Presentation on theme: "Multiphase and Reactive Flow Modelling"— Presentation transcript:
1 Multiphase and Reactive Flow Modelling BMEGEÁTMW07K. G. SzabóDept. of Hydraulic and Water Management Engineering,Faculty of Civil EngineeringSpring semester, 2012
2 Introduction Physical phenomena Major concepts, definitions, notions and terminologyEquilibrium vs. non-equilibrium modelsLagrangean vs. Eulerian descriptionDimensionless numbersModelling strategiesUpdate me!
3 Basic notions and terminology Ordinary phases:SolidLiquidGaseouspreserve volumeCondensed phasesexpandspreserves shapeFluid phasesdeformThere also exist extraordinary phases, like plastics and similarly complex materialsThe property of fluidity serves in the definition of fluids
4 Properties and models of solids Properties of solids:Mass (inertia), position, translationExtension (density, volume), rotation, inertial momentumElastic deformations (small, reversible and linear), deformation and stress fieldsInelastic deformations (large, irreversible and nonlinear), dislocations, failure etc.Modelled features:MechanicsStatics: mechanical equilibrium is necessaryDynamics: governed by deviation from mechanical equilibriumThermodynamics of solidsMass point modelRigid body modelThe simplest continuum modelEven more complex models
5 Models and properties of fluids Key properties of fluids:Large, irreversible deformationsDensity, pressure, viscosity, thermal conductivity, etc. (are these properties or states?)Features to be modelled:StaticsHydrostatics: definition of fluid (inhomogeneous [pressure and density])Thermostatics: thermal equilibrium (homogenous state)DynamicsMechanical dynamics: motion governed by deviation from equilibrium of forcesThermodynamics of fluids:Deviation from global thermodynamic equilibrium often governs processes multiphase, multi-component systemsLocal thermodynamic equilibrium is (almost always) maintainedOnly continuum models are appropriate!
6 Modelling Simple Fluids Inside the fluid:Transport equationsMass, momentum and energy balances5 PDE’s forConstitutive equationsAlgebraic equations forBoundary conditionsOn explicitly or implicitly specified surfacesInitial conditionsPrimary (direct)field variables(p,T) represaentattionSecondary (indirect) field variables
7 Thermodynamical representations Representation (independent variables)TD potentialenthropy and volume (s,1/ρ)internal energytemperature and volume (T,1/ρ)free energyenthropy and pressure (s,p)enthalpytemperature and pressure (T,p)free enthalpyAll of these are equivalent: can be transformed to each other by appropriate formulæUse the one which is most practicable: e.g., (s,p) in acoustics: s = const ρ(s,p) ρ(p). We prefer (T,p)Note
8 Some models of fluids Stoksean fluid compressible In both of these, the heat transport problem can be solved separately (one-way coupling):Mutually coupled thermo-hydraulic equations:Non-Newtonian behaviour etc.compressible(or barotropic) fluidfluid dynamical equationsheat transport equation (1 PDE)general simple fluidfluid dynamical equationsheat transport equationmodels for complex fluids
10 Typical phase diagrams of a pure material: (crystal structures) may existSeveral solid phasesTODO: 1/rho=spec volume, viscosity, chemical potential. Write equations of equilibrium.In equilibrium 1, 2 or 3 phases can exist togetherComplete mechanical and thermal equilibrium
11 Conditions of local phase equilibrium in a contact point in case of a pure material 2 phases:T(1)=T(2)=:Tp(1)=p(2)=:pμ(1)(T,p)= μ(2)(T,p)Locus of solution: a line Ts(p) or ps(T), the saturation temperature or pressure (e.g. ‘boiling point´).3 phases:T(1)=T(2) =T(3)=:Tp(1)=p(2)=p(3) =:pμ(1)(T,p)= μ(2)(T,p) = μ(3)(T,p)Locus of solution: a point (Tt,pt), the triple point.This should be recast to the multicomponent case
12 Multiple componentsAlmost all systems have more than 1 chemical componentsPhases are typically multi-component mixturesConcentration(s): measure(s) of compositionThere are lot of practical concentrations in use, e.g.Mass fraction (we prefer this!)Volume fraction (good only if volume is conserved upon mixing!)Concentration fields appear as new primary field variables in the equation: One of them (usually that of the solvent) is redundant, not used.TODO: concentrations
