011.

26

thus need simplifying assumptions regarding the macro-scale, such as homogeneous turbulent conditions, or simple shear flow. The first group of methods is usually employed when a computational domain is large and/or a number of different phenomena revealing themselves on different scales are involved. For example, a force interaction of dispersed and continuous phases is modelled through a drag force that is calculated based on empirical correlations. Heat and mass transfer in such a case are also described by empirical equations. The situation becomes even more complicated when concentration of a dispersed phase is high. Then dispersed phase components interact with each other. Those interactions lead to generation of additional stresses in a flow, causing a change in flow pattern. If particles are solid then kinetic theory of granular media can be employed for modelling dispersed phase dynamics. Models where both continuous and dispersed phases are represented as two interacting interpenetrating continua are classified as two-fluid models and often used in engineering practice. Such models are incorporated into commercial CFD codes (e.g., fluent and CFX) and widely used for computing of large-scale technological devices (e.g., fluidized bed chemical reactors, hydraulic or pneumatic conveying pipelines, etc.). Alternatively, the equations of motion of the dispersed phase are solved in Lagrangian coordinates. In this case motion of each particle is tracked. Collisions of a tracked particle with others are accounted for assuming that it moves through a cloud formed by other particles. It is assumed that particle–particle collisions are binary and mutual orientations of colliding particles are random. Postcollision particle velocities are calculated based on momentum conservation for particle pair. This approach is more accurate than the two-fluid model, but limited to relatively low particle concentration and computationally expensive. If the second phase is not dispersed down to small size particles or droplets the two-fluid approach cannot be used. If both phases are immiscible fluids then dynamics of each fluid is modelled by solving the corresponding conservation equations. The models for each fluid flow are coupled through no-slip conditions and equality of stress on the fluid/fluid boundary. That boundary is tracked by one of the known techniques (e.g., volume of fluid method). The option of computing immiscible fluid flows is provided by modern commercial CFD codes. Examples of successful application of such an approach are modelling a bubbly flow in a capillary channel, a liquid film flow on a surface, etc. Direct Numerical Simulation (DNS) is a direction that is rapidly developing during the last 20 years. DNS methods suppose solving the conservation equations for all phases composing the system directly, without introducing simplifying assumptions. For example, in a case of a fluid–solids flow a dispersed phase is treated as a moving boundary of a complicated changing configuration. There are a number of different DNS methods. A DNS method is often considered as a technique for solving problems on the meso-scale that is assumed to be a minimum scale representing an important property of a flow system. An example of such a property is an apparent viscosity of a slurry or an emulsion. The meso-scale in this case is a characteristic size of a computational domain that is sufficient to calculate the apparent viscosity based on accurate modelling of dynamics of interacting carrying and dispersed phases. A DNS method should not require a model of turbulence, that is, such a method should allow resolving Navier–Stokes equation from a micro-scale (significantly smaller than the inner turbulence scale) to a relatively large-scale (e.g., a few percent of a tube radius for a pipe flow). Some known DNS methods are based on direct solution of the conservation equations for all components of a given flow system.

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Unfortunately, these equations are strongly non-linear and characterized by poor convergence and numerical stability. The other group of DNS methods is based on ideas borrowed from statistical mechanics. A fluid is represented as a system of particles, characterized by a probability density in a 6-dimensional space (3 coordinates in the geometrical space and 3 coordinates in the velocity space). Dynamics of such a system is described by a known Boltzmann equation. It was proven that the Boltzmann equation can be reduced to the Navier–Stokes equation, therefore the Boltzmann equation can be used for modelling fluid flows. Examples of methods based on such an approach are: the Lattice Boltzmann Method (LBM), its predecessor Lattice Gas Automata (LGA), and Dissipative Particle Dynamics (DPD) approach. LBM and LGA methods employ a fixed grid. The velocity in each node is discretized to a number of fixed directions. The simplified kinetic equation formulated for such a grid allows obtaining an approximate solution of the Boltzmann equation and the Navier– Stokes equation, respectively. The Lattice–Boltzmann technique can be applied to modelling flows in a domain with very complex boundary conditions. The LBM equations are free of drawbacks associated with strong nonlinearity of Navier–Stokes equation. LBM has been successfully used for modelling multiphase flows, especially on a micro-scale. The DPD is an off-lattice mesoscopic simulation method which involves a set of particles randomly moving in continuous space. Each particle moves under the action of three pairwise-additive forces: a conservative force, the dissipative force, and the random force. The DPD technique has an advantage over other methods when it is necessary to relate the macroscopic nonNewtonian flow properties of a fluid to its microscopic structure. Though DNS methods are prospective for accurate modelling of flows on a mesoscopic level their applicability are often limited to small computational domains. Modern technological equipment is often characterized by enormous dimensions. DNS methods may serve as an excellent tool for deriving correlations or models used as sub-models for macro-scale CFD codes. In an ideal world, micro, meso, and macro-scale simulations are tightly connected to provide a multiscale approach for truly predictive modelling of large-scale industrial processes. The challenges in multi-scale modelling are the formulation of generic coarse and fine graining techniques to meaningfully connect simulations at vastly different length and time scales. The topic of our article series on CFD of multiphase flow covers given the above considerations a broad spectrum of methods and applications. The articles presented can be considered as examples of developments and applications of CFD techniques. Their common theme is solution of engineering problems arising in process industries. We hope that the series will be interesting for scientists from industry and academia as well as for practicing engineers involved in simulations of multiphase systems.

Multicomponent Flow Mass transfer deals with situations in which there is more than one component present in a system; for instance, situations involving chemical reactions, dissolution, or mixing phenomena25. A simple example of such a MultiComponent system is a binary (two component) solution consisting of a 25

Lecture Notes from CBE 6333, Levicky.

Figure 11

Binary System of Gases

28

solute in an excess of chemically different solvent. In a multicomponent system, the velocity of different components is in general different. For example, in Figure 11 pure gas A is present on the left and pure gas B on the right. When the wall separating the two gases is removed and the gases begin to mix, A will flow from left to right and B from right to left clearly the velocities of A and B will be different. The velocity of particles (molecules) of component A (relative to the laboratory frame of reference) will be denoted vA. Then, in this frame of reference, the molar flux NA of species A (units: moles of A/(area time) ) is

NA cA vA

Eq. 2.6

where cA is the molar concentration of A (moles of A/volume). This could be used to calculate how many moles of A flow through an area Ac per unit time (see Figure 12) where the flux is assumed to be normal to the area Ac. Then the amount of A carried across the area Ac per unit time is Amount of A carried through Ac per unit time = NA AC = cA vA Ac (moles/time). Since the volume swept out by the flow of A per unit time equals vA AC (see Figure 12), the above expression is seen to equal this rate of volumetric "sweeping" times cA, the amount of A per volume. More generally, for arbitrary direction of NA and a differential area element dB, the rate of A transport through dB would be (see Figure 12), flux of A through dB = - cAvA . n dB (moles/time) n is the outward unit normal vector to dB. One can understand this by realizing that - vA . n dB is the volumetric flowrate of A species (volume/time) passing across dB from Figure 12 Volumetric Flux "outside" to "inside", where "outside" is pointed at by the unit normal vector n. Multiplying the volumetric flowrate -vA . n dB by the number of moles of A per volume, cA, equals the moles of A passing through dB per unit time. cA is related to the total molar concentration c (c is moles of particles, irrespective of particle type, per volume) where xA is the mole fraction of A. Summing over the mole fractions of all species must produce unity (n equals the total number of different species present in solution). Similarly, we can also define a mass flux of A, nA (units: mass of A/(area time) ), nA = ρA vA. Here, vA is still the velocity of species A. ρA is the mass concentration of A (mass of A per volume of solution).

cA x Ac ,

n

xi 1 i 1

,

A A ,

n

i 1

i

1

Eq. 2.7

As previously stated, in general each chemical species "i" in a multicomponent mixture has a different velocity vi. However, it will nevertheless prove convenient to define an average velocity of the bulk fluid, a velocity that represents an average over all the vi's. In general, three types of average velocities are employed: mass average velocity v (v is what is usually dealt with in Fluid Mechanics), molar average velocity V, and volume average velocity vo. We will only deal with the first two average velocities.

29

n

n

v ωi v i , V x i v i i 1

i 1

n

, ρ ρi i 1

n

,

c ci

Eq. 2.8

i 1

Why bother with two different average velocities? The mass average velocity is what is needed in equations such as the Navier Stokes equations, which deal with momentum, a property that depends on how much mass is in motion. Thus, the amount of momentum per unit volume of a flowing multicomponent mixture is ρv ( ρv = mv/Volume, where m is the total mass traveling with velocity v; m/Volume = ρ); thus momentum must be calculated using the mass average velocity v. Similarly, the Equation of Continuity expresses conservation of mass, and is similarly written in terms of v. The physical laws expressed by these equations (conservation of momentum, conservation of mass) do not depend on the moles of particles involved, but they do depend on the mass of the particles. On the other hand, when dealing with mass transfer, we will see that it is Figure 13 Volume Swept common to write some of the basic equations in terms of V as well as v. The reason for using V, in addition to v, is convenience. For instance, if in a particular problem there is no bulk flow of particles from one location to another so that, during the mass transfer process the number of particles at each point in space stays the same, then V = 0. Setting V to zero simplifies the mathematics. Figure 11 at the beginning provides an example. Imagine that, in their separated state as drawn, A and B are both ideal gases at the same pressure p and temperature T. Then, from the ideal gas equation, the molar concentration of A and B is the same, cA = cB = c = p/RT (R = gas constant). The equality of cA and cB to the total concentration c is appropriate because the gases are pure; thus in each compartment the concentration of the gas (A or B) must also equal the total concentration c. After the separating wall is removed, particles of A and B will mix until a uniform composition is achieved throughout the vessel. In the final state, assuming the gases remain ideal when mixed, the value of p and T will remain the same as in the unmixed state and therefore the total concentration c also remains the same, c = p/RT (p is now the total pressure, a sum of the partial pressures of A and B). Thus, in the final mixed state, the number of particles per volume c (here a sum of particles of A and B types) is the same as the number of particles per volume in the initial unmixed state. Thus mixing produced no net transfer of particles from one side of the vessel to the other, it only mixed the different particle types together. Under these conditions, when there is not net transfer of particles from one part of a system to another, V = 0. In contrast, for the same mixing process, in general v will not be zero. For example, imagine that mass of A particles is twice as large as that of B particles. Then in the initial unmixed state the left hand side of the vessel (filled with A) contains more mass, and the density (mass/volume) of the gas A is higher than that of B even though its concentration (particles/volume) is the same. Once A and B mix, however, the density everywhere will become uniform. For this uniformity to be achieved mass must have been transferred from the A side to the B side; therefore, in contrast to the molar average velocity V, the mass average velocity v was not zero during the mixing process.

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Integral and Differential Balances on Chemical Species We will refer to the species under consideration as species A. Following a derivation that parallels that employed for the other conservation laws (divergence theorem), the first step in the derivation of a conservation law on the amount of species A is to perform a balance for a closed control volume V'. V' is enclosed by a closed surface B ( see Figure 14). The amount of species A inside V' can change either due to convection through the boundary B, or by generation/consumption of A due to a chemical reaction. In words, the conservation for species A can stated as:

Accumulation of A in V' = Convection of A into V' + Generation of A by chemical reactions

Eq. 2.9

2.9.1.1 Molar Basis An integral molar balance on species A, performed over the control volume V', is written as

d c A dV c A v A . n dB R A dV dt V B V

Eq. 2.10

where n is the outward unit normal vector to surface B, not to be confused with the mass flux ni = ρi vi of species i. On the left side, cA is concentration of A in moles per volume; thus cAdV' is the number of moles of A in a differential volume dV'. Integrating (i.e. summing) this term over the entire control volume V' yields the total number of moles of A in V'; the time derivative of this integral is the rate of change of moles of A inside V' (units: moles/time). Thus, the left hand term is just the rate of accumulation of A in V', expressed in molar units. The accumulation term equals the rate at which A is convected into V' (1st term on right) plus the rate at which A is generated inside V' by a homogeneous chemical reaction (2nd term on right). The 2nd term on the right in equation (11) represents production of A by homogeneous reactions. A "homogeneous" reaction is one that Figure 14 Divergence Theorem applied to occurs throughout the interior of V'. In contrast, a chemical species heterogeneous chemical reaction would be one that occurs only at an interface; for instance, between a solid and a liquid phase; and is not distributed throughout the entire volume. The molar reaction rate RA has units of moles/(volume time) and represents the rate at which moles of A are produced or consumed by all homogeneous reactions. RAdV' is the number of moles of A produced inside a volume element dV' per unit time (units: moles/time). Summing this production over the entire control volume leads to the total molar rate of production of A, inside V', due to homogeneous chemical reactions. Equation (11), by assumption, did not include any generation of A due to heterogeneous reactions. Clearly, if in V' there was a large interface at which a heterogeneous reaction leads to production of A, one would have to add that term to equation (11). The term would typically have the form of a rate of production of A per area (moles / (area time)) times the total area of the reacting surface. However, it may also be that a heterogeneous reaction is actually more conveniently modeled as homogeneous. Because the reaction occurs only at the interface between a particle and the liquid, it is heterogeneous. However, since the particles are dispersed throughout V', one could think of the

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reaction rate on a per volume basis (i.e. moles produced per volume of solution per time) as opposed to a per area basis (moles produced per surface area of particles per time). 1. As done previously for the other balances, one can use the Divergence Theorem to convert the surface integral of the convection term (1st term on right) into a volume integral, 2. Move the d/dt derivative inside the accumulation integral since the integration limits are time independent (the limits do not depend on time because a fixed control volume is considered, whose shape and location do not change; this assumption can be relaxed at the expense of a somewhat more complicated mathematical expression), 3. Combine all terms under a common volume integral to obtain,

c A .c A v A R A dV 0 t

V

Eq. 2.11

The only way to ensure that equation (12) evaluates to zero for an arbitrary control volume V' is to require that

c A .c A v A R A 0 t

or

c A .c A v A R A t

Eq. 2.12

Equation (14) is the differential molar balance on species A. It states that the rate of accumulation of moles of A at a point in space (left hand side) is equals the rate at which moles of A are convected into that point (1st term on right), plus the rate at which moles of A are produced at that point by chemical reactions (2nd term on right). These physical interpretations can be verified by tracing the origin of the terms back to the corresponding terms in the integral balance, equation (11). 2.9.1.2 Mass Basis Similarly, For mass balance on species A,

d ρ A dV ρ A v A . n dB rA dV dt V B V

or

ρ A .ρ A v A rA t

Eq. 2.13

The differential equation of continuity (total mass balance) derived in fluid mechanics for single component systems. It also applies to multicomponent systems in which chemical reactions happen. The prove this is straightforward. n n ρi . ρi v i ri i 1 t i 1 i 1 n

Eq. 2.14

This equation states the law of mass conservation in multicomponent systems, even if chemical reactions are present. The total accumulation of mass at a point (left hand side) can only occur by convection of mass to that point (right hand side). For multicomponent systems whose density ρ is constant (i.e. ρ does not vary from point to point irrespective of variations that may be present in temperature, pressure, or composition).

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Diffusion Fluxes The diffusion flux of species A is that portion of its total flux that is not attributed to bulk flow (as represented by the mass or molar average velocities). More precisely, jA, the mass diffusive flux of A, is defined as

jA = total M ass flux of A - mass flux of A due to bulk motion n A - A v A v A - A v A ( v A - v) J A = total M olar flux of A - molar flux of A due to bulk motion

Eq. 2.15

N A - cA V cA ( v A - V ) Thermodynamics also states that, for a system with n components, n + 1 intensive variables are sufficient to fully specify the equilibrium state. Since binary fluid mixtures are being considered, for which n = 2, three intensive variables are needed. It will be convenient to choose temperature T, pressure p, and μT (the chemical potential per mass of species i by μi). In the absence of equilibrium, one or more of these variables will vary with location in a way that non-equilibrium gradients in ΔT, Δp, and ΔμT exist. To move toward equilibrium, the system will transfer heat and masses of the different species around so as to eliminate these gradients. For instance, heat flux will occur from hot to cold to equalize the temperature, and mass fluxes of individual chemical species will occur so as to equalize each species' total (μi + yi) potentials. Non-equilibrium pressure gradients can be normalized by bulk flow of material from high to low pressure regions. One possible way the system can eliminate non-equilibrium gradients in ΔT, Δp, and ΔμT is by diffusion of the various chemical species; it is then logical to assume that the diffusive fluxes will be functions of these gradients, with steeper gradients producing greater fluxes. Let's consider the diffusive mass flux jA, assumed to be a function of the gradients such that jA = jA(ΔT, Δp, ΔμT). When the gradients ΔT, Δp, and ΔμT are not too large, one could perform a Taylor series expansion of jA (around equilibrium) in the gradients and truncate it after the first order terms. Such an expansion would lead to the following mathematical relation for jA:

jA C1μ T C2T C3p

Eq. 2.16

The proportionality factors Ci are functions of T, p, and ΔT, but not of the non-equilibrium gradients in these quantities (this is because, recalling Taylor Series expansions, these factors are to be evaluated at equilibrium, the "point" around which the expansion is being formed. However, at equilibrium, the non-equilibrium gradients are zero). Equation (16) must be constrained to obey requirements imposed by thermodynamics. In particular, using the second law of thermodynamics, it can be shown that C3 must equal zero 26. Therefore, (Eq. 16) simplifies to

jA C1μ T C2T

Eq. 2.17

A species' chemical potential, at some point in the mixture, can be viewed as a function of the pressure, temperature, and composition at that point. For a binary mixture, this means that μA = μA(T, p, ωA) and μB = μB(T, p, ωA)27. Landau & Lifshitz, Fluid Mechanics, Pergamon Press, pp 187 and 222, 1959. An implicit assumption is being made that thermodynamic relations such as μ A = μA(T, p, ωA), which are strictly applicable to systems at equilibrium, apply even though equilibrium does not exist throughout the 26

27

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K D

p AB KT D AB p T (μ A μ B ) (μ A μ B ) jA C1 C 2 T C1 p T T ωA p,ω A T, Pressure Diffusion

Thermal Diffusion

D AB

(μ A μ B ) C1 (y A - y B ) ω A C 1 T T, p Forced (Gravity) Diffusion Ordinary (Mass) Diffusion (Fick's Law)

Eq. 2.18 If the diffusion coefficient DAB and chemical potentials are known (i.e. from experimental measurement), C1 can be evaluated. Eq. 2.18 becomes

ρD K jA AB T T T

ρD ABK p p - ρD ABωA - C1 (y A - y B ) - p

Eq. 2.19

The product DAB kT is called the thermal diffusion coefficient, kT the thermal diffusion ratio, and DAB kP may be called the baro-diffusion coefficient. Equation (19) shows that diffusive flux of mass of species A, in a binary solution of A and B, can arise from four different contributions. Fick's Law Having briefly outlined the four causes of mass diffusion spatial variations in composition, external potential, temperature, and pressure it is useful to highlight the most common scenario in which only ordinary diffusion is of importance. The discussion will be specialized to binary solutions that obey Fick's Law. The restriction to binary solutions is not as limiting as it may seem. Indeed, even when more than two components are present, as long as the solutions are sufficiently dilute the diffusion of solute species can be modeled as for a binary system. This is because when one of the species (the solvent) is present in vast excess, with all the rest (the solutes) in trace amounts, the diffusion of each solute species can be treated as if it was in pure solvent alone. Under these dilute conditions a solute particle will not "see" any of the other solute particles, and so its diffusion will not be affected by their presence but only by the solvent. Such a situation is effectively a two component problem, the solute of interest plus the solvent. Fick's Law can be written in several common forms such as Mass diffusion flux and reference bulk average velocity v, or Molar diffusion flux; reference bulk average velocity V:

system. Qualitatively, this assumption can be expected to hold over sufficiently short length scales over which only insignificant variations in temperature, pressure, and composition occur, so that the values of these quantities are well defined. Since thermodynamic quantities are only needed at a “point” (i.e. over very short lengths), from a practical perspective this consideration is not limiting. Also, note that ω B is not an independent thermodynamic variable since, for a binary mixture, ωB = 1 - ωA.

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jA ρ A ( v A v) ρ D ABωA J A c A ( v A - V ) jA

M c D ABx A MAMB

Eq. 2.20

Species' Balances for Systems Obeying Fick's Law The differential molar and mass balances on a species, are general in the sense that they are independent of the number of components present or any models of diffusion. For example, they do not require ordinary diffusion to obey Fick's Law. These equations are only based on the statement that the rate at which the amount of A can change at a point in space equals the rate at which A is convected into that point plus the rate at which it is generated by chemical reactions. However, in their present form, these equations can be inconvenient because they contain the species' velocity vA, which must be known in order to accomplish the usual goal of solving for the mass or molar concentrations. One way to model vA is to consider it as consisting of the two contributions previously encountered: 1. one due to bulk flow of the fluid mixture (convection) 2. one due to the diffusion of species A (mass diffusion) Separation of the total flux of A into diffusive and bulk convection contributions is motivated by convenience. The mass average velocity v is easy to measure, and can be obtained by direct calculation from the differential equations of fluid mechanics (e.g. Navier Stokes equations for Newtonian fluids with ρ and μ constant). Typically, the diffusive fluxes are modeled well by Fick's Law for solutions in which only ordinary diffusion is present and which are either binary or, as discussed earlier, dilute. Equations can be specialized to fluids that follow Fick's Law as

Mass balance : Molar balance :

ρ A .D ABωA - v.ρ A ρ A.v rA t c A .J A - .c A v R A .cD ABx A - .c A R A t

Eq. 2.21

Further information can be obtained in handout “Multicomponent Systems”, CBE 6333, [Levicky].

Multiscale Modeling Multiscale modeling refers to a style of modeling in which multiple models at different scales are used simultaneously to describe a system. The different models usually focus on different scales of resolution. They sometimes originate from physical laws of different nature, for example, one from continuum mechanics and one from molecular dynamics. In this case, one speaks of multi-physics modeling even though the terminology might not be fully accurate. The need for multiscale modeling comes usually from the fact that the available macroscale models are not accurate enough, and the microscale models are not efficient enough and/or offer too much information. By combining both viewpoints, one hopes to arrive at a reasonable compromise between accuracy and efficiency. The subject of multiscale modeling consists of three closely related components: multiscale analysis, multiscale models and multiscale algorithms. Multiscale analysis tools allow us to understand the relation between models at different scales of resolutions. Multiscale models allow us to formulate models that couple together models at different scales. Multiscale algorithms allow us to use

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multiscale ideas to design computational algorithms 28. Traditional Approaches to Modeling Traditional approaches to modeling focus on one scale. Macroscale models require constitutive relations which are almost always obtained empirically, by guessing. Making the right guess often requires and represents far-reaching physical insight, as we see from the work of Newton and Landau, for example. It also means that for complex systems, the guessing game can be quite hard and less productive, as we have learned from our experience with modeling complex fluids. The other extreme is to work with a microscale model, such as the first principle of quantum mechanics. As was declared by Dirac back in 1929, the right physical principle for most of what we are interested in is already provided by the principles of quantum mechanics, there is no need to look further. There are no empirical parameters in the quantum many-body problem. We simply have to input the atomic numbers of all the participating atoms, then we have a complete model which is sufficient for chemistry, much of physics, material science, biology, etc. Dirac also recognized the daunting mathematical difficulties with such an approach after all, we are dealing with a quantum many-body problem. With each additional particle, the dimensionality of the problem is increased by three. For this reason, direct applications of the first principle are limited to rather simple systems without much happening at the macroscale. Take for example, the incompressible fluids. The fundamental laws are simply that of the conservation of mass and momentum, which, after introducing the notion of stress, can be expressed as follows:

. u 0

,

u ρ (u.) u .τ t

Eq. 2.22

where u is the velocity field, ρ is the density of the fluid which is assumed to be constant and τ is the stress tensor. To close this system of equations, we need an expression for the stress tensor in terms of the velocity field. Here is where the guessing game begins. The standard approach is to ask: What is the simplest expression that is consistent with

symmetry (Galilean, translational/rotational invariance); laws of physics, particularly the second law of thermodynamics; experimental data.

In this case, Galilean invariance implies that τ does not depend on u . The next simplest thing is to say that τ is a function (in fact, linear function) of ∇u . This gives us

τ pI μ (u (u)T )

Eq. 2.23

where p is the pressure, μ is the kinematic viscosity coefficient, which in this context is the only parameter that carries the information specific to the microscopic constituents of the system, I is the identity tensor. Second law of thermodynamics requires that μ ≥ 0 . Substituting the above expression for τ into the momentum equation, we obtain the celebrated Navier-Stokes equation. Even though the reasoning that went into the derivation of the Navier-Stokes equation is exceedingly simple, the model itself has proven to be extremely successful in describing virtually all phenomena that we encounter for simple fluids, i.e. fluids that are made up of molecules with a simple molecular structure. Partly for this reason, the same approach has been followed in modeling complex fluids, 28

Weinan E and Jianfeng Lu, Scholarpedia, “Multiscale modeling”, 2011.

36

such as polymeric fluids. Unfortunately the success there is quite limited. The first problem is that simplicity is largely lost. In order to model the complex rheological properties of polymer fluids, one is forced to make more complicated constitutive assumptions with more and more parameters. The second difficulty is that accuracy is not guaranteed. For polymer fluids we are often interested in understanding how the conformation of the polymer interacts with the flow. This kind of Figure 15 Illustration of the multi-physics hierarchy information is missing in the kind of empirical approach described above. A more rigorous approach is to derive the constitutive relation from microscopic models, such as atomistic models, by taking the hydrodynamic limit. This is an example of the multiscale analysis approach. For simple fluids, this will result in the same Navier-Stokes equation we derived earlier, now with a formula for μ in terms of the output from the microscopic model. But for complex fluids, this would result in rather different kinds of models. This is one of the starting points of multiscale modeling (see Figure 15). Multiscale Modeling In the multiscale approach, one uses a variety of models at different levels of resolution and complexity to study one system. The different models are linked together either analytically or numerically. For example, one may study the mechanical behavior of solids using both the atomistic and continuum models at the same time, with the constitutive relations needed in the continuum model computed from the atomistic model. The hope is that by using such a multi-scale (and multiphysics) approach, one might be able to strike a balance between accuracy (which favors using more detailed and microscopic models) and feasibility (which favors using less detailed, more macroscopic models)29. Sequential Multiscale Modeling In sequential multiscale modeling, one has a macroscale model in which some details of the constitutive relations are precomputed using microscale models. For example, if the macroscale model is the gas dynamics equation, then an equation of state is needed. This equation of state can be precomputed using kinetic theory. When performing molecular dynamics simulation using empirical potentials, one assumes a functional form of the empirical potential, the parameters in the potential are precomputed using quantum mechanics. Sequential multiscale modeling is mostly limited to the case when only a few parameters are passed between the macro and microscale models. For this reason, it is also called parameter passing. This does not have to be the case though: It has been demonstrated that constitutive relations which are functions of up to 6 variables can be effectively precomputed if sparse representations are used.

It should be noted that this diagram is slightly misleading: Quantum mechanics is valid not just at the microscale, it also applies at the macroscale; only that much simpler models are already quite sufficient at the macroscale 29

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Concurrent Multiscale Modeling In concurrent multiscale modeling, the quantities needed in the macroscale model are computed onthe-fly from the microscale models as the computation proceeds. In this setup, the macro- and microscale models are used concurrently. Take again the example of molecular dynamics. If one wants to compute the inter-atomic forces from the first principle instead of modeling them empirically, then it is much more efficient to do this on-the-fly. Precomputing the inter-atomic forces as functions of the positions of all the atoms in the system is not practical since there are too many independent variables. On the other hand, in a typical simulation, one only probes an extremely small portion of the potential energy surface. Concurrent coupling allows one to evaluate these forces at the locations where they are needed. Two types of Multiscale Problems The first type are problems where some interesting events, such as chemical reactions, singularities or defects, are happening locally. In this situation, we need to use a microscale model to resolve the local behavior of these events, and we can use macroscale models elsewhere. The second type are problems for which some constitutive information is missing in the macroscale model, and coupling with the microscale model is required in order to supply this missing information. We refer to the first type as type A problems and the second type as type B problems. Modeling Approach defined based on Length Scale Alternatively, another modeling approach is developed which is based length scale on fluid and particle systems30. It was originally presented by Anderson and Jackson31. The idea was that modeling complexity increases as more effects associated with time and length scales are included in the simulation. Depending on the length scales, various combinations of modeling scales can be suggested. These are classified as Micro, Meso, and Macro scale models. In a Micro scale model, trajectories of individual particles are calculated through the equation of particle motion and the fluid length scale is the same as the particle size or even smaller. At the same time, instantaneous flow field around individual particles is calculated. In the Meso-Scale Model, both solid and fluid phases are considered as inter penetrating bands. The conservation equations are solved over a mesh of cells. The size of the cells is small enough to capture main features of the flow, like bubble motions and clusters, and large enough (essentially larger than the size of individual particles) to allow averaging of properties (porosity, interactions, etc.) over the cells. In the Macro Scale Model, the fluid length scale is in the order of the flow field. This means that motions of the fluid and the assemblage of particles are treated in one dimension based on overall quantities32. It is also possible to develop some intermediate models in which the length scales of fluid and solid phases are different. For example, the length scale of solid phase can be kept at the Micro Scale while changing the length scale of fluid phase to meso or macro. Under these conditions, the affective interactions in the larger scale can be calculated by averaging the information in the smaller scale. In general, in multi‐scale modeling, the smaller scale model takes into account various interactions (i.e, fluid–particle, and particle–particle) in detail. These interaction details can be used with some assumptions and averaging to develop closure laws for calculating the effective interactions (e.g.,

Coupled CFD-DEM Modeling: Formulation, Implementation and Application to Multiphase Flows, First Edition. Hamid Reza Norouzi, Reza Zarghami, Rahmat Sotudeh-Gharebagh and Navid Mostoufi. John Wiley & Sons, Ltd. 31 Anderson, T.B. and Jackson, R. (1967),”Fluid mechanical description of fluidized beds. Equations of motion”, Industrial and Engineering Chemistry Fundamentals, 6(4), 527–539. 32 Tsuji, Y. (2007) Multi‐scale modeling of dense phase gas–particle flow. Chemical Engineering Science, 62(13), 3410–3418. 30

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drag force) in the larger scale model33. This allows capture of the essential information needed on the larger scale. Alternatively, calculation of effective interactions can be performed through the local experimental data, if available. Combination of fluid/particle motion with different modeling scales can be calculated by averaging the information in the smaller scale can provide different modeling approaches, as sketched in Figure 16 with detailed below. 2.10.6.1 Micro Approach (Fluid–Micro, Particle-Micro) In this approach, the fluid flow around particles is estimated by the Navier–Stokes equation. Since the forces acting on particles are calculated by integrating stresses on the surface of the particle, the empirical correlation for drag and lift forces are not required. This approach is used in cases where particle inertial force is relatively small (e.g., liquid–particle flow) or the fluid lubricating effect on particles is rather significant (e.g., dense‐phase liquid–particle flow). A typical example of such an approach, is the direct numerical simulation–discrete element method (DNS‐DEM). 2.10.6.2 Meso Approach (Fluid–Meso, Particle-Meso) This is referred to as the two‐fluid model (TFM), in addition to the real fluid, the assemblage of particles is also considered to be the second continuum phase. The flow field is divided into a number of small cells to capture motions of both phases, provided that the cell size is larger than the particle size. The two continuous phases are modeled by applying laws of momentum and mass conservations in each fluid cell, leading to averaged Navier–Stokes and continuity equations. Capability of the TFM in capturing the solid phase motion greatly depends on the closure laws used for this phase. These closure laws always involve some simplifications or are obtained by semi‐ empirical correlations. While this approach is preferred in commercial packages for its computational simplicity, its effectiveness depends on the constitutive equations and is not easily applicable to all flow conditions. The TFM has been successfully utilized to obtain the flow behavior of various non‐reacting and reacting multiphase flows in laboratory, pilot, and industrial scales. 2.10.6.3 Macro Approach (Fluid–Macro, Particle-Macro) This approach provides a one‐dimensional (1D) description of gas‐particle flows. The main output of such a model is the pressure drop, which is considered as the sum of pressure drops due to flow of fluid and particles. Usually, a formula for the single phase flow, such as Darcy–Weisbach equation, is used for the fluid pressure drop and that of particles is balanced with the fluid drag formula from the momentum balance. This approach would also allow the calculation of averaged flow properties by empirical correlations that are essential in design and analysis of industrial processes. A typical example of such approach, is the two‐phase model (TPM) in fluidization. In this model, conservation equations are written for bubbles and emulsion, both having the length scale of the system in a fluidized bed. 2.10.6.4 Macro‐Micro Approach (Fluid–Macro, Particle-Micro) In this approach, shown in Figure 16 by 1D‐DEM, the fluid forces acting on particles are calculated from empirical correlations (e.g., drag and lift) while translational and rotational motions of particles are described based on Newton’s and Euler’s second laws. At very low concentration of particles, effect of particles on the fluid motion can be neglected. However, at higher concentrations, closure laws should be modified to account for the closeness of surrounding particles. Generally, in this approach the flow field, which is considered to change in one dimension, is not divided into cells and additional pressure drop is taken into account to reflect the effect of particles on the fluid motion.

Van der Hoef, M., Ye, M., van Sint Annaland, M., Andrews, A., Sundaresan, S., and Kuipers, J. ,”Multiscale modeling of gas‐fluidized beds”, Advances in Chemical Engineering, 31, 65–149, 2006. 33

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2.10.6.5 Meso‐Micro Approach (Fluid–Meso, Particle-Micro) In this method, referred to as CFDDEM and shown in Figure 16, the flow field is divided into cells with a size larger than the particle size but still less than the flow field. Effect of motion of particles on the flow of fluid is considered by the volume fraction of each phase and momentum exchange through the drag force. This approach is the focus of this book and is explained in detail in the following sections.

Figure 16

1-D DEM

CFD - DEM

Length Scale

Modeling Scales in Fluid-Particle Systems

Block-Spectral Method of Solution The traditional treatment of this kind of problems would be to use empirically based models to count for the effects of the small scale features and to solve an up-scaled problem on a coarse mesh34. The generality of the solutions following this kind of approaches are of course limited by the very empirical nature of these fine-scale models. Here we are interested in developing a new multi-scale methodology, called ‘Block-Spectral Method’. The main intended attribute of the new approach is that the same numerical discretization scheme and integration method are used for both the coarse (macro) and fine (micro) scales, so that the numerical resolution is consistently and completely dictated by the mesh scales. A blocking of the fine resolution domain is introduced to facilitate the two basic but competing requirements:

34

high resolution for fine scale flow features; avoidance of having to have fine meshes for a large domain

“Osney Thermo-Fluids Laboratory “, University of Oxford.

