Version 4.3b

Introduction toPar ticle Tracing Module

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Part No. CM022702

I n t r o d u c t i o n t o t h e P a r t i c l e T r a c i n g M o d u l e 19982013 COMSOL

Protected by U.S. Patents 7,519,518; 7,596,474; and 7,623,991. Patents pending.

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Version: May 2013 COMSOL 4.3b

Contents

Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .i

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

The Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

Physics List by Space Dimension and Preset Study Type . . . . . . . . 6

Modelin

The Mo

Compu

Static M | i

g Tools . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

del Library . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

ting Particle Trajectories through a Laminar ixer. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

ii |

Introduction

Particle tracing provides a Lagrangian description of a problem by solving ordinary differential equations using Newtons law of motion. Newtons law of motion requires specification of the particle mass, and all forces acting on the particle. The forces acting on particles can be divided into two categories, those due to external fields and due to interactions between particles. Forces due to external fields are typically computed from a finite element model, using the physics interfaces available in COFor each particof the positionsolved for eachon each particlIf particle-partto the total foruntil the specifTracing Modutrajectories, thparticle motionmovement, ansystems.

The Applicat

CHARGED PACharged particions or small ioforces that affe The electric

potential ornegative chparticles witelectric forc

The magnesignificantlyIntroduction | 1

MSOL Multiphysics.le, an ordinary differential equation is solved for each component vector. This means that three ordinary differential equations are particle in 3D and two in 2D. At each time step, the forces acting e are queried from the external fields at the current particle position. icle interaction forces are included in the model then they are added ce. The particle position is then updated, and the process repeats ied end time for the simulation is reached. Since the Particle le uses a very general formulation for computing particle e particle tracing user interfaces can be used to model charged in electromagnetic fields, large scale planetary and galactic

d particle motion in laminar, turbulent, and multiphase fluid

ions

RTICLES

les in the context of this discussion refers to electrons, individual n clusters in electric and magnetic fields. There are three primary ct such particles: force, which arises either due to a gradient in the electric due to a time-varying magnetic vector potential. Particles with arge move in the opposite direction to the electric field and h positive charge move in the direction of the electric field. The e does work on the particles.tic force, which does no work on the charged particles but can alter their trajectory. The magnetic force often results in

2 | Introduction

banana orbits for charged particles, causing them to orbit around magnetic field lines with a distance proportional to their mass.

Collisional forces, which do work on the particles. This corresponds to the charged particles colliding with a background gas. The higher the background pressure, the more important the collisional forces.

If the number density of charged species is less than around 1013 1/m3, the effect of the particles on the fields can be neglected. This allows the fields to be computed independently from the particle trajectories. The fields are then used to compute the ethat the particland computatiIf the density oinclude the Coparticle-particlrequirements iincluding the Cparticles, solveCharged particstrike surfaces undesirable de

Photomultiplier. A first electrode, secoprocess continues lectric, magnetic, and collisional forces on the particles. The fact e trajectories can be computed in their own study allows efficient onally inexpensive iterative solver to be used.f charged particles is suitably high then it may be necessary to ulomb force which acts between the particles. When e interactions are included in a model the computational ncrease and scale with the number of particles squared. When oulomb force, it is often best to start with a small number of

the model, and then assess whether or not the effect is important.les can acquire significant energy from electric fields, and when they secondary particles may be ejected. This may be desirable or pending on the specific application.

single incident particle enters the modeling domain on the left side. As it strikes the ndary elections are ejected, which are then accelerated to the second electrode. This across all 8 stages leading to an exponential growth in the electron density.

PARTICLES IN FLUIDSMotion of microscopic and macroscopic sized particles is typically dominated by the drag force acting on particles immersed in a fluid. There are two phases in the system: a discrete phase consisting of bubbles, particles, or droplets, and a continuous phase in which the particles are immersed. In order for the particle tracing approach to be valid, the system should be a dilute or dispersed flow. This means that the volume fraction of the discrete phase should be much smaller than the volume fraction of the continuous phase, generally less than 1%. When the volume fraction of the particles is not small, the fluid system is categorized as a dense flow andIt is importantdisplace the flu

Sparse FlowIn a sparse flowvice versa. Thia system, it is ucompute the trthe stationary vand then the p

Comet tail plot of pshow the deviationIntroduction | 3

a different modeling approach is required. to realize that with the particle tracing approach, particles do not id they occupy.

, the continuous phase affects the motion of the particles but not s is often referred to as oneway coupling. When modeling such sually most efficient to solve for the continuous phase first, then ajectories of the dispersed phases. For example, in the figure below elocity field and pressure were first solved using a stationary study, article trajectories were computed in a time-dependent study.

article trajectories through a laminar static mixer (colored). In addition, Poincar maps of particle trajectories from their initial position (blue & red).

4 | Introduction

Dilute FlowIn a dilute flow the continuous phase affects the motion of the particles, and the particle motion in turn disrupts the continuous phase. This is often referred to as twoway coupling. In order to model this effect the continuous phase and disperse phase must be computed simultaneously. Thus, the computational demand is significantly higher when modeling dilute flows than sparse flows.