13 Notations to be used (or at least attempted) Phase index (upper):(p) or(s), (l), (g), (v), (f) for solid, liquid, gas, fluid, vapourComponent index (lower): kCoordinate index (lower): i, j or tExamples:Partial differentiation:Note
14 Material properties in multicomponent mixtures One needs constitutional equations for each phaseThese algebraic equations depend also on the concentrationsFor each phase (p) one needs to know:the equation of statethe viscositythe thermal conductivitythe diffusion coefficients
15 Conditions of local phase equilibrium in a contact point in case of multiple components Suppose N phases and K components:Thermal and mechanical equilibrium on the interfaces:T(1)=T(2) =T(3)=:Tp(1)=p(2)=p(3)=:pMass balance for each component among all phases:(N-1)K equations for 2+N(K-1) unknownsThis should be recast to the multicomponent case
16 Phase equilibrium in a multi-component mixture Gibbs’ Rule of Phases, in equilibrium:If there is no (global) TD equilibrium: additional phases may also existin transient metastable state orspatially separated, in distant pointsTD limit on the # of phases
17 MiscibilityThe number of phases in a given system is also influenced by the miscibility of the components:Gases always mix →Typically there is at most 1 contiguous gas phaseLiquids maybe miscible or immiscible →Liquids may separate into more than 1 phases(e.g. polar water + apolar oil)Surface tension (gas-liquid interface)Interfacial tension (liquid-liquid interface)(In general: Interfacial tension on fluid-liquid interfaces)Solids typically remain granular
18 Topology of phases and interfaces A phase may beContiguous(more than 1 contiguous phases can coexist)Dispersed:solid particles, droplets or bubblesof small sizeusually surrounded by a contiguous phaseCompoundInterfaces are2D interface surfaces separating 2 phasesgas-liquid: surfaceliquid-liquid: interfacesolid-fluid: wall1D contact lines separating 3 phases and 3 interfaces0D contact points with 4 phases, 6 interfaces and 4 contact linesTopological limit on the # of phases (always local)
19 Special Features to Be Modelled Multiple components →chemical reactionsmolecular diffusion of constituentsMultiple phases → inter-phase processesmomentum transport,mass transport andenergy (heat) transferacross interfaces.(Local deviation from total TD equilibrium is typical)
20 Class 3 outline Balance equations Mass balance — equation of continuityComponent balanceAdvectionMolecular diffusionChemical reactions
21 extensive quantity: Fdensity: φ=F/V=ρ∙fspecific value f=F/mmolar value f=F/nmolecular value F*=F/N
22 Differential forms of balance equations Conservation of F:equations for the densitygeneralonly convective fluxequation for the specific valueThese forms describe passive advection of F
24 Incomplete without class notes !Diffusion — continuedfurther diffusion modelsthe advection—diffusion equationsChemical reactionsthe advection—diffusion—reaction equationsstochiometric equationreaction heatchemical equilibriumreaction kineticsfrozen and fast reactions
25 Further diffusion models Thermodiffusion and/or barodiffusionOccur(s) athigh concentrationshigh T and/or p gradientsFor a binary mixture:coefficient of thermodiffusioncoefficient of barodiffusionAnalogous cross effects appear in the heat conduction equation
26 Further diffusion models Nonlinear diffusion modelCross effect among species’ diffusionValid athigh concentrationsmore than 2 componentslow T and/or p gradients(For a binary mixture it falls back to Fick’s law.)Make D caligraphic. C.f. flug 8.9
27 The advection–diffusion equations local rate of changeadvective fluxdiffusive fluxThe concentrations are conserved but not passive quantities
28 The advection–diffusion–reaction equations local rate of changeadvective fluxdiffusive fluxreactive source termsThe concentrations are not conserved quantities
29 Incomplete without class notes !Mathematical description of interfacesimplicit descriptionparametric description (homework)normal, tangent, curvatureinterface motionTransport through interfacescontinuity and jump conditionsmass balanceheat balanceforce balance
30 Interfaces and their motion Description of interface surfaces:parametricallyby implicit function(the explicit description is the common case of the two)Moving phase interface: (only!) the normal velocity component makes senseNew primary(?) field variablesIncomplete without class notes!