40

The block spectral approach can be simply illustrated by comparing a direct solution and a block spectral solution. The method has been shown to lead to a significant gain in solving micro-scale problems (up to 102 reduction of degrees of freedom). An important perspective is that the methodology would enable to resolve the kind of the micro-scale problems currently intractable [L. He]35-36.

35 36

He L, “Block-spectral Mapping for Multi-scale Solution”, Journal of Computational Physics, Vol.250, 2013. He L, “Block-spectral Approach to Film-cooling Modelling”, Journal of Turbomachinery, Vol.134, 2012.

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3 Case Studies for Composite Fluid Case Study 1 - Liquid-Particle Suspension Consider a binary system of solid particles suspended in a Newtonian liquid. We denote the continuous fluid phase by subscript c and the dispersed particle phase by subscript d. We assume that both phases are incompressible, that the suspension is non-reactive, i.e., there is no mass transfer between the two phases, and that surface tension between solid and liquid is negligible. Both the densities ρc and ρd are thus constants, and

Γc Γd 0 Mc Md 0

Eq. 3.1

The mutual momentum equation can now be written as

M Mc Md

and A Ac Ad and nˆ nˆ d nˆ c

Eq. 3.2

Case Study 2 - Two Fluid Flow Starting from a literature search, customizing the model adding specific constitutive equations and boundary conditions. The results in terms of both concentration distribution and pressure gradient were compared to experimental data available in literature37-38-39, over a wide range of operating conditions: average solids concentration between 10% and 40% by volume; uniform particle size between 90 and 520 μm; slurry velocities between 1 m/s and 5.5 m/s; and pipe diameters between 50 and 150 mm. The Two-Fluid model was obtained by adding the following two features to the original model, necessary to correctly reproduce the flow: Mixture Viscosity A correlation for the viscosity of the mixture, used to define the particle Reynolds number, is implemented. Among the different expressions available in literature, use is made of that of Mooney, which best fits the experimental data

ηα p μ m ρ c ν1,c exp 1 α /α p pm

Eq. 3.3

in which μ is the intrinsic viscosity, taken equal to 2.5 as suggested for spherical particles; α m is the maximum packing concentration, taken as 0.7; ρc is the density of the carrier fluid phase; ν1,c and is the laminar kinematic viscosity of the carrier fluid phase. Drag force The drag force law is related to the particle Reynolds number according to the Standard Drag Law Pipeline Flow Using ANSYS-CFX. Ind. Eng. Chem. Res. 48(17), 8159-8171. Roco, M.C., Shook, C.A., ”Modeling of Slurry Flow: The Effect of Particle Size”, Canadian J. Chem. Eng. 1983. 39 Gillies, R.G., Shook, C.A., Xu, J., ”Modelling Heterogeneous Slurry Flow at High Velocities”, Canadian J. Chem. Eng. 82(5), 1060-1065-2004. 37 38

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correlation, but the particle Reynolds number is defined with respect to the viscosity of the mixture instead of that of the carrier fluid phase; therefore, where and are the particle diameter and the slip velocity vector respectively. This modification is necessary to describe the phenomenon whenever in some cells the solid volume fraction approaches the maximum packing one. At the pipe wall, no slip conditions are imposed to the carrier fluid phase, and a logarithmic law wall function is applied in the near wall-region. The proper wall boundary conditions for the solid phase are still a matter of discussion in literature. Two alternatives have been considered. At first, a zero-flux condition is applied to the particles. Afterwards, to account for particle-particle and particle-wall interactions, a Bagnold-type shear stress is imposed. In particular, the following term, derived from the model of Shook and Bartosik40, is introduced in the momentum equation of the particle phase:

α 1/3 8.3018 2 pm τB ρ d 1 2.317 p p α p Re

1.5

τ w,c ρ v c 1,c

2

Eq. 3.4

Which Re = dpUs/ν1, c is the bulk Reynolds number, defined with respect to the pipe diameter dp and the slurry bulk-mean velocity Us; ρp is the density of the particles; τ w, c is the wall shear stress of the liquid phase. Figure 17 demonstrates the geometry of the problem; the computational domain covers only half of the pipe section due to a substantial symmetry of the phenomenon. A fully-developed turbulent flow profile is applied at the inlet. No slip is assumed between the phases. The inlet Figure 17 Contour plots for particle volume fraction volume fraction of the solids is taken as uniform. At the outlet, the normal gradients of all variables and the value of the pressure are set to zero. The length of the computational domain, equal to 100 pipe diameters, is sufficient to ensure the reaching of fully-developed flow conditions. When imposing a zero-flux condition at the wall to the particle phase, the predictions of the Two- Fluid model show good agreement to the experimental evidence in terms of solid volume fraction distribution. As an example, Figure 18 reports the results for the flow conditions considered by [Gillies]41 i.e., and mean solid volume Figure 18 Sketch of the problem Shook, C.A., Bartosik, A.S.,”Particle-wall stresses in vertical slurry flows”, Power Tech. 81(2), 117-124, 1994. Gillies, R.G., Shook, C.A., Xu, J., ”Modelling Heterogeneous Slurry Flow at High Velocities”, Canadian J. Chem. Eng., 2004. 40 41

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fraction from 12% to 41%. The contour plots of Figure 17 highlight the gradual accumulation of the particles as the mean solid volume fraction increases, phenomenon that can be correctly reproduced applying the above mentioned modifications to the original model. The solid volume fraction profiles along the vertical diameter (AB in Figure 18) appear in quantitative agreement to the experimental data.

Case Study 3 - Unsteady MHD 2-Phase Flow of Fluid-Particle Suspension between Two Concentric Cylinders Statement of Problem The problem of two-phase unsteady Magneto-Hydro-dynamic (MHD) flow between two concentric cylinders of infinite length has been analyzed by [Jha & Aperea]42 when the outer cylinder is impulsively started. The system of partial differential equations describing the flow problem is formulated taking the viscosity of the particle phase into consideration. Unified closed form expressions are obtained for the velocities and the skin frictions for both cases of the applied magnetic field being fixed to either the fluid or the moving outer cylinder. 3.3.1.1 Literature Survey and Background The study of fluid/particle mixture is important in petroleum transport, powder technology, fluidization transport of solid particles by a liquid and liquid slurries in chemical and nuclear processing and metalized liquid fuel slurries for rocketry. Its relevance is also seen in the field of mining, agriculture and food technologies. The flow of an electrically conducting fluid through a channel or a circular pipe in the presence of a transverse magnetic field is encountered in a variety of applications such as magneto-hydrodynamic (MHD) generators, pumps, accelerators, and flow meters. The annular geometry is widely employed in the gas cooled nuclear reactors in which the cylindrical fissionable fuel elements are placed axially in vertical coolants channel within the graphite moderators and the cooling gas is flowing along the channel parallel to the fuel elements and also in drilling operation of oil and gas well. [Makinde]43 studied the steady, axisymmetric, MHD flow of a viscous, Newtonian, incompressible, electrically conducting fluid through an isotropic, homogenous porous medium located in the annular zone between two concentric rotating cylinders in the presence of a radial magnetic field. The unsteady hydro magnetic Generalized Couette flow and heat transfer characteristics of a reactive variable viscosity incompressible electrically conducting third grade fluid in a channel with asymmetric convective cooling at the walls in the presence of uniform transverse magnetic field was also investigated. The high possibility of particle entrainment in these flows made the formulation of the equation of motion for a two-phase fluid/particle system a necessity in order to understand the role of particles on the flow. [Saffman]44 observed the effect of dust particles on the laminar flow of an incompressible fluid with mass concentration of dust particles. The equations he obtained neglected the interaction between the individual dust particles and also assumed that in the continuum hypothesis, the cloud of dust particles form a pseudo-fluid. Using Saffman’s model, [Nag et al.]45 studied the Coquette flow of a dusty gas between two parallel Basant K. Jha and Clement A. Aperea, “Unsteady MHD two-phase Coquette flow of fluid-particle suspension in an annulus”, Published by the AIP Publishing LLC, 2011. 43 O. D. Makinde, O. A. B´eg, and H. S. Takhar, “Magneto-hydrodynamic viscous flow in a porous medium cylindrical annulus with an applied radial magnetic field,” Int. J. Appl. Math. and Mech. 5, 68–81 (2009). 44 P. G. Saffman, “On the stability of laminar flow of a dusty gas,” J. Fluid Mech. 13, 120–128 (1962). 45S. K. Nag, R. N. Jana, and N. Datta, “Couettte flow of a dusty gas,” Acta Mech 33, 179–187 (1979). 42

44

infinite plates for both the impulsive and the accelerated start of one of the plates. [Singh and Dube]46 studied the laminar flow of an unsteady, incompressible viscous fluid with uniform distribution of dust particles through a circular pipe under the influence of pressure gradient. All of these studies excluded the particle-phase viscous effects. However, [Rahmatulin]47 developed a model which examines the unstable movement of the interface between the fluid and the particles and thereby included the particle-phase viscosity term in his model. In this model, each phase of the suspension has a local velocity and material properties, also, the phases are assumed to interact mutually in a continuous manner and the phases are homogeneous and evenly distributed per unit volume of the mixture. The following partial differential equations describes the unstable movements of the two-phase incompressible fluids for a plane channel geometry according to Rahmatulin.

V1 ρ f1 N f1μ1ΔV1 K(V2 V1 ) t V2 ρ f 2 N f 2μ 2 ΔV2 K(V1 V2 ) t

V1 0

, ,

V2 0

Eq. 3.5

where K , f are described by48 and subscript 1 and 2 denote each streams. The problem is solved using a combination of the Laplace transform technique, D’ Alemberts and the Riemann-Sum approximation methods. The solution obtained is validated by comparisons with the closed form solutions obtained for the steady states which has been derived separately. The governing equations are also solved using the implicit finite difference method to verify the present proposed method. The variation of the velocity and the skin friction with the dimensionless parameters occurring in the problem are illustrated graphically and discussed for both phases. The flow of an electrically conducting fluid through a channel or a circular pipe in the presence of a transverse magnetic field is encountered in a variety of applications such as Magneto-Hydro-Dynamic (MHD) generators, pumps, accelerators, and flow meters. The annular geometry is widely employed in the gas cooled nuclear reactors in which the cylindrical fissionable fuel elements are placed axially in vertical coolants channel within the graphite moderators and the cooling gas is flowing along the channel parallel to the fuel elements49, as well as in drilling operation of oil and gas well. Mathematical Formulation Consider the motion of an unsteady, laminar, viscous fluid/particle suspension between two concentric cylinders of infinite length. The fluid phase is assumed to be electrically conducting, while the particles are assumed to be electrically non-conducting. The z-axis is assumed to be on the axis of the cylinders in the horizontal direction and r´ is on the radial direction. The fluid-particle J. Singh and S. N. Dube, “Unsteady flow of a dusty fluid through a circular pipe,” Indian J. Pure Appl. Math. 6, 69–79, (1973). 47 H. A. Rahmatulin, “Osnovi gidrodinamiki vzaimopronikayu¸sih dvijeniy,” Prikladnaya Matematika i Makanika 20, 56–65, (1956). 48 Basant K. Jha and Clement A. Aperea, “Unsteady MHD two-phase Couette flow of fluid-particle suspension in an annulus”, Published by the AIP Publishing LLC, 2011. 49 S. K. Singh, B. K. Jha, and A. K. Singh, “Natural convection in vertical concentric annuli under a radial magnetic field,” Heat and Mass Transfer 32, 399–401 (1997). 46

45

suspension exists in the region a ≤ r´ ≤ b between the two cylinders in the presence of magnetic field acting perpendicular to the flow direction, where a and b are the radii of the inner and the outer cylinders respectively. Initially, the fluid, the particles and the two cylinders were at rest. However, when t > 0, the outer cylinder begins to move in its own plane with a velocity Ut´n, where U is a constant and the inner cylinder remains at rest. Since the cylinders are of infinite length and the flow is time dependent fully developed one-dimensional flow, all physical variables are functions of r´ and t´. The magnetic Reynolds number is assumed to be small so that the induced magnetic field and the Hall effects of MHD are negligible. No applied, polarization voltage exists (Ē = 0) i.e. no energy is being added or subtracted from the system50. It is further assumed that the fluid and the particles are

Figure 19

Schematic diagram of the problem

interacting as a continuum. Figure 19 for the schematic diagram of the problem. Under the assumptions made in the present problem and a cylindrical geometry, the equation of motion for the problem is

σB 02 u1 μ1 2 u1 1 u1 f1 u1 K(u2 u1 ) t ρ1 r2 r r ρ1 u2 μ 2 2 u2 1 u2 f2 K(u1 u2 ) t ρ 2 r2 r r

Eq. 3.6

Where f1 and f2 are substance quantity of mixture in unit volume (f1 + f2 = 1), K´ is dimensionless

50

G. Sutton and A. Sherman, Engineering Magneto-hydrodynamics (McGraw-Hill New York, 1965).

46

interaction coefficient, and subscripts 1 and 2 represents the fluid and the particle respectively. This equation is valid when the magnetic field is fixed relative to the fluid. If the magnetic field is also moving with the same velocity as the moving outer cylinder, we must account for the relative motion. Thus, eq. (3) turn to

2 u1 1 u1 σB 02 u1 1f1 2 K(u2 u1 ) (u1 - UGt n ) t r r ρ1 r where

Eq. 3.7

0 B0 is fixed realtive to fluid G 1 B0 is fixed realtive to moving cylinder

Considering the impulsive motion of the outer cylinder which corresponds to n = 0 and substituting the dimensionless quantities provided in51, the following dimensionless equations result,

u1 2 u1 1 u1 2 K(u 2 u1 ) M 2 (u1 - G) t r r r 2 u1 1 u1 u 2 f ν Rf Rν 2 K(u1 u 2 ) where R f 2 and R ν 2 t r r f1 ν1 r subject to : u1 u 2 0 1 r λ and

t0

u1 u 2 0 r 1 and

t0

u1 u 2 1 r

t0

and

Eq. 3.8

Analytical Approach Taking the Laplace transform of eqs. (8) and (9) we have the following ordinary differential equations We choose the D’Alembert’s method to obtain the solutions for these equation by multiply eq. (11) by A and add it to eq. (12). Details of transformation are given in52 and omitted here for simplicity

d 2 u 1 1 du 1 M 2G 2 (s M K)u1 Ku 2 dr 2 r dr s 2 d u 2 1 du1 (s K) K u2 u1 0 2 dr r dr R f R ν Rf R ν

where u1 u1 (r, t)e dt 0

st

Eq. 3.9

,

u 2 u 2 (r, t)est dt

and s 0

0

Basant K. Jha and Clement A. Aperea, “Unsteady MHD two-phase Couette flow of fluid-particle suspension in an annulus”, Published by the AIP Publishing LLC, 2011. 51

52

See previous.

47

N e εt 1 ikπ u1 (r, t) u1 (r, ε) Real u1 (r, ε )(1) k t 2 t k 1 N e εt 1 ikπ u 2 (r, t) u 2 (r, ε) Real u 2 (r, ε )(1) k t 2 t k 1

Eq. 3.10

where Real refers to the real part of’, i2 = −1 is the imaginary number, N is the number of terms used in the Riemann-sum approximation and ε is the real part of the Bromwich contour that issued in inverting Laplace transforms. The Riemann-Sum approximation for the Laplace inversion involves a single summation for the numerical process. Its accuracy depends on the value of ε and the truncation error dictated by N. The quantity εt = 4.7 is found to be more appropriate, since other tested values of εt seem to need longer computational time. Comparison with Numerical The momentum equations for fluid and dust particles with under the initial and boundary conditions are solved numerically using implicit finite difference method53, and the results presented in Table 4. The procedure involves the replacement of the partial differential into the finite difference equations at the grid point. The time derivative is replaced by the backward difference formula, while the spatial derivatives are replaced by central difference formula. The above equations are solved by Thomas algorithm by manipulating into a system of linear algebraic equations in the tridiagonal form. At each time step, the process of numerical integration for every dependent variable starts from the first neighboring grid point of the outer surface of the inner cylinder at r = 1 and proceeds towards the inner surface of the outer cylinder at r = λ. The process of computation is advanced until a steady state is approached by satisfying the following convergence criterion:

A

i, j1

A i, j

M A max

106

Eq. 3.11

Here Ai, j stands for the velocity field of the fluid and dust particle, M* is the number of interior grid points and |A|max is the maximum absolute value of Ai, j. In the numerical computation special attention is needed to specify Δt to get steady state solution as rapid as possible, yet small enough to avoid instabilities. In conclusion, we have employed the combination of the Laplace transform technique, D’Alembert’s and Riemann-sum approximation methods to obtain the solutions of unsteady MHD two-phase flow of fluid-particle suspension in the annular gap of two concentric cylinders. We have considered the effect of various dimensionless parameters occurring in the problem such as the Hartmann number M, the viscosity ratio Rν and the ratio of the substance quantity of mixture Rf on the flow field. The skin frictions at the opposite cylinders are seen to be of opposite sign; at the outer surface of the inner cylinder it is positive and negative at the inner surface of the outer cylinder. The result obtained demonstrates the reliability of the Riemann-Sum approximation method as comparisons made with those obtained using the implicit finite difference method shows consistency between them.

Obtained using the Riemann-sum approximation method and that obtained using finite difference method for both the fluid and the particle phases when t = 0.2, G = 1, K = 50, Rf = 0.25 and M = 2. 53

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Table 4

Comparison of numerical velocity values (Riemann Sum vs. Finite Difference)

Case Study 4 - Simulation of Compressible 3-Phase Flows in an Oil Reservoir Problem Statement Oil extraction represents an important investment and the control of a rational exploitation of a field means mastering various scientific techniques including the understanding of the dynamics of fluids in place. A theoretical investigation of the dynamic behavior of an oil reservoir during its exploitation is presented by [Ahmadi, et. al.]54. More exactly, the mining process consists in introducing a miscible gas into the oil phase of the field by means of four injection wells which are placed on four corners of the reservoir while the production well is situated in the middle of this one. So, a mathematical model of multiphase multi-component flows in porous media was presented and the cell-centered finite volume method was used as discretization scheme of the considered model equations. It ensues from the analysis of the contour representation of respective saturations of oil, gas and water phases that the conservation law of pore volume is well respected. Besides, the more one moves away from the injection wells towards the production well; the lower is the pressure value. However, an increase of this model variable value was noticed during production period. Furthermore, a significant accumulated flow of oil was observed at the level of the production well, whereas the aqueous and gaseous phases are there present in weak accumulated flow. The considered model so allows the prediction of the dynamic behavior of the studied reservoir and highlights the achievement of the exploitation process aim55. Malik El’Houyoun Ahmadi, Hery Tiana Rakotondramiarana, Rakotonindrainy, “Modeling and Simulation of Compressible Three-Phase Flows in an Oil Reservoir: Case Study of Tsimiroro Madagascar”, American Journal of Fluid Dynamics 2014, 4(4): 181-193. 54

Malik El’Houyoun Ahmadi, Hery Tiana Rakotondramiarana, Rakotonindrainy, “Modeling and Simulation of Compressible Three-Phase Flows in an Oil Reservoir: Case Study of Tsimiroro Madagascar”, American Journal of Fluid Dynamics 2014, 4(4): 181-193. 55

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Mathematical Modeling The “black-oil” model is considered as constituted by three fluid phases (oil, gas, and water) in each of them can be present the following three components: a lighter component (gas) which can be at the same time in the oil phase and in the gas phase, a heavier component which can only be in the oil phase, and a component water which can only be in the water phase 56. The capillary pressures are assumed to be negligible57-58. Accordingly, all the present phases in the reservoir have the same pressure. Moreover, the studied medium is considered as isotropic so that the components of the permeability tensor have the same values in all directions. As there is a mass transfer between the oil and gas phases, mass conservation is not satisfied for these two phases. However, the total mass of each component is conserved. Besides, as the water phase is completely separated from the other phases and the component water is only present in the water phase, the mass conservation is well respected for this phase. Hence, the equations of the black-oil model can be formulated as follows. Mass conservation related to the component water can be written as:

( ρ w ) (ρ w u w ) q w t

Eq. 3.12

whereas for the heavier component, one can write:

( ρ 0oSo ) (ρ 0ou o ) q o t

Eq. 3.13

and the lighter component is governed by:

( ρ GoSo ρ gSg ) t

(ρ Go u o ρ g u g ) q g

Eq. 3.14

where ρGo and ρOo are the densities of the lighter and the heavier components in the oil phase respectively. Equation implies that, the lighter component (gas) can be at the same time in the oil and the gas phases. The velocity of each phase (α = w, o, g) is governed by the generalized Darcy’s law and can be calculated by the following relationship:

uα

k rα K (pα ρα g) μα

Eq. 3.15

in which, K denotes the tensor of the absolute porousness’s of the porous media, whereas g denotes the gravity acceleration vector. The system of mentioned equations is completed by the following 56 Z. Chen, G. Huen,

Y. Ma, “Computational Methods for multiphase flows in porous media”, SIAM Ed., Philadelphia PA, USA, 2006. 57 Krogstad, S., Lie, K.A., Nilsen, H.M., Natvig, J.R.., Skaflestad, B., and Aarnes J.E., “A Multiscale Mixed FiniteElement Solver for Three-Phase Black-Oil Flow”., Proc., Society of Petroleum Engineers (SPE) Reservoir Simulation Symposium, The Woodlands, Texas, USA, 2009. 58 Hoteit, H., and Firoozabadi A., 2006, “Compositional modeling of discrete fracture media without transfer functions by the discontinuous Galerkin and mixed methods,” SPE journal, 11(03), 341-352.

50

closing relationships. Conservation of pore volume (the sum of the saturations in a pore is equal to unity):

So Sg Sw 1

Eq. 3.16

Constraints on the tube pressures (oil-water) pcow and (gas-oil) pcgo:

p cow p o p w 0 p cgo p g p o 0

Eq. 3.17

Temporal and Spatial Discretization Methods An implicit Euler scheme is used for the time discretization while a cell-centered finite volume scheme was used for spatial discretization. Indeed, the developed model is constituted by conservation equations. Since the finite volume schemes are conservative, they are better suited for solving the considered system of equations. All the borders of the reservoir are considered impervious. No flow either goes out or enters anywhere but places where wells are positioned. As depicted in Figure 20, four injection wells are placed in the four corners of the reservoir while a production well is placed in its center. A condition of pressure in each cell containing a well with perforation is imposed. While the injection of gas in the reservoir and in each cell containing an injection wells being the simulation object, gas saturation is considered equal to one. Figure 20 Sketch of the reservoir with the four injection wells at Further information can be the corners and the production well in the center obtained from [Malik, al.]59. The Newton-Raphson method was used for the linearization of the above nonlinear system of discretized equations. Afterward, the obtained linear system can be solved by classical methods of resolution of linear systems. In the present work, iterative Generalized Minimum Residual (GMRES) Method was used for this purpose. Results and Discussion Results simulating a three-phase oil-water-gas three components model whose lighter component may be simultaneously in the oil phase as well as in the gas phase. They show the evolution of the pressure, saturation and cumulative flow rate generated during the oil extraction. A spatial decay of pressure is observed while a temporal pressure increase occurs. The pressure imposed on the injection wells is higher than the pressure within the reservoir; the aforementioned spatial pressure decay that is observed following the fluid displacement front towards the center may be due to a Malik El’Houyoun Ahamadi, Hery Tiana Rakotondramiarana, Rakotonindrainy, “Modeling and Simulation of Compressible Three-Phase Flows in an Oil Reservoir: Case Study of Tsimiroro Madagascar”, American Journal of Fluid Dynamics 2014, 4(4): 181-193. 59

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pressure drop in the gas flow through the porous medium60. Moreover, the energy transferred by the gas to fluids that are in place (oil and water) justifies the temporal pressure increase. However, the pressure is still quite enough to push the oil to the production well and ease its drawing out. Indeed, the purpose of the gas injection in the reservoir is not only to increase the pressure, but also to make the oil less viscous to facilitate mobility for its extraction. As can be seen from Figure 21 which show the variations of cumulative flow in the production well, a good amount of oil is produced. Such result conforms to the goal sought by the extraction process. The miscibility of the oil in the gas, combined with the fact that the oil is much lighter than the existing water, promotes such production. There is also a very low quantity of produced water which is lower compared to that of the produced gas, which itself is significantly less than that of produced oil. It can be justified by the fact that not only there may be an expansion of the gas during production, but the gas is also much lighter than water.

Figure 21

Cumulative flow in the production well for a production day

Case Study 5 - Effects of Mass Transfer & Mixture of Non-Ideality on Multiphase Flow Problem Statement In the chemical and petrochemical industries, Multi-Component phases commonly undergo composition changes due to various phenomena resulting in the transfer of species from one phase to another, or conversion of species through chemical reactions61. On the other hand, the thermophysical properties of a multi-component system are strongly dependent on its compositions. This is due to the fact that the properties of a multi-component mixture are not necessarily equal to the weighted average of the corresponding properties of its constituting pure components. This results in a severe interaction between hydrodynamic and thermodynamic behavior of a multi-component system. This interaction seems to have more effects on the hydrodynamic behavior of multiSee previous. Irani, Mohammad; Bozorgmehry Boozarjomehry, Ramin*+; Pishvaie, Sayed Mahmoud Reza and Tavasoli, Ahmad, “Investigating the Effects of Mass Transfer and Mixture Non-Ideality on Multiphase Flow Hydrodynamics Using CFD Methods”, Iran. J. Chem. Chem. Eng., Vol. 29, No. 1, 2010. 60 61

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component multiphase systems. Here we concern with the effects of the non-ideal behavior of phases on their hydrodynamic behaviors, have been studied based on a CFD framework in which the properties of each phase are rigorously modeled as a function of temperature, pressure and concentration of phase constituting components. The CFD framework is developed based on Eulerian - Eulerian model. The proposed framework can be used in modeling and simulation of multiphase flow of non-ideal mixtures. Mathematical Model The variation of liquid holdup with time and position is obtained by solving the continuity equations for the liquid and gas phases. The continuity equation for the flowing liquid and gas is written in terms of the accumulation and convection terms balanced by the total mass transferred to and from the other phases (written in terms of interphase fluxes for gas-liquid equations, discussed later). Since gas and liquid phases do not interpenetrate into each other in the reactor, the VOF approach is used. In this approach, the motion of all phases is modeled by formulating local, instantaneous conservation equations for mass and momentum62. The variation of velocity with time and position is calculated by solving the momentum balance equation. The momentum equations can be written in terms of accumulation and convection terms on the left-hand side, and the gravity, pressure gradient and viscous stresses terms on the right-hand side, as the pressure and velocity are assumed to be equal in both phases. The properties appearing in the transport equations are determined by their averaging based on phase volume-fraction. The continuity and momentum equations for a phase, ‘q’, in a multiphase flow problem is as follows:

(α q ρ q ) .(α q ρ q v) Spq , t

Spq Sqp ,

(ρv) .(ρvv) -p . μ v v T ρg i t

N

α q 1

q

1 n

,

ρ ρqαq i 1

n

, μ μ qαq i 1

Eq. 3.18

The variation of velocity with time and position is calculated by solving the momentum balance equation. The momentum equations can be written in terms of accumulation and convection terms on the left-hand side, and the gravity, pressure gradient and viscous stresses terms on the right-hand side, as the pressure and velocity are assumed to be equal in both phases. The properties appearing in the transport equations are determined by their averaging based on phase volume-fraction. 3.5.2.1 Bulk Species Transport The dynamic variation in the liquid and gas phase species concentrations are obtained by solving the unsteady state species mass balance equations, consisting of accumulation, convection, and interphase transport for the gas and liquid phases written as

(α g Cig ) .(α g Cig v - Dig α g Cig ) α g N igl t (α i Cil ) .(α l Cil v - Dil α lCil ) α l N igl t

Eq. 3.19

Hirt C. W., Nichols, B.D., Volume of Fluid (VOF) Method for the Dynamics of Free Boundaries, J. Comp. Phys., 39, p. 201 (1981). 62

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3.5.2.2 Interphase mass transfer Observations have indicated that the rates of mass transfers are closely related to the diffusion at the interface that is related to the concentration gradients at the interface, too. In real problems, however, we have usually no direct way to measure the concentration gradients at the interface. One of the approaches that can be used to estimate the concentration gradient is the approximation of various elements of concentration gradient in each phase using Finite Difference approach. In fact mass transfer coefficient based on Film theory is originally obtained through this approach. According to this approach various elements of concentration gradients of phase 'q' can be obtained as follows:

Ciq x j

Ciq Ciq

Eq. 3.20

Δx j

Where Ciq is the concentration of i-th component in phase q right at the interface and C*iq is the concentration of this component when phase q is at equilibrium with the other phase in the mixture. This is based on the fact that in a multiphase system, they are assumed to be at equilibrium right at their interface. For a mixture containing vapor and liquid the equilibrium concentration of various components can be obtained through isothermal flash calculations which are presented at all chemical engineering thermodynamic text books63-64. Details of flash calculation algorithm and equations were given in65. The concentration of various species in vapor and liquid phases are obtained based on Eqs. (4.5), respectively. Having obtained equilibrium concentrations, one can obtain the flux of species transfer (q Ni) and the rate of inter-phase mass transfer (Spq) through Eqs. (8.9), respectively, in which Mi is molecular weight for i-th species. Calculated flux or component i ( q Ni) in one phase is a source or sink for the same component in the other phase because there is no accumulation at the interface.

N Di q i

Ciq Ciq Δz j

,

N ip -Niq

n

,

Spq N iq M i

Eq. 3.21

i 1

Simulation Procedure The transport equations (Eq. (1.1) and (4.5)) were discretized by control volume formulation. UPWIND scheme was used for discretization66. A segregated implicit solver method with implicit linearization was used to solve discretized momentum equations. These equations have been obtained through the application of the first-order upwind scheme and for the pressure velocity coupling, the SIMPLE scheme has been used. For the pressure equation, the pressure staggering option (PRESTO) scheme was used. The program first reads the structured data from pre-processing section (in which the mesh representing the equipment has been built), before it goes into two nested iteration loops. Inner loop iterations are performed within each time step using the equations corresponding to the discretized version of the proposed model, while the outer loop goes through simulation times until it gets to the final time or steady state whichever happens sooner. At each time step, before going into the inner loop the fluid properties in each cell are calculated. In the inner loop, all the discretized equations are solved in three steps. In the first step, the physical Smith J.M, Van Ness H.C., “Introduction to Chemical Engineering Thermodynamics”, 6th Edition, McGraw-Hill. Walas S.M, “Phase Equilibria in Chemical Engineering”, Butterworth, Storeham, USA (1985). 65 See 44. 66 Patankar S.V., “Numerical Heat Transfer and Fluid Flow”, Taylor and Francis, (1980). 63 64

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properties such as density is updated based on the current solution. If the calculation has just begun, the fluid properties will be updated based on the initialized solution. In the second step, the flash calculation is performed in order to obtain the equilibrium concentrations based on which the source terms of the species concentrations and continuity equations are obtained. In the third step, equations of continuity and momentum are solved and after obtaining the velocity and pressure fields, equations corresponding to species concentration are solved in order to obtain the profiles of the concentration of various species. In this step, with the help of Eulerian-Eulerian approach (VOF approach), the trajectory of interface between two phases (liquid and gas) is determined. At the end of this step, convergence checking based on the norm of errors is done. Due to the nonlinearity and interactions of various equations, the convergence is usually achieved after several iterations at each time step. In order to get stable and meaningful results the time step must be very small (in the order of 10−4 s). However, as time goes on, and various states of the system (e.g., velocities and species concentration) obtain their corresponding smooth profile throughout the system, the time step can be gradually increased. This is due to the fact that dependence of various physical properties (e.g., density, specific heat…) on species concentration increases the amount of interaction and coupling of equations. It should also be noted that, in this mechanism the time step could not get values beyond 10-3s. In general, the time-stepping strategy depends on the number of iterations by time step needed to ensure very low residuals values (less than 10−7 for concentration and 10-5 for momentum and continuity). Results and Discussion The initial condition of the simulation at which the liquid height measured from bottom was 7cm. At the same time, the concentration of octane in gas phase and propane in liquid phase were set to zero. It was also assumed that there is no movement in the system and hence the velocity was set to zero for the whole domain. As time goes on, species are transferred between phases, this leads to a time varying concentration profiles in both phases and a general velocity field for the whole fluid (see Figure 22). Octane was transferred from liquid phase to gas phase and concentration of octane in liquid was decreased whereas concentration of octane in gas was increased. On the other hand, Propane dissolved in liquid phase which leads to its concentration decrease in gas phase, can be seen right at the interface. The propane concentration has its least value for gas phase and the largest value for the liquid phase . As a result of mass transfer in the interface, velocity in this region is higher than others. Density of gas phase was increased near the interface because concentration of octane was increased. In contrast, density of liquid phase was decreased near interface because concentration of propane was increased; (for further information please see67).

Irani, Mohammad; Bozorgmehry Boozarjomehry, Ramin*+; Pishvaie, Sayed Mahmoud Reza and Tavasoli, Ahmad, “Investigating the Effects of Mass Transfer and Mixture Non-Ideality on Multiphase Flow Hydrodynamics Using CFD Methods”, Iran. J. Chem. Chem. Eng., Vol. 29, No. 1, 2010. 67

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t = 3650

t = 0.0

Figure 22

Contour of gas volume fraction at different time levels

Concluding Remarks In the present work, a CFD framework has been proposed to simulate the multiphase mass transfer problems in chemical processes. For this purpose, a numerical method based on a macroscopic model and the finite volume method was applied. The proposed CFD framework is able to solve multiphase mass transfer problems with high interaction of thermodynamic and hydrodynamic behavior of the system. The proposed framework makes it possible to take into account the interacting effects of mixture non-ideality, mass transfer and hydrodynamics on multiphase system in a more realistic way. None of the analysis and studies that have been done on the hydrodynamic of multiphase systems has covered these effects till now, the major reason why these issues have not been covered till now was the fact that none of the available commercial CFD applications has a readymade frame work for such an analysis.

