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January 2008

Modeling of dispersed phase by Lagrangianapproach in Fluent

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Modeling of Dispersed Phase by Lagrangian Approach in Fluent

Theory and simulation of dispersed-phase multiphase flows, Autumn 2007 – Spring 2008

11 March 2008Kari Myöhänenkari.myohanen@lut.fi

Presentation Outline

• Introduction• Modeling options and limitations in Fluent• Model theory• Solution strategies• Example calculation

Introduction

• The discrete phase model (DPM) in Fluent follows the Euler-Lagrange approach.• The fluid phase (gas or liquid, “continuous phase”) is treated as a continuum by

solving the time-averaged Navier-Stokes equations (Eulerian reference frame).• The dispersed phase is solved by tracking a number of particles through the

calculated flow field of continuous phase (Lagrangian reference frame). • The particles may be taken to represent solid particles in gas or liquid, liquid droplets

in gas or bubbles in liquid. • The dispersed phase can exchange momentum, mass and energy with the fluid

phase.

Discrete Phase Modeling Options in Fluent

Fluent provides the following discrete phase modeling options:• Calculation of the particle trajectories using a Lagrangian formulation that includes:

– Discrete phase inertia– Hydrodynamic drag– Force of gravity– Other forces

• pressure gradient, thermophoretic, rotating reference frame, brownian motion, Saffman lift, and user defined forces

• Steady state and transient flows.• Turbulent dispersion of particles.• Heating and cooling of the discrete phase.• Vaporization and boiling of liquid droplets.• Combusting particles, including volatile evolution and char combustion to simulate

coal combustion.• Optional two-way coupling of the continuous phase flow and the discrete phase.• Wall film modeling.• Spray model (droplet collision and breakup).

Limitations in Fluent

• Particle-particle interactions are neglected.– Assumption: dispersed phase is sufficiently dilute.– Fluent manual provides a hand rule ”volume fraction usually less than 10-12%”.– In general, this limit is far too high and does not fulfill the requirement of ratio

between the momentum response time and collisional time V/ C < 1(see lecture notes, session 1).

– The DPM model is however often used for dense dispersed flows as well. Care should be taken when interpreting the results.

• The steady state DPM model cannot be applied for continuous suspension of particles– The particle streams should have well-defined entrance and exit conditions.– For cases, in which the particles are suspended indefinetely in the continuum (e.g.

stirred tanks), the unsteady DPM modeling should be used instead.• If the dispersed phase model is used with Eulerian-Eulerian multiphase model the

coupling is defined with the primary phase only.• Several restrictions when using DPM model with other Fluent models

– Limitations with parallel computing, streamwise periodic flows, combustion models, sliding meshes, etc. See Fluent manual for details.

Regimes of Dispersed Two-Phase Flows

Sommerfeld (2000), based on Elghobashi (1994).

fluid particle fluid particle fluid particle particle

Momentum Equation

The force balance of particle in Lagrangian reference frame defines the movement of the particles.

The momentum equation for i-direction:

Drag Gravity Additional accelerationdue to other forces(force/unit particle mass)

Acceleration

Drag Coefficient

For smooth spherical particles, Fluent uses equation by Morsi and Alexander (1972):

The constants a1, a2 and a3 are determined for different ranges of Re:

For nonspherical particles, the equation by Haider and Levenspiel (1989) is used:

Shape factorSurface area of sphere with same volumeActual surface area

Comparison of Drag Coefficient Equations

• The discrete phase and the continuous phase can be coupled in a number of ways. In Fluent, the one-way or two-way coupling are possible to model.

• One-way coupling– The continuous phase affects the discrete phase, but there is no reverse effect.– In Fluent, this is referred as “uncoupled approach”.– The discrete phase is solved once after the continuous phase flow has been

solved.• Two-way coupling

– Both phases affect each other (exchange of momentum, mass and energy).– In Fluent, this is referred as “coupled approach”.– The continuous phase flow field is impacted by the discrete phase and the

calculations of the continuous phase and dispersed phase equations are alternated until the solution is converged (hopefully).

• Three-way coupling– Particle disturbance of the fluid locally affects

another particle’s motion, e.g. drafting of a trailing particle.• Four-way coupling

– Particle collisions affect motion of individual particles.