31 Description of an interface by an implicit function
32 Equation of motion of an interface given by implicit function Equation of interfacePath of the point that remains on the interface (but not necessarily a fluid particle)DifferentiateFor any such point the normal velocity component must be the samePropagation speed and velocity of the interfaceOnly the normal component makes sense
33 Parametric description of an interface and its motion Homework: Try to set it up analogously
34 Mass balance through an interface Steps of the derivation:describe in a reference frame that moves with the interface (e.g. keep the position of the origin on the interface)velocities inside the phases in the moving framemass fluxes in the moving frameflatten the control volume onto the interfaceIncomplete without class notes!
35 The kinematical boundary conditions Incomplete without class notes!This condition does not follow from mass conservationThe tangential eqs do not follow from the principle of conservation of mass!!continuity of velocityconservation of interface
36 Diffusion through an interface Mass flux of component k in the co-moving reference frame:Case of conservation of component mass:on a pure interface (no surface phase, no surfactants)without surface reactions (not a reaction front)The component flux through the interface:The tangential eqs do not follow from the principle of conservation of mass!!
38 Momentum balance through an interface Effects due tosurface tensionsurface viscositysurface compressibilitymass transfer
39 Incomplete without class notes Surface tensionThe origin and interpretation of surface tensionIncomplete without class notes!
40 Dynamical boundary conditions with surface/interfacial tension Fluids in restnormal component:Moving fluids without interfacial mass transfertangential components:The viscous stress tensor:Modifies the thermodynamic phase equilibrium conditionsIncomplete without class notes!
41 The heat conduction equation The equationFourier’s formula(thermodiffusion not included!)Volumetric heat sources:viscous dissipationdirect heatingchemical reaction heatBoundary conditionsThermal equilibriumHeat flux:continuity (simplest)latent heat (phase transition of pure substance)Even more complex cases:chemical component diffusionchemical reactions on surfacedirect heating of surfaceJump conditions
42 Boundary conditions on moving interfaces Physical balance equations imply conditions on the interface elements:continuity conditionsjump conditionsOther conditions prescribed to obtain a well set mathematical modelWith and without mass transferIncomplete without class notes!
43 Approaches of fine models Phase-by-phaseSeparate sets of governing equations for each phaseEach phase is treated as a simple fluidDescribing/capturing moving interfacesPrescribing jump conditions at the interfacesOne-fluidA single set of governing equation for all phasesComplicated constitutional equationsDescribing/capturing moving interfacesJumps on the interfaces are described as singular source terms in the governing equations1-fluid = embedded interface
44 Phase-by-phase mathematical models A separate phase domain for each phaseA separate set of balance equations for each phase domain, for the primary field variables introduced for the single phase problems, supplemented by the constitutional relations describing the material properties of the given phaseThe sub-model for the motion of phase domains and phase boundaries (further primary model variables)Prescribing the moving boundary conditions: coupling among the field variables of the neighbouring phase domains and the interface variablesII/13Bp6f1
45 The one-fluid mathematical model A single fluid domainCharacteristic function for each phaseMaterial properties expressed by the properties of individual phases and the characteristic functionsA single set of balance equations for the primary field variables introduced for the single phase problems, supplemented by discrete source terms describing interface processesThe sub-model for the motion of phase domains and phase boundaries (further primary model variables)II/13Bp6f1
46 The sub-models of phase motion (interface sub-models) The choice of the mathematical level sub-model is influenced by the available effective numerical methods.