Case Study 6 - Numerical Study of Turbulent Two-Phase Coquette Flow

Motivation and Literature Survey The motivation behind this work is the need to understand bubble generation mechanisms due to interactions between free surface and turbulent boundary layers as commonly seen near ship walls as investigated by [Ovsyannikov, et al.]68. As a recognized problem, we consider a turbulent plane coquette flow with vertical parallel sidewalls and an air-water interface established by gravity in the vertical direction. Two-phase coquette flow has been described in the literature in different flow setups. Most studies were limited to cases of low Reynolds number and the evolution of a single bubble/droplet. Deformation and breakup of a single droplet in a plane coquette flow at low Reynolds number has been studied experimentally, theoretically, and numerically. In another example of the two-phase coquette flow is the two-layers of immiscible fluids which are set between moving horizontal walls. Due to viscosity difference between fluids, there is a jump in the tangential velocity 68 A.Y. Ovsyannikov, D. Kimy, A. Mani AND P. Moin, “Numerical study of turbulent two-phase Couette flow”, Center

for Turbulence Research Annual Research Briefs 2014.

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gradient across the interface which induces instabilities at the fluid interface [Coward]69, [Charru & Hinch]70. At high Reynolds number, there are only a few studies of two-phase Couette flow. [Iwasaki]71 studied the dynamics of a single immiscible drop in turbulent gas ow between two moving walls. [Fulgosi]72 and [Liu]73 performed direct numerical simulation (DNS) of interface evolution in Couette flow between two moving horizontal walls. One interesting case of a two-phase Couette flow in a turbulent regime is when the initial interface is set to be orthogonal to the moving vertical walls. In such a setup, the interaction between the fluid interface and the turbulent boundary layer is a key phenomenon. At sufficiently high Reynolds, Weber and Froude numbers, shear-induced interfacial waves can break, which leads to the formation of air cavities. These air cavities will be further fragmented by turbulence to smaller bubbles. These complexities make two-phase Couette flow at high Reynolds number a challenging problem for both experiments and numerical analysis. Capturing the small-scale ow and interface features requires high-resolution experimental techniques and very expensive DNS calculations. Only two numerical studies Kim74-75 have been performed for investigation of air entrainment and bubble generation in two-phase Couette flow. The current work is a continuation of our recent studies (Kim), where we performed numerical simulations of the interface breakup in two-phase Couette flow at Reynolds number of approximately 13000 and Weber number of approximately 42000 (surface tension coefficient was much smaller than the realistic value for an air-water system). The effect of Froude number on the interface breakup and bubble generation was studied in Kim et al. (2012). The second paper of Kim et al. (2013) was mostly devoted to the development and assessment of a mass conservative interfacecapturing method based on a geometric volume-of-fluid (VOF) approach, and one simulation of the two-phase Couette flow was performed for a Reynolds number of 12000 and a Weber number of 200. However, there have been no studies with one-to-one matched experiments and numerical simulations. Objectives The primary objective of this work is to perform a numerical simulation of a two-phase Couette flow with flow parameters very close to those of the laboratory experiment conducted by collaborators. The bubble formation rate, bubble size distribution, and the effect of interface on the modulation of turbulence are the main characteristics of this flow type, and our investigations are focused on understanding these characteristics. Problem Statement We perform a simulation of a two-phase system with realistic air/water density and viscosity ratios. The density of liquid is ρliq = 1000 kg/m3 and the density of gas is ρgas = 1.2 kg/m3. However, to reduce 69 Coward, A. V., Renardy, Y. Y.,

Renardy, M. & Richards, J. R., “Temporal evolution of periodic disturbances in twolayer Couette flow”, J. Computational Physics, 132 (2), pp. 346-361, 1997. 70 Charru, F. & Hinch, J. E., “Phase diagram of interfacial instabilities in a two-layer Couette flow and mechanism of the long-wave instability”, J. Fluid Mech. 414, pp. 195-223, 2000. 71 Iwasaki, T., Nishimura, K., Tanaka, M. & Hagiwara, Y. ,”Direct numerical simulation of turbulent Couette flow with immiscible droplets”, Int. J. Heat Fluid Flow 22, pp. 332-342, 2001. 72 Fulgosi, M., Lakehal, D., Banerjee, S. & De Angelis, V. ,”Direct numerical simulation of turbulence in a sheared air-water ow with a deformable interface”, J. Fluid Mech. 482, pp. 319-345, 2003. 73 Liu, S., Kermani, A., Shen, L. & Yue, D. K. P.,” Investigation of coupled air-water turbulent boundary layers using direct numerical simulations”. Phys. Fluids 21 (6), pp. 62-108, 2009. 74 Kim, D., Mani, A. & Moin, P., “Numerical simulation of wave breaking in turbulent two-phase Couette flow”, Annual Research Briefs, Center for Turbulence Research, Stanford University, pp. 171-178-2012. 75 Kim, D., Mani, A. & Moin, P., “Numerical simulation of bubble formation by breaking waves in turbulent twophase Couette flow”, Annual Research Briefs, pp. 37-46, 2013.

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values. Thus the viscosity of the gaseous phase is μgas = 7.2 x105 Pa-s and the viscosity of the liquid phase is μliq = 4x103 Pa - s. The surface tension coefficient σ and gravity acceleration g take realistic values of 0.07N/m and 9.81m/s2, respectively. Figure 23 depicts the schematic of the flow configuration and domain size. The computational domain is a rectangular box with sizes 2_h, 2h and 6h in the stream wise (x), wall-normal (y) and span wise (z) directions, respectively. Here h = 2 cm is the half-distance between walls. Initially, the interface is located at a plane z = 4h, hence a height of liquid layer Hliq is 4h and a height of gas layer Hgas is 2h. The sidewalls are moving in the opposite directions with speed of Uw = 1.6m/s. Based on the chosen parameters, the following main dimensionless parameters are:

Re

ρ liq U w h μ liq

, We

Figure 23

ρliq U 2w h σ

, Fr

Schematic illustration of the flow geometry

Uw gh

, Ar

H liq h

Eq. 3.22

which are Reynolds, Weber, Froude numbers, and aspect ratio (ratio of water depth to h), respectively. All non-dimensional groups are determined using properties of the liquid phase. In the current study, Re = 8000, We = 730, Fr = 3.6 and Ar = 4. Governing Equations and Numerical Method The governing equations describing the motion of two immiscible, incompressible New tonian fluids are the conservation of mass and momentum. The first equation is given in terms of the volume fraction function as

ψ .(ψu) 0 t ρu .(ρuu) -p .τ ρg Fst t

Eq. 3.23

where u is the velocity, p is the pressure, ρ is the density, τ = μ(⩢ u +⩢ uT ) is the shear stress tensor, μ is the dynamic viscosity, g is the acceleration due to gravity and Fst = σκδ∑n is the surface tension force. Here, σ is the surface tension coefficient, δ∑ is the Dirac delta function, κ is the interface curvature, and n is the interface normal vector. The volume fraction function is equal to 1 in the gas phase and 0 in the liquid phase. The interface is then represented by the volume fraction values, 0

> η. Therefore, we choose conventional DNS resolution as in single-phase flows. We use a Cartesian grid with uniform mesh spacing in x and z directions, and stretched mesh in the y direction according to

59

2(j 1) h tanh γ( 1 N y 1 yj tanh ( )

Eq. 3.25

where the stretching parameter ϒ is 2.9. Based on the viscous length scale in the liquid phase the grid resolution is Δx+ =13, Δy+min= 0.2, Δy+max =13 and Δz+ =7. Such discretization results in at least 4 grid points per Hinze scale (8 points per bubble diameter) and 18 grid points per viscous sublayer. Finally, the time step is given from the stability constraint due to surface tension as it is more restrictive than the stability conditions due to convection and gravity terms (the viscous terms are treated implicitly):

Δt

ρ liq ρ gas 4ππ

-6 Δy3/2 min 10 s

Eq. 3.26

Results Figure 24 (a-d) show instantaneous snapshots of the interface (given by iso surface ψ = 0.5). On the

(a) t* = 5

(b) t* = 10

(c) t* = 15

(d) t* = 60

Figure 24

Snapshots of the air-water interface at different times

60

free surface, shear-induced oblique wave structures are observed (Figure 24(a)), then the interfacial waves grow in amplitude (Figure 24(b)), leading to breakup of the interface (Figure 24(c)). The air cavities are found underneath the free surface, trapped between the breaking interfacial waves. These air cavities are subsequently fragmented into air bubbles in the water. Finally, Figure 24(d) shows the interface at the fully developed state. Error! Reference source not found. shows time- and tream wise-averaged flow for two cases76: (a) a two-phase Couette flow and (b) a single-phase Couette water flow at the same Reynolds number of 8000. The color contours correspond to the stream wise component of the mean velocity, ū, while the wall-normal and span wise components, (_v, _ w), are shown as vector plot. The maximum Root-Mean-Square (RMS) value of in-plane velocities represents (after 100 flow-through times) approximately 6-7% of the maximum stream wise velocity for both cases and does not diminish with time. For the single-phase Couette flow, secondary flow is represented by four large eddies, while for the two-phase case the secondary flow is more complex due to the effects of the interface. In our simulations the length of the computational domain, 2πh, was chosen as in classical simulations of pressure-driven channel ow. However, it is known from studies of single-phase Couette flow [Lee & Kim 1991; Komminaho et al. 1996; Papavassiliou & Hanratty 1997] that the domain length should be much longer. In this work we focus on a study of interface/boundary layer interaction and leave the study of Couette flow in longer channels for future research. Mean stream wise velocity profiles measured at z/h = 1.5, z/h = 2.5, z/h = 3.5 and z/h = 4, locations are shown in Figure 2677. In addition to two-phase Couette results, we

Figure 25

Turbulent statistics: time- and stream wise-averaged velocity field

Color plot shows the mean stream wise velocity; vector plot depicts mean in-plane flow. (a) Two-phase Couette flow at Re = 8000, We = 730 and Fr = 3.6; (b) Single-phase Couette flow at Re = 8000. 77 (a) mean stream wise velocity; (b) mean stream wise velocity in viscous units. Lines represent profiles at different depths: ---- z/h = 4; ,- - - z/h = 3.5; -.-.-.- , z/h = 2.5;-..-..-..-, z/h = 1.5. Symbols ˚ correspond to DNS results (also averaged in the vertical direction) for single-phase Couette water flow at the same Reynolds number. 76

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show the result of simulation for single-phase Couette flow at the same Reynolds number of 8000. Figure 26 (a) shows velocity profiles in non-dimensional units, U/Uw versus y/h. Figure 26 (b) shows the same data in non-dimensional viscous units according to U+ = (U - Uw)/uτ and y+ = (h y)/δν. As seen from Figure 26 (b), there is no collapse of the results for two-phase Couette flow with the log-law even for a velocity profile quite far from the interface at z/h = 1.5. Even for single-phase Couette flow, a particular profile of stream wise velocity does not necessarily collapse with the loglaw owing to the presence of persistent roller structures. Only after averaging of velocity in a vertical direction does the velocity profile for single-phase Couette flow matches with the log-law.

Figure 26

Turbulent statistics for two-phase Couette flow

Influence of the water depth Figure 2778 shows a comparison of the fully developed air-water interface from our previous results 79. (Figure 27(a)) and the current simulation (Figure 27(b)). The results on the left figure were obtained for an Aspect Ratio of 1.57, whereas in the current simulation the Aspect Ratio is 4. As seen, the simulation for the higher value of the aspect ratio predicts much less air entrainment. To confirm our hypothesis of the influence of the water depth on bubble density, we are currently running simulations for low and high values of the aspect ratio (1.57 and 8) for the same non-dimensional parameters: Reynolds, Weber and Froude numbers, as in the simulation presented in this work. Conclusions In this work we performed a numerical simulation of a two-phase Couette flow at a Reynolds number of 8000, Froude number of 3.6, Weber number of 730 and at realistic air-to-water density and viscosity ratios. Except for Reynolds number and water depth, the parameters of simulation correspond to experiments currently being performed at the University of Maryland. The VOF method has been used to predict interface dynamics. To achieve statistically state two-phase simulations of Couette flow require much longer time compared to simulations of single-phase Couette flow. Complication of turbulent intensity was found near the interface, whereas diminished

(a) Results for an Aspect Ratio of 1.57 and Re = 12000, We = 206, Fr = 3:8 (Kim et al. 2013); (b) Present results for an Aspect Ratio of 4 and Re = 8000, We = 730, Fr = 3.6 79 Kim, D., Mani, A. & Moin, P. ,”Numerical simulation of bubble formation by breaking waves in turbulent twophase Couette flow”, Annual Research Briefs, pp. 37{46. Center for Turbulence Research, Stanford University, pp. 37-46, 2013. 78

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turbulent intensity was observed in the core region. It was found that the bubble density depends on the water depth. To validate this result, in our ongoing work we compare numerical results obtained for different aspect ratios with experimental data. In addition, a simulation is being carried out with deeper water at an aspect ratio Hliq=h of 8 and the same non-dimensional parameters, Re = 8000, We = 730, Fr = 3:6.

Figure 27

Air-water interface at fully developed state

Case Study 7 - Slug Flow in Horizontal Air and Water Pipe Flow Motivation and Statement of Problem The growth of liquid slugs in oil and gas pipelines is a vast and costly problem for the oil firms. A pressure drop in oil production is the main source of the problem that leads to terrain-induced slug flow in the pipeline between the production platform and wellhead platform. This type of slug flow condition can create huge transient surges. The transient nature of the slugs if not appropriately considered might become climacteric and can hasten the material’s fatigue with the risk of pipe damage and maintenance costs. Recently, [Z. I. Al-Hashimy, et al.]80 has been simulated the slug flow regime in an air-water horizontal pipe flow. The variables identified to characterize the slug regime are the slug length and slug initiation. The volume of fluid method was employed assuming unsteady, immiscible air/water flow, constant fluid properties and coaxial flow. The simulated pipe segment was 8 m long and had a 0.074 m internal diameter. Three cases of airwater volume fractions have been investigated, where the water flow rate was pre-set at 0.0028 m3/s, and the air flow rate was varied at three dissimilar values of 0.0105, 0.0120 and 0.015 m3/s. These flow rates were converted to superficial velocities and used as boundary conditions at the inlet of the pipe. The simulation was validated by bench marking with a Baker chart, and it had successfully predicted the slug parameters. The computational fluid dynamics simulation results revealed that the slug length and pressure were increasing as the air superficial velocity increased. The slug initiation position was observed to end up being shifted to a closer position to the inlet. It was believed that the strength of the slug was high at the initiation stage and reduced as the slug Zahid I. Al-Hashimy1, Hussain H. Al-Kayiem, Rune W. Time & Zena K. Kadhim, “Numerical Characterization Of Slug Flow In Horizontal Air/Water Pipe Flow”, Int. J. Comp. Meth. And Exp. Meas., Vol. 4, No. 2 (2016) 114– 130. 80

63

progressed to the end of the pipe. The pressure gradient of the flow was realized to increase as the gas flow rate was increasing, which in turn was a result of the higher mean velocity. Slug Flow and Slug Formation in Pipe Slug flow refers to the phenomenon at which 2-phase liquid-gas movements exist in pipelines over a broad range of intermediate flow rates, generating improper disorder resulting from the actions of the liquid and gas plugs, known as slugs. The plug distribution of liquids and gases in slug flows are highly unique but intermittent, basically because of the nature of the terrain, gas/liquid velocity fluctuations, pigging, etc. A slug unit consists of an aerated liquid slug as well as an accompanying gas bubble, controlled within a liquid film of varying thicknesses. The actual thickness of the film in most cases differs from the minimum value at the front of the following slug towards the maximum value at the rear of the preceding slug. Consequently, the slug length may remain steady along the direction of travel while the pressure drops systematically across the sections of the pipe81. The slug formation is a three-step process that is depicted in Figure 28. Originally, the flow is stratified where the gas is at the top of the pipe and the liquid at the bottom. As the gas passes over a wave, there is a pressure drop, then a pressure recovery, creating a small force upward within the wave. Under the conditions that this upward force is sufficient to raise the wave until it extends to the top of the pipe, the flow is considered as slug flow. Once the wave reaches the top of the pipe, it forms into the familiar slug shape with a nose and tail. The slug is forced forward by the gas and thus, can travel at a greater velocity than the liquid film.

Slug Unit Figure 28

Hydrodynamic slug formation (Courtesy of Z. I. Al-Hashimy et al.)

The actual slug movement can be explained by changing the liquid slugs and the gas bubbles moving above the liquid films, which in turn combine to develop what is known as a slug unit. The slug frequency is described by the number of slugs passing a particular point along the pipeline over a particular period of time. Amongst the slug flow characteristics, the slug frequency is an essential component which relates to significant operational difficulties, like the flooding of downstream facilities, severe pipe vibration, pipeline structural instability and wellhead pressure fluctuation. It is generally known that pipe corrosion is substantially impacted by a high slug frequency. Baker Chart Flow regime maps for the 2-phase flow in a horizontal pipe have been intensively researched by many researchers. [Baker]82 presented a map of a two-phase flow in a horizontal pipe by using various fluids in addition to demonstrating distinct phases of mass fluxes along with corresponding fluid properties such as density and surface tension (see Figure 29). The Baker chart also features two dimensionless parameters, l and y, to enable its application for various gas/vapor-liquid combinations different from the standard one (air-water at atmospheric pressure and room Orell, A., “Experimental validation of a simple model for gas–liquid slug flow in horizontal pipes”. Chemical Engineering Science, 60(5), pp. 1371–1381, 2005. 82 Baker, O., “Simultaneous flow of oil and gas”. Oil and Gas Journal. 53, pp. 185–195, 1954 81

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temperature) for which both parameters equate to unity. The Baker chart was used as a reference for the simulation of the horizontal slug flow regimes in the present study.

Figure 29

Baker chart where (.) Operating conditions of water–air two-phase flow

Problem Formulation In this present work, the Volume of Fluid (VOF) method has been implemented employing the commercial software Star-CCM+ to simulate the horizontal sections of a pipe for air-water slug flow. The objective has been to investigate the volume fraction profile and pressure drop along 8 m of the pipe length will be predicted by the simulation. The [Baker] chart was adopted to justify the slug presence in the simulation by computing the superficial mass velocities of the water and air. The simulated horizontal slug flow patterns observed through visualizations of the phase distributions were qualitatively compared against the flow regimes expected by the [Baker] chart. variation, with time, in seven different cross sections along the pipe. In addition, the pressure drop along 8 m of the pipe length will be predicted by the simulation. The [Baker] chart was adopted to justify the slug presence in the simulation by computing the superficial mass velocities of the water and air. The simulated horizontal slug flow patterns observed through visualizations of the phase distributions were qualitatively compared against the flow regimes expected by the [Baker] chart. The [Baker] flow regime map, as demonstrated in Figure 29, shows the standardized boundaries of the various flow pattern regions as functions of the mass flux of the gas phase, G, and the ratio of the mass flux of the water phase and air, L/G. Where, G was the mass flux of the gas phase (kg/m2 s) = (gas mass flow rate/tube cross-sectional area) and L was the mass flux of water phase (kg/m2s) =

65

(water mass flow rate/tube cross-sectional area). The dimensionless parameters, l and y, had been added so that the chart could be utilized for any gas/liquid combination that differed from the standard combination. The standard combinations, at which both parameters, l and y, equate to unity, which are water and air flow under atmospheric pressure and at room temperature. Consequently, for the present application, where the fluids were air and water, the values of l and y are equal to 1.0. By taking into account, the predicted values for l and y, the pattern of the two-phase flows with any gas/liquid at other pressures and temperatures can be forecasted using the same chart. The parameters l and y were can be calculated from

w l w

1

w l

2

3 , g l a w

0.5

Eq. 3.27

Where σ, μ, ρ are the surface tension, viscosity and density, respectively. The subscripts ‘a’ and ‘w’ refers to the air and water, respectively, at normal temperature and atmospheric pressure; whereas, the subscripts ‘g’ and ‘l’ refers to the vapor and liquid conditions of the fluid being considered. 3.7.4.1 Boundary Condition The required boundary conditions depend on the physical models used, where water was designated as the primary phase and air as the secondary phase for all cases [Al-Hashimy, et al.]83. The water and air were considered incompressible. The most suitable boundary condition for external faces in incompressible water was the velocity inlet. The outlet was considered as a pressure-outlet boundary. The boundary conditions used are illustrated schematically in Figure 30. The velocity in a multiphase flows is defined as the ratio of the velocity and the volume fraction of the considered phase in a multiphase system. Actual velocity of phase = (velocity of phase)/(volume fraction of phase). The average velocity of the flow varied depending on the volume fraction of each phase; which, when defined as the area fraction of a phase, is expected to change in space and time. The average velocity for the gas phase and liquid phase, which are called the gas superficial velocity and liquid superficial velocity, can be given by:

Figure 30

Boundary condition for water-air slug flow through a pipe

Zahid I. Al-Hashimy1, Hussain H. Al-Kayiem, Rune W. Time & Zena K. Kadhim, “Numerical Characterization Of Slug Flow In Horizontal Air/Water Pipe Flow”, Int. J. Comp. Meth. And Exp. Meas., Vol. 4, No. 2 (2016) 114– 130. 83

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USG

UG AG UG a G A

, USW

UWAW UWa W A

Eq. 3.28

There were two methods used in specifying the inlet boundary conditions for the simulation of the slug flow84. The first method imposed perturbations at the inlet so that the volume fraction of the liquid phase entered the pipe as a function of time. The second method, the pipe was initially assumed to be filled with stratified air and water with 50% volume percentage and zero velocity. For the present simulation, the initial and inlet region were: the upper half of the pipe was occupied by 50% void fraction of the gas phase, aa and the lower half by 50% volume fraction of the water phase, aw. Then, the field function was used to define the inlet water volume fraction as a function of time. Volume of Fluid (VOF) The VOF model has been utilized in this study to track the interface between the gas-liquid phases in order to define the slug flow regime. The VOF technique exhibits an immense capability in tracking the interface between the two phases using a color function. The color function is Ca = 1 for the entire ak fluids and Ca = 0 for the void; thus, the interface is located at 0 < Ca < 1. In this method, all the cells should be occupied by a single fluid or a combination of fluids because the VOF does not allow for any void cells85. The water phase was selected as the primary phase. The tracking of the interface between the phases was accomplished by the solution of a continuity equation for the volume fraction of the secondary phase. The volume fraction equation was not solved for the primary phase; the primary-phase (water phase) volume fraction was computed as in Eq. (3.6).

aW aG 1

Eq. 3.29

Since the control volume at the interface location was occupied by fluids, the fluid properties, particularly the viscosity and density, changed abruptly with the interface motion. The mixture properties for the density and viscosity appearing in the momentum and mass equations were calculated as:

ρ m a Wρ W a GρG

μ m a Wμ W a Gμ G

Eq. 3.30

Using conservation of mass and momentum, the local volume fraction of the water was given by the following continuity equation:

a W (a W U W ) (a W VW ) (a W WW ) 0 t x y z (ρ m U i ) (ρ m U i U j ) (p) t x i x i xi

Ui Uj μ ρ u u m m i j ρmgi F xj xi

Eq. 3.31

Where the first term, on the left-hand side, denotes the rate of the momentum increasing per unit volume, and the second term denotes the change of the momentum due to convection per unit volume. On the right side, the first term represents the pressure gradient, the second term represents Al-Hashimy, Z.I., Al-Kayiem, H.H., Kadhim, Z.K. & Mohmmed, A.O., “Numerical simulation and pressure drop prediction of slug flow in oil/gas pipelines”. WIT Transactions on Engineering Sciences, 89, 2015. 85 Ranade, VV., “Computational Flow Modeling for Chemical Reactor Engineering”, Vol. 5. Academic press, 2001. 84

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the gravitational force, and the third term represents the viscous effect. The external force per unit volume is given by the last term in the above equation and can be modelled using the continuum surface force (CSF) model. For turbulence modeling, an SST k−w model in conjunction with implicit time integration has been used. Details of methodology is available in [Z. I. Al-Hashimy, et al.]86.

Figure 31

Slug initiation of the air-water slug flow

Results and Discussion 3.7.6.1 Slug Initiation The determination of the slug initiation had been according to the presence of the first slug in the flow field. The simulation results at the three air superficial velocities are shown in Figure 31. It is obvious that the first slug initiation was faster since the air superficial velocity was higher. It was reveals the time and location of the slug initiation at a constant water superficial velocity of 0.651 m/s and three different air superficial velocities of 2.443, 2.792, and 2.49 m/s. The slug initiation position was transferred to a shorter distance from the inlet. 3.7.6.2 Slug Length For the slug body length and slug frequency, which is a reciprocal of the slug unit period, they could actually be considered as the mean number of slugs per unit time as observed by a fixed observer. The measurements of the average slug body by selecting the X coordinate of both the front and rear ends are shown in Figure 32. The reference line indicated in each case is located about 4 m downstream of the inlet. The slug length was estimated as LS = Xfront − Xrear . The slug length was measured from the reference line up to the front of the slug. There was a proportional relation between the slug length and the air superficial velocity. Prediction results of the slug length revealed that when the air superficial velocity increased, the generated slugs became longer compared to the

Zahid I. Al-Hashimy1, Hussain H. Al-Kayiem, Rune W. Time & Zena K. Kadhim, “Numerical Characterization Of Slug Flow In Horizontal Air/Water Pipe Flow”, Int. J. Comp. Meth. And Exp. Meas., Vol. 4, No. 2 (2016) 114– 130. 86

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case of the lower superficial velocity of air. Table 5 shows the predicted slug length at various air superficial velocities. Air/Water

Slug length (m)

Case 1

Mixture Velocity (m/s) 3.049

0.449

Initial Time (s) 0.64

Crossing Time (s) 2.02

Case 2

3.443

1.037

0.48

1.942

Case 3

4.141

1.5

0.373

1.641

Table 5

Figure 32

Slug length at different air-water velocities

Slug length calculation of air-water slug flow

3.7.6.3 Slug Volume Fraction The volume fraction in the slug body or gas void fraction is a crucial parameter for the design of multiphase pipelines and the associated separation equipment; while, the phase composition is proportional to its volume fraction. Figure 33 has provided the results of the simulation for the predicted void fraction of the air-water slug flow regime at various cross sections from 1 to 7 m along of the horizontal pipe, for the first three times. The distribution of water and air in the horizontal slug flow can be vividly noticed. The red color refers to the water phase while the dark blue color refers to the air phase, and the line between both colors display the presence of an interface. The best approximation of the slug flow regime is observed compared with the slug flow regime in the Baker char. The water slugs touched the upper part of the pipe and performed complete slug regime. As stated before, t = 0 represents the initial conditions where the flow along the pipe was stratified flow, in which the upper part was occupied by air, and the lower part was occupied by water due to the gravitational effect. However, the water phase was steady until the generation of the first wave crest because of the sinusoidal perturbation at the inlet; large waves were initiated, which heightened steadily, filling the cross section of the pipe at time 0.5 s. The long slug was observed at 0.75 s, and it

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continued to grow along the pipe at the downstream sections. Generally, the void fraction increased with the increase in the gas velocity. In conclusion, the internal air-water two-phase slug flow behavior in a horizontal pipe were examined and described the numerical procedure used to simulate the case. Slug initiation and growth was effectively predicted by the 3D transient VOF model combined with the homogeneous k−ω turbulence model. The volume fraction profile and pressure variation with time in seven different cross sections along the pipe were examined.

Figure 33

Cross section of the fluid domain for the extraction of volume fraction for Case 3

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4 Chemical Reaction Chemical reaction engineering (reaction engineering or reactor engineering) is a specialty in chemical engineering or industrial chemistry dealing with chemical reactors. Frequently the term relates specifically to catalytic reaction systems where either a homogeneous or heterogeneous catalyst is present in the reactor. Sometimes a reactor per se is not present by itself, but rather is integrated into a process, for example in reactive separations vessels, retorts, certain fuel cells, and photocatalytic surfaces. The issue of solvent effects on reaction kinetics is also considered as an integral part. It is studying and optimizing chemical reactions in order to define the best reactor design. Hence, the interactions of flow phenomena, mass transfer, heat transfer, and reaction kinetics are of prime importance in order to relate reactor performance to feed composition and operating conditions. Although originally applied to the petroleum and petrochemical industries, its general methodology combining reaction chemistry and chemical engineering concepts allows to optimize a variety of systems where modeling or engineering of reactions is needed. Chemical reaction engineering approaches are indeed tailored for the development of new processes and the improvement of existing technologies87.

Overview of Chemical Reaction Engineering

Every industrial chemical process is designed to produce economically a desired product from a variety of starting materials through a succession of treatment steps.

Recycle Figure 34

Typical Chemical Process

Figure 34 shows a typical situation where the raw materials undergo a number of physical treatment steps to put them in the form in which they can be reacted chemically. Then they pass through the reactor. The products of the reaction must then undergo further physical treatment-separations, purifications, etc. for the final desired product to be obtained88. Variables Affecting the Rate of Reaction Many variables may affect the rate of a chemical reaction. In homogeneous systems the temperature, pressure, and composition are obvious variables. In heterogeneous systems more than one phase is involved; hence, the problem becomes more complex. Material may have to move from phase to phase during reaction; hence, the rate of mass transfer can become important. For example, in the burning of a coal briquette the diffusion of oxygen through the gas film surrounding the particle, and From Wikipedia. Marek Ściążko, “Overview of Chemical Reaction Engineering”, Based on: Octave Levenspiel, Chemical Reaction Engineering, 3rd Edition, 2013. 87 88

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through the ash layer at the surface of the particle, can play an important role in limiting the rate of reaction. In addition, the rate of heat transfer may also become a factor. Consider, for example, an exothermic reaction taking place at the interior surfaces of a porous catalyst pellet. If the heat released by reaction is not removed fast enough, a severe non-uniform temperature distribution can occur within the pellet, which in turn will result in differing point rates of reaction. These heat and mass transfer effects become increasingly important the faster the rate of reaction, and in very fast reactions, such as burning flames, they become rate controlling. Thus, heat and mass transfer may play important roles in determining the rates of heterogeneous reactions. Definition of Reaction Rate We next ask how to define the rate of reaction in meaningful and useful ways. To answer this, let us adopt a number of definitions of rate of reaction, all interrelated and all intensive rather than extensive measures. But first we must select one reaction component for consideration and define the rate in terms of this component i. If the rate of change in number of moles of this component due to reaction is dNildt, then the rate of reaction in its various forms is defined as follows:

Based on unit volume of Reacting fluid : 1 dN i moles i formed V dt (volume of fluid)(time) Based on unit mass of solid in fluid/soli d : ri

1 dN i moles i formed W dt (volume of solid)(time) Based on unit interfacia l surface in two fluid systemor basesd on unit surface of solid in gas/solid systems : ri

1 dN i moles i formed S dt (surface)(time) Based on unit volume of solid in gas/solid systems: 1 dN i moles i formed ri Vs dt (volume of solid)(time) ri

Based on unit volume of reactor if different from the rate based on unit volume of fluid : ri

1 dN i moles i formed Vr dt (volume of reactor)(time)

Eq. 4.1

Speed of Chemical Reactions Some reactions occur very rapidly; others very, very slowly. For example, in the production of polyethylene, one of our most important plastics, or in the production of gasoline from crude petroleum, we want the reaction step to be complete in less than one second, while in waste water treatment, reaction may take days and days to do the job. Figure 35 indicates the relative rates at which reactions occur. To give you an appreciation of the relative rates or relative values between what goes on in sewage treatment plants and in rocket engines, this is equivalent to 1 sec to 3 yr. With such a large ratio, of course the design of reactors will be quite different in these cases.

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Figure 35

Moles of A disappearing Rate of reactions

Classification of Reactions There are many ways of classifying chemical reactions. In chemical reaction engineering probably the most useful scheme is the breakdown according to the number and types of phases involved, the big division being between the homogeneous and heterogeneous systems. A reaction is homogeneous if it takes place in one phase alone. A reaction is heterogeneous if it requires the presence of at least two phases to proceed at the rate that it does. It is immaterial whether the reaction takes place in one, two, or more phases; at an interface; or whether the reactants and products are distributed among the phases or are all contained within a single phase. All that counts is that at least two phases are necessary for the reaction to proceed as it does. The Common Types of Chemical Reactions Several general types of chemical reactions can occur based on what happens when going from reactants to products89. The more common types of chemical reactions are as follows:

Combination or Synthesis Decomposition Single Displacement Double Displacement Combustion Redox Organic

4.1.5.1 Combination Chemical Reactions In combination reactions, two or more reactants form one product. The reaction of sodium and chlorine to form sodium chloride,

2Na(s) Cl2 (g) 2NaCl(s) and the burning of coal (carbon) to give carbon dioxide,

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Dummies, a Wiley Brand.

Eq. 4.2

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C(s) O2 (g) CO2 (g)

Eq. 4.3

are examples of combination reactions. Depending on conditions or the relative amounts of the reactants, more than one product can be formed in a combination reaction. 4.1.5.2 Decomposition Chemical Reactions Decomposition reactions are really the opposite of combination reactions. In decomposition reactions, a single compound breaks down into two or more simpler substances (elements and/or compounds).The decomposition of water into hydrogen and oxygen gases,

2H2O(l) 2H2 (g) O2 (g)

Eq. 4.4

and the decomposition of hydrogen peroxide to form oxygen gas and water,

2H2O2 (l) 2H2O(l) O2 (g)

Eq. 4.5

are examples of decomposition reactions. 4.1.5.3 Single Displacement Chemical Reactions In single displacement reactions, a more active element displaces (kicks out) another less active element from a compound. For example, if you put a piece of zinc metal into a copper(II) sulfate solution, the zinc displaces the copper, as shown in this equation:

Zn(s) CuSO 4 (aq) ZnSO4 (aq) Cu(s)

Eq. 4.6

The notation (aq) indicates that the compound is dissolved in water in an aqueous solution. Because zinc replaces copper in this case, it’s said to be more active. If you place a piece of copper in a zinc sulfate solution, nothing will happen. The following table shows the activity series of some common metals. Notice that because zinc is more active in the table, it will replace copper, just as the preceding equation shows. 4.1.5.4 Double Displacement Chemical Reactions In single displacement reactions, only one chemical species is displaced. In double displacement reactions, or metathesis reactions, two species (normally ions) are displaced. Most of the time, reactions of this type occur in a solution, and either an insoluble solid (precipitation reactions) or water (neutralization reactions) will be formed. 4.1.5.5 Precipitation Reactions If you mix a solution of potassium chloride and a solution of silver nitrate, a white insoluble solid is formed in the resulting solution. The formation of an insoluble solid in a solution is called precipitation. Here is the molecular equation for this double-displacement reaction:

KCl(aq) AgNO3 (aq) AgCl(s) KNO3 (aq) The white insoluble solid that’s formed is silver chloride.