Coupling

Two-Way Coupling in Fluent

Momentum exchange

Drag Other interaction forces

Heat exchange(without chemical reactions) Vaporization

and pyrolysisSensible heat

Mass exchange

Particle Types and Laws in Fluent

Particle type Description Requirements Laws activated

Inert inert/heating or cooling Available for all models 1, 6

Droplet heating/evaporation/boiling

Energy equation.Minimum two chemical species or the nonpremixed or partially premixed combustion model.Gas phase density by ideal law.

1, 2, 3, 6

Combusting heating; evolution of volatiles/swelling; heterogeneous surface reaction

Energy equation.Minimum three chemical species or the nonpremixed combustion model.Gas phase density by ideal law.

1, 4, 5, 6

Multicomponent multicomponent droplets/particles

Energy equation.Min. two chemical species.Use volume weighted mixing law to define define particle mixture density.

7

Law 1: Particle temperature below vaporization temperature.Law 2: Droplet vaporization.Law 3: Droplet boiling.Law 4: Devolatilization of combusting particle.Law 5: Surface combustion.Law 6: Volatile fraction of the particle consumed.Law 7: Multicomponent particle definition

Example of Laws Applied for a Drying Droplet

Tem

pera

ture

Particle time

Law 1:Inert heatingbefore vaporization

Law 2:Vaporization

Law 3:Boiling

Tbp

Tvap

Tinjection

Law 6:Volatile fractionconsumed

Different energy and mass transfer equations are applied during different laws.

Mass transfer (molar flux of vapor)

Mass and Energy Transfer of Drying Droplet

Law 1:Inert heating before vaporizationLaw 6: Volatile fraction consumed

Law 2: Vaporization

Law 3: Boiling

Heat transferConvection Radiation

Vapor concentration at droplet surface / bulk gas

Vapor pressuremust be correctly defined

Diffusion coefficientgiven by user

Particle temperature is constant.Energy required for vaporization appears as energy sink for gas phase

Evaporation

Mass transfer without radiationwith radiation

Heat transfer

Particle-Wall Interaction

Different particle boundary conditions can be defined for walls, inlets and outlets:

volatile fractionflashes to vapor

Escape Reflect Trap

For particle reflection, a restitution coefficient e is specified:

Normal component:

Tangential component:

Turbulent Dispersion of Particles

In Fluent, the dispersion of particles due to continuous phase turbulence can be modeled by• a stochastic tracking model (random walk model, eddy interaction model), or • a particle cloud model.

In the random walk model, the instantaneous continuous phase velocity is formedof mean velocity and fluctuating component:

• The fluctuating component varies randomly during a particle track.• Each particle injection is tracked repeatedly in order to generate a statistically

meaningful sampling.

The cloud model uses statistical methods to trace the turbulent dispersion of particlesabout a mean trajectory• Mean trajectory is calculated from the ensemble average of the equations of motion

for the particles represented in the cloud.• Distribution of particles inside the cloud is represented by a Gaussian PDF.

Eddy Interaction Model

The stochastic tracking model in Fluent is based on eddy interaction model. The discrete particle is assumed to interact with a succession of eddies. Each eddy is characterized by• a Gaussian distributed random velocity fluctuation u’i

• a time scale (life time of eddy) e• a length scale (size of eddy) LeDuring interaction, the fluctuating velocity is kept constant. The interaction lasts until time exceeds the eddy lifetime or the eddy crossing time. Literature presents several theories for determining the above values (see Graham and James (1996)). The following presents the equations used in Fluent with k- turbulence model.

Fluid Lagrangian integral time Coefficient CL defined by user. Default value CL = 0.15.

Characteristic life time of eddy

Eddy length scale(based on Karema(2008))

Notice: in literature, the length scale and life time are often linked:In Fluent, this seems to be:

23 /

Lek

CL

kL

e

e

21 3

2 kL

e

e

Eddy crossing time Velocity response time 18

2pp d

Fluctuating velocity

For k- turbulence model:

= Gaussian distributed random number (standard normal distribution)

or alternatively random variation: rlnTLe

r = uniform random number [0...1]. Notice: Le Trln 1

Injection Setup

Particle injections can be defined by various methods:

• Single: a particle stream is injected from a single point.• Group: particle streams are injected along a line.• Cone: streams are injected in a hollow conical pattern.• Solid cone.• Surface: particle streams are injected from a surface

(one stream from each cell face).• Atomizer: streams are injected by using various predefined

atomizer models.• File: injection locations and initial conditions are defined

in an external file.