Dispersed FlowIn addition to the effects mentioned above, particle-particle interactions may also need to be takParticleparticlimitations app Hard sphere

with respectCoulomb fo

The compubecause eveThere is noother partic

If the particto use a veroption is se

Modeling AdvContinuum madvection and the more numratio of the ratmethods cannoIn typically mitemperature, tPclet numbertrajectories, onremoving the ranything from is added to thea Brownian forbe purely diffuthe backgrounPclet numberen into account. This is often referred to as four-way coupling. le interactions may be included in models but the following ly: collisions are not supported. Forces must vary continuously to the distance between particles as in, for example, the rce between charged particles.

tation time scales as the number of particles squared. This is ry particle interacts with every other particle over all distances. way to implement a cut-off scheme, where particles only see les within a certain radius.le-particle interaction law is highly nonlinear, it may be necessary y small time step. This is particularly true if the Lennard-Jones lected.

ection and Diffusionethods have one major drawback when it comes to modeling the diffusion of a particulates in a fluid. The higher the Pclet number, erically unstable the method becomes. The Pclet number is the e of advection to the rate of diffusion. In general, continuum t handle systems where the Pclet number is above around 1000.

crofluidic systems, for 100 nm diameter particles in water at room he Pclet number is on the order of 108. Handling such a large with continuum methods is clearly not possible. Particle the other hand, are computed in a Lagrangian reference frame, estriction of the Pclet number. The Pclet number can be 0 to infinity without introducing numerical instabilities. Advection particles via the drag force. Diffusion is added to particles by adding ce. If the background velocity field is zero then particle motion will sive (zero Pclet number). If the Brownian force is neglected and d velocity is nonzero, the motion will be pure advection (infinite ).

MATHEMATICAL PARTICLE TRACINGParticle tracing is also of interest in cases where the particles are neither charged nor immersed on a fluid. Mechanical systems may be modeled using a particle based approach, with the equations of motion specified by Newtons second law, a Lagrangian or a Hamiltonian. Often it is easier to write down an expression for the Lagrangian or Hamiltonian for particles rather than deriving the equations of motion. The Hamiltonian formulation solves for both the particle position and the particle momentum.

Motion of stars in Introduction | 5

a galaxy using the user defined particle-particle interaction option.

6 | Introduction

Physics List by Space Dimension and Preset Study Type

CHARGED PAThe Charged Pbranch in the Melectrons. Theelastic collisionusing the Cou

PARTICLE TRAThe Particle TFlow branch inbackground flumagnetic, and also possible toparticle-particl

MATHEMATICThe Mathematunderlying maParticle TracinTracing interfaa Lagrangian othe Lagrangian

PHYSICS ICON TAG SPACE DIMENSION

AVAILABLE PRESET STUDY TYPE

AC/DC

Charged Particle Tracing cpt 3D, 2D, 2D axisymmetric

time dependent

Fluid Flow

Particle Tracing f

Mathemat

Mathematical ParRTICLE TRACINGarticle Tracing user interface ( ) is found under the AC/DC odel Wizard, and can be used to model the trajectories of ions and

re are predefined forces for the electric force, magnetic force, and force. In addition, you can model particle-particle interaction

lomb force.

CING FOR FLUID FLOWracing for Fluid Flow user interface ( ), found under the Fluid the Model Wizard, computes the motion of particles in a id. Particle motion can be driven by drag, gravity, and electric,

acoustophoretic forces. User-defined forces can also be added. It is compute the particle mass and temperature as well as

e interactions.

AL PARTICLE TRACINGical Particle Tracing user interface ( ) gives access to the thematical formalism on which the Charged Particle Tracing and g for Fluid Flow interfaces are built. The Mathematical Particle ce also allows for specification of particle motion in terms of either r Hamiltonian. Often it is easier to write down an expression for or Hamiltonian for particles rather than deriving the equations of

or Fluid Flow fpt 3D, 2D, 2D axisymmetric

time dependent

ics

ticle Tracing pt 3D, 2D, 2D axisymmetric

time dependent

motion. The Hamiltonian formulation solves for both the particle position and the particle momentum, so the number of degrees of freedom doubles when the Hamiltonian formulation is activated.

Boundary Conditions

There are six dwhen they makscattering, andIntroduction | 7

ifferent options available for the boundary conditions for particles e contact with a wall: Bounce, Freeze, Stick, Disappear, Diffuse General reflection.

The Freeze option (default) fixes the particle position and velocity at the instant a wall is struck. So, the particle position no longer changes after contact with the wall, and the particle velocity remains at the same value as when the particle struck the wall. This boundary condition is typically used to recover the velocity or energy distribution of charged particles at the instant contact was made with the wall.The Bounce option specularly reflects from the wall such that the particle momentum is conserved. This option is typically used when tracing microscopic particles in a fluid.The Stick option fixes the particle position at the instant the wall is struck. The particle velocity is set to zero. This can be used if the velocity or energy of the particles striking a wall is not of interest.The Disappear option means that the particle is not displayed once it has made contact with

8 | Introduction

the wall. This option should be used if display of the particle location after contact with the wall is not of interest.The Diffuse scattering option bounces particles off a wall according to Knudsens cosine law. That is, the probability a particle bouncing off the surface in a given direction within a solid angle d is given by cos(q)d, where q is the angle between the direction of the particle and the wall normal. The total particle momentum is conserved.The General reflection option allows an arbitrary velocity to be specified after a particle makes contact with the wall. This can either be done in Cartesian coordinates or can be functioNote that the toption.The above bouconditional baapplies the bouexpression optto something n

SECONDARY PSecondary partparticle strikes per incident pabe User definea constant speein