Eq. 4.7

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4.1.5.6 Neutralization Reactions The other type of double-displacement reaction is the reaction between an acid and a base. This double-displacement reaction, called a neutralization reaction, forms water. Take a look at the mixing solutions of sulfuric acid (auto battery acid) and sodium hydroxide (lye). Here is the molecular equation for this reaction:

H2SO 4 (aq) 2NaOH(aq) Na 2SO 4 (aq) 2H2 O(l)

Eq. 4.8

4.1.5.7 Combustion Chemical Reactions Combustion reactions occur when a compound, usually one containing carbon, combines with the oxygen gas in the air. This process is commonly called burning. Heat is the most-useful product of most combustion reactions. Here’s the equation that represents the burning of propane:

C3H8 (g) 5O2 (g) 3CO2 (g) 4H 2 O(l)

Eq. 4.9

Propane belongs to a class of compounds called hydrocarbons, compounds composed only of carbon and hydrogen. The product of this reaction is heat. Combustion reactions are also a type of redox reaction. 4.1.5.8 Redox Chemical Reactions Redox reactions, or reduction-oxidation reactions, are reactions in which electrons are exchanged:

2Na(s) Cl 2 (g) 2 NaCl(s) C(s) O 2 (g) CO 2 (g)

Eq. 4.10

Zn(s) CuSO 4 (aq) ZnSO 4 (aq) Cu(s) The preceding reactions are examples of other types of reactions (such as combination, combustion, and single-replacement reactions), but they’re all redox reactions. They all involve the transfer of electrons from one chemical species to another. Redox reactions are involved in combustion, rusting, photosynthesis, respiration, batteries, and more. 4.1.5.9 Organic Reaction Organic reactions occur between organic molecules (molecules containing carbon and hydrogen). Since there is a virtually unlimited number of organic molecules, the scope of organic reactions is very large. However, many of the characteristics of organic molecules are determined by functional groups small groups of atoms that react in predictable ways. Chemkin Chemkin is a proprietary software tool for solving complex chemical kinetics problems. It is used worldwide in the combustion, chemical processing, microelectronics and automotive industries, and also in atmospheric science. It was originally developed at Sandia National Laboratories and is now developed by a US company, Reaction Design. CHEMKIN solves thousands of reaction combinations to develop a comprehensive understanding of a particular process, which might involve multiple chemical species, concentration ranges, and gas temperatures. Chemical kinetics simulation software allows for a more time-efficient investigation of a potential new process compared to direct laboratory investigation. It is a plug-in chemistry solver that can be linked to other computational software packages, such as ANSYS’ FLUENT CFD software, to add accuracy, speed and stability to

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calculations using finite-rate, multi-step reaction kinetics. Detailed reaction kinetics are frequently required to represent combustion and materials process reactions accurately. Such accuracy typically means use of multi-step reaction kinetics. However, using detailed kinetics often causes problems for other software simulation packages, because of computational stiffness, where small changes in one variable may cause large changes in another. In the past, stiff chemistry simulations were characterized by long computational times and unstable solutions that often did not converge. CHEMKIN-CFD tackles these challenges through advanced and more accurate numerical methods, allowing you to incorporate more detailed kinetics descriptions into your simulations.

CFD Applied To Chemical Reaction Engineering

Chemical reaction engineering specifically deals with chemical reactors. It is also referred to as reaction and/or reactor engineering90. The primary goal of chemical reaction engineering is to optimize transport processes, such as heat transfer, mass transfer and mixing, to improve product yield/conversion and reactor operation safety. Simply said, it helps maximize yield while minimizing costs, such as those associated with feedstock, energy input, heat removal or cooling, stirring or agitation, pumping to increase Phase Interactions pressure and/or frictional pressure loss. From the standpoint of CFD, reaction engineering is the application of transport phenomenon and Reaction chemical kinetics knowledge to Flow Mode Types industrial systems. Transport phenomenon is key and defines Reaction which property is important. Design Chemical kinetics, or the study of rates of chemical processes, is founded on the experimental study of how different conditions influence the speed of a chemical Flow reaction and its mechanism and Transport Distribution transition states; and on the development of mathematical models representing the reaction’s characteristics. Figure 36 Essential Aspects of Chemical Reactor Design Essential aspects of a chemical reactor design are shown in Figure 36. Reactor Design and CFD Designing a reactor involves a number of key considerations, including: Reactants/products phase or state comprising solid, gas, liquid or aqueous/dissolved in water Reaction type, including single, multiple and parallel series or polymerization 90

White paper issued by Siemens PLM Software.

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Catalyst identification, which may take in flow distribution and mixing Species transport Operation mode, such as batch, semi-batch or continuous Considering underlying transport processes, such as fluid flow, heat transfer, mass transfer and reactions, is beneficial as CFD simulation can add substantial value to these characteristics. There are many steps in a reactor design process that enable you to take the discovery of a new chemical with novel properties from concept to commercial production, including:

Conceptualizing how you analyze new chemistry Lab scale to accommodate analysis of kinetics, catalysis, thermodynamics, material properties and toxicity Reactor selection involving the analysis of flow regime, heat release, residence time distribution (RTD), liquid hourly space velocity (LHSV) as well as gas hourly space velocity (GHSV) Engineering with idealized models for analysis of plug flow or continuously-stirred tank reactors (CSTR), volume sizing and overall heat transfer Preliminary design of vessel configuration, internals, baffles and coils Scale up simulations, including pilot scale design, scale up parameters and design space exploration Final design to encompass the extrapolation from scale up rules, flow modeling, safety, risk and runaway analysis and dynamic modeling

You can use an idealized reactor model that has been sized to lab or bench scale to initiate the engineering process and predict key reactor behavior variables, including reaction parameters, material properties, toxicity, ideal operating conditions, optimal catalysts and preliminary reactor dimensions. A well-known traditional approach adopted in chemical engineering to circumvent the intrinsic difficulties in obtaining the “complete velocity distribution map” is the characterization of non-ideal flow patterns by means of residence time distribution (RTD) experiments where typically the response of a piece of process equipment is measured due to a disturbance of the inlet concentration of a tracer. Applications of CFD may be divided into broad categories; those involving single-phase systems and those involving multiphase systems. Within single-phase systems, a further distinction can be made between systems involving laminar flows, turbulent flows, flows with complex rheology, and fast chemical reactions. In many processes encountered in industrial practice, multiphase flows are encountered and, it can be stated that, because of the inherent complexity of such flows, general applicable models and related CFD codes are nonexistent. Gas-Phase Reacting Flow Models The most basic types of flow models are for gas phase reacting flows in which different species (feedstocks) enter the reactor non-premixed, completely mixed or partly premixed. Some models are regarded as tabulated chemistry models to reduce computation times, while others make use of detailed chemistry. A simple glass furnace is a representative example in which air and fuel that is not premixed enters the domain. The simplest model is an eddy break-up combustion model which highlights the flame zone and the region of NOx formation within the overall combustion chamber. NOx is a generic term for mono-nitrogen oxides NO (nitric-oxide) and NO2 (nitrogen dioxide). Liquid-Phase Reactions Liquid-phase reactions significantly differ from gas-phase reactions because the liquid’s rate of diffusion is much lower than its viscosity; therefore, liquid-phase reactions can be strongly influenced

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by scalar gradients. The most important difference in this scenario is the phenomenon of micromixing, or mixing at the molecular scale. In Figure 37, the reactor’s physical configuration will determine whether the reaction will preferentially form product S or, alternatively, product R. In this schematic, for example, the location at which species B is added, either at the vessel’s top or close to the impeller, can significantly alter the results.

Figure 37

Physical reactor configurations essential in predicting liquid phase micro-mixing reactions

Basic Equations of Chemically Reacting Flows in CFD

Basic equations for chemically reacting flows are analyzed conferring to [Sheng-Tao Yu]91. Chemical reactions are modeled by finite rate chemistry, and species equations are incorporated with the momentum and energy equations to model reactive flows. Three physical processes are involved in a reacting flow:

1. Fluid Dynamics, 2. Thermodynamics, 3. Chemical Reactions. The fluid dynamics process is the balance between the temporal evolution and the spatial convection of the flow properties due to conservation of mass, momentum, and energy. The thermodynamics of the reactive fluid include microscopic heat transfer between gas molecules, work done by pressure, and the associated volume change. And, chemical reactions determine the generation/destruction of chemical species under the constraint of mass conservation. Each of the above processes could be either evolving or in equilibrium. For the evolving condition, each above process has its own space and time scales, and they are very different from that of other processes. Such differences in space

91 Sheng-Tao Yu,”

Basic Equations of Chemically Reacting Flows For Computational Fluid Dynamics”, Wayne State University, Detroit, Michigan.

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and time scales, on one hand, could allow simplification in the theoretical model. On the other hand, they could be the source of numerical difficulties. Flow and Reaction Interactions The interactions between chemical reactions and fluid dynamics is best described by the Damköhler number (Da). The Damköhler numbers are dimensionless numbers used in chemical engineering to relate the chemical reaction timescale (reaction rate) to the transport phenomena rate occurring in a system as

Da

Typical time required for Mixing Typical time requires for Chemical Reaction

Eq. 4.11

Da >> 1 - As the Damkohler number approaches infinity, the chemical reactions are much faster compared to fluid dynamics. Thus, the chemical composition can be treated as a state variable governed by the chemical equilibrium theory, i.e., free energy minimization. In this case, the above thermal equilibrium needs be modified to include the effect of free energy minimization. Thus the fluid dynamics becomes the only evolving process. The discussion of this condition, however, is out of the scope of the present work. Da

Multiphase Flow Ideen Sadrehaghighi, Ph.D.

Multiphase Flow

Multicomponent Flow

Chemical Reaction

1

ANNAPOLIS, MD

2

Contents 1

2

Introduction .................................................................................................................................. 7

1.2 Multiphase Flow ..................................................................................................................................................... 7 1.2.1 Solids Phase ......................................................................................................................... 8 1.2.2 Liquids Phase ....................................................................................................................... 8 1.2.3 Gases Phase ......................................................................................................................... 8 1.2.4 Phase Rule ........................................................................................................................... 8 1.2.4.1 Pure Substances (one component) ................................................................................ 9 1.3 Multicomponent Flow .......................................................................................................................................... 9 1.4 Multiscale Flow .................................................................................................................................................... 10

Multi /Phase/Component/Scale/ Flows ........................................................................... 12

2.1 Equations of Multiphase Flow ....................................................................................................................... 12 2.2 Multiphase Coupling .......................................................................................................................................... 13 2.3 Examples of Multiphase Flow ........................................................................................................................ 13 2.4 Modeling Approach Defined Based on Interface Physics ................................................................... 14 2.4.1 VOF Model ......................................................................................................................... 16 2.4.2 Eulerian Multiphase Model ............................................................................................... 16 2.4.2.1 Equations of Eulerian 2-Phase Model .......................................................................... 16 2.4.2.2 Simplification Applied to 2 Phase Flows ...................................................................... 18 2.4.2.3 2-Phase Flow Instability Mechanisms .......................................................................... 18 2.4.2.4 3-Phase Flow................................................................................................................. 18 2.4.2.5 Poly-Dispersed Flow ..................................................................................................... 18 2.4.2.6 Inhomogeneous Multiphase Flow ................................................................................ 18 2.4.2.7 Homogeneous Multiphase Flow ................................................................................... 19 2.4.2.8 Multi-Component Multiphase Flow ............................................................................. 19 2.4.2.9 Volume of Fraction ....................................................................................................... 19 2.4.2.10 Free Surface Flow .................................................................................................... 19 2.4.2.11 Surface Tension ....................................................................................................... 19 2.5.3 Mixture Model ..................................................................................................................... 19 2.4.3 Dispersed Phase Model (DPM) .......................................................................................... 19 2.4.4 Porous bed Model ............................................................................................................. 21 2.5 Guidelines for Selecting a Multiphase Model ........................................................................................... 21 2.6 Volume Averaging Formulation .................................................................................................................... 22 2.7 Constitutive Relations ....................................................................................................................................... 23 2.8 Some Thought in Multiphase CFD for Industrial Processes .............................................................. 25 2.9 Multicomponent Flow ....................................................................................................................................... 27 2.9.1 Integral and Differential Balances on Chemical Species ................................................... 30 2.9.1.1 Molar Basis ................................................................................................................... 30 2.9.1.2 Mass Basis..................................................................................................................... 31 2.9.2 Diffusion Fluxes.................................................................................................................. 32 2.9.3 Fick's Law ........................................................................................................................... 33 2.9.4 Species' Balances for Systems Obeying Fick's Law ............................................................ 34 2.10 Multiscale Modeling ..................................................................................................................................... 34 2.10.1 Traditional Approaches to Modeling ................................................................................. 35 2.10.2 Multiscale Modeling .......................................................................................................... 36

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2.10.3 Sequential Multiscale Modeling ........................................................................................ 36 2.10.4 Concurrent Multiscale Modeling ....................................................................................... 37 2.10.5 Two types of Multiscale Problems..................................................................................... 37 2.10.6 Modeling Approach defined based on Length Scale ......................................................... 37 2.10.6.1 Micro Approach (Fluid–Micro, Particle-Micro) ....................................................... 38 2.10.6.2 Meso Approach (Fluid–Meso, Particle-Meso) ......................................................... 38 2.10.6.3 Macro Approach (Fluid–Macro, Particle-Macro) .................................................... 38 2.10.6.4 Macro‐Micro Approach (Fluid–Macro, Particle-Micro)........................................... 38 2.10.6.5 Meso‐Micro Approach (Fluid–Meso, Particle-Micro) ............................................. 39 2.10.7 Block-Spectral Method of Solution.................................................................................... 39

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3.1 3.2

Case Studies for Composite Fluid ........................................................................................ 41

Case Study 1 - Liquid-Particle Suspension ............................................................................................... 41 Case Study 2 - Two Fluid Flow....................................................................................................................... 41 3.2.1 Mixture Viscosity ............................................................................................................... 41 3.2.2 Drag force .......................................................................................................................... 41 3.3 Case Study 3 - Unsteady MHD 2-Phase Flow of Fluid-Particle Suspension between Two Concentric Cylinders ...................................................................................................................................................... 43 3.3.1 Statement of Problem ....................................................................................................... 43 3.3.1.1 Literature Survey and Background ............................................................................... 43 3.3.2 Mathematical Formulation ................................................................................................ 44 3.3.3 Analytical Approach ........................................................................................................... 46 3.3.4 Comparison with Numerical .............................................................................................. 47 3.4 Case Study 4 - Simulation of Compressible 3-Phase Flows in an Oil Reservoir ........................ 48 3.4.1 Problem Statement ........................................................................................................... 48 3.4.2 Mathematical Modeling .................................................................................................... 49 3.4.3 Temporal and Spatial Discretization Methods .................................................................. 50 3.4.4 Results and Discussion....................................................................................................... 50 3.5 Case Study 5 - Effects of Mass Transfer & Mixture of Non-Ideality on Multiphase Flow ...... 51 3.5.1 Problem Statement ........................................................................................................... 51 3.5.2 Mathematical Model ......................................................................................................... 52 3.5.2.1 Bulk Species Transport ................................................................................................. 52 3.5.2.2 Interphase mass transfer.............................................................................................. 53 3.5.3 Simulation Procedure ........................................................................................................ 53 3.5.4 Results and Discussion....................................................................................................... 54 3.5.5 Concluding Remarks .......................................................................................................... 55 3.6 Case Study 6 - Numerical Study of Turbulent Two-Phase Coquette Flow................................... 55 3.6.1 Motivation and Literature Survey ..................................................................................... 55 3.6.2 Objectives .......................................................................................................................... 56 3.6.3 Problem Statement ........................................................................................................... 56 3.6.4 Governing Equations and Numerical Method ................................................................... 57 3.6.5 Initial and Boundary Conditions ........................................................................................ 58 3.6.6 Grid Resolution and Time Step Requirement .................................................................... 58 3.6.6.1 Turbulent length scale .................................................................................................. 58 3.6.6.2 Interface length scale ................................................................................................... 58 3.6.7 Results ............................................................................................................................... 59 3.6.8 Influence of the water depth............................................................................................. 61 3.6.9 Conclusions ........................................................................................................................ 61

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3.7 Case Study 7 - Slug Flow in Horizontal Air and Water Pipe Flow ................................................... 62 3.7.1 Motivation and Statement of Problem ............................................................................. 62 3.7.2 Slug Flow and Slug Formation in Pipe................................................................................ 63 3.7.3 Baker Chart ........................................................................................................................ 63 3.7.4 Problem Formulation......................................................................................................... 64 3.7.4.1 Boundary Condition ...................................................................................................... 65 3.7.5 Volume of Fluid (VOF)........................................................................................................ 66 3.7.6 Results and Discussion....................................................................................................... 67 3.7.6.1 Slug Initiation ................................................................................................................ 67 3.7.6.2 Slug Length ................................................................................................................... 67 3.7.6.3 Slug Volume Fraction .................................................................................................... 68

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Chemical Reaction .................................................................................................................... 70 4.1 Overview of Chemical Reaction Engineering........................................................................................... 70 4.1.1 Variables Affecting the Rate of Reaction ........................................................................... 70 4.1.2 Definition of Reaction Rate ............................................................................................... 71 4.1.3 Speed of Chemical Reactions............................................................................................. 71 4.1.4 Classification of Reactions ................................................................................................. 72 4.1.5 The Common Types of Chemical Reactions....................................................................... 72 4.1.5.1 Combination Chemical Reactions ................................................................................. 72 4.1.5.2 Decomposition Chemical Reactions ............................................................................. 73 4.1.5.3 Single Displacement Chemical Reactions ..................................................................... 73 4.1.5.4 Double Displacement Chemical Reactions ................................................................... 73 4.1.5.5 Precipitation Reactions................................................................................................. 73 4.1.5.6 Neutralization Reactions .............................................................................................. 74 4.1.5.7 Combustion Chemical Reactions .................................................................................. 74 4.1.5.8 Redox Chemical Reactions............................................................................................ 74 4.1.5.9 Organic Reaction .......................................................................................................... 74 4.1.6 Chemkin ............................................................................................................................. 74 4.2 CFD Applied To Chemical Reaction Engineering ................................................................................... 75 4.2.1 Reactor Design and CFD .................................................................................................... 75 4.2.2 Gas-Phase Reacting Flow Models ...................................................................................... 76 4.2.3 Liquid-Phase Reactions ...................................................................................................... 76 4.3 Basic Equations of Chemically Reacting Flows in CFD ........................................................................ 77 4.3.1 Flow and Reaction Interactions ......................................................................................... 78 4.3.2 Governing Equations ......................................................................................................... 78

List of Tables Table 1 Single and Multi-Phase flow vs Single and Multi-Component ........................................................ 14 Table 2 Modeling available for Multi-Phase flows............................................................................................... 15 Table 3 Suggested Multiphase Models for some common processes (Courtesy of Andresen et al.) ....................................................................................................................................................................................................... 21 Table 4 Comparison of numerical velocity values (Riemann Sum vs. Finite Difference).................... 48 Table 5 Slug length at different air-water velocities .......................................................................................... 68

List of Figures Figure 1

Example of Multi-Phase flow ........................................................................................................................ 7

5

Figure 2 Carbon Dioxide (CO2) pressure-temperature phase diagram showing the triple point and critical point of CO2 ................................................................................................................................................................ 9 Figure 3 Multi-gas separated by a wall .................................................................................................................... 10 Figure 4 Theories and methods for different temporal and spatial scales................................................ 10 Figure 5 Description of Multiphase flow ................................................................................................................. 12 Figure 6 Coupling in multiphase flows ..................................................................................................................... 13 Figure 7 Shock wave generation in a gaseous medium due to a high pressure and supersonic jet flow. The image of shock wave is captured using synchrotron x-radiography. ......................................... 14 Figure 8 Solid and Fluid in 2-Phase flow and Transport analysis between them .................................. 15 Figure 9 Schematic guide for the selection of Multiphase Models................................................................ 20 Figure 10 Average Volume V and three phases α, β, γ ....................................................................................... 22 Figure 11 Binary System of Gases .............................................................................................................................. 27 Figure 12 Volumetric Flux ............................................................................................................................................. 28 Figure 13 Volume Swept ................................................................................................................................................ 29 Figure 14 Divergence Theorem applied to chemical species.......................................................................... 30 Figure 15 Illustration of the multi-physics hierarchy ........................................................................................ 36 Figure 16 Modeling Scales in Fluid-Particle Systems ......................................................................................... 39 Figure 17 Contour plots for particle volume fraction ........................................................................................ 42 Figure 18 Sketch of the problem ................................................................................................................................. 42 Figure 19 Schematic diagram of the problem....................................................................................................... 45 Figure 20 Sketch of the reservoir with the four injection wells at the corners and the production well in the center ................................................................................................................................................................... 50 Figure 21 Cumulative flow in the production well for a production day ................................................... 51 Figure 22 Contour of gas volume fraction at different time levels ............................................................... 55 Figure 23 Schematic illustration of the flow geometry ..................................................................................... 57 Figure 24 Snapshots of the air-water interface at different times................................................................ 59 Figure 25 Turbulent statistics: time- and stream wise-averaged velocity field...................................... 60 Figure 26 Turbulent statistics for two-phase Couette flow ............................................................................. 61 Figure 27 Air-water interface at fully developed state...................................................................................... 62 Figure 28 Hydrodynamic slug formation (Courtesy of Z. I. Al-Hashimy et al.) ......................................... 63 Figure 29 Baker chart where (.) Operating conditions of water–air two-phase flow ......................... 64 Figure 30 Boundary condition for water-air slug flow through a pipe....................................................... 65 Figure 31 Slug initiation of the air-water slug flow ............................................................................................ 67 Figure 32 Slug length calculation of air-water slug flow .................................................................................. 68 Figure 33 Cross section of the fluid domain for the extraction of volume fraction for Case 3 .......... 69 Figure 34 Typical Chemical Process .......................................................................................................................... 70 Figure 35 Moles of A disappearing Rate of reactions ........................................................................................ 72 Figure 36 Essential Aspects of Chemical Reactor Design ................................................................................. 75 Figure 37 Physical reactor configurations essential in predicting liquid .................................................. 77

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1 Introduction Multiphase Flow The term Multi-phase flow is used to refer to any fluid flow consisting of more than one phase or component. For brevity and because they are covered in other texts, we exclude those circumstances in which the components are well mixed above the molecular level. Consequently, the flows considered here have some level of phase or component separation at a scale well above the molecular level. This still leaves an enormous spectrum of different multiphase flows. One could classify them according to the state of the different phases or components and therefore refer to gas/solids flows, or liquid/solids flows or gas/particle flows or bubbly flows and so on. Many texts exist that limit their attention in this way2. Consequently, the flows considered here have some level of phase separation at a scale well above the molecular level. Some treatises are defined in terms of a specific type of fluid flow and deal with low Reynolds number suspension flows, dusty gas dynamics and so on. Others focus attention on a specific application such as slurry flows, cavitation flows, aerosols, debris flows, fluidized beds and etc. Again there are many such texts and here, we attempt to identify the basic fluid mechanical phenomena and to illustrate those phenomena with examples from a broad range of applications and types of flow (see Figure 1). Virtually every processing technology must deal with multiphase flow, from activating pumps and turbines to electro photographic processes. Clearly the ability to predict the fluid flow behavior of these processes is central to the efficiency and Figure 1 Example of Multi-Phase flow effectiveness of those processes. For example, the effective flow of toner is a major factor in the quality and speed of electro-photographic printers. Multi-Phase flows are also a pervasive feature of our environment whether one considers rain, snow, fog, avalanches, mud slides, sediment transport, debris flows, and countless other natural phenomena. Very critical biological and medical flows are also multiphase, from blood flow to the bends to lithotripsy to laser surgery cavitation and so on. No single list can adequately illustrate the diversity; consequently any attempt at a comprehensive treatment of multiphase flows is flawed unless it focuses on common phenomenological themes and avoids the temptation to deviate into lists of observations. Be aware that there are situations in which the Multi-Phase homogeneous and Multi-Component cases are overlap and hard to distinguished. The difference is the Multi-Component model assumes they mix into a single phase, which can be represented by a bulk density, viscosity etc. and the components are mixed on a microscopic scale. On the other hand, Multi-Phase homogeneous means you have multiple phases (e.g., gas and liquid) and they are separated on a resolvable scale. Some commercial software (i.e., Ansys CFX®) simulating it as multiphase homogenous flow it solves for volume fractions, after that calculates density (mean density?) and then solves momentum equations. If it is multicomponent than it solves for mass fractions, after that calculates mean density and momentum and so on in the end. 2

Christopher E. Brennen, “Fundamentals of Multiphase Flows”, Cambridge University Press 2005.

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Two general topologies of multiphase flow can be usefully identified at the outset, namely Disperse flows and Separated flows. By disperse flows we mean those consisting of finite particles, drops or bubbles (the disperse phase) distributed in a connected volume of the continuous phase. On the other hand separated flows consist of two or more continuous streams of different fluids separated by interfaces. In multiphase flows, solid phases are denoted by the subscript S, liquid phases by the subscript L and gas phases by the subscript G. Some of the main characteristics of these three types of phases are as follows: Solids Phase In a multiphase flow, the solid phase is in the form of lumps or particles which are carried along in the flow. The characteristics of the movement of the solid are strongly dependent on the size of the individual elements and on the motions of the associated fluids. Very small particles follow the fluid motions, whereas larger particles are less responsive. Liquids Phase In a multiphase flow containing a liquid phase, the liquid can be the continuous phase containing dispersed elements of solids (particles), gases (bubbles) or other liquids (drops). The liquid phase can also be discontinuous, as in the form of drops suspended in a gas phase or in another liquid phase. Another important property of liquid phases relates to wettability. When a liquid phase is in contact with a solid phase (such as a channel wall) and is adjacent to another phase which is also in contact with the wall, there exists at the wall a triple interface, and the angle subtended at this interface by the liquid-gas and liquid-solid interface is known as the Contact Angle. Gases Phase As a fluid, a gas has the same properties as a liquid in its response to forces. However, it has the important additional property of being (in comparison to liquids and solids) highly compressible. Notwithstanding this property, many multiphase flows containing gases can be treated as essentially incompressible, particularly if the pressure is reasonably high and the Mach Number, with respect to the gas phase, is low (e.g., < 0.2). Phase Rule Gibbs's phase rule3 was proposed by Josiah Willard Gibbs in his landmark paper titled On the Equilibrium of Heterogeneous Substances. The rule applies to non-reactive multi-component heterogeneous systems in thermodynamic equilibrium and is given by the equality

F CP2 where F number of degrees of freedom Eq. 1.1

C number of components P number of phases

The number of degrees of freedom is the number of independent intensive variables, i.e. the largest number of thermodynamic parameters such as temperature or pressure that can be varied simultaneously and arbitrarily without affecting one another. An example of one-component system is a system involving one pure chemical, while two-component systems, such as mixtures of water and ethanol, have two chemically independent components, and so on. 3

Gibbs, J. W., Scientific Papers (Dover, New York, 1961).

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1.2.4.1 Pure Substances (one component) For pure substances C = 1 so that F = 3 − P. In a single phase (P = 1) condition of a pure component system, two variables (F = 2), such as temperature and pressure, can be chosen independently to be any pair of values consistent with the phase. However, if the temperature and pressure combination ranges to a point where the pure component undergoes a separation into two phases (P = 2), F decreases from 2 to 1. When the system enters the two-phase region, it becomes no longer possible to independently control temperature and pressure. If the pressure is increased by compression, (see Figure 2) some of the gas condenses and the temperature goes up for CO2. If the temperature is decreased by cooling, some of the gas condenses, decreasing the pressure. Throughout both processes, the temperature and pressure stay in the relationship shown by this boundary curve unless one phase is entirely consumed by evaporation or condensation, or unless the critical point is reached. As long as there are two phases, Figure 2 Carbon Dioxide (CO2) pressure-temperature there is only one degree of freedom, which phase diagram showing the triple point and critical point corresponds to the position along the of CO2 phase boundary curve. The critical point is the black dot at the end of the liquid–gas boundary. As this point is approached, the liquid and gas phases become progressively more similar until, at the critical point, there is no longer a separation into two phases. Above the critical point and away from the phase boundary curve, F = 2 and the temperature and pressure can be controlled independently. Hence there is only one phase, and it has the physical properties of a dense gas, but is also referred to as a supercritical fluid. Of the other two-boundary curves, one is the solid–liquid boundary or melting point curve which indicates the conditions for equilibrium between these two phases, and the other at lower temperature and pressure is the solid–gas boundary. Even for a pure substance, it is possible that three phases, such as solid, liquid and vapor, can exist together in equilibrium (P = 3). If there is only one component, there are no degrees of freedom (F = 0) when there are three phases. Therefore, in a single-component system, this three-phase mixture can only exist at a single temperature and pressure, which is known as a triple point. In the diagram for CO2 (see Figure 2), the triple point is the point at which the solid, liquid and gas phases come together, at 5.2 bar and 217 K. It is also possible for other sets of phases to form a triple point, for example in the water system there is a triple point where ice I, ice III and liquid can coexist4.

Multicomponent Flow The multi-component model assumes they mix into a single phase, which can be represented by a bulk density, viscosity etc. The components are mixed on a microscopic scale. The multi-component flow (species transport) refers to flow that the components are mixed at molecular level and can be

4

From Wikipedia, the free encyclopedia.

10

characterized by a single velocity and temperature field for all species5. A simple example of such a multicomponent system is a binary (two component) solution consisting of a solute in an excess of chemically different solvent. (see Figure 3).

Figure 3

Multi-gas separated by a wall

Multiscale Flow Multiscale modeling refers to a style of modeling in which multiple models at different scales are used simultaneously to describe a system. The different models usually focus on different scales of resolution. They sometimes originate from physical laws of different nature, for example, one from continuum mechanics and one from molecular dynamics. In this case, one speaks of multi-physics modeling even though the terminology might not be fully accurate. The need for multiscale modeling comes usually from the fact that the available macroscale models are not accurate enough, and the microscale models are not efficient enough and/or offer too much information. By combining both viewpoints, one hopes to arrive at a reasonable compromise between accuracy and efficiency. The subject of multiscale modeling consists of three closely related components: multiscale analysis, multiscale models and multiscale algorithms. Multiscale analysis tools allow us to understand the relation between models at different scales of resolutions. Multiscale models allow us to formulate models that couple together models at different scales. Multiscale algorithms allow us to use multiscale ideas to design computational algorithms.

Figure 4

Theories and methods for different temporal and spatial scales

Associate Professor Britt M. Halvorsen Amaranath S. Kumara, ”Computational Fluid Dynamics (CFD) and Multiphase Flow Modelling”, Telemark University College, Porsgrunn, Norway. 5

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Figure 4 summarizes theories and typical numerical methods for different temporal and spatial scales. When the continuum assumption breaks down, the fluid has to be described by atomistic point of view, such as the molecular dynamics as a microscopic method or statistical rules for molecular groups, i.e. kinetic theories, as the mesoscopic methods for a larger scale. If the characteristic length is smaller than 1 nm (1 Nano-meter is 1×10−9 m) or the characteristic time is shorter than 1 fs (1 femtosecond is equal to 10−15 seconds), the quantum effect may be not negligible for the concerned system and the quantum mechanics has to be brought in to describe the transport as a result. In fact modeling from a smaller scale may lead to a more accurate description of the problem, but will bring much more computational cost as well. Therefore we may have to find an appropriate tradeoff for our concerned fluid behaviors in engineering6.

6 Shiyi

Chen, Moran Wang, Zhenhua Xia, “Multiscale fluid mechanics and modeling”, Procedia IUTAM 10 ( 2014).

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2 Multi /Phase/Component/Scale/ Flows Up to this point we were dealing with single phase flows. To get matters complicated, we now concern ourselves with multi-phase flows which exist in many industrial applications such as Oil & Gas, Power Generation, Biomedical, Automotive, Chemical Processing and Aerospace among others. Multiphase flows refer to flows of several fluids in the domain of interest. In general, we associate fluid phases with gases, liquids or solids and as such some simple examples of multi-phase flows are: air bubbles rising in a glass of water, sand particles carried by wind, rain drops in air. In fact, the definition of ‘phase’ can be generalized and applied to other fluid characteristics such as size and shape, density, temperature, etc. With this broader definition, multiple phases can be used to represent the entire size distribution of particles in several size groups or ‘phases’ of a multi-phase model. In fluid mechanics, multiphase flow is simultaneous flow of (a) materials with different states or phases (i.e. gas, liquid or solid), or (b) materials with different chemical properties but in the same state or phase (i.e. liquid-liquid systems such as oil droplets in water). Generally, a multiphase fluid is composed of two or more distinct phases which themselves may be fluids, gases or Figure 5 Description of Multiphase flow solids, and has the characteristic properties of a fluid. Within the discipline of multi-phase flow dynamics the present status is quite different from that of the single phase flows. The theoretical background of the single phase flows is well established and apparently the only outstanding practical problem that still remains unsolved is turbulence, or perhaps more generally, problems associated with flow stability. Generally, a phase is a class of matter, with a definable boundary and a particular dynamic response to the surrounding flow and potential field. Phases are generally identified by solid, liquid or gaseous states of matter but can also refer to other forms: Materials with different chemical properties but in the same state or phase (i.e. liquid-liquid, such as, oil-water). The fluid system is defined by a primary and multiple secondary phases (See Figure 5). There may be several secondary phase denoting particles with different sizes.

Equations of Multiphase Flow While it is rather straightforward to derive the equations of the conservation of mass, momentum and energy for an arbitrary mixture, no general counterpart of the Navier-Stokes equation for multiphase flows have been found. Using a proper averaging procedure it is however quite possible to derive a set of equations of multiphase flow which in principle correctly describes the dynamics of any multiphase system and is subject only to very general assumptions7. A direct consequence of the complexity and diversity of these flows is that the discipline of multiphase fluid dynamics is and may long remain a prominently experimental branch of fluid mechanics. Preliminary small scale model testing followed by a trial and error stage with the full scale system is still the only conceivable solution for many practical engineering problems involving multiphase flows. Inferring the necessary constitutive relations from measured data and verifying the final results are of vital importance also within those approaches for which theoretical modeling and subsequent numerical solution is considered feasible. 7

Multiphase flow Dynamics, Theory and Numeric.