For each injection, the following data are defined:• Particle type (inert, droplet, combusting, multicomponent)• Material (from database)• Initial conditions (particle size, velocity, etc.)• Destination species for reacting particles.• Evaporating material for combusting particles.

DPM Concentration

Fluent can report a ”DPM concentration” in a coupled calculation. This is a totalconcentration of the discrete phase in a continuous cell.

The mass flow of a particle track is determined based on particle mass andmass flow at the particle injection and particle mass at current location.The particle mass can change due to evaporation and other phase changes.

The discrete phase concentration inside a cell can be determined from theresidence time and mass flow.

Inside a cell, the particle stream is tracked with n particle time steps. Theresidence time of one particle track is the sum of these time steps.

The total concentration is summed over all particle tracks.

The particle-particle interaction is neglected, thus when multipleparticle tracks cross the cell, the calculated concentration can exceed the bulk density of solids or even solid density (volume fraction of solids above 1). These results are notphysically sensible but they can show areas, where the particleloading is high and the assumption of dilute flow is not valid.

t0

tNmp

Solution Strategies: Particle Tracking

• The particle tracks are calculated in steps. The ”step length factor” determines approximately the number of steps per fluid cell. The default value is 5, but it should preferably be higher: 10 – 20.

• Increasing the step length factor (i.e. decreasing the step length) can improve stability of heat and mass exchange (e.g. when calculating vaporization).

• The ”max. number of steps” limits the number of calculated time steps. This should be large enough so that the particles can travel from entrance to exit.

• If particles remain suspended in the model (tracking incomplete), then steady state solution is questionable and transient tracking should be used instead. The transient calculations in Fluent can be performed in a number of ways and combinations. This presentation is focused on steady state calculation.

Solution Strategies: Two-Way Coupling

• The solution of the continuous field without coupling is usually the starting point. In most cases, the continuous flow does not have to be fully converged before the coupling is started, because the particle tracks will have a large effect on the continuous flow.

• In a coupled calculation, additional source terms appear in discretized flow equations of continuous phase. During particle tracking, each particle is seeing a ”fresh” cell and makes no notice of particles already visited and marked the cell with their source terms. This leads to overprediction of the source terms and bad convergence behaviour with evaporation, combustion and radiation.

– Use solution limits to limit the temperature in the domain.• Increasing the number of trajectories (especially with random walk model) will

smooth the particle source terms, which should help convergence.• The discrete phase source terms can be under-relaxed (e.g. 0.5). The flow

equations may need to be under-relaxed as well (energy and species).• The number of continuous phase calculations between the trajectory

calculations can either be small (< 3) or high (>15). In the first choice, the dispersed and continuous flow are closer coupled and the solution of both should slowly convergence. In the second choice, the flows are decoupled and the solution of continuous field remains ”better converged” and the calculation is more stable. In the latter case, the continuous phase may appear to be converged, but the discrete phase is not.

• If the dispersed phase is not dilute, then convergence is very difficult to achieve in coupled calculations.

Calculatecontinuous

phase

Calculateparticletracks

Updatesourceterms

Modeling Example

The model geometry is shown below. Hot air flows in a 200 mm diameter duct.Wet limestone particles are injected from the top of the duct(inlet d = 50 mm) at location 500 mm before a 90° bend.

Air inlet: 10 m/s, 270°C, D= 0.2 mParticle inlet: 0.1 kg/s, 0.1 m/s, dp=200 µm, p=2700 kg/m3, H2O=30%

Average volume fraction ofsolids in the duct:

dilute, two-way coupling(but only as average)

Mesh

Gas Properties

Solid Properties (Limestone)

Model Parameters

Solution of Continuous Phase

• The continuous phase was first solved without the particles.• The convergence was good.