48 Front tracking methods on a fixed grid by connected marker points(Suits the parametric mathematical description)In 3D: triangulated unstructured grid represents the surfaceTasks to solve:Advecting the frontInteraction with the grid (efficient data structures are needed!)Merging and splitting (hard!)II/13Bp11f7
49 MAC (Marker-And-Cell method) An interface reconstruction — front capturing — model (the primary variable is the characteristic function of the phase domain, the interface is reconstructed from this information)The naive numerical implementation of the mathematical transport equation :1st (later 2nd) order upwind differential schemeErrors (characteristic to other methods as well!):numerical diffusion in the 1st ordernumerical oscillation in higher ordersIncomplete without class notes!Due to the discontinuities of the function
50 Incomplete without class notes MACIncomplete without class notes!
51 VOF (Volume-Of-Fluid method) 1D version (1st order explicit in time):Gives a sharp interface, conserves massRequires special algorithmic handlingThe scheme of evolution:
52 VOF in 2D and 3D SLIC: Simple Line Interface Construction PLIC: Piecewise Linear Interface ConstructionHirt & Nichols
53 Numerical steps of VOF Interface reconstruction within the cell determine nseveral schemesposition straight interfaceInterface advectionseveral schemes exist, goals:conserve mass exactlyavoid diffusionavoid oscillationsCompute the surface tension force in the Navier–Stokes eqs.several schemes
54 Implementation of VOF in Any number of phases can be presentThe transport equation for is adapted to allowvariable density of phasesmass transport between phasesContact angle model at solid walls is coupledSpecial (`open channel´) boundary conditions for VOFSurface tension is implemented as a continuous surface force in the momentum equationFor the flux calculations ANSYS FLUENT can use one of the following schemes:Geometric Reconstruction: PLIC, adapted to non-structured gridsDonor-Acceptor: Hirt & Nichols, for quadrilateral or hexahedral grid onlyCompressive Interface Capturing Scheme for Arbitrary Meshes (CICSAM): a general purpose sheme for sharp jumps (e.g. high ratios of viscosities) for arbitrary meshesAny of its standard schemes (probably diffuse and oscillate)References in FLUENT Users Guide: [Youngs ], , [Ubbink ]
55 The level set method [hu: nívófelület-módszer] the interface is implicitF is continuousstandard advection schemes work finethe curvature can be obtained easilythe effect of surface tension within a cell can be computed
56 The level set methodIf then the computational demand can be substantially decreased
57 Signed distance function as an implicit level-set function What kind of function is it? Signed distance from the interface!Alas, is not conserved.Generating F: τ is pseudo-time (t is not changed)Apply alternatively!Unfortunately, mass is not conserved in the numeric implementation.A better numeric scheme
58 Only first order accurate in h Numerical implementation of the interfacial source terms in the transport equationsMissing function graph!With ε = 1.5h, the interface forces are smeared out to a three-cell thick bandOnly first order accurate in h
59 For example, the normal jump condition due to surface tension can be expressed as an embedded singular source term in the Navier–Stokes equation:contribution to a single cell in a finite volume model:Other source terms (latent heat, mass flux) in the transport equations can be treated analogously.C.f. VOF
61 Evaluation criteria for comparison Not only for VOF and Level SetAbility toconserve mass/volume exactlynumerical stabilitykeep interfaces sharp (avoid numerical diffusion and oscillation)Ability and complexity to modelmore than 2 phasesphase transitionscompressible fluid phasesDemands on resourcesnumber of equationsgrid spacinggrid structuretime steppingdifferentiation schemesLimitations of applicabilitygrid typesdifferential schemesaccuracy
62 Recommended booksStanley Osher, Ronald Fedkiw: Level Set Methods and Dynamic Implicit Surfaces Applied Mathematical Sciences, Vol. 153 (Springer, 2003). ISBNDetails on the level set methodGrétar Tryggvason, Ruben Scardovelli, Stéphane Zaleski: Direct Numerical Simulations of Gas–Liquid Multiphase Flows (Cambridge, 2011). ISBNModern solutions in VOF and front trackingISBN:
63 SPH Smoothed Particle Hydrodynamics The other extreme — a meshless method: The fluid is entirely modelled by moving representative fluid particles — fully LagrangianThere are nomesh cellsinterfacesPDEfield variablesEverything is described via ODE’s
66 Liquid vs. liquid-gas simulation AirEntrapped airVacuumVoid bubble
67 Evaluation of SPH Advantages Conceptually easy Best suits problems in which inertia dominates (violent motion, transients, impacts)FSI modellingwith free surface or liquid–gas interfaceInterface develops naturallyComputationally fastEasy to paralleliseCan be adapted to GPU’sDisadvantagesHigh number of particlesHard to achieve incompressibilitySome important boundary conditions are not realised so far