13

Multiphase Coupling Phase coupling, in terms of momentum, energy, and mass, is a basic concept in the description of any multiphase flow. The coupling can occur through exchange of momentum, energy, and mass among phases as shown in Figure 8. In principle, fluid‐particulate properties can be described by position, velocity, size, temperature, and species concentration of fluid and/or particle. While the phenomenological description of multiphase flow can be applied to classify flow characteristics, it also can be used to determine appropriate numerical formulations. In various modes of coupling, depending on the contribution of phases and phenomena, different coupling schemes can be adapted. This may allow independent treatment of phases or simultaneous integration of momentum, heat, and mass exchanges between phases. In general, modeling complexity increases as more effects associated with time and length scales are included in the simulation. In general, coupling depends on particle size, relative velocity, volume fraction. Three ways that coupling could be presents as shown in Figure 6:

One Way

Two Way

Four Way

1. One-way coupling: Sufficiently dilute such that fluid feels no effect from presence of Figure 6 Coupling in multiphase flows particles. Particles move in dynamic response to fluid motion. 2. Two-way coupling: Enough particles are present such that momentum exchange between dispersed and carrier phase interfaces alters dynamics of the carrier phase. 3. Four-way coupling: Flow is dense enough that dispersed phase collisions are significant momentum exchange mechanism8.

Examples of Multiphase Flow While the modeling and numerical simulation of multiphase and multicomponent flows poses far greater challenges than that of single-phase and single-component flows, their accessibility in nature is numerous. Rain and snow, and a vaporing tea pot is among prime example of multiphase flow. Others include, Spray drying, Pollution control, Pneumatic transport, Slurry transport, Fluidized beds, Spray forming, Plasma spray coating, Abrasive water jet cutting, Pulverized coal fired furnaces, Solid propellant rockets, Fire suppression and controls9. These challenges are due to interfaces between phases and large or discontinuous property variations across interfaces between phases and/or components. High-pressure and supersonic multiphase and multicomponent jet flow is one of the most challenging problems in multiphase flow due to the complexity of the dynamics of the jet. For example, the presence of cavitation and gas entrapment inside the nozzle orifice can greatly affect 8 9

Ken Kiger, “Multiphase Turbulent Flow”, UMCP presentation. Grétar Tryggvason, “Multiphase Flow Modeling”, spring 2010.

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the development and formation of the external jet. Another case involves high-pressure fuel spray has never been recognized as supersonic under typical fuel injection conditions10-11. Recently, MacPhee et al12 have used a synchrotron x-radiography and a fast x-ray detector to record the time evolution of the transient fuel sprays from a high-pressure injector. In their experiment, the propagation of the spray-induced shock waves in a gaseous medium were captured and the complex nature of the spray Figure 7 Shock wave generation in a gaseous medium due hydrodynamics were revealed. They to a high pressure and supersonic jet flow. The image of have found out that under injection shock wave is captured using synchrotron x-radiography. conditions similar to those in operating engines, the fuel jets can exceed supersonic speeds and result an oblique shock wave in the gaseous medium, see Figure 7. However, the effect of this shock wave to the atomization of the fuel and the combustion processes is currently not known. There are 4 distinctive new attitude which cover the following flow regimes13 in Table 1.

Single - Phase

Multi - Phase

Table 1

Single - Component

Multi - Component

Water flow Pure Nitrogen flow Steam bubble in H20 Freon-Freon Vapor flow Ice Slurry flow

Air flow H20 + Oil blends Coal particles in air Sand particle in H20

Single and Multi-Phase flow vs Single and Multi-Component

Modeling Approach Defined Based on Interface Physics

Genuine models for multiphase flows have been developed mainly following two different approaches. Within the Eulerian Approach all phases are treated formally as fluids which obey normal one phase equations of motion in the unobservable ’mesoscopic’ level (e.g., in the size scale of suspended particles) with appropriate boundary conditions specified at phase boundaries. The macroscopic flow equations are derived from these mesoscopic equations using an averaging T. Nakahira, M. Komori, K. Nishida, and K. Tsujimura. Shock W aves, K. Takayama, Ed., 2:1271–1276, 1992 H.H. Shi, K. Takayama, and O Onodera. JSME Intl. J. Ser. B, 37:509, 1994. 12 A.G. MacPhee, M.W. Tate, C.F. Powell, Y. Yue, M.J. Renzi, A. Ercan, S. Narayanan, E. Fontes, J. Walther, J. Schaller, S.M. Gruner, and J. Wang, ”X-ray imaging of shock waves generated by high pressure fuel spray”, Science, 295:1261–1263, 2002. 13 Randy S. Lagumbay, “Modeling and Simulation of Multiphase/Multicomponent Flows”, A thesis submitted to the Faculty of the Graduate School of the University of Colorado in partial fulfillment of the requirements for the degree of Doctor of Philosophy Department of Mechanical Engineering -2006. 10 11

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procedure of some kind. This averaging procedure can be carried out in several alternative ways such as time averaging, volume averaging and ensemble averaging. Various combinations of these basic methods can also been considered. Irrespective of the method used, the averaging procedure leads to equations of the same generic form, namely the form of the original physical equations with a few extra terms. These extra terms include the interactions (change of mass, momentum etc.) at

Interface

Figure 8

Solid and Fluid in 2-Phase flow and Transport analysis between them

phase boundaries and terms analogous to the ordinary Reynolds stresses in the turbulent single phase flow equations. Each averaging procedure may however provide a slightly different view in the physical interpretation of these additional terms and, consequently, may suggest different approach for solving the closure problem that is invariably associated with the solution of these equations. The manner, in which the various possible interaction mechanisms are naturally divided between these additional terms, may also depend on the averaging procedure being used. The advantage of the Eulerian method is its generality means that in principle it can be applied to any multiphase system, irrespective of the number and nature of the phases. A drawback of the straightforward Eulerian approach is that it often leads to a very complicated set of flow equations and closure relations. In some cases, however, it is possible to use a simplified formulation of the full Eulerian approach, namely Mixture Model (or Algebraic Slip Model). The mixture model may be applicable, e.g., for a relatively homogeneous suspension of one or more species of dispersed phase that closely follow the Approaches

Model

Definitions

Eulerian

Volume of Fluid Model (VOF)

Eulerian

Eulerian Model

Eulerian

Mixture Model

Lagrangian

Dispersed Phase Model (DPM)

Direct method of predicting interface shape between immiscible phases Model resulting from averaging of VOF model applicable to dispersed flows Simplification of Euler model; applicable when inertia of dispersed phase is small Lagrangian particle/bubble/droplet tracking

Table 2

Modeling available for Multi-Phase flows

Flow Regions Stratified Flow Dispersed Flow Dispersed Flow Dilute Flow

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motion of the continuous carrier fluid. For such a system the mixture model includes the continuity equation and the momentum equations for the mixture, and the continuity equations for each dispersed phase. The slip velocities between the continuous phase and the dispersed phases are inferred from approximate algebraic balance equations. This reduces the computational effort considerably, especially when several dispersed phases are considered. Another common approach is the so called Lagrangian Method which is mainly restricted to particulate suspensions. Within that approach only the fluid phase is treated as continuous while the motion of the discontinuous particulate phase is obtained by integrating the equation of motion of individual particles along their trajectories. Table 2 represents the different modeling approaches. VOF Model The VOF Model is designed to track the position of the interface between two or more immiscible fluids. Tracking is accomplished by solution of phase continuity equation resulting volume fraction abrupt change points out the interface location. A mixture fluid momentum equation is solved using mixture material properties. Thus the mixture fluid material properties experience jump across the interface. Turbulence and energy equations are also solved for mixture fluid. Surface tension and wall adhesion effects can be taken into account. VOF is an Eulerian fixed-grid technique. Eulerian Multiphase Model Eulerian Multiphase Model is a result of averaging of NS equations over the volume including arbitrary particles + continuous phase. The result is a set of conservation equations for each phase (continuous phase + N particle “media”). Both phases coexist simultaneously: conservation equations for each phase contain single-phase terms (pressure gradient, thermal conduction etc.). Interfacial terms express interfacial momentum (drag), heat and mass exchange. These are nonlinearly proportional to degree of mechanical (velocity difference between phases), thermal (temperature difference). Hence equations are harder to converge. Within the Eulerian-Eulerian Model, certain inter phase transfer terms used in momentum, heat, and other interphase transfer models, can be modeled using either the Particle Model, the Mixture Model or the Free Surface Model. In particular, the calculation of the interfacial area density, used for all inhomogeneous transfer models for a given fluid pair, is calculated according to one of these models. The available options depend on the morphology of each phase of the pair (for example, continuous, dispersed, etc.), and as settings in (homogeneous options, free surface model option). In the Eulerian Multiphase Model, the phases are treated as interpenetrating continua coexisting in the flow domain. Equations for conservation of mass, momentum and energy are solved for each phase. The share of the flow domain occupied by each phase is given by its volume fraction and each phase has its own velocity, temperature and physical properties. Interactions between phases due to differences in velocity and temperature are taken into account via the inter-phase transfer terms in the transport equations. Eulerian multi-phase modelling provides a general framework for all types of multi-phase flows; both dispersed (e.g. bubble, droplet, and particle flows) and stratified (e.g. freesurface flows) flows can be modelled. Comparing the Eulerian multi-phase with the Lagrangian twophase method, the former has the advantage of being computationally more efficient in situations where the phases are widely dispersed and/or when the dispersed phase volume fraction is high. For free-surface flows, similar advantages could be found in the Eulerian model relative to the previously developed approach. However, the free surface calculated will be less sharp in comparison with the VOF method. 2.4.2.1 Equations of Eulerian 2-Phase Model While it is rather straightforward to derive the equations of the conservation of mass, momentum and energy for an arbitrary mixture, no general counterpart of the Navier-Stokes equation for multiphase flows have been found. Using a proper averaging procedure it is however quite possible

17

to derive a set of Equations of Multi-Phase Flow which in principle correctly describes the dynamics of any multiphase system and is subject only to very general assumptions14. A direct consequence of the complexity and diversity of these flows is that the discipline of multiphase fluid dynamics is and may long remain a prominently experimental branch of fluid mechanics. Preliminary small scale model testing followed by a trial and error stage with the full scale system is still the only conceivable solution for many practical engineering problems involving multiphase flows. Inferring the necessary constitutive relations from measured data and verifying the final results are of vital importance also within those approaches for which theoretical modeling and subsequent numerical solution is considered feasible. The six-equations, one pressure model is currently implemented, supplemented with the High Reynolds number k-ε model for turbulence. A set of distinct mass, momentum and energy conservation equations is solved for each phase, and the phases are coupled via momentum and heat transfer terms. The pressure is assumed to be the same in each phase. Sub-models are provided to describe the interphase exchange terms and close the equations. In the following, we first present the fundamental equations for generic phase k (where k could be either the continuous or dispersed phase), before going on to present the models and sub-models which are implemented to obtain closure. The fundamental equations for the Eulerian two-phase model are: N - phase jk m kj ) (α k ρ k ) .(α k ρ k u k ) Ι k (m t j 1 j k

where

α k volume fraction , I k the rate of mass transfer to the phase k from the other phases N - phase

per unit total volume & since mass as a whole must be conserved

I k 1

ρ k density , u k mean phase velocity and

Dispersed

α

k k Continous

k

0

1

(α k ρ k u k ) .(α k ρ k u k u k ) - k p k k g .[ k ( k kt )] M k (Eint ) k t where k and kt molecular and turbulent stress respectively p pressure to be assumed to be equal for both phases M k inter - phase momentum trasfer per unit volume (Fint ) k internal forces g gravity vector Eq. 2.1

The inter-phase momentum transfer represents the sum of all the forces the phases exert on one another and satisfies Mc = - Md. The internal forces represent forces within a phase. In the current form, they are limited to particle-particle interaction forces in the dispersed phase. The derivation

14

Multiphase flow Dynamics, Theory and Numeric.

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for energy equation and interphase momentum transfer can be applied through CD-adapco® methodology manual. 2.4.2.2 Simplification Applied to 2 Phase Flows Certain statement can be made for simplification of the equation with regard to the case investigated. For example in gas-particle two-phase flows, when the concentration of the dispersed phase is low, certain assumptions may be made which simplify considerably the equations to solve. The gas and particle flows are then linked only via the interaction terms. One may therefore uncouple the full system of equations into two subsystems: one for the gas phase, whose homogeneous part reduces to the Euler equations; and a second system for the particle motion, whose homogeneous part is a degenerate hyperbolic system. The equations governing the gas phase flow may be solved using a high-resolution scheme, while the equations describing the motion of the dispersed phase are treated by a donor-cell method using the solution of a particular Riemann problem. Coupling is then achieved via the right-hand-side terms15. 2.4.2.3 2-Phase Flow Instability Mechanisms The objective here is to review the main kinds of instabilities occurring in two-phase flows. It complements previous reviews, putting all two-phase flow instabilities in the same context and updating the information including coherently the data accumulated in recent years. In the first section, a description of the main mechanisms involved in the occurrence of two-phase flow instabilities is made. In order to get a clear picture of the phenomena taking place in two-phase flow systems it is necessary to introduce some common terms used in this field. The first distinction should be made between microscopic and macroscopic instabilities. The term microscopic instabilities is used for the phenomena occurring locally at the liquid–gas interface; for example, the Helmholtz and Taylor instabilities, bubble collapse, etc. The treatment of this kind of instabilities is out of the scope of this work. On the other hand, the macroscopic instabilities involve the entire twophase flow system. In this review, the main focus is kept on macroscopic phenomena. The most popular classification, introduced in [Bouré]16, divides two-phase flow instabilities in static and dynamic. In the first case, the threshold of the unstable behavior can be predicted from the steadystate conservation laws. On the other hand, to describe the behavior of dynamic instabilities it is necessary to take into account different dynamic effects, such as the propagation time, the inertia, compressibility, etc. In addition, the term compound instability is normally used when several of the basic mechanisms, described later, interact with each other17. 2.4.2.4 3-Phase Flow It is possible to have more than one dispersed phase in a continuous phase. For example, certain regimes of water-oil-gas flow in an oil pipeline may involve both oil droplets and gas bubbles immersed in a continuous water phase. 2.4.2.5 Poly-Dispersed Flow The above dispersed flow examples assume a single mean particle diameter for the dispersed phases. Poly-dispersed flows involve dispersed phases of different mean diameters. 2.4.2.6 Inhomogeneous Multiphase Flow Inhomogeneous multiphase flow refers to the case where separate velocity fields and other relevant R. Saurel, A. Forestier, D. Veyret And J. C. Loraud, “A Finite Volume Scheme For Two-Phase Compressible Flows”, International Journal For Numerical Methods In Fluids, Vol. 18, 803-819 (1994). 16 J. Bouré, A. Bergles, L. Tong, Review of two-phase flow instabilities, Nucl. Eng. Des. 25 (1973) 165–192. 17 Leonardo Carlos Ruspini, Christian Pablo Marcel, Alejandro Clausse, “Two-phase flow instabilities: A review”, International Journal of Heat and Mass Transfer 71 (2014) 521–548. 15

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fields exist for each fluid. The pressure field is shared by all fluids. The fluids interact via interphase transfer terms. The Particle and Mixture Models are both inhomogeneous multiphase models. 2.4.2.7 Homogeneous Multiphase Flow Homogeneous multiphase flow is a limiting case of Eulerian-Eulerian multiphase flow where all fluids share the same velocity fields, as well as other relevant fields such as temperature, turbulence, etc. The pressure field is also shared by all fluids. 2.4.2.8 Multi-Component Multiphase Flow It is possible to combine the notions of multicomponent and multiphase flows. In this case, more than one fluid is present, and each such fluid may be a mixture of chemical species mixed at molecular length scales. An example is air bubbles in water in which ozone gas is dissolved in both the gaseous and liquid phases. In this case, mass transfer of common species may occur by diffusion across the phase interface. 2.4.2.9 Volume of Fraction Multi-Phase modeling employs the notion of interpenetrating continua. Although phases are mixed at length scales much larger than molecular, they are also assumed to be mixed at length scales smaller than you want to resolve. Thus, each phase is assumed to be present in principle in each control volume, and assigned a volume fraction equal to the fraction of the control volume occupied by that phase. 2.4.2.10 Free Surface Flow Free Surface flow refers to a multiphase situation where the fluids (commonly water and air) are separated by a distinct resolvable interface. 2.4.2.11 Surface Tension Surface tension is a force that exists at a free surface interface which acts to minimize the surface area of the interface. It gives rise to effects such as a pressure discontinuity at the interface and capillary effects at adhesive walls. 2.5.3 Mixture Model Mixture Model is a simplified Eulerian approach for modeling n-phase flows. The simplification is based on the assumption that the Stokes number is small (particle and primary fluid velocity is nearly equal in both magnitude and direction). Solves the mixture momentum equation (for mass-averaged mixture velocity) and prescribes relative velocities to describe the dispersed phases. Interphase exchange terms depend on relative (slip) velocities which are algebraically determined based on the assumption that St is defined here as a volume average instead of a time average as the usual Reynolds stress. It also contains momentum fluxes that arise both from the turbulent fluctuations of the mesoscopic flow and from the fluctuations of the velocity of phase due to the presence of other phases. Consequently, it does not necessarily vanish even if the mesoscopic flow is laminar. The so called transfer integrals, Ϻα, Гα are defined as

Γα

1 ρα (uα u A ).nˆ αdA V Aα

Mα

1 ραuα (uα u A ).nˆ αdA V Aα

Eq. 2.5

nˆ α unit outward normal vector of phase α Constitutive Relations Eqn. (5.4) are, in principle, exact equations for the averaged quantities. So far, they do not contain much information about the dynamics of the particular system to be described. That information must be provided by a set of system dependent constitutive relations which specify the material properties of each phase, the interactions between different phases and the (pseudo)turbulent stresses of each phase in the presence of other phases. These relations finally render the set of equations in a closed form where solution is feasible. At this point we do not attempt to elaborate in detail the possible strategies for attaining the constitutive relations in specific cases, but simply state the basic principles that should be followed in inferring such relations. The unknown terms that appear in the averaged equations (5.4) such as the transfer integrals and stress terms that still contain macroscopic quantities, should be replaced by new terms. Typically, constitutive relations are given in a form where these new terms include free parameters which are supposed to be determined experimentally. In some cases constitutive laws can readily be derived from the properties of the mixture, or from the properties of the pure phase. For example, the incompressibility of the pure phase α implies the constitutive relation ρα = constant. Similarly, the equation of state Pα = Cρα, where C = constant for the pure phase, implies Pα = Cρα. In most cases, however, the constitutive relations must be either extracted from experiments, derived analytically under suitable simplifying assumptions, or postulated.

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Including a given physical mechanism in the model by imposing proper constitutive relations is not always straightforward even if adequate experimental and theoretical information is available. In particular, making specific assumptions concerning one of the unknown quantities may induce the constraints on other terms. For example, the transfer integrals Гα and Mα contain the effect of exchange of mass and momentum between the phases. According to Eqn. (5.4), the quantity Гα gives the rate of mass transfer per unit volume through the phase boundary Aα into phase α from the other phases. In a reactive mixture, where phase α is changed into phase, the mass transfer term Гα might be given in terms of the experimental rate of the chemical reaction α → γ, correlated to the volume fractions ϕα and ϕγ, and to the temperature of the mixture T. Similarly, the quantity Mα gives the rate of momentum transfer per unit volume into phase α through the phase boundary Aα. The second integral on the right side of Eqn. (5.4) contains the transfer of momentum carried by the mass exchanged between phases. It is obvious that this part of the momentum transfer integral Mα must be consistently correlated with the mass transfer integral Гα. Similarly, the first integral on the right side of Eqn. (5.4) contains the change of momentum of phase α due to stresses imposed on the phase boundary by the other phases. Physically, this term contains forces such as buoyancy which may be correlated to average pressures and gradients of volume fractions, and viscous drag which might be correlated to volume fractions and average velocity differences. For instance in a liquid-particle suspension, the average stress inside solid particles depends on the hydrodynamic forces acting on the surface of the particles. The choice of e.g., drag force correlation between fluid and particles should therefore influence the choice of the stress correlation for the particulate phase. While this particular problem can be solved exactly for some idealized cases, there seems to be no general solution available. Perhaps the most intricate term which phase boundary Aα. The second integral on the right side of Eqn. (5.4) contains the transfer of momentum carried by the mass exchanged between phases. It is obvious that this part of the momentum transfer integral Mα must be consistently correlated with the mass transfer integral Гα. Similarly, the first integral on the right side of Eqn. (5.4) contains the change of momentum of phase α due to stresses imposed on the phase boundary by the other phases. Physically, this term contains forces such as buoyancy which may be correlated to average pressures and gradients of volume fractions, and viscous drag which might be correlated to volume fractions and average velocity differences. For instance in a liquid-particle suspension, the average stress inside solid particles depends on the hydrodynamic forces acting on the surface of the particles. The choice of, e.g., drag force correlation between fluid and particles should therefore influence the choice of the stress correlation for the particulate phase. While this particular problem can be solved exactly for some idealized cases, there seems to be no general solution available. Perhaps the most intricate term this is to be correlated to the averaged quantities through constitutive relations is the tensor given by Eqn. (5.4). It contains the momentum transfer inside phase α which arises from the genuine turbulence of phase α and from the velocity fluctuations due to presence of other phases, and which are present also in the case that the flow is laminar in the mesoscopic scale. Moreover, the truly turbulent fluctuations of phase α may be substantially modulated by the other phases. Bearing in mind the intricacies that are encountered in modeling turbulence in single phase flows, it is evident that inferring realistic constitutive relations for tensor remains as a considerable challenge. It may, however, be attempted, e.g., for fluid-particle suspensions by generalizing the corresponding models for single phase flows, such as turbulence energy dissipation models, large-eddy simulations or direct

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numerical simulations.

Some Thought in Multiphase CFD for Industrial Processes CFD is a rapidly evolving discipline oriented on developing computational tools for solving problems related to transport processes: fluid mechanics, heat and mass transfer, reactive flow, and multiphase flow, [Eskin and Derksen]24 . In narrow terms CFD is the numerical solution of the mass, momentum, and energy conservation equations with properly defined boundary conditions. Those equations may be supplemented with (Newtonian or non-Newtonian) constitutive equations and equations of state for compressible fluids. In broader terms CFD also involves modelling (parameterization) of phenomena at length and time scales that are too small to be fully resolved computationally; the three most prominent examples being turbulence, multiple phases flows, and reactive flows. In strongly turbulent flows, the spectrum of length and time scales is simply too wide to be completely resolved in a single computation. Models for small-scale turbulence are used to alleviate the computational burden and make simulations of large-scale industrial turbulent flows possible. Multiphase flows usually take the form of a continuous phase that carries one or more dispersed phases. The solid particles, or droplets, or gas bubbles that constitute the dispersed phases are often too small to be fully resolved; their impact on the macroscopic flow patterns needs to be modelled. A similar multiscale issue relates to chemically reacting flow where mixing at the micro (molecular)-scale defines the rate of chemical reactions. The most important issue in predictive modelling of chemical industrial processes is how to deal with their multiphase character. Process equipment (chemical reactors, burners, mixers, crystallizers, hydro and pneumatic conveying pipe lines, fluidized beds, flotation cells) usually operates with multiple phases, modelling of which is much more complicated than that of a single phase flow. In dependence on the phases composing the flow system, the geometry of the flow domain and the process conditions (flow rates, agitation speeds), an abundance of flow regimes and flow phenomena can be distinguished. Resolving and predicting these in a numerical simulation is a clear and grand challenge. Key in virtually any simulation effort is to distinguish between the relevant and irrelevant physics and model what is relevant. Though, general mathematical descriptions of multiphase processes are known, it is practically impossible to solve all the conservation equations numerically without simplifications. There are the two major groups of approaches, which are currently used in engineering and science: 1. Methods based on a simplified model representation of some processes involved in a certain multiphase system. 2. Methods of direct numerical simulation. The simplified model representation focuses on the macroscale processes and global flow patterns, and uses simplifying assumptions and models to represent micro-scale effects. Direct simulations aim at fully resolving the micro-scale including the behavior (motion, deformation, breakup, coalescence, aggregation) of individual dispersed phase particles (solids, droplets, bubbles). Given the high resolution at the micro-scale, direct simulations are only able to simulate small volumes and

24 Dmitry Eskin and Jos Derksen, “Introduction To A

Series Of Featured Article-Multiphase Computational Fluid Dynamics For Industrial Processes”, The Canadian Journal Of Chemical Engineering, Volume 89, April 2011.

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thus need simplifying assumptions regarding the macro-scale, such as homogeneous turbulent conditions, or simple shear flow. The first group of methods is usually employed when a computational domain is large and/or a number of different phenomena revealing themselves on different scales are involved. For example, a force interaction of dispersed and continuous phases is modelled through a drag force that is calculated based on empirical correlations. Heat and mass transfer in such a case are also described by empirical equations. The situation becomes even more complicated when concentration of a dispersed phase is high. Then dispersed phase components interact with each other. Those interactions lead to generation of additional stresses in a flow, causing a change in flow pattern. If particles are solid then kinetic theory of granular media can be employed for modelling dispersed phase dynamics. Models where both continuous and dispersed phases are represented as two interacting interpenetrating continua are classified as two-fluid models and often used in engineering practice. Such models are incorporated into commercial CFD codes (e.g., fluent and CFX) and widely used for computing of large-scale technological devices (e.g., fluidized bed chemical reactors, hydraulic or pneumatic conveying pipelines, etc.). Alternatively, the equations of motion of the dispersed phase are solved in Lagrangian coordinates. In this case motion of each particle is tracked. Collisions of a tracked particle with others are accounted for assuming that it moves through a cloud formed by other particles. It is assumed that particle–particle collisions are binary and mutual orientations of colliding particles are random. Postcollision particle velocities are calculated based on momentum conservation for particle pair. This approach is more accurate than the two-fluid model, but limited to relatively low particle concentration and computationally expensive. If the second phase is not dispersed down to small size particles or droplets the two-fluid approach cannot be used. If both phases are immiscible fluids then dynamics of each fluid is modelled by solving the corresponding conservation equations. The models for each fluid flow are coupled through no-slip conditions and equality of stress on the fluid/fluid boundary. That boundary is tracked by one of the known techniques (e.g., volume of fluid method). The option of computing immiscible fluid flows is provided by modern commercial CFD codes. Examples of successful application of such an approach are modelling a bubbly flow in a capillary channel, a liquid film flow on a surface, etc. Direct Numerical Simulation (DNS) is a direction that is rapidly developing during the last 20 years. DNS methods suppose solving the conservation equations for all phases composing the system directly, without introducing simplifying assumptions. For example, in a case of a fluid–solids flow a dispersed phase is treated as a moving boundary of a complicated changing configuration. There are a number of different DNS methods. A DNS method is often considered as a technique for solving problems on the meso-scale that is assumed to be a minimum scale representing an important property of a flow system. An example of such a property is an apparent viscosity of a slurry or an emulsion. The meso-scale in this case is a characteristic size of a computational domain that is sufficient to calculate the apparent viscosity based on accurate modelling of dynamics of interacting carrying and dispersed phases. A DNS method should not require a model of turbulence, that is, such a method should allow resolving Navier–Stokes equation from a micro-scale (significantly smaller than the inner turbulence scale) to a relatively large-scale (e.g., a few percent of a tube radius for a pipe flow). Some known DNS methods are based on direct solution of the conservation equations for all components of a given flow system.

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Unfortunately, these equations are strongly non-linear and characterized by poor convergence and numerical stability. The other group of DNS methods is based on ideas borrowed from statistical mechanics. A fluid is represented as a system of particles, characterized by a probability density in a 6-dimensional space (3 coordinates in the geometrical space and 3 coordinates in the velocity space). Dynamics of such a system is described by a known Boltzmann equation. It was proven that the Boltzmann equation can be reduced to the Navier–Stokes equation, therefore the Boltzmann equation can be used for modelling fluid flows. Examples of methods based on such an approach are: the Lattice Boltzmann Method (LBM), its predecessor Lattice Gas Automata (LGA), and Dissipative Particle Dynamics (DPD) approach. LBM and LGA methods employ a fixed grid. The velocity in each node is discretized to a number of fixed directions. The simplified kinetic equation formulated for such a grid allows obtaining an approximate solution of the Boltzmann equation and the Navier– Stokes equation, respectively. The Lattice–Boltzmann technique can be applied to modelling flows in a domain with very complex boundary conditions. The LBM equations are free of drawbacks associated with strong nonlinearity of Navier–Stokes equation. LBM has been successfully used for modelling multiphase flows, especially on a micro-scale. The DPD is an off-lattice mesoscopic simulation method which involves a set of particles randomly moving in continuous space. Each particle moves under the action of three pairwise-additive forces: a conservative force, the dissipative force, and the random force. The DPD technique has an advantage over other methods when it is necessary to relate the macroscopic nonNewtonian flow properties of a fluid to its microscopic structure. Though DNS methods are prospective for accurate modelling of flows on a mesoscopic level their applicability are often limited to small computational domains. Modern technological equipment is often characterized by enormous dimensions. DNS methods may serve as an excellent tool for deriving correlations or models used as sub-models for macro-scale CFD codes. In an ideal world, micro, meso, and macro-scale simulations are tightly connected to provide a multiscale approach for truly predictive modelling of large-scale industrial processes. The challenges in multi-scale modelling are the formulation of generic coarse and fine graining techniques to meaningfully connect simulations at vastly different length and time scales. The topic of our article series on CFD of multiphase flow covers given the above considerations a broad spectrum of methods and applications. The articles presented can be considered as examples of developments and applications of CFD techniques. Their common theme is solution of engineering problems arising in process industries. We hope that the series will be interesting for scientists from industry and academia as well as for practicing engineers involved in simulations of multiphase systems.

Multicomponent Flow Mass transfer deals with situations in which there is more than one component present in a system; for instance, situations involving chemical reactions, dissolution, or mixing phenomena25. A simple example of such a MultiComponent system is a binary (two component) solution consisting of a 25

Lecture Notes from CBE 6333, Levicky.

Figure 11

Binary System of Gases

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solute in an excess of chemically different solvent. In a multicomponent system, the velocity of different components is in general different. For example, in Figure 11 pure gas A is present on the left and pure gas B on the right. When the wall separating the two gases is removed and the gases begin to mix, A will flow from left to right and B from right to left clearly the velocities of A and B will be different. The velocity of particles (molecules) of component A (relative to the laboratory frame of reference) will be denoted vA. Then, in this frame of reference, the molar flux NA of species A (units: moles of A/(area time) ) is

NA cA vA

Eq. 2.6

where cA is the molar concentration of A (moles of A/volume). This could be used to calculate how many moles of A flow through an area Ac per unit time (see Figure 12) where the flux is assumed to be normal to the area Ac. Then the amount of A carried across the area Ac per unit time is Amount of A carried through Ac per unit time = NA AC = cA vA Ac (moles/time). Since the volume swept out by the flow of A per unit time equals vA AC (see Figure 12), the above expression is seen to equal this rate of volumetric "sweeping" times cA, the amount of A per volume. More generally, for arbitrary direction of NA and a differential area element dB, the rate of A transport through dB would be (see Figure 12), flux of A through dB = - cAvA . n dB (moles/time) n is the outward unit normal vector to dB. One can understand this by realizing that - vA . n dB is the volumetric flowrate of A species (volume/time) passing across dB from Figure 12 Volumetric Flux "outside" to "inside", where "outside" is pointed at by the unit normal vector n. Multiplying the volumetric flowrate -vA . n dB by the number of moles of A per volume, cA, equals the moles of A passing through dB per unit time. cA is related to the total molar concentration c (c is moles of particles, irrespective of particle type, per volume) where xA is the mole fraction of A. Summing over the mole fractions of all species must produce unity (n equals the total number of different species present in solution). Similarly, we can also define a mass flux of A, nA (units: mass of A/(area time) ), nA = ρA vA. Here, vA is still the velocity of species A. ρA is the mass concentration of A (mass of A per volume of solution).

cA x Ac ,

n

xi 1 i 1

,

A A ,

n

i 1

i

1

Eq. 2.7

As previously stated, in general each chemical species "i" in a multicomponent mixture has a different velocity vi. However, it will nevertheless prove convenient to define an average velocity of the bulk fluid, a velocity that represents an average over all the vi's. In general, three types of average velocities are employed: mass average velocity v (v is what is usually dealt with in Fluid Mechanics), molar average velocity V, and volume average velocity vo. We will only deal with the first two average velocities.

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n

n

v ωi v i , V x i v i i 1

i 1

n

, ρ ρi i 1

n

,

c ci

Eq. 2.8

i 1

Why bother with two different average velocities? The mass average velocity is what is needed in equations such as the Navier Stokes equations, which deal with momentum, a property that depends on how much mass is in motion. Thus, the amount of momentum per unit volume of a flowing multicomponent mixture is ρv ( ρv = mv/Volume, where m is the total mass traveling with velocity v; m/Volume = ρ); thus momentum must be calculated using the mass average velocity v. Similarly, the Equation of Continuity expresses conservation of mass, and is similarly written in terms of v. The physical laws expressed by these equations (conservation of momentum, conservation of mass) do not depend on the moles of particles involved, but they do depend on the mass of the particles. On the other hand, when dealing with mass transfer, we will see that it is Figure 13 Volume Swept common to write some of the basic equations in terms of V as well as v. The reason for using V, in addition to v, is convenience. For instance, if in a particular problem there is no bulk flow of particles from one location to another so that, during the mass transfer process the number of particles at each point in space stays the same, then V = 0. Setting V to zero simplifies the mathematics. Figure 11 at the beginning provides an example. Imagine that, in their separated state as drawn, A and B are both ideal gases at the same pressure p and temperature T. Then, from the ideal gas equation, the molar concentration of A and B is the same, cA = cB = c = p/RT (R = gas constant). The equality of cA and cB to the total concentration c is appropriate because the gases are pure; thus in each compartment the concentration of the gas (A or B) must also equal the total concentration c. After the separating wall is removed, particles of A and B will mix until a uniform composition is achieved throughout the vessel. In the final state, assuming the gases remain ideal when mixed, the value of p and T will remain the same as in the unmixed state and therefore the total concentration c also remains the same, c = p/RT (p is now the total pressure, a sum of the partial pressures of A and B). Thus, in the final mixed state, the number of particles per volume c (here a sum of particles of A and B types) is the same as the number of particles per volume in the initial unmixed state. Thus mixing produced no net transfer of particles from one side of the vessel to the other, it only mixed the different particle types together. Under these conditions, when there is not net transfer of particles from one part of a system to another, V = 0. In contrast, for the same mixing process, in general v will not be zero. For example, imagine that mass of A particles is twice as large as that of B particles. Then in the initial unmixed state the left hand side of the vessel (filled with A) contains more mass, and the density (mass/volume) of the gas A is higher than that of B even though its concentration (particles/volume) is the same. Once A and B mix, however, the density everywhere will become uniform. For this uniformity to be achieved mass must have been transferred from the A side to the B side; therefore, in contrast to the molar average velocity V, the mass average velocity v was not zero during the mixing process.