Uncoupled Mean Particle Tracks

• The mean particle tracks were solved without two-way coupling.• The particle tracks are thus calculated only once after the continuous phase was solved.• The following images present particle tracks colored by mass, which indicates evaporation.

Initial mass 1.13E-8 kgFully evaporated 7.92E-9 kg

Uncoupled Turbulent Tracks

• Random walk model with 50 stochastic tracks (total 2400) was used with default CL = 0.15.• Uncoupled solution, ie. one-way coupled calculation of dispersed phase.• Turbulence effects are fairly small, but can be noticed in the track images.

Initial mass 1.13E-8 kgFully evaporated 7.92E-9 kg

Solution of Coupled Calculation

• Two-way coupled solution did not converge well.• Different step length factors, under-relaxation parameters and number of continuous phase

iterations were tried.• In the final calculations, the step length factor was 20 and the number of continuous phase

iterations between dispersed phase calculations was 20. The residuals were indicating poor convergence.

Coupled Particle Tracks

• The particle tracks show that some of the particle streams circulate for long times before reaching the outlet.

• The solution of flow is much different from uncoupled solution.• The images do not show all particle tracks.

Initial mass 1.13E-8 kgFully evaporated 7.92E-9 kg

Effect on Continuous Flow Field

• In the coupled calculation, the particle tracks affect the continuous phase flow.• In this case, the effect is considerable.

DPM Concentration

• The DPM concentration shows the total concentration of dispersed phase.• Results indicate that in the bend, the dispersed phase is not dilute ( max = 0.094).• Reaching a converged solution in this case would be impossible.

» The results should be utilized with caution.

Visualization of Results

• Different process variables can be easily visualized: pressure, velocities, temperature, concentration of species, turbulence variables, ...

Summary

• The DPM model in Fluent can be used for studying one-way or two-way coupled dilute dispersed flows, including effects of turbulence.

• The basic model is easy to use and physics are clear and simple.• The limitations of the DPM model should be carefully considered when

analyzing the results.• The model neglects particle-particle interaction, thus it is valid for dilute

dispersed phase only.• The one-way coupling is valid for very dilute flow only. The two-way coupled

solution can be much different from the one-way coupled solution. • The average flow can be dilute, but it can contain regions, in which the

dispersed phase is dense. In these regions, the model results are false. Moreover, the convergence is poor, if the dispersed phase is dense and the momentum, mass and energy exchange to continuous phase is strong.

• Despite the limitations, the DPM model can be (and is) successfully used for modeling various applications.

References

• Bakker, A. (2006). Lecture notes, Computational Fluid Dynamics, Dartmouth College. http://www.bakker.org/dartmouth06/engs150/.

• Elghobashi, S. (1994). On predicting particle-laden turbulent flows, Appl. Sci. Res. 52, pp. 309–329.

• Fluent 6.3 Documentation (2008).• Fluent Training Material (2008). http://www.fluentusers.com.• Graham D. I. and James P.W. (1996). Turbulent dispersion of particles using eddy interaction

models. Int. J. Multiphase Flow, 22-1, pp 157-175.• Haider, A. and Levenspiel, O. (1989). Drag Coefficient and Terminal Velocity of Spherical and

Nonspherical Particles.Powder Technology, 58, pp. 63–70.• Jalali, P. (2007). Lecture notes, Theory and simulation of dispersed-phase multiphase flows,

Lappeenranta University of Technology. http://www2.et.lut.fi/ttd/Dispersed2007/Dispersed.htm• Karema, H. (2008). Discussions with Hannu Karema (Process Flow), January 2008.• Loth, E. (2008). Computational Fluid Dynamics of Bubbles, Drops and Particles (draft).

http://www.ae.uiuc.edu/~loth/CUP/Loth.htm• Morsi, S. and Alexander A. (1972), An investigation of particle trajectories in two-phase flow

systems, Journal of Fluid Mechanics 55, pp. 193–208.• Sommerfeld, M. (2000). Theoretical and Experimental Modelling of Particulate Flows. Lecture

Series 2000-06, von Karman Institute for Fluid Dynamics. http://www-mvt.iw.uni-halle.de/download.php?id=571340,326,2