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Integral and Differential Balances on Chemical Species We will refer to the species under consideration as species A. Following a derivation that parallels that employed for the other conservation laws (divergence theorem), the first step in the derivation of a conservation law on the amount of species A is to perform a balance for a closed control volume V'. V' is enclosed by a closed surface B ( see Figure 14). The amount of species A inside V' can change either due to convection through the boundary B, or by generation/consumption of A due to a chemical reaction. In words, the conservation for species A can stated as:

Accumulation of A in V' = Convection of A into V' + Generation of A by chemical reactions

Eq. 2.9

2.9.1.1 Molar Basis An integral molar balance on species A, performed over the control volume V', is written as

d c A dV c A v A . n dB R A dV dt V B V

Eq. 2.10

where n is the outward unit normal vector to surface B, not to be confused with the mass flux ni = ρi vi of species i. On the left side, cA is concentration of A in moles per volume; thus cAdV' is the number of moles of A in a differential volume dV'. Integrating (i.e. summing) this term over the entire control volume V' yields the total number of moles of A in V'; the time derivative of this integral is the rate of change of moles of A inside V' (units: moles/time). Thus, the left hand term is just the rate of accumulation of A in V', expressed in molar units. The accumulation term equals the rate at which A is convected into V' (1st term on right) plus the rate at which A is generated inside V' by a homogeneous chemical reaction (2nd term on right). The 2nd term on the right in equation (11) represents production of A by homogeneous reactions. A "homogeneous" reaction is one that Figure 14 Divergence Theorem applied to occurs throughout the interior of V'. In contrast, a chemical species heterogeneous chemical reaction would be one that occurs only at an interface; for instance, between a solid and a liquid phase; and is not distributed throughout the entire volume. The molar reaction rate RA has units of moles/(volume time) and represents the rate at which moles of A are produced or consumed by all homogeneous reactions. RAdV' is the number of moles of A produced inside a volume element dV' per unit time (units: moles/time). Summing this production over the entire control volume leads to the total molar rate of production of A, inside V', due to homogeneous chemical reactions. Equation (11), by assumption, did not include any generation of A due to heterogeneous reactions. Clearly, if in V' there was a large interface at which a heterogeneous reaction leads to production of A, one would have to add that term to equation (11). The term would typically have the form of a rate of production of A per area (moles / (area time)) times the total area of the reacting surface. However, it may also be that a heterogeneous reaction is actually more conveniently modeled as homogeneous. Because the reaction occurs only at the interface between a particle and the liquid, it is heterogeneous. However, since the particles are dispersed throughout V', one could think of the

31

reaction rate on a per volume basis (i.e. moles produced per volume of solution per time) as opposed to a per area basis (moles produced per surface area of particles per time). 1. As done previously for the other balances, one can use the Divergence Theorem to convert the surface integral of the convection term (1st term on right) into a volume integral, 2. Move the d/dt derivative inside the accumulation integral since the integration limits are time independent (the limits do not depend on time because a fixed control volume is considered, whose shape and location do not change; this assumption can be relaxed at the expense of a somewhat more complicated mathematical expression), 3. Combine all terms under a common volume integral to obtain,

c A .c A v A R A dV 0 t

V

Eq. 2.11

The only way to ensure that equation (12) evaluates to zero for an arbitrary control volume V' is to require that

c A .c A v A R A 0 t

or

c A .c A v A R A t

Eq. 2.12

Equation (14) is the differential molar balance on species A. It states that the rate of accumulation of moles of A at a point in space (left hand side) is equals the rate at which moles of A are convected into that point (1st term on right), plus the rate at which moles of A are produced at that point by chemical reactions (2nd term on right). These physical interpretations can be verified by tracing the origin of the terms back to the corresponding terms in the integral balance, equation (11). 2.9.1.2 Mass Basis Similarly, For mass balance on species A,

d ρ A dV ρ A v A . n dB rA dV dt V B V

or

ρ A .ρ A v A rA t

Eq. 2.13

The differential equation of continuity (total mass balance) derived in fluid mechanics for single component systems. It also applies to multicomponent systems in which chemical reactions happen. The prove this is straightforward. n n ρi . ρi v i ri i 1 t i 1 i 1 n

Eq. 2.14

This equation states the law of mass conservation in multicomponent systems, even if chemical reactions are present. The total accumulation of mass at a point (left hand side) can only occur by convection of mass to that point (right hand side). For multicomponent systems whose density ρ is constant (i.e. ρ does not vary from point to point irrespective of variations that may be present in temperature, pressure, or composition).

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Diffusion Fluxes The diffusion flux of species A is that portion of its total flux that is not attributed to bulk flow (as represented by the mass or molar average velocities). More precisely, jA, the mass diffusive flux of A, is defined as

jA = total M ass flux of A - mass flux of A due to bulk motion n A - A v A v A - A v A ( v A - v) J A = total M olar flux of A - molar flux of A due to bulk motion

Eq. 2.15

N A - cA V cA ( v A - V ) Thermodynamics also states that, for a system with n components, n + 1 intensive variables are sufficient to fully specify the equilibrium state. Since binary fluid mixtures are being considered, for which n = 2, three intensive variables are needed. It will be convenient to choose temperature T, pressure p, and μT (the chemical potential per mass of species i by μi). In the absence of equilibrium, one or more of these variables will vary with location in a way that non-equilibrium gradients in ΔT, Δp, and ΔμT exist. To move toward equilibrium, the system will transfer heat and masses of the different species around so as to eliminate these gradients. For instance, heat flux will occur from hot to cold to equalize the temperature, and mass fluxes of individual chemical species will occur so as to equalize each species' total (μi + yi) potentials. Non-equilibrium pressure gradients can be normalized by bulk flow of material from high to low pressure regions. One possible way the system can eliminate non-equilibrium gradients in ΔT, Δp, and ΔμT is by diffusion of the various chemical species; it is then logical to assume that the diffusive fluxes will be functions of these gradients, with steeper gradients producing greater fluxes. Let's consider the diffusive mass flux jA, assumed to be a function of the gradients such that jA = jA(ΔT, Δp, ΔμT). When the gradients ΔT, Δp, and ΔμT are not too large, one could perform a Taylor series expansion of jA (around equilibrium) in the gradients and truncate it after the first order terms. Such an expansion would lead to the following mathematical relation for jA:

jA C1μ T C2T C3p

Eq. 2.16

The proportionality factors Ci are functions of T, p, and ΔT, but not of the non-equilibrium gradients in these quantities (this is because, recalling Taylor Series expansions, these factors are to be evaluated at equilibrium, the "point" around which the expansion is being formed. However, at equilibrium, the non-equilibrium gradients are zero). Equation (16) must be constrained to obey requirements imposed by thermodynamics. In particular, using the second law of thermodynamics, it can be shown that C3 must equal zero 26. Therefore, (Eq. 16) simplifies to

jA C1μ T C2T

Eq. 2.17

A species' chemical potential, at some point in the mixture, can be viewed as a function of the pressure, temperature, and composition at that point. For a binary mixture, this means that μA = μA(T, p, ωA) and μB = μB(T, p, ωA)27. Landau & Lifshitz, Fluid Mechanics, Pergamon Press, pp 187 and 222, 1959. An implicit assumption is being made that thermodynamic relations such as μ A = μA(T, p, ωA), which are strictly applicable to systems at equilibrium, apply even though equilibrium does not exist throughout the 26

27

33

K D

p AB KT D AB p T (μ A μ B ) (μ A μ B ) jA C1 C 2 T C1 p T T ωA p,ω A T, Pressure Diffusion

Thermal Diffusion

D AB

(μ A μ B ) C1 (y A - y B ) ω A C 1 T T, p Forced (Gravity) Diffusion Ordinary (Mass) Diffusion (Fick's Law)

Eq. 2.18 If the diffusion coefficient DAB and chemical potentials are known (i.e. from experimental measurement), C1 can be evaluated. Eq. 2.18 becomes

ρD K jA AB T T T

ρD ABK p p - ρD ABωA - C1 (y A - y B ) - p

Eq. 2.19

The product DAB kT is called the thermal diffusion coefficient, kT the thermal diffusion ratio, and DAB kP may be called the baro-diffusion coefficient. Equation (19) shows that diffusive flux of mass of species A, in a binary solution of A and B, can arise from four different contributions. Fick's Law Having briefly outlined the four causes of mass diffusion spatial variations in composition, external potential, temperature, and pressure it is useful to highlight the most common scenario in which only ordinary diffusion is of importance. The discussion will be specialized to binary solutions that obey Fick's Law. The restriction to binary solutions is not as limiting as it may seem. Indeed, even when more than two components are present, as long as the solutions are sufficiently dilute the diffusion of solute species can be modeled as for a binary system. This is because when one of the species (the solvent) is present in vast excess, with all the rest (the solutes) in trace amounts, the diffusion of each solute species can be treated as if it was in pure solvent alone. Under these dilute conditions a solute particle will not "see" any of the other solute particles, and so its diffusion will not be affected by their presence but only by the solvent. Such a situation is effectively a two component problem, the solute of interest plus the solvent. Fick's Law can be written in several common forms such as Mass diffusion flux and reference bulk average velocity v, or Molar diffusion flux; reference bulk average velocity V:

system. Qualitatively, this assumption can be expected to hold over sufficiently short length scales over which only insignificant variations in temperature, pressure, and composition occur, so that the values of these quantities are well defined. Since thermodynamic quantities are only needed at a “point” (i.e. over very short lengths), from a practical perspective this consideration is not limiting. Also, note that ω B is not an independent thermodynamic variable since, for a binary mixture, ωB = 1 - ωA.

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jA ρ A ( v A v) ρ D ABωA J A c A ( v A - V ) jA

M c D ABx A MAMB

Eq. 2.20

Species' Balances for Systems Obeying Fick's Law The differential molar and mass balances on a species, are general in the sense that they are independent of the number of components present or any models of diffusion. For example, they do not require ordinary diffusion to obey Fick's Law. These equations are only based on the statement that the rate at which the amount of A can change at a point in space equals the rate at which A is convected into that point plus the rate at which it is generated by chemical reactions. However, in their present form, these equations can be inconvenient because they contain the species' velocity vA, which must be known in order to accomplish the usual goal of solving for the mass or molar concentrations. One way to model vA is to consider it as consisting of the two contributions previously encountered: 1. one due to bulk flow of the fluid mixture (convection) 2. one due to the diffusion of species A (mass diffusion) Separation of the total flux of A into diffusive and bulk convection contributions is motivated by convenience. The mass average velocity v is easy to measure, and can be obtained by direct calculation from the differential equations of fluid mechanics (e.g. Navier Stokes equations for Newtonian fluids with ρ and μ constant). Typically, the diffusive fluxes are modeled well by Fick's Law for solutions in which only ordinary diffusion is present and which are either binary or, as discussed earlier, dilute. Equations can be specialized to fluids that follow Fick's Law as

Mass balance : Molar balance :

ρ A .D ABωA - v.ρ A ρ A.v rA t c A .J A - .c A v R A .cD ABx A - .c A R A t

Eq. 2.21

Further information can be obtained in handout “Multicomponent Systems”, CBE 6333, [Levicky].

Multiscale Modeling Multiscale modeling refers to a style of modeling in which multiple models at different scales are used simultaneously to describe a system. The different models usually focus on different scales of resolution. They sometimes originate from physical laws of different nature, for example, one from continuum mechanics and one from molecular dynamics. In this case, one speaks of multi-physics modeling even though the terminology might not be fully accurate. The need for multiscale modeling comes usually from the fact that the available macroscale models are not accurate enough, and the microscale models are not efficient enough and/or offer too much information. By combining both viewpoints, one hopes to arrive at a reasonable compromise between accuracy and efficiency. The subject of multiscale modeling consists of three closely related components: multiscale analysis, multiscale models and multiscale algorithms. Multiscale analysis tools allow us to understand the relation between models at different scales of resolutions. Multiscale models allow us to formulate models that couple together models at different scales. Multiscale algorithms allow us to use

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multiscale ideas to design computational algorithms 28. Traditional Approaches to Modeling Traditional approaches to modeling focus on one scale. Macroscale models require constitutive relations which are almost always obtained empirically, by guessing. Making the right guess often requires and represents far-reaching physical insight, as we see from the work of Newton and Landau, for example. It also means that for complex systems, the guessing game can be quite hard and less productive, as we have learned from our experience with modeling complex fluids. The other extreme is to work with a microscale model, such as the first principle of quantum mechanics. As was declared by Dirac back in 1929, the right physical principle for most of what we are interested in is already provided by the principles of quantum mechanics, there is no need to look further. There are no empirical parameters in the quantum many-body problem. We simply have to input the atomic numbers of all the participating atoms, then we have a complete model which is sufficient for chemistry, much of physics, material science, biology, etc. Dirac also recognized the daunting mathematical difficulties with such an approach after all, we are dealing with a quantum many-body problem. With each additional particle, the dimensionality of the problem is increased by three. For this reason, direct applications of the first principle are limited to rather simple systems without much happening at the macroscale. Take for example, the incompressible fluids. The fundamental laws are simply that of the conservation of mass and momentum, which, after introducing the notion of stress, can be expressed as follows:

. u 0

,

u ρ (u.) u .τ t

Eq. 2.22

where u is the velocity field, ρ is the density of the fluid which is assumed to be constant and τ is the stress tensor. To close this system of equations, we need an expression for the stress tensor in terms of the velocity field. Here is where the guessing game begins. The standard approach is to ask: What is the simplest expression that is consistent with

symmetry (Galilean, translational/rotational invariance); laws of physics, particularly the second law of thermodynamics; experimental data.

In this case, Galilean invariance implies that τ does not depend on u . The next simplest thing is to say that τ is a function (in fact, linear function) of ∇u . This gives us

τ pI μ (u (u)T )

Eq. 2.23

where p is the pressure, μ is the kinematic viscosity coefficient, which in this context is the only parameter that carries the information specific to the microscopic constituents of the system, I is the identity tensor. Second law of thermodynamics requires that μ ≥ 0 . Substituting the above expression for τ into the momentum equation, we obtain the celebrated Navier-Stokes equation. Even though the reasoning that went into the derivation of the Navier-Stokes equation is exceedingly simple, the model itself has proven to be extremely successful in describing virtually all phenomena that we encounter for simple fluids, i.e. fluids that are made up of molecules with a simple molecular structure. Partly for this reason, the same approach has been followed in modeling complex fluids, 28

Weinan E and Jianfeng Lu, Scholarpedia, “Multiscale modeling”, 2011.

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such as polymeric fluids. Unfortunately the success there is quite limited. The first problem is that simplicity is largely lost. In order to model the complex rheological properties of polymer fluids, one is forced to make more complicated constitutive assumptions with more and more parameters. The second difficulty is that accuracy is not guaranteed. For polymer fluids we are often interested in understanding how the conformation of the polymer interacts with the flow. This kind of Figure 15 Illustration of the multi-physics hierarchy information is missing in the kind of empirical approach described above. A more rigorous approach is to derive the constitutive relation from microscopic models, such as atomistic models, by taking the hydrodynamic limit. This is an example of the multiscale analysis approach. For simple fluids, this will result in the same Navier-Stokes equation we derived earlier, now with a formula for μ in terms of the output from the microscopic model. But for complex fluids, this would result in rather different kinds of models. This is one of the starting points of multiscale modeling (see Figure 15). Multiscale Modeling In the multiscale approach, one uses a variety of models at different levels of resolution and complexity to study one system. The different models are linked together either analytically or numerically. For example, one may study the mechanical behavior of solids using both the atomistic and continuum models at the same time, with the constitutive relations needed in the continuum model computed from the atomistic model. The hope is that by using such a multi-scale (and multiphysics) approach, one might be able to strike a balance between accuracy (which favors using more detailed and microscopic models) and feasibility (which favors using less detailed, more macroscopic models)29. Sequential Multiscale Modeling In sequential multiscale modeling, one has a macroscale model in which some details of the constitutive relations are precomputed using microscale models. For example, if the macroscale model is the gas dynamics equation, then an equation of state is needed. This equation of state can be precomputed using kinetic theory. When performing molecular dynamics simulation using empirical potentials, one assumes a functional form of the empirical potential, the parameters in the potential are precomputed using quantum mechanics. Sequential multiscale modeling is mostly limited to the case when only a few parameters are passed between the macro and microscale models. For this reason, it is also called parameter passing. This does not have to be the case though: It has been demonstrated that constitutive relations which are functions of up to 6 variables can be effectively precomputed if sparse representations are used.

It should be noted that this diagram is slightly misleading: Quantum mechanics is valid not just at the microscale, it also applies at the macroscale; only that much simpler models are already quite sufficient at the macroscale 29

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Concurrent Multiscale Modeling In concurrent multiscale modeling, the quantities needed in the macroscale model are computed onthe-fly from the microscale models as the computation proceeds. In this setup, the macro- and microscale models are used concurrently. Take again the example of molecular dynamics. If one wants to compute the inter-atomic forces from the first principle instead of modeling them empirically, then it is much more efficient to do this on-the-fly. Precomputing the inter-atomic forces as functions of the positions of all the atoms in the system is not practical since there are too many independent variables. On the other hand, in a typical simulation, one only probes an extremely small portion of the potential energy surface. Concurrent coupling allows one to evaluate these forces at the locations where they are needed. Two types of Multiscale Problems The first type are problems where some interesting events, such as chemical reactions, singularities or defects, are happening locally. In this situation, we need to use a microscale model to resolve the local behavior of these events, and we can use macroscale models elsewhere. The second type are problems for which some constitutive information is missing in the macroscale model, and coupling with the microscale model is required in order to supply this missing information. We refer to the first type as type A problems and the second type as type B problems. Modeling Approach defined based on Length Scale Alternatively, another modeling approach is developed which is based length scale on fluid and particle systems30. It was originally presented by Anderson and Jackson31. The idea was that modeling complexity increases as more effects associated with time and length scales are included in the simulation. Depending on the length scales, various combinations of modeling scales can be suggested. These are classified as Micro, Meso, and Macro scale models. In a Micro scale model, trajectories of individual particles are calculated through the equation of particle motion and the fluid length scale is the same as the particle size or even smaller. At the same time, instantaneous flow field around individual particles is calculated. In the Meso-Scale Model, both solid and fluid phases are considered as inter penetrating bands. The conservation equations are solved over a mesh of cells. The size of the cells is small enough to capture main features of the flow, like bubble motions and clusters, and large enough (essentially larger than the size of individual particles) to allow averaging of properties (porosity, interactions, etc.) over the cells. In the Macro Scale Model, the fluid length scale is in the order of the flow field. This means that motions of the fluid and the assemblage of particles are treated in one dimension based on overall quantities32. It is also possible to develop some intermediate models in which the length scales of fluid and solid phases are different. For example, the length scale of solid phase can be kept at the Micro Scale while changing the length scale of fluid phase to meso or macro. Under these conditions, the affective interactions in the larger scale can be calculated by averaging the information in the smaller scale. In general, in multi‐scale modeling, the smaller scale model takes into account various interactions (i.e, fluid–particle, and particle–particle) in detail. These interaction details can be used with some assumptions and averaging to develop closure laws for calculating the effective interactions (e.g.,

Coupled CFD-DEM Modeling: Formulation, Implementation and Application to Multiphase Flows, First Edition. Hamid Reza Norouzi, Reza Zarghami, Rahmat Sotudeh-Gharebagh and Navid Mostoufi. John Wiley & Sons, Ltd. 31 Anderson, T.B. and Jackson, R. (1967),”Fluid mechanical description of fluidized beds. Equations of motion”, Industrial and Engineering Chemistry Fundamentals, 6(4), 527–539. 32 Tsuji, Y. (2007) Multi‐scale modeling of dense phase gas–particle flow. Chemical Engineering Science, 62(13), 3410–3418. 30

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drag force) in the larger scale model33. This allows capture of the essential information needed on the larger scale. Alternatively, calculation of effective interactions can be performed through the local experimental data, if available. Combination of fluid/particle motion with different modeling scales can be calculated by averaging the information in the smaller scale can provide different modeling approaches, as sketched in Figure 16 with detailed below. 2.10.6.1 Micro Approach (Fluid–Micro, Particle-Micro) In this approach, the fluid flow around particles is estimated by the Navier–Stokes equation. Since the forces acting on particles are calculated by integrating stresses on the surface of the particle, the empirical correlation for drag and lift forces are not required. This approach is used in cases where particle inertial force is relatively small (e.g., liquid–particle flow) or the fluid lubricating effect on particles is rather significant (e.g., dense‐phase liquid–particle flow). A typical example of such an approach, is the direct numerical simulation–discrete element method (DNS‐DEM). 2.10.6.2 Meso Approach (Fluid–Meso, Particle-Meso) This is referred to as the two‐fluid model (TFM), in addition to the real fluid, the assemblage of particles is also considered to be the second continuum phase. The flow field is divided into a number of small cells to capture motions of both phases, provided that the cell size is larger than the particle size. The two continuous phases are modeled by applying laws of momentum and mass conservations in each fluid cell, leading to averaged Navier–Stokes and continuity equations. Capability of the TFM in capturing the solid phase motion greatly depends on the closure laws used for this phase. These closure laws always involve some simplifications or are obtained by semi‐ empirical correlations. While this approach is preferred in commercial packages for its computational simplicity, its effectiveness depends on the constitutive equations and is not easily applicable to all flow conditions. The TFM has been successfully utilized to obtain the flow behavior of various non‐reacting and reacting multiphase flows in laboratory, pilot, and industrial scales. 2.10.6.3 Macro Approach (Fluid–Macro, Particle-Macro) This approach provides a one‐dimensional (1D) description of gas‐particle flows. The main output of such a model is the pressure drop, which is considered as the sum of pressure drops due to flow of fluid and particles. Usually, a formula for the single phase flow, such as Darcy–Weisbach equation, is used for the fluid pressure drop and that of particles is balanced with the fluid drag formula from the momentum balance. This approach would also allow the calculation of averaged flow properties by empirical correlations that are essential in design and analysis of industrial processes. A typical example of such approach, is the two‐phase model (TPM) in fluidization. In this model, conservation equations are written for bubbles and emulsion, both having the length scale of the system in a fluidized bed. 2.10.6.4 Macro‐Micro Approach (Fluid–Macro, Particle-Micro) In this approach, shown in Figure 16 by 1D‐DEM, the fluid forces acting on particles are calculated from empirical correlations (e.g., drag and lift) while translational and rotational motions of particles are described based on Newton’s and Euler’s second laws. At very low concentration of particles, effect of particles on the fluid motion can be neglected. However, at higher concentrations, closure laws should be modified to account for the closeness of surrounding particles. Generally, in this approach the flow field, which is considered to change in one dimension, is not divided into cells and additional pressure drop is taken into account to reflect the effect of particles on the fluid motion.

Van der Hoef, M., Ye, M., van Sint Annaland, M., Andrews, A., Sundaresan, S., and Kuipers, J. ,”Multiscale modeling of gas‐fluidized beds”, Advances in Chemical Engineering, 31, 65–149, 2006. 33

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2.10.6.5 Meso‐Micro Approach (Fluid–Meso, Particle-Micro) In this method, referred to as CFDDEM and shown in Figure 16, the flow field is divided into cells with a size larger than the particle size but still less than the flow field. Effect of motion of particles on the flow of fluid is considered by the volume fraction of each phase and momentum exchange through the drag force. This approach is the focus of this book and is explained in detail in the following sections.

Figure 16

1-D DEM

CFD - DEM

Length Scale

Modeling Scales in Fluid-Particle Systems

Block-Spectral Method of Solution The traditional treatment of this kind of problems would be to use empirically based models to count for the effects of the small scale features and to solve an up-scaled problem on a coarse mesh34. The generality of the solutions following this kind of approaches are of course limited by the very empirical nature of these fine-scale models. Here we are interested in developing a new multi-scale methodology, called ‘Block-Spectral Method’. The main intended attribute of the new approach is that the same numerical discretization scheme and integration method are used for both the coarse (macro) and fine (micro) scales, so that the numerical resolution is consistently and completely dictated by the mesh scales. A blocking of the fine resolution domain is introduced to facilitate the two basic but competing requirements:

34

high resolution for fine scale flow features; avoidance of having to have fine meshes for a large domain

“Osney Thermo-Fluids Laboratory “, University of Oxford.

40

The block spectral approach can be simply illustrated by comparing a direct solution and a block spectral solution. The method has been shown to lead to a significant gain in solving micro-scale problems (up to 102 reduction of degrees of freedom). An important perspective is that the methodology would enable to resolve the kind of the micro-scale problems currently intractable [L. He]35-36.

35 36

He L, “Block-spectral Mapping for Multi-scale Solution”, Journal of Computational Physics, Vol.250, 2013. He L, “Block-spectral Approach to Film-cooling Modelling”, Journal of Turbomachinery, Vol.134, 2012.

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3 Case Studies for Composite Fluid Case Study 1 - Liquid-Particle Suspension Consider a binary system of solid particles suspended in a Newtonian liquid. We denote the continuous fluid phase by subscript c and the dispersed particle phase by subscript d. We assume that both phases are incompressible, that the suspension is non-reactive, i.e., there is no mass transfer between the two phases, and that surface tension between solid and liquid is negligible. Both the densities ρc and ρd are thus constants, and

Γc Γd 0 Mc Md 0

Eq. 3.1

The mutual momentum equation can now be written as

M Mc Md

and A Ac Ad and nˆ nˆ d nˆ c

Eq. 3.2

Case Study 2 - Two Fluid Flow Starting from a literature search, customizing the model adding specific constitutive equations and boundary conditions. The results in terms of both concentration distribution and pressure gradient were compared to experimental data available in literature37-38-39, over a wide range of operating conditions: average solids concentration between 10% and 40% by volume; uniform particle size between 90 and 520 μm; slurry velocities between 1 m/s and 5.5 m/s; and pipe diameters between 50 and 150 mm. The Two-Fluid model was obtained by adding the following two features to the original model, necessary to correctly reproduce the flow: Mixture Viscosity A correlation for the viscosity of the mixture, used to define the particle Reynolds number, is implemented. Among the different expressions available in literature, use is made of that of Mooney, which best fits the experimental data

ηα p μ m ρ c ν1,c exp 1 α /α p pm

Eq. 3.3

in which μ is the intrinsic viscosity, taken equal to 2.5 as suggested for spherical particles; α m is the maximum packing concentration, taken as 0.7; ρc is the density of the carrier fluid phase; ν1,c and is the laminar kinematic viscosity of the carrier fluid phase. Drag force The drag force law is related to the particle Reynolds number according to the Standard Drag Law Pipeline Flow Using ANSYS-CFX. Ind. Eng. Chem. Res. 48(17), 8159-8171. Roco, M.C., Shook, C.A., ”Modeling of Slurry Flow: The Effect of Particle Size”, Canadian J. Chem. Eng. 1983. 39 Gillies, R.G., Shook, C.A., Xu, J., ”Modelling Heterogeneous Slurry Flow at High Velocities”, Canadian J. Chem. Eng. 82(5), 1060-1065-2004. 37 38

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correlation, but the particle Reynolds number is defined with respect to the viscosity of the mixture instead of that of the carrier fluid phase; therefore, where and are the particle diameter and the slip velocity vector respectively. This modification is necessary to describe the phenomenon whenever in some cells the solid volume fraction approaches the maximum packing one. At the pipe wall, no slip conditions are imposed to the carrier fluid phase, and a logarithmic law wall function is applied in the near wall-region. The proper wall boundary conditions for the solid phase are still a matter of discussion in literature. Two alternatives have been considered. At first, a zero-flux condition is applied to the particles. Afterwards, to account for particle-particle and particle-wall interactions, a Bagnold-type shear stress is imposed. In particular, the following term, derived from the model of Shook and Bartosik40, is introduced in the momentum equation of the particle phase:

α 1/3 8.3018 2 pm τB ρ d 1 2.317 p p α p Re

1.5

τ w,c ρ v c 1,c

2

Eq. 3.4

Which Re = dpUs/ν1, c is the bulk Reynolds number, defined with respect to the pipe diameter dp and the slurry bulk-mean velocity Us; ρp is the density of the particles; τ w, c is the wall shear stress of the liquid phase. Figure 17 demonstrates the geometry of the problem; the computational domain covers only half of the pipe section due to a substantial symmetry of the phenomenon. A fully-developed turbulent flow profile is applied at the inlet. No slip is assumed between the phases. The inlet Figure 17 Contour plots for particle volume fraction volume fraction of the solids is taken as uniform. At the outlet, the normal gradients of all variables and the value of the pressure are set to zero. The length of the computational domain, equal to 100 pipe diameters, is sufficient to ensure the reaching of fully-developed flow conditions. When imposing a zero-flux condition at the wall to the particle phase, the predictions of the Two- Fluid model show good agreement to the experimental evidence in terms of solid volume fraction distribution. As an example, Figure 18 reports the results for the flow conditions considered by [Gillies]41 i.e., and mean solid volume Figure 18 Sketch of the problem Shook, C.A., Bartosik, A.S.,”Particle-wall stresses in vertical slurry flows”, Power Tech. 81(2), 117-124, 1994. Gillies, R.G., Shook, C.A., Xu, J., ”Modelling Heterogeneous Slurry Flow at High Velocities”, Canadian J. Chem. Eng., 2004. 40 41

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fraction from 12% to 41%. The contour plots of Figure 17 highlight the gradual accumulation of the particles as the mean solid volume fraction increases, phenomenon that can be correctly reproduced applying the above mentioned modifications to the original model. The solid volume fraction profiles along the vertical diameter (AB in Figure 18) appear in quantitative agreement to the experimental data.

Case Study 3 - Unsteady MHD 2-Phase Flow of Fluid-Particle Suspension between Two Concentric Cylinders Statement of Problem The problem of two-phase unsteady Magneto-Hydro-dynamic (MHD) flow between two concentric cylinders of infinite length has been analyzed by [Jha & Aperea]42 when the outer cylinder is impulsively started. The system of partial differential equations describing the flow problem is formulated taking the viscosity of the particle phase into consideration. Unified closed form expressions are obtained for the velocities and the skin frictions for both cases of the applied magnetic field being fixed to either the fluid or the moving outer cylinder. 3.3.1.1 Literature Survey and Background The study of fluid/particle mixture is important in petroleum transport, powder technology, fluidization transport of solid particles by a liquid and liquid slurries in chemical and nuclear processing and metalized liquid fuel slurries for rocketry. Its relevance is also seen in the field of mining, agriculture and food technologies. The flow of an electrically conducting fluid through a channel or a circular pipe in the presence of a transverse magnetic field is encountered in a variety of applications such as magneto-hydrodynamic (MHD) generators, pumps, accelerators, and flow meters. The annular geometry is widely employed in the gas cooled nuclear reactors in which the cylindrical fissionable fuel elements are placed axially in vertical coolants channel within the graphite moderators and the cooling gas is flowing along the channel parallel to the fuel elements and also in drilling operation of oil and gas well. [Makinde]43 studied the steady, axisymmetric, MHD flow of a viscous, Newtonian, incompressible, electrically conducting fluid through an isotropic, homogenous porous medium located in the annular zone between two concentric rotating cylinders in the presence of a radial magnetic field. The unsteady hydro magnetic Generalized Couette flow and heat transfer characteristics of a reactive variable viscosity incompressible electrically conducting third grade fluid in a channel with asymmetric convective cooling at the walls in the presence of uniform transverse magnetic field was also investigated. The high possibility of particle entrainment in these flows made the formulation of the equation of motion for a two-phase fluid/particle system a necessity in order to understand the role of particles on the flow. [Saffman]44 observed the effect of dust particles on the laminar flow of an incompressible fluid with mass concentration of dust particles. The equations he obtained neglected the interaction between the individual dust particles and also assumed that in the continuum hypothesis, the cloud of dust particles form a pseudo-fluid. Using Saffman’s model, [Nag et al.]45 studied the Coquette flow of a dusty gas between two parallel Basant K. Jha and Clement A. Aperea, “Unsteady MHD two-phase Coquette flow of fluid-particle suspension in an annulus”, Published by the AIP Publishing LLC, 2011. 43 O. D. Makinde, O. A. B´eg, and H. S. Takhar, “Magneto-hydrodynamic viscous flow in a porous medium cylindrical annulus with an applied radial magnetic field,” Int. J. Appl. Math. and Mech. 5, 68–81 (2009). 44 P. G. Saffman, “On the stability of laminar flow of a dusty gas,” J. Fluid Mech. 13, 120–128 (1962). 45S. K. Nag, R. N. Jana, and N. Datta, “Couettte flow of a dusty gas,” Acta Mech 33, 179–187 (1979). 42

44

infinite plates for both the impulsive and the accelerated start of one of the plates. [Singh and Dube]46 studied the laminar flow of an unsteady, incompressible viscous fluid with uniform distribution of dust particles through a circular pipe under the influence of pressure gradient. All of these studies excluded the particle-phase viscous effects. However, [Rahmatulin]47 developed a model which examines the unstable movement of the interface between the fluid and the particles and thereby included the particle-phase viscosity term in his model. In this model, each phase of the suspension has a local velocity and material properties, also, the phases are assumed to interact mutually in a continuous manner and the phases are homogeneous and evenly distributed per unit volume of the mixture. The following partial differential equations describes the unstable movements of the two-phase incompressible fluids for a plane channel geometry according to Rahmatulin.

V1 ρ f1 N f1μ1ΔV1 K(V2 V1 ) t V2 ρ f 2 N f 2μ 2 ΔV2 K(V1 V2 ) t

V1 0

, ,

V2 0

Eq. 3.5

where K , f are described by48 and subscript 1 and 2 denote each streams. The problem is solved using a combination of the Laplace transform technique, D’ Alemberts and the Riemann-Sum approximation methods. The solution obtained is validated by comparisons with the closed form solutions obtained for the steady states which has been derived separately. The governing equations are also solved using the implicit finite difference method to verify the present proposed method. The variation of the velocity and the skin friction with the dimensionless parameters occurring in the problem are illustrated graphically and discussed for both phases. The flow of an electrically conducting fluid through a channel or a circular pipe in the presence of a transverse magnetic field is encountered in a variety of applications such as Magneto-Hydro-Dynamic (MHD) generators, pumps, accelerators, and flow meters. The annular geometry is widely employed in the gas cooled nuclear reactors in which the cylindrical fissionable fuel elements are placed axially in vertical coolants channel within the graphite moderators and the cooling gas is flowing along the channel parallel to the fuel elements49, as well as in drilling operation of oil and gas well. Mathematical Formulation Consider the motion of an unsteady, laminar, viscous fluid/particle suspension between two concentric cylinders of infinite length. The fluid phase is assumed to be electrically conducting, while the particles are assumed to be electrically non-conducting. The z-axis is assumed to be on the axis of the cylinders in the horizontal direction and r´ is on the radial direction. The fluid-particle J. Singh and S. N. Dube, “Unsteady flow of a dusty fluid through a circular pipe,” Indian J. Pure Appl. Math. 6, 69–79, (1973). 47 H. A. Rahmatulin, “Osnovi gidrodinamiki vzaimopronikayu¸sih dvijeniy,” Prikladnaya Matematika i Makanika 20, 56–65, (1956). 48 Basant K. Jha and Clement A. Aperea, “Unsteady MHD two-phase Couette flow of fluid-particle suspension in an annulus”, Published by the AIP Publishing LLC, 2011. 49 S. K. Singh, B. K. Jha, and A. K. Singh, “Natural convection in vertical concentric annuli under a radial magnetic field,” Heat and Mass Transfer 32, 399–401 (1997). 46

45

suspension exists in the region a ≤ r´ ≤ b between the two cylinders in the presence of magnetic field acting perpendicular to the flow direction, where a and b are the radii of the inner and the outer cylinders respectively. Initially, the fluid, the particles and the two cylinders were at rest. However, when t > 0, the outer cylinder begins to move in its own plane with a velocity Ut´n, where U is a constant and the inner cylinder remains at rest. Since the cylinders are of infinite length and the flow is time dependent fully developed one-dimensional flow, all physical variables are functions of r´ and t´. The magnetic Reynolds number is assumed to be small so that the induced magnetic field and the Hall effects of MHD are negligible. No applied, polarization voltage exists (Ē = 0) i.e. no energy is being added or subtracted from the system50. It is further assumed that the fluid and the particles are

Figure 19

Schematic diagram of the problem

interacting as a continuum. Figure 19 for the schematic diagram of the problem. Under the assumptions made in the present problem and a cylindrical geometry, the equation of motion for the problem is

σB 02 u1 μ1 2 u1 1 u1 f1 u1 K(u2 u1 ) t ρ1 r2 r r ρ1 u2 μ 2 2 u2 1 u2 f2 K(u1 u2 ) t ρ 2 r2 r r

Eq. 3.6

Where f1 and f2 are substance quantity of mixture in unit volume (f1 + f2 = 1), K´ is dimensionless

50

G. Sutton and A. Sherman, Engineering Magneto-hydrodynamics (McGraw-Hill New York, 1965).

46

interaction coefficient, and subscripts 1 and 2 represents the fluid and the particle respectively. This equation is valid when the magnetic field is fixed relative to the fluid. If the magnetic field is also moving with the same velocity as the moving outer cylinder, we must account for the relative motion. Thus, eq. (3) turn to

2 u1 1 u1 σB 02 u1 1f1 2 K(u2 u1 ) (u1 - UGt n ) t r r ρ1 r where

Eq. 3.7

0 B0 is fixed realtive to fluid G 1 B0 is fixed realtive to moving cylinder

Considering the impulsive motion of the outer cylinder which corresponds to n = 0 and substituting the dimensionless quantities provided in51, the following dimensionless equations result,

u1 2 u1 1 u1 2 K(u 2 u1 ) M 2 (u1 - G) t r r r 2 u1 1 u1 u 2 f ν Rf Rν 2 K(u1 u 2 ) where R f 2 and R ν 2 t r r f1 ν1 r subject to : u1 u 2 0 1 r λ and

t0

u1 u 2 0 r 1 and

t0

u1 u 2 1 r

t0

and

Eq. 3.8

Analytical Approach Taking the Laplace transform of eqs. (8) and (9) we have the following ordinary differential equations We choose the D’Alembert’s method to obtain the solutions for these equation by multiply eq. (11) by A and add it to eq. (12). Details of transformation are given in52 and omitted here for simplicity

d 2 u 1 1 du 1 M 2G 2 (s M K)u1 Ku 2 dr 2 r dr s 2 d u 2 1 du1 (s K) K u2 u1 0 2 dr r dr R f R ν Rf R ν

where u1 u1 (r, t)e dt 0

st

Eq. 3.9

,

u 2 u 2 (r, t)est dt

and s 0

0

Basant K. Jha and Clement A. Aperea, “Unsteady MHD two-phase Couette flow of fluid-particle suspension in an annulus”, Published by the AIP Publishing LLC, 2011. 51

52

See previous.

47

N e εt 1 ikπ u1 (r, t) u1 (r, ε) Real u1 (r, ε )(1) k t 2 t k 1 N e εt 1 ikπ u 2 (r, t) u 2 (r, ε) Real u 2 (r, ε )(1) k t 2 t k 1

Eq. 3.10

where Real refers to the real part of’, i2 = −1 is the imaginary number, N is the number of terms used in the Riemann-sum approximation and ε is the real part of the Bromwich contour that issued in inverting Laplace transforms. The Riemann-Sum approximation for the Laplace inversion involves a single summation for the numerical process. Its accuracy depends on the value of ε and the truncation error dictated by N. The quantity εt = 4.7 is found to be more appropriate, since other tested values of εt seem to need longer computational time. Comparison with Numerical The momentum equations for fluid and dust particles with under the initial and boundary conditions are solved numerically using implicit finite difference method53, and the results presented in Table 4. The procedure involves the replacement of the partial differential into the finite difference equations at the grid point. The time derivative is replaced by the backward difference formula, while the spatial derivatives are replaced by central difference formula. The above equations are solved by Thomas algorithm by manipulating into a system of linear algebraic equations in the tridiagonal form. At each time step, the process of numerical integration for every dependent variable starts from the first neighboring grid point of the outer surface of the inner cylinder at r = 1 and proceeds towards the inner surface of the outer cylinder at r = λ. The process of computation is advanced until a steady state is approached by satisfying the following convergence criterion:

A

i, j1

A i, j

M A max

106

Eq. 3.11

Here Ai, j stands for the velocity field of the fluid and dust particle, M* is the number of interior grid points and |A|max is the maximum absolute value of Ai, j. In the numerical computation special attention is needed to specify Δt to get steady state solution as rapid as possible, yet small enough to avoid instabilities. In conclusion, we have employed the combination of the Laplace transform technique, D’Alembert’s and Riemann-sum approximation methods to obtain the solutions of unsteady MHD two-phase flow of fluid-particle suspension in the annular gap of two concentric cylinders. We have considered the effect of various dimensionless parameters occurring in the problem such as the Hartmann number M, the viscosity ratio Rν and the ratio of the substance quantity of mixture Rf on the flow field. The skin frictions at the opposite cylinders are seen to be of opposite sign; at the outer surface of the inner cylinder it is positive and negative at the inner surface of the outer cylinder. The result obtained demonstrates the reliability of the Riemann-Sum approximation method as comparisons made with those obtained using the implicit finite difference method shows consistency between them.

Obtained using the Riemann-sum approximation method and that obtained using finite difference method for both the fluid and the particle phases when t = 0.2, G = 1, K = 50, Rf = 0.25 and M = 2. 53

48

Table 4

Comparison of numerical velocity values (Riemann Sum vs. Finite Difference)

Case Study 4 - Simulation of Compressible 3-Phase Flows in an Oil Reservoir Problem Statement Oil extraction represents an important investment and the control of a rational exploitation of a field means mastering various scientific techniques including the understanding of the dynamics of fluids in place. A theoretical investigation of the dynamic behavior of an oil reservoir during its exploitation is presented by [Ahmadi, et. al.]54. More exactly, the mining process consists in introducing a miscible gas into the oil phase of the field by means of four injection wells which are placed on four corners of the reservoir while the production well is situated in the middle of this one. So, a mathematical model of multiphase multi-component flows in porous media was presented and the cell-centered finite volume method was used as discretization scheme of the considered model equations. It ensues from the analysis of the contour representation of respective saturations of oil, gas and water phases that the conservation law of pore volume is well respected. Besides, the more one moves away from the injection wells towards the production well; the lower is the pressure value. However, an increase of this model variable value was noticed during production period. Furthermore, a significant accumulated flow of oil was observed at the level of the production well, whereas the aqueous and gaseous phases are there present in weak accumulated flow. The considered model so allows the prediction of the dynamic behavior of the studied reservoir and highlights the achievement of the exploitation process aim55. Malik El’Houyoun Ahmadi, Hery Tiana Rakotondramiarana, Rakotonindrainy, “Modeling and Simulation of Compressible Three-Phase Flows in an Oil Reservoir: Case Study of Tsimiroro Madagascar”, American Journal of Fluid Dynamics 2014, 4(4): 181-193. 54

Malik El’Houyoun Ahmadi, Hery Tiana Rakotondramiarana, Rakotonindrainy, “Modeling and Simulation of Compressible Three-Phase Flows in an Oil Reservoir: Case Study of Tsimiroro Madagascar”, American Journal of Fluid Dynamics 2014, 4(4): 181-193. 55

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Mathematical Modeling The “black-oil” model is considered as constituted by three fluid phases (oil, gas, and water) in each of them can be present the following three components: a lighter component (gas) which can be at the same time in the oil phase and in the gas phase, a heavier component which can only be in the oil phase, and a component water which can only be in the water phase 56. The capillary pressures are assumed to be negligible57-58. Accordingly, all the present phases in the reservoir have the same pressure. Moreover, the studied medium is considered as isotropic so that the components of the permeability tensor have the same values in all directions. As there is a mass transfer between the oil and gas phases, mass conservation is not satisfied for these two phases. However, the total mass of each component is conserved. Besides, as the water phase is completely separated from the other phases and the component water is only present in the water phase, the mass conservation is well respected for this phase. Hence, the equations of the black-oil model can be formulated as follows. Mass conservation related to the component water can be written as:

( ρ w ) (ρ w u w ) q w t

Eq. 3.12

whereas for the heavier component, one can write:

( ρ 0oSo ) (ρ 0ou o ) q o t

Eq. 3.13

and the lighter component is governed by:

( ρ GoSo ρ gSg ) t

(ρ Go u o ρ g u g ) q g

Eq. 3.14

where ρGo and ρOo are the densities of the lighter and the heavier components in the oil phase respectively. Equation implies that, the lighter component (gas) can be at the same time in the oil and the gas phases. The velocity of each phase (α = w, o, g) is governed by the generalized Darcy’s law and can be calculated by the following relationship:

uα

k rα K (pα ρα g) μα

Eq. 3.15

in which, K denotes the tensor of the absolute porousness’s of the porous media, whereas g denotes the gravity acceleration vector. The system of mentioned equations is completed by the following 56 Z. Chen, G. Huen,

Y. Ma, “Computational Methods for multiphase flows in porous media”, SIAM Ed., Philadelphia PA, USA, 2006. 57 Krogstad, S., Lie, K.A., Nilsen, H.M., Natvig, J.R.., Skaflestad, B., and Aarnes J.E., “A Multiscale Mixed FiniteElement Solver for Three-Phase Black-Oil Flow”., Proc., Society of Petroleum Engineers (SPE) Reservoir Simulation Symposium, The Woodlands, Texas, USA, 2009. 58 Hoteit, H., and Firoozabadi A., 2006, “Compositional modeling of discrete fracture media without transfer functions by the discontinuous Galerkin and mixed methods,” SPE journal, 11(03), 341-352.

50

closing relationships. Conservation of pore volume (the sum of the saturations in a pore is equal to unity):

So Sg Sw 1

Eq. 3.16

Constraints on the tube pressures (oil-water) pcow and (gas-oil) pcgo:

p cow p o p w 0 p cgo p g p o 0

Eq. 3.17

Temporal and Spatial Discretization Methods An implicit Euler scheme is used for the time discretization while a cell-centered finite volume scheme was used for spatial discretization. Indeed, the developed model is constituted by conservation equations. Since the finite volume schemes are conservative, they are better suited for solving the considered system of equations. All the borders of the reservoir are considered impervious. No flow either goes out or enters anywhere but places where wells are positioned. As depicted in Figure 20, four injection wells are placed in the four corners of the reservoir while a production well is placed in its center. A condition of pressure in each cell containing a well with perforation is imposed. While the injection of gas in the reservoir and in each cell containing an injection wells being the simulation object, gas saturation is considered equal to one. Figure 20 Sketch of the reservoir with the four injection wells at Further information can be the corners and the production well in the center obtained from [Malik, al.]59. The Newton-Raphson method was used for the linearization of the above nonlinear system of discretized equations. Afterward, the obtained linear system can be solved by classical methods of resolution of linear systems. In the present work, iterative Generalized Minimum Residual (GMRES) Method was used for this purpose. Results and Discussion Results simulating a three-phase oil-water-gas three components model whose lighter component may be simultaneously in the oil phase as well as in the gas phase. They show the evolution of the pressure, saturation and cumulative flow rate generated during the oil extraction. A spatial decay of pressure is observed while a temporal pressure increase occurs. The pressure imposed on the injection wells is higher than the pressure within the reservoir; the aforementioned spatial pressure decay that is observed following the fluid displacement front towards the center may be due to a Malik El’Houyoun Ahamadi, Hery Tiana Rakotondramiarana, Rakotonindrainy, “Modeling and Simulation of Compressible Three-Phase Flows in an Oil Reservoir: Case Study of Tsimiroro Madagascar”, American Journal of Fluid Dynamics 2014, 4(4): 181-193. 59

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pressure drop in the gas flow through the porous medium60. Moreover, the energy transferred by the gas to fluids that are in place (oil and water) justifies the temporal pressure increase. However, the pressure is still quite enough to push the oil to the production well and ease its drawing out. Indeed, the purpose of the gas injection in the reservoir is not only to increase the pressure, but also to make the oil less viscous to facilitate mobility for its extraction. As can be seen from Figure 21 which show the variations of cumulative flow in the production well, a good amount of oil is produced. Such result conforms to the goal sought by the extraction process. The miscibility of the oil in the gas, combined with the fact that the oil is much lighter than the existing water, promotes such production. There is also a very low quantity of produced water which is lower compared to that of the produced gas, which itself is significantly less than that of produced oil. It can be justified by the fact that not only there may be an expansion of the gas during production, but the gas is also much lighter than water.

Figure 21

Cumulative flow in the production well for a production day

Case Study 5 - Effects of Mass Transfer & Mixture of Non-Ideality on Multiphase Flow Problem Statement In the chemical and petrochemical industries, Multi-Component phases commonly undergo composition changes due to various phenomena resulting in the transfer of species from one phase to another, or conversion of species through chemical reactions61. On the other hand, the thermophysical properties of a multi-component system are strongly dependent on its compositions. This is due to the fact that the properties of a multi-component mixture are not necessarily equal to the weighted average of the corresponding properties of its constituting pure components. This results in a severe interaction between hydrodynamic and thermodynamic behavior of a multi-component system. This interaction seems to have more effects on the hydrodynamic behavior of multiSee previous. Irani, Mohammad; Bozorgmehry Boozarjomehry, Ramin*+; Pishvaie, Sayed Mahmoud Reza and Tavasoli, Ahmad, “Investigating the Effects of Mass Transfer and Mixture Non-Ideality on Multiphase Flow Hydrodynamics Using CFD Methods”, Iran. J. Chem. Chem. Eng., Vol. 29, No. 1, 2010. 60 61

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component multiphase systems. Here we concern with the effects of the non-ideal behavior of phases on their hydrodynamic behaviors, have been studied based on a CFD framework in which the properties of each phase are rigorously modeled as a function of temperature, pressure and concentration of phase constituting components. The CFD framework is developed based on Eulerian - Eulerian model. The proposed framework can be used in modeling and simulation of multiphase flow of non-ideal mixtures. Mathematical Model The variation of liquid holdup with time and position is obtained by solving the continuity equations for the liquid and gas phases. The continuity equation for the flowing liquid and gas is written in terms of the accumulation and convection terms balanced by the total mass transferred to and from the other phases (written in terms of interphase fluxes for gas-liquid equations, discussed later). Since gas and liquid phases do not interpenetrate into each other in the reactor, the VOF approach is used. In this approach, the motion of all phases is modeled by formulating local, instantaneous conservation equations for mass and momentum62. The variation of velocity with time and position is calculated by solving the momentum balance equation. The momentum equations can be written in terms of accumulation and convection terms on the left-hand side, and the gravity, pressure gradient and viscous stresses terms on the right-hand side, as the pressure and velocity are assumed to be equal in both phases. The properties appearing in the transport equations are determined by their averaging based on phase volume-fraction. The continuity and momentum equations for a phase, ‘q’, in a multiphase flow problem is as follows:

(α q ρ q ) .(α q ρ q v) Spq , t

Spq Sqp ,

(ρv) .(ρvv) -p . μ v v T ρg i t

N

α q 1

q

1 n

,

ρ ρqαq i 1

n

, μ μ qαq i 1

Eq. 3.18

The variation of velocity with time and position is calculated by solving the momentum balance equation. The momentum equations can be written in terms of accumulation and convection terms on the left-hand side, and the gravity, pressure gradient and viscous stresses terms on the right-hand side, as the pressure and velocity are assumed to be equal in both phases. The properties appearing in the transport equations are determined by their averaging based on phase volume-fraction. 3.5.2.1 Bulk Species Transport The dynamic variation in the liquid and gas phase species concentrations are obtained by solving the unsteady state species mass balance equations, consisting of accumulation, convection, and interphase transport for the gas and liquid phases written as

(α g Cig ) .(α g Cig v - Dig α g Cig ) α g N igl t (α i Cil ) .(α l Cil v - Dil α lCil ) α l N igl t

Eq. 3.19

Hirt C. W., Nichols, B.D., Volume of Fluid (VOF) Method for the Dynamics of Free Boundaries, J. Comp. Phys., 39, p. 201 (1981). 62

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3.5.2.2 Interphase mass transfer Observations have indicated that the rates of mass transfers are closely related to the diffusion at the interface that is related to the concentration gradients at the interface, too. In real problems, however, we have usually no direct way to measure the concentration gradients at the interface. One of the approaches that can be used to estimate the concentration gradient is the approximation of various elements of concentration gradient in each phase using Finite Difference approach. In fact mass transfer coefficient based on Film theory is originally obtained through this approach. According to this approach various elements of concentration gradients of phase 'q' can be obtained as follows:

Ciq x j

Ciq Ciq

Eq. 3.20

Δx j

Where Ciq is the concentration of i-th component in phase q right at the interface and C*iq is the concentration of this component when phase q is at equilibrium with the other phase in the mixture. This is based on the fact that in a multiphase system, they are assumed to be at equilibrium right at their interface. For a mixture containing vapor and liquid the equilibrium concentration of various components can be obtained through isothermal flash calculations which are presented at all chemical engineering thermodynamic text books63-64. Details of flash calculation algorithm and equations were given in65. The concentration of various species in vapor and liquid phases are obtained based on Eqs. (4.5), respectively. Having obtained equilibrium concentrations, one can obtain the flux of species transfer (q Ni) and the rate of inter-phase mass transfer (Spq) through Eqs. (8.9), respectively, in which Mi is molecular weight for i-th species. Calculated flux or component i ( q Ni) in one phase is a source or sink for the same component in the other phase because there is no accumulation at the interface.

N Di q i

Ciq Ciq Δz j

,

N ip -Niq

n

,

Spq N iq M i

Eq. 3.21

i 1

Simulation Procedure The transport equations (Eq. (1.1) and (4.5)) were discretized by control volume formulation. UPWIND scheme was used for discretization66. A segregated implicit solver method with implicit linearization was used to solve discretized momentum equations. These equations have been obtained through the application of the first-order upwind scheme and for the pressure velocity coupling, the SIMPLE scheme has been used. For the pressure equation, the pressure staggering option (PRESTO) scheme was used. The program first reads the structured data from pre-processing section (in which the mesh representing the equipment has been built), before it goes into two nested iteration loops. Inner loop iterations are performed within each time step using the equations corresponding to the discretized version of the proposed model, while the outer loop goes through simulation times until it gets to the final time or steady state whichever happens sooner. At each time step, before going into the inner loop the fluid properties in each cell are calculated. In the inner loop, all the discretized equations are solved in three steps. In the first step, the physical Smith J.M, Van Ness H.C., “Introduction to Chemical Engineering Thermodynamics”, 6th Edition, McGraw-Hill. Walas S.M, “Phase Equilibria in Chemical Engineering”, Butterworth, Storeham, USA (1985). 65 See 44. 66 Patankar S.V., “Numerical Heat Transfer and Fluid Flow”, Taylor and Francis, (1980). 63 64

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properties such as density is updated based on the current solution. If the calculation has just begun, the fluid properties will be updated based on the initialized solution. In the second step, the flash calculation is performed in order to obtain the equilibrium concentrations based on which the source terms of the species concentrations and continuity equations are obtained. In the third step, equations of continuity and momentum are solved and after obtaining the velocity and pressure fields, equations corresponding to species concentration are solved in order to obtain the profiles of the concentration of various species. In this step, with the help of Eulerian-Eulerian approach (VOF approach), the trajectory of interface between two phases (liquid and gas) is determined. At the end of this step, convergence checking based on the norm of errors is done. Due to the nonlinearity and interactions of various equations, the convergence is usually achieved after several iterations at each time step. In order to get stable and meaningful results the time step must be very small (in the order of 10−4 s). However, as time goes on, and various states of the system (e.g., velocities and species concentration) obtain their corresponding smooth profile throughout the system, the time step can be gradually increased. This is due to the fact that dependence of various physical properties (e.g., density, specific heat…) on species concentration increases the amount of interaction and coupling of equations. It should also be noted that, in this mechanism the time step could not get values beyond 10-3s. In general, the time-stepping strategy depends on the number of iterations by time step needed to ensure very low residuals values (less than 10−7 for concentration and 10-5 for momentum and continuity). Results and Discussion The initial condition of the simulation at which the liquid height measured from bottom was 7cm. At the same time, the concentration of octane in gas phase and propane in liquid phase were set to zero. It was also assumed that there is no movement in the system and hence the velocity was set to zero for the whole domain. As time goes on, species are transferred between phases, this leads to a time varying concentration profiles in both phases and a general velocity field for the whole fluid (see Figure 22). Octane was transferred from liquid phase to gas phase and concentration of octane in liquid was decreased whereas concentration of octane in gas was increased. On the other hand, Propane dissolved in liquid phase which leads to its concentration decrease in gas phase, can be seen right at the interface. The propane concentration has its least value for gas phase and the largest value for the liquid phase . As a result of mass transfer in the interface, velocity in this region is higher than others. Density of gas phase was increased near the interface because concentration of octane was increased. In contrast, density of liquid phase was decreased near interface because concentration of propane was increased; (for further information please see67).

Irani, Mohammad; Bozorgmehry Boozarjomehry, Ramin*+; Pishvaie, Sayed Mahmoud Reza and Tavasoli, Ahmad, “Investigating the Effects of Mass Transfer and Mixture Non-Ideality on Multiphase Flow Hydrodynamics Using CFD Methods”, Iran. J. Chem. Chem. Eng., Vol. 29, No. 1, 2010. 67

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t = 3650

t = 0.0

Figure 22

Contour of gas volume fraction at different time levels

Concluding Remarks In the present work, a CFD framework has been proposed to simulate the multiphase mass transfer problems in chemical processes. For this purpose, a numerical method based on a macroscopic model and the finite volume method was applied. The proposed CFD framework is able to solve multiphase mass transfer problems with high interaction of thermodynamic and hydrodynamic behavior of the system. The proposed framework makes it possible to take into account the interacting effects of mixture non-ideality, mass transfer and hydrodynamics on multiphase system in a more realistic way. None of the analysis and studies that have been done on the hydrodynamic of multiphase systems has covered these effects till now, the major reason why these issues have not been covered till now was the fact that none of the available commercial CFD applications has a readymade frame work for such an analysis.

Case Study 6 - Numerical Study of Turbulent Two-Phase Coquette Flow

Motivation and Literature Survey The motivation behind this work is the need to understand bubble generation mechanisms due to interactions between free surface and turbulent boundary layers as commonly seen near ship walls as investigated by [Ovsyannikov, et al.]68. As a recognized problem, we consider a turbulent plane coquette flow with vertical parallel sidewalls and an air-water interface established by gravity in the vertical direction. Two-phase coquette flow has been described in the literature in different flow setups. Most studies were limited to cases of low Reynolds number and the evolution of a single bubble/droplet. Deformation and breakup of a single droplet in a plane coquette flow at low Reynolds number has been studied experimentally, theoretically, and numerically. In another example of the two-phase coquette flow is the two-layers of immiscible fluids which are set between moving horizontal walls. Due to viscosity difference between fluids, there is a jump in the tangential velocity 68 A.Y. Ovsyannikov, D. Kimy, A. Mani AND P. Moin, “Numerical study of turbulent two-phase Couette flow”, Center

for Turbulence Research Annual Research Briefs 2014.

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gradient across the interface which induces instabilities at the fluid interface [Coward]69, [Charru & Hinch]70. At high Reynolds number, there are only a few studies of two-phase Couette flow. [Iwasaki]71 studied the dynamics of a single immiscible drop in turbulent gas ow between two moving walls. [Fulgosi]72 and [Liu]73 performed direct numerical simulation (DNS) of interface evolution in Couette flow between two moving horizontal walls. One interesting case of a two-phase Couette flow in a turbulent regime is when the initial interface is set to be orthogonal to the moving vertical walls. In such a setup, the interaction between the fluid interface and the turbulent boundary layer is a key phenomenon. At sufficiently high Reynolds, Weber and Froude numbers, shear-induced interfacial waves can break, which leads to the formation of air cavities. These air cavities will be further fragmented by turbulence to smaller bubbles. These complexities make two-phase Couette flow at high Reynolds number a challenging problem for both experiments and numerical analysis. Capturing the small-scale ow and interface features requires high-resolution experimental techniques and very expensive DNS calculations. Only two numerical studies Kim74-75 have been performed for investigation of air entrainment and bubble generation in two-phase Couette flow. The current work is a continuation of our recent studies (Kim), where we performed numerical simulations of the interface breakup in two-phase Couette flow at Reynolds number of approximately 13000 and Weber number of approximately 42000 (surface tension coefficient was much smaller than the realistic value for an air-water system). The effect of Froude number on the interface breakup and bubble generation was studied in Kim et al. (2012). The second paper of Kim et al. (2013) was mostly devoted to the development and assessment of a mass conservative interfacecapturing method based on a geometric volume-of-fluid (VOF) approach, and one simulation of the two-phase Couette flow was performed for a Reynolds number of 12000 and a Weber number of 200. However, there have been no studies with one-to-one matched experiments and numerical simulations. Objectives The primary objective of this work is to perform a numerical simulation of a two-phase Couette flow with flow parameters very close to those of the laboratory experiment conducted by collaborators. The bubble formation rate, bubble size distribution, and the effect of interface on the modulation of turbulence are the main characteristics of this flow type, and our investigations are focused on understanding these characteristics. Problem Statement We perform a simulation of a two-phase system with realistic air/water density and viscosity ratios. The density of liquid is ρliq = 1000 kg/m3 and the density of gas is ρgas = 1.2 kg/m3. However, to reduce 69 Coward, A. V., Renardy, Y. Y.,

Renardy, M. & Richards, J. R., “Temporal evolution of periodic disturbances in twolayer Couette flow”, J. Computational Physics, 132 (2), pp. 346-361, 1997. 70 Charru, F. & Hinch, J. E., “Phase diagram of interfacial instabilities in a two-layer Couette flow and mechanism of the long-wave instability”, J. Fluid Mech. 414, pp. 195-223, 2000. 71 Iwasaki, T., Nishimura, K., Tanaka, M. & Hagiwara, Y. ,”Direct numerical simulation of turbulent Couette flow with immiscible droplets”, Int. J. Heat Fluid Flow 22, pp. 332-342, 2001. 72 Fulgosi, M., Lakehal, D., Banerjee, S. & De Angelis, V. ,”Direct numerical simulation of turbulence in a sheared air-water ow with a deformable interface”, J. Fluid Mech. 482, pp. 319-345, 2003. 73 Liu, S., Kermani, A., Shen, L. & Yue, D. K. P.,” Investigation of coupled air-water turbulent boundary layers using direct numerical simulations”. Phys. Fluids 21 (6), pp. 62-108, 2009. 74 Kim, D., Mani, A. & Moin, P., “Numerical simulation of wave breaking in turbulent two-phase Couette flow”, Annual Research Briefs, Center for Turbulence Research, Stanford University, pp. 171-178-2012. 75 Kim, D., Mani, A. & Moin, P., “Numerical simulation of bubble formation by breaking waves in turbulent twophase Couette flow”, Annual Research Briefs, pp. 37-46, 2013.

57

values. Thus the viscosity of the gaseous phase is μgas = 7.2 x105 Pa-s and the viscosity of the liquid phase is μliq = 4x103 Pa - s. The surface tension coefficient σ and gravity acceleration g take realistic values of 0.07N/m and 9.81m/s2, respectively. Figure 23 depicts the schematic of the flow configuration and domain size. The computational domain is a rectangular box with sizes 2_h, 2h and 6h in the stream wise (x), wall-normal (y) and span wise (z) directions, respectively. Here h = 2 cm is the half-distance between walls. Initially, the interface is located at a plane z = 4h, hence a height of liquid layer Hliq is 4h and a height of gas layer Hgas is 2h. The sidewalls are moving in the opposite directions with speed of Uw = 1.6m/s. Based on the chosen parameters, the following main dimensionless parameters are:

Re

ρ liq U w h μ liq

, We

Figure 23

ρliq U 2w h σ

, Fr

Schematic illustration of the flow geometry

Uw gh

, Ar

H liq h

Eq. 3.22

which are Reynolds, Weber, Froude numbers, and aspect ratio (ratio of water depth to h), respectively. All non-dimensional groups are determined using properties of the liquid phase. In the current study, Re = 8000, We = 730, Fr = 3.6 and Ar = 4. Governing Equations and Numerical Method The governing equations describing the motion of two immiscible, incompressible New tonian fluids are the conservation of mass and momentum. The first equation is given in terms of the volume fraction function as

ψ .(ψu) 0 t ρu .(ρuu) -p .τ ρg Fst t

Eq. 3.23

where u is the velocity, p is the pressure, ρ is the density, τ = μ(⩢ u +⩢ uT ) is the shear stress tensor, μ is the dynamic viscosity, g is the acceleration due to gravity and Fst = σκδ∑n is the surface tension force. Here, σ is the surface tension coefficient, δ∑ is the Dirac delta function, κ is the interface curvature, and n is the interface normal vector. The volume fraction function is equal to 1 in the gas phase and 0 in the liquid phase. The interface is then represented by the volume fraction values, 0

> η. Therefore, we choose conventional DNS resolution as in single-phase flows. We use a Cartesian grid with uniform mesh spacing in x and z directions, and stretched mesh in the y direction according to

59

2(j 1) h tanh γ( 1 N y 1 yj tanh ( )

Eq. 3.25

where the stretching parameter ϒ is 2.9. Based on the viscous length scale in the liquid phase the grid resolution is Δx+ =13, Δy+min= 0.2, Δy+max =13 and Δz+ =7. Such discretization results in at least 4 grid points per Hinze scale (8 points per bubble diameter) and 18 grid points per viscous sublayer. Finally, the time step is given from the stability constraint due to surface tension as it is more restrictive than the stability conditions due to convection and gravity terms (the viscous terms are treated implicitly):

Δt

ρ liq ρ gas 4ππ

-6 Δy3/2 min 10 s

Eq. 3.26

Results Figure 24 (a-d) show instantaneous snapshots of the interface (given by iso surface ψ = 0.5). On the

(a) t* = 5

(b) t* = 10

(c) t* = 15

(d) t* = 60

Figure 24

Snapshots of the air-water interface at different times

60

free surface, shear-induced oblique wave structures are observed (Figure 24(a)), then the interfacial waves grow in amplitude (Figure 24(b)), leading to breakup of the interface (Figure 24(c)). The air cavities are found underneath the free surface, trapped between the breaking interfacial waves. These air cavities are subsequently fragmented into air bubbles in the water. Finally, Figure 24(d) shows the interface at the fully developed state. Error! Reference source not found. shows time- and tream wise-averaged flow for two cases76: (a) a two-phase Couette flow and (b) a single-phase Couette water flow at the same Reynolds number of 8000. The color contours correspond to the stream wise component of the mean velocity, ū, while the wall-normal and span wise components, (_v, _ w), are shown as vector plot. The maximum Root-Mean-Square (RMS) value of in-plane velocities represents (after 100 flow-through times) approximately 6-7% of the maximum stream wise velocity for both cases and does not diminish with time. For the single-phase Couette flow, secondary flow is represented by four large eddies, while for the two-phase case the secondary flow is more complex due to the effects of the interface. In our simulations the length of the computational domain, 2πh, was chosen as in classical simulations of pressure-driven channel ow. However, it is known from studies of single-phase Couette flow [Lee & Kim 1991; Komminaho et al. 1996; Papavassiliou & Hanratty 1997] that the domain length should be much longer. In this work we focus on a study of interface/boundary layer interaction and leave the study of Couette flow in longer channels for future research. Mean stream wise velocity profiles measured at z/h = 1.5, z/h = 2.5, z/h = 3.5 and z/h = 4, locations are shown in Figure 2677. In addition to two-phase Couette results, we

Figure 25

Turbulent statistics: time- and stream wise-averaged velocity field

Color plot shows the mean stream wise velocity; vector plot depicts mean in-plane flow. (a) Two-phase Couette flow at Re = 8000, We = 730 and Fr = 3.6; (b) Single-phase Couette flow at Re = 8000. 77 (a) mean stream wise velocity; (b) mean stream wise velocity in viscous units. Lines represent profiles at different depths: ---- z/h = 4; ,- - - z/h = 3.5; -.-.-.- , z/h = 2.5;-..-..-..-, z/h = 1.5. Symbols ˚ correspond to DNS results (also averaged in the vertical direction) for single-phase Couette water flow at the same Reynolds number. 76

61

show the result of simulation for single-phase Couette flow at the same Reynolds number of 8000. Figure 26 (a) shows velocity profiles in non-dimensional units, U/Uw versus y/h. Figure 26 (b) shows the same data in non-dimensional viscous units according to U+ = (U - Uw)/uτ and y+ = (h y)/δν. As seen from Figure 26 (b), there is no collapse of the results for two-phase Couette flow with the log-law even for a velocity profile quite far from the interface at z/h = 1.5. Even for single-phase Couette flow, a particular profile of stream wise velocity does not necessarily collapse with the loglaw owing to the presence of persistent roller structures. Only after averaging of velocity in a vertical direction does the velocity profile for single-phase Couette flow matches with the log-law.

Figure 26

Turbulent statistics for two-phase Couette flow

Influence of the water depth Figure 2778 shows a comparison of the fully developed air-water interface from our previous results 79. (Figure 27(a)) and the current simulation (Figure 27(b)). The results on the left figure were obtained for an Aspect Ratio of 1.57, whereas in the current simulation the Aspect Ratio is 4. As seen, the simulation for the higher value of the aspect ratio predicts much less air entrainment. To confirm our hypothesis of the influence of the water depth on bubble density, we are currently running simulations for low and high values of the aspect ratio (1.57 and 8) for the same non-dimensional parameters: Reynolds, Weber and Froude numbers, as in the simulation presented in this work. Conclusions In this work we performed a numerical simulation of a two-phase Couette flow at a Reynolds number of 8000, Froude number of 3.6, Weber number of 730 and at realistic air-to-water density and viscosity ratios. Except for Reynolds number and water depth, the parameters of simulation correspond to experiments currently being performed at the University of Maryland. The VOF method has been used to predict interface dynamics. To achieve statistically state two-phase simulations of Couette flow require much longer time compared to simulations of single-phase Couette flow. Complication of turbulent intensity was found near the interface, whereas diminished

(a) Results for an Aspect Ratio of 1.57 and Re = 12000, We = 206, Fr = 3:8 (Kim et al. 2013); (b) Present results for an Aspect Ratio of 4 and Re = 8000, We = 730, Fr = 3.6 79 Kim, D., Mani, A. & Moin, P. ,”Numerical simulation of bubble formation by breaking waves in turbulent twophase Couette flow”, Annual Research Briefs, pp. 37{46. Center for Turbulence Research, Stanford University, pp. 37-46, 2013. 78

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turbulent intensity was observed in the core region. It was found that the bubble density depends on the water depth. To validate this result, in our ongoing work we compare numerical results obtained for different aspect ratios with experimental data. In addition, a simulation is being carried out with deeper water at an aspect ratio Hliq=h of 8 and the same non-dimensional parameters, Re = 8000, We = 730, Fr = 3:6.

Figure 27

Air-water interface at fully developed state

Case Study 7 - Slug Flow in Horizontal Air and Water Pipe Flow Motivation and Statement of Problem The growth of liquid slugs in oil and gas pipelines is a vast and costly problem for the oil firms. A pressure drop in oil production is the main source of the problem that leads to terrain-induced slug flow in the pipeline between the production platform and wellhead platform. This type of slug flow condition can create huge transient surges. The transient nature of the slugs if not appropriately considered might become climacteric and can hasten the material’s fatigue with the risk of pipe damage and maintenance costs. Recently, [Z. I. Al-Hashimy, et al.]80 has been simulated the slug flow regime in an air-water horizontal pipe flow. The variables identified to characterize the slug regime are the slug length and slug initiation. The volume of fluid method was employed assuming unsteady, immiscible air/water flow, constant fluid properties and coaxial flow. The simulated pipe segment was 8 m long and had a 0.074 m internal diameter. Three cases of airwater volume fractions have been investigated, where the water flow rate was pre-set at 0.0028 m3/s, and the air flow rate was varied at three dissimilar values of 0.0105, 0.0120 and 0.015 m3/s. These flow rates were converted to superficial velocities and used as boundary conditions at the inlet of the pipe. The simulation was validated by bench marking with a Baker chart, and it had successfully predicted the slug parameters. The computational fluid dynamics simulation results revealed that the slug length and pressure were increasing as the air superficial velocity increased. The slug initiation position was observed to end up being shifted to a closer position to the inlet. It was believed that the strength of the slug was high at the initiation stage and reduced as the slug Zahid I. Al-Hashimy1, Hussain H. Al-Kayiem, Rune W. Time & Zena K. Kadhim, “Numerical Characterization Of Slug Flow In Horizontal Air/Water Pipe Flow”, Int. J. Comp. Meth. And Exp. Meas., Vol. 4, No. 2 (2016) 114– 130. 80

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progressed to the end of the pipe. The pressure gradient of the flow was realized to increase as the gas flow rate was increasing, which in turn was a result of the higher mean velocity. Slug Flow and Slug Formation in Pipe Slug flow refers to the phenomenon at which 2-phase liquid-gas movements exist in pipelines over a broad range of intermediate flow rates, generating improper disorder resulting from the actions of the liquid and gas plugs, known as slugs. The plug distribution of liquids and gases in slug flows are highly unique but intermittent, basically because of the nature of the terrain, gas/liquid velocity fluctuations, pigging, etc. A slug unit consists of an aerated liquid slug as well as an accompanying gas bubble, controlled within a liquid film of varying thicknesses. The actual thickness of the film in most cases differs from the minimum value at the front of the following slug towards the maximum value at the rear of the preceding slug. Consequently, the slug length may remain steady along the direction of travel while the pressure drops systematically across the sections of the pipe81. The slug formation is a three-step process that is depicted in Figure 28. Originally, the flow is stratified where the gas is at the top of the pipe and the liquid at the bottom. As the gas passes over a wave, there is a pressure drop, then a pressure recovery, creating a small force upward within the wave. Under the conditions that this upward force is sufficient to raise the wave until it extends to the top of the pipe, the flow is considered as slug flow. Once the wave reaches the top of the pipe, it forms into the familiar slug shape with a nose and tail. The slug is forced forward by the gas and thus, can travel at a greater velocity than the liquid film.

Slug Unit Figure 28

Hydrodynamic slug formation (Courtesy of Z. I. Al-Hashimy et al.)

The actual slug movement can be explained by changing the liquid slugs and the gas bubbles moving above the liquid films, which in turn combine to develop what is known as a slug unit. The slug frequency is described by the number of slugs passing a particular point along the pipeline over a particular period of time. Amongst the slug flow characteristics, the slug frequency is an essential component which relates to significant operational difficulties, like the flooding of downstream facilities, severe pipe vibration, pipeline structural instability and wellhead pressure fluctuation. It is generally known that pipe corrosion is substantially impacted by a high slug frequency. Baker Chart Flow regime maps for the 2-phase flow in a horizontal pipe have been intensively researched by many researchers. [Baker]82 presented a map of a two-phase flow in a horizontal pipe by using various fluids in addition to demonstrating distinct phases of mass fluxes along with corresponding fluid properties such as density and surface tension (see Figure 29). The Baker chart also features two dimensionless parameters, l and y, to enable its application for various gas/vapor-liquid combinations different from the standard one (air-water at atmospheric pressure and room Orell, A., “Experimental validation of a simple model for gas–liquid slug flow in horizontal pipes”. Chemical Engineering Science, 60(5), pp. 1371–1381, 2005. 82 Baker, O., “Simultaneous flow of oil and gas”. Oil and Gas Journal. 53, pp. 185–195, 1954 81

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temperature) for which both parameters equate to unity. The Baker chart was used as a reference for the simulation of the horizontal slug flow regimes in the present study.

Figure 29

Baker chart where (.) Operating conditions of water–air two-phase flow

Problem Formulation In this present work, the Volume of Fluid (VOF) method has been implemented employing the commercial software Star-CCM+ to simulate the horizontal sections of a pipe for air-water slug flow. The objective has been to investigate the volume fraction profile and pressure drop along 8 m of the pipe length will be predicted by the simulation. The [Baker] chart was adopted to justify the slug presence in the simulation by computing the superficial mass velocities of the water and air. The simulated horizontal slug flow patterns observed through visualizations of the phase distributions were qualitatively compared against the flow regimes expected by the [Baker] chart. variation, with time, in seven different cross sections along the pipe. In addition, the pressure drop along 8 m of the pipe length will be predicted by the simulation. The [Baker] chart was adopted to justify the slug presence in the simulation by computing the superficial mass velocities of the water and air. The simulated horizontal slug flow patterns observed through visualizations of the phase distributions were qualitatively compared against the flow regimes expected by the [Baker] chart. The [Baker] flow regime map, as demonstrated in Figure 29, shows the standardized boundaries of the various flow pattern regions as functions of the mass flux of the gas phase, G, and the ratio of the mass flux of the water phase and air, L/G. Where, G was the mass flux of the gas phase (kg/m2 s) = (gas mass flow rate/tube cross-sectional area) and L was the mass flux of water phase (kg/m2s) =

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(water mass flow rate/tube cross-sectional area). The dimensionless parameters, l and y, had been added so that the chart could be utilized for any gas/liquid combination that differed from the standard combination. The standard combinations, at which both parameters, l and y, equate to unity, which are water and air flow under atmospheric pressure and at room temperature. Consequently, for the present application, where the fluids were air and water, the values of l and y are equal to 1.0. By taking into account, the predicted values for l and y, the pattern of the two-phase flows with any gas/liquid at other pressures and temperatures can be forecasted using the same chart. The parameters l and y were can be calculated from

w l w

1

w l

2

3 , g l a w

0.5

Eq. 3.27

Where σ, μ, ρ are the surface tension, viscosity and density, respectively. The subscripts ‘a’ and ‘w’ refers to the air and water, respectively, at normal temperature and atmospheric pressure; whereas, the subscripts ‘g’ and ‘l’ refers to the vapor and liquid conditions of the fluid being considered. 3.7.4.1 Boundary Condition The required boundary conditions depend on the physical models used, where water was designated as the primary phase and air as the secondary phase for all cases [Al-Hashimy, et al.]83. The water and air were considered incompressible. The most suitable boundary condition for external faces in incompressible water was the velocity inlet. The outlet was considered as a pressure-outlet boundary. The boundary conditions used are illustrated schematically in Figure 30. The velocity in a multiphase flows is defined as the ratio of the velocity and the volume fraction of the considered phase in a multiphase system. Actual velocity of phase = (velocity of phase)/(volume fraction of phase). The average velocity of the flow varied depending on the volume fraction of each phase; which, when defined as the area fraction of a phase, is expected to change in space and time. The average velocity for the gas phase and liquid phase, which are called the gas superficial velocity and liquid superficial velocity, can be given by:

Figure 30

Boundary condition for water-air slug flow through a pipe

Zahid I. Al-Hashimy1, Hussain H. Al-Kayiem, Rune W. Time & Zena K. Kadhim, “Numerical Characterization Of Slug Flow In Horizontal Air/Water Pipe Flow”, Int. J. Comp. Meth. And Exp. Meas., Vol. 4, No. 2 (2016) 114– 130. 83

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USG

UG AG UG a G A

, USW

UWAW UWa W A

Eq. 3.28

There were two methods used in specifying the inlet boundary conditions for the simulation of the slug flow84. The first method imposed perturbations at the inlet so that the volume fraction of the liquid phase entered the pipe as a function of time. The second method, the pipe was initially assumed to be filled with stratified air and water with 50% volume percentage and zero velocity. For the present simulation, the initial and inlet region were: the upper half of the pipe was occupied by 50% void fraction of the gas phase, aa and the lower half by 50% volume fraction of the water phase, aw. Then, the field function was used to define the inlet water volume fraction as a function of time. Volume of Fluid (VOF) The VOF model has been utilized in this study to track the interface between the gas-liquid phases in order to define the slug flow regime. The VOF technique exhibits an immense capability in tracking the interface between the two phases using a color function. The color function is Ca = 1 for the entire ak fluids and Ca = 0 for the void; thus, the interface is located at 0 < Ca < 1. In this method, all the cells should be occupied by a single fluid or a combination of fluids because the VOF does not allow for any void cells85. The water phase was selected as the primary phase. The tracking of the interface between the phases was accomplished by the solution of a continuity equation for the volume fraction of the secondary phase. The volume fraction equation was not solved for the primary phase; the primary-phase (water phase) volume fraction was computed as in Eq. (3.6).

aW aG 1

Eq. 3.29

Since the control volume at the interface location was occupied by fluids, the fluid properties, particularly the viscosity and density, changed abruptly with the interface motion. The mixture properties for the density and viscosity appearing in the momentum and mass equations were calculated as:

ρ m a Wρ W a GρG

μ m a Wμ W a Gμ G

Eq. 3.30

Using conservation of mass and momentum, the local volume fraction of the water was given by the following continuity equation:

a W (a W U W ) (a W VW ) (a W WW ) 0 t x y z (ρ m U i ) (ρ m U i U j ) (p) t x i x i xi

Ui Uj μ ρ u u m m i j ρmgi F xj xi

Eq. 3.31

Where the first term, on the left-hand side, denotes the rate of the momentum increasing per unit volume, and the second term denotes the change of the momentum due to convection per unit volume. On the right side, the first term represents the pressure gradient, the second term represents Al-Hashimy, Z.I., Al-Kayiem, H.H., Kadhim, Z.K. & Mohmmed, A.O., “Numerical simulation and pressure drop prediction of slug flow in oil/gas pipelines”. WIT Transactions on Engineering Sciences, 89, 2015. 85 Ranade, VV., “Computational Flow Modeling for Chemical Reactor Engineering”, Vol. 5. Academic press, 2001. 84

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the gravitational force, and the third term represents the viscous effect. The external force per unit volume is given by the last term in the above equation and can be modelled using the continuum surface force (CSF) model. For turbulence modeling, an SST k−w model in conjunction with implicit time integration has been used. Details of methodology is available in [Z. I. Al-Hashimy, et al.]86.

Figure 31

Slug initiation of the air-water slug flow

Results and Discussion 3.7.6.1 Slug Initiation The determination of the slug initiation had been according to the presence of the first slug in the flow field. The simulation results at the three air superficial velocities are shown in Figure 31. It is obvious that the first slug initiation was faster since the air superficial velocity was higher. It was reveals the time and location of the slug initiation at a constant water superficial velocity of 0.651 m/s and three different air superficial velocities of 2.443, 2.792, and 2.49 m/s. The slug initiation position was transferred to a shorter distance from the inlet. 3.7.6.2 Slug Length For the slug body length and slug frequency, which is a reciprocal of the slug unit period, they could actually be considered as the mean number of slugs per unit time as observed by a fixed observer. The measurements of the average slug body by selecting the X coordinate of both the front and rear ends are shown in Figure 32. The reference line indicated in each case is located about 4 m downstream of the inlet. The slug length was estimated as LS = Xfront − Xrear . The slug length was measured from the reference line up to the front of the slug. There was a proportional relation between the slug length and the air superficial velocity. Prediction results of the slug length revealed that when the air superficial velocity increased, the generated slugs became longer compared to the

Zahid I. Al-Hashimy1, Hussain H. Al-Kayiem, Rune W. Time & Zena K. Kadhim, “Numerical Characterization Of Slug Flow In Horizontal Air/Water Pipe Flow”, Int. J. Comp. Meth. And Exp. Meas., Vol. 4, No. 2 (2016) 114– 130. 86

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case of the lower superficial velocity of air. Table 5 shows the predicted slug length at various air superficial velocities. Air/Water

Slug length (m)

Case 1

Mixture Velocity (m/s) 3.049

0.449

Initial Time (s) 0.64

Crossing Time (s) 2.02

Case 2

3.443

1.037

0.48

1.942

Case 3

4.141

1.5

0.373

1.641

Table 5

Figure 32

Slug length at different air-water velocities

Slug length calculation of air-water slug flow

3.7.6.3 Slug Volume Fraction The volume fraction in the slug body or gas void fraction is a crucial parameter for the design of multiphase pipelines and the associated separation equipment; while, the phase composition is proportional to its volume fraction. Figure 33 has provided the results of the simulation for the predicted void fraction of the air-water slug flow regime at various cross sections from 1 to 7 m along of the horizontal pipe, for the first three times. The distribution of water and air in the horizontal slug flow can be vividly noticed. The red color refers to the water phase while the dark blue color refers to the air phase, and the line between both colors display the presence of an interface. The best approximation of the slug flow regime is observed compared with the slug flow regime in the Baker char. The water slugs touched the upper part of the pipe and performed complete slug regime. As stated before, t = 0 represents the initial conditions where the flow along the pipe was stratified flow, in which the upper part was occupied by air, and the lower part was occupied by water due to the gravitational effect. However, the water phase was steady until the generation of the first wave crest because of the sinusoidal perturbation at the inlet; large waves were initiated, which heightened steadily, filling the cross section of the pipe at time 0.5 s. The long slug was observed at 0.75 s, and it

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continued to grow along the pipe at the downstream sections. Generally, the void fraction increased with the increase in the gas velocity. In conclusion, the internal air-water two-phase slug flow behavior in a horizontal pipe were examined and described the numerical procedure used to simulate the case. Slug initiation and growth was effectively predicted by the 3D transient VOF model combined with the homogeneous k−ω turbulence model. The volume fraction profile and pressure variation with time in seven different cross sections along the pipe were examined.

Figure 33

Cross section of the fluid domain for the extraction of volume fraction for Case 3

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4 Chemical Reaction Chemical reaction engineering (reaction engineering or reactor engineering) is a specialty in chemical engineering or industrial chemistry dealing with chemical reactors. Frequently the term relates specifically to catalytic reaction systems where either a homogeneous or heterogeneous catalyst is present in the reactor. Sometimes a reactor per se is not present by itself, but rather is integrated into a process, for example in reactive separations vessels, retorts, certain fuel cells, and photocatalytic surfaces. The issue of solvent effects on reaction kinetics is also considered as an integral part. It is studying and optimizing chemical reactions in order to define the best reactor design. Hence, the interactions of flow phenomena, mass transfer, heat transfer, and reaction kinetics are of prime importance in order to relate reactor performance to feed composition and operating conditions. Although originally applied to the petroleum and petrochemical industries, its general methodology combining reaction chemistry and chemical engineering concepts allows to optimize a variety of systems where modeling or engineering of reactions is needed. Chemical reaction engineering approaches are indeed tailored for the development of new processes and the improvement of existing technologies87.

Overview of Chemical Reaction Engineering

Every industrial chemical process is designed to produce economically a desired product from a variety of starting materials through a succession of treatment steps.

Recycle Figure 34

Typical Chemical Process

Figure 34 shows a typical situation where the raw materials undergo a number of physical treatment steps to put them in the form in which they can be reacted chemically. Then they pass through the reactor. The products of the reaction must then undergo further physical treatment-separations, purifications, etc. for the final desired product to be obtained88. Variables Affecting the Rate of Reaction Many variables may affect the rate of a chemical reaction. In homogeneous systems the temperature, pressure, and composition are obvious variables. In heterogeneous systems more than one phase is involved; hence, the problem becomes more complex. Material may have to move from phase to phase during reaction; hence, the rate of mass transfer can become important. For example, in the burning of a coal briquette the diffusion of oxygen through the gas film surrounding the particle, and From Wikipedia. Marek Ściążko, “Overview of Chemical Reaction Engineering”, Based on: Octave Levenspiel, Chemical Reaction Engineering, 3rd Edition, 2013. 87 88

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through the ash layer at the surface of the particle, can play an important role in limiting the rate of reaction. In addition, the rate of heat transfer may also become a factor. Consider, for example, an exothermic reaction taking place at the interior surfaces of a porous catalyst pellet. If the heat released by reaction is not removed fast enough, a severe non-uniform temperature distribution can occur within the pellet, which in turn will result in differing point rates of reaction. These heat and mass transfer effects become increasingly important the faster the rate of reaction, and in very fast reactions, such as burning flames, they become rate controlling. Thus, heat and mass transfer may play important roles in determining the rates of heterogeneous reactions. Definition of Reaction Rate We next ask how to define the rate of reaction in meaningful and useful ways. To answer this, let us adopt a number of definitions of rate of reaction, all interrelated and all intensive rather than extensive measures. But first we must select one reaction component for consideration and define the rate in terms of this component i. If the rate of change in number of moles of this component due to reaction is dNildt, then the rate of reaction in its various forms is defined as follows:

Based on unit volume of Reacting fluid : 1 dN i moles i formed V dt (volume of fluid)(time) Based on unit mass of solid in fluid/soli d : ri

1 dN i moles i formed W dt (volume of solid)(time) Based on unit interfacia l surface in two fluid systemor basesd on unit surface of solid in gas/solid systems : ri

1 dN i moles i formed S dt (surface)(time) Based on unit volume of solid in gas/solid systems: 1 dN i moles i formed ri Vs dt (volume of solid)(time) ri

Based on unit volume of reactor if different from the rate based on unit volume of fluid : ri

1 dN i moles i formed Vr dt (volume of reactor)(time)

Eq. 4.1

Speed of Chemical Reactions Some reactions occur very rapidly; others very, very slowly. For example, in the production of polyethylene, one of our most important plastics, or in the production of gasoline from crude petroleum, we want the reaction step to be complete in less than one second, while in waste water treatment, reaction may take days and days to do the job. Figure 35 indicates the relative rates at which reactions occur. To give you an appreciation of the relative rates or relative values between what goes on in sewage treatment plants and in rocket engines, this is equivalent to 1 sec to 3 yr. With such a large ratio, of course the design of reactors will be quite different in these cases.

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Figure 35

Moles of A disappearing Rate of reactions

Classification of Reactions There are many ways of classifying chemical reactions. In chemical reaction engineering probably the most useful scheme is the breakdown according to the number and types of phases involved, the big division being between the homogeneous and heterogeneous systems. A reaction is homogeneous if it takes place in one phase alone. A reaction is heterogeneous if it requires the presence of at least two phases to proceed at the rate that it does. It is immaterial whether the reaction takes place in one, two, or more phases; at an interface; or whether the reactants and products are distributed among the phases or are all contained within a single phase. All that counts is that at least two phases are necessary for the reaction to proceed as it does. The Common Types of Chemical Reactions Several general types of chemical reactions can occur based on what happens when going from reactants to products89. The more common types of chemical reactions are as follows:

Combination or Synthesis Decomposition Single Displacement Double Displacement Combustion Redox Organic

4.1.5.1 Combination Chemical Reactions In combination reactions, two or more reactants form one product. The reaction of sodium and chlorine to form sodium chloride,

2Na(s) Cl2 (g) 2NaCl(s) and the burning of coal (carbon) to give carbon dioxide,

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Dummies, a Wiley Brand.

Eq. 4.2

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C(s) O2 (g) CO2 (g)

Eq. 4.3

are examples of combination reactions. Depending on conditions or the relative amounts of the reactants, more than one product can be formed in a combination reaction. 4.1.5.2 Decomposition Chemical Reactions Decomposition reactions are really the opposite of combination reactions. In decomposition reactions, a single compound breaks down into two or more simpler substances (elements and/or compounds).The decomposition of water into hydrogen and oxygen gases,

2H2O(l) 2H2 (g) O2 (g)

Eq. 4.4

and the decomposition of hydrogen peroxide to form oxygen gas and water,

2H2O2 (l) 2H2O(l) O2 (g)

Eq. 4.5

are examples of decomposition reactions. 4.1.5.3 Single Displacement Chemical Reactions In single displacement reactions, a more active element displaces (kicks out) another less active element from a compound. For example, if you put a piece of zinc metal into a copper(II) sulfate solution, the zinc displaces the copper, as shown in this equation:

Zn(s) CuSO 4 (aq) ZnSO4 (aq) Cu(s)

Eq. 4.6

The notation (aq) indicates that the compound is dissolved in water in an aqueous solution. Because zinc replaces copper in this case, it’s said to be more active. If you place a piece of copper in a zinc sulfate solution, nothing will happen. The following table shows the activity series of some common metals. Notice that because zinc is more active in the table, it will replace copper, just as the preceding equation shows. 4.1.5.4 Double Displacement Chemical Reactions In single displacement reactions, only one chemical species is displaced. In double displacement reactions, or metathesis reactions, two species (normally ions) are displaced. Most of the time, reactions of this type occur in a solution, and either an insoluble solid (precipitation reactions) or water (neutralization reactions) will be formed. 4.1.5.5 Precipitation Reactions If you mix a solution of potassium chloride and a solution of silver nitrate, a white insoluble solid is formed in the resulting solution. The formation of an insoluble solid in a solution is called precipitation. Here is the molecular equation for this double-displacement reaction:

KCl(aq) AgNO3 (aq) AgCl(s) KNO3 (aq) The white insoluble solid that’s formed is silver chloride.

Eq. 4.7

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4.1.5.6 Neutralization Reactions The other type of double-displacement reaction is the reaction between an acid and a base. This double-displacement reaction, called a neutralization reaction, forms water. Take a look at the mixing solutions of sulfuric acid (auto battery acid) and sodium hydroxide (lye). Here is the molecular equation for this reaction:

H2SO 4 (aq) 2NaOH(aq) Na 2SO 4 (aq) 2H2 O(l)

Eq. 4.8

4.1.5.7 Combustion Chemical Reactions Combustion reactions occur when a compound, usually one containing carbon, combines with the oxygen gas in the air. This process is commonly called burning. Heat is the most-useful product of most combustion reactions. Here’s the equation that represents the burning of propane:

C3H8 (g) 5O2 (g) 3CO2 (g) 4H 2 O(l)

Eq. 4.9

Propane belongs to a class of compounds called hydrocarbons, compounds composed only of carbon and hydrogen. The product of this reaction is heat. Combustion reactions are also a type of redox reaction. 4.1.5.8 Redox Chemical Reactions Redox reactions, or reduction-oxidation reactions, are reactions in which electrons are exchanged:

2Na(s) Cl 2 (g) 2 NaCl(s) C(s) O 2 (g) CO 2 (g)

Eq. 4.10

Zn(s) CuSO 4 (aq) ZnSO 4 (aq) Cu(s) The preceding reactions are examples of other types of reactions (such as combination, combustion, and single-replacement reactions), but they’re all redox reactions. They all involve the transfer of electrons from one chemical species to another. Redox reactions are involved in combustion, rusting, photosynthesis, respiration, batteries, and more. 4.1.5.9 Organic Reaction Organic reactions occur between organic molecules (molecules containing carbon and hydrogen). Since there is a virtually unlimited number of organic molecules, the scope of organic reactions is very large. However, many of the characteristics of organic molecules are determined by functional groups small groups of atoms that react in predictable ways. Chemkin Chemkin is a proprietary software tool for solving complex chemical kinetics problems. It is used worldwide in the combustion, chemical processing, microelectronics and automotive industries, and also in atmospheric science. It was originally developed at Sandia National Laboratories and is now developed by a US company, Reaction Design. CHEMKIN solves thousands of reaction combinations to develop a comprehensive understanding of a particular process, which might involve multiple chemical species, concentration ranges, and gas temperatures. Chemical kinetics simulation software allows for a more time-efficient investigation of a potential new process compared to direct laboratory investigation. It is a plug-in chemistry solver that can be linked to other computational software packages, such as ANSYS’ FLUENT CFD software, to add accuracy, speed and stability to

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calculations using finite-rate, multi-step reaction kinetics. Detailed reaction kinetics are frequently required to represent combustion and materials process reactions accurately. Such accuracy typically means use of multi-step reaction kinetics. However, using detailed kinetics often causes problems for other software simulation packages, because of computational stiffness, where small changes in one variable may cause large changes in another. In the past, stiff chemistry simulations were characterized by long computational times and unstable solutions that often did not converge. CHEMKIN-CFD tackles these challenges through advanced and more accurate numerical methods, allowing you to incorporate more detailed kinetics descriptions into your simulations.

CFD Applied To Chemical Reaction Engineering

Chemical reaction engineering specifically deals with chemical reactors. It is also referred to as reaction and/or reactor engineering90. The primary goal of chemical reaction engineering is to optimize transport processes, such as heat transfer, mass transfer and mixing, to improve product yield/conversion and reactor operation safety. Simply said, it helps maximize yield while minimizing costs, such as those associated with feedstock, energy input, heat removal or cooling, stirring or agitation, pumping to increase Phase Interactions pressure and/or frictional pressure loss. From the standpoint of CFD, reaction engineering is the application of transport phenomenon and Reaction chemical kinetics knowledge to Flow Mode Types industrial systems. Transport phenomenon is key and defines Reaction which property is important. Design Chemical kinetics, or the study of rates of chemical processes, is founded on the experimental study of how different conditions influence the speed of a chemical Flow reaction and its mechanism and Transport Distribution transition states; and on the development of mathematical models representing the reaction’s characteristics. Figure 36 Essential Aspects of Chemical Reactor Design Essential aspects of a chemical reactor design are shown in Figure 36. Reactor Design and CFD Designing a reactor involves a number of key considerations, including: Reactants/products phase or state comprising solid, gas, liquid or aqueous/dissolved in water Reaction type, including single, multiple and parallel series or polymerization 90

White paper issued by Siemens PLM Software.

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Catalyst identification, which may take in flow distribution and mixing Species transport Operation mode, such as batch, semi-batch or continuous Considering underlying transport processes, such as fluid flow, heat transfer, mass transfer and reactions, is beneficial as CFD simulation can add substantial value to these characteristics. There are many steps in a reactor design process that enable you to take the discovery of a new chemical with novel properties from concept to commercial production, including:

Conceptualizing how you analyze new chemistry Lab scale to accommodate analysis of kinetics, catalysis, thermodynamics, material properties and toxicity Reactor selection involving the analysis of flow regime, heat release, residence time distribution (RTD), liquid hourly space velocity (LHSV) as well as gas hourly space velocity (GHSV) Engineering with idealized models for analysis of plug flow or continuously-stirred tank reactors (CSTR), volume sizing and overall heat transfer Preliminary design of vessel configuration, internals, baffles and coils Scale up simulations, including pilot scale design, scale up parameters and design space exploration Final design to encompass the extrapolation from scale up rules, flow modeling, safety, risk and runaway analysis and dynamic modeling

You can use an idealized reactor model that has been sized to lab or bench scale to initiate the engineering process and predict key reactor behavior variables, including reaction parameters, material properties, toxicity, ideal operating conditions, optimal catalysts and preliminary reactor dimensions. A well-known traditional approach adopted in chemical engineering to circumvent the intrinsic difficulties in obtaining the “complete velocity distribution map” is the characterization of non-ideal flow patterns by means of residence time distribution (RTD) experiments where typically the response of a piece of process equipment is measured due to a disturbance of the inlet concentration of a tracer. Applications of CFD may be divided into broad categories; those involving single-phase systems and those involving multiphase systems. Within single-phase systems, a further distinction can be made between systems involving laminar flows, turbulent flows, flows with complex rheology, and fast chemical reactions. In many processes encountered in industrial practice, multiphase flows are encountered and, it can be stated that, because of the inherent complexity of such flows, general applicable models and related CFD codes are nonexistent. Gas-Phase Reacting Flow Models The most basic types of flow models are for gas phase reacting flows in which different species (feedstocks) enter the reactor non-premixed, completely mixed or partly premixed. Some models are regarded as tabulated chemistry models to reduce computation times, while others make use of detailed chemistry. A simple glass furnace is a representative example in which air and fuel that is not premixed enters the domain. The simplest model is an eddy break-up combustion model which highlights the flame zone and the region of NOx formation within the overall combustion chamber. NOx is a generic term for mono-nitrogen oxides NO (nitric-oxide) and NO2 (nitrogen dioxide). Liquid-Phase Reactions Liquid-phase reactions significantly differ from gas-phase reactions because the liquid’s rate of diffusion is much lower than its viscosity; therefore, liquid-phase reactions can be strongly influenced

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by scalar gradients. The most important difference in this scenario is the phenomenon of micromixing, or mixing at the molecular scale. In Figure 37, the reactor’s physical configuration will determine whether the reaction will preferentially form product S or, alternatively, product R. In this schematic, for example, the location at which species B is added, either at the vessel’s top or close to the impeller, can significantly alter the results.

Figure 37

Physical reactor configurations essential in predicting liquid phase micro-mixing reactions

Basic Equations of Chemically Reacting Flows in CFD

Basic equations for chemically reacting flows are analyzed conferring to [Sheng-Tao Yu]91. Chemical reactions are modeled by finite rate chemistry, and species equations are incorporated with the momentum and energy equations to model reactive flows. Three physical processes are involved in a reacting flow:

1. Fluid Dynamics, 2. Thermodynamics, 3. Chemical Reactions. The fluid dynamics process is the balance between the temporal evolution and the spatial convection of the flow properties due to conservation of mass, momentum, and energy. The thermodynamics of the reactive fluid include microscopic heat transfer between gas molecules, work done by pressure, and the associated volume change. And, chemical reactions determine the generation/destruction of chemical species under the constraint of mass conservation. Each of the above processes could be either evolving or in equilibrium. For the evolving condition, each above process has its own space and time scales, and they are very different from that of other processes. Such differences in space

91 Sheng-Tao Yu,”

Basic Equations of Chemically Reacting Flows For Computational Fluid Dynamics”, Wayne State University, Detroit, Michigan.

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and time scales, on one hand, could allow simplification in the theoretical model. On the other hand, they could be the source of numerical difficulties. Flow and Reaction Interactions The interactions between chemical reactions and fluid dynamics is best described by the Damköhler number (Da). The Damköhler numbers are dimensionless numbers used in chemical engineering to relate the chemical reaction timescale (reaction rate) to the transport phenomena rate occurring in a system as

Da

Typical time required for Mixing Typical time requires for Chemical Reaction

Eq. 4.11

Da >> 1 - As the Damkohler number approaches infinity, the chemical reactions are much faster compared to fluid dynamics. Thus, the chemical composition can be treated as a state variable governed by the chemical equilibrium theory, i.e., free energy minimization. In this case, the above thermal equilibrium needs be modified to include the effect of free energy minimization. Thus the fluid dynamics becomes the only evolving process. The discussion of this condition, however, is out of the scope of the present work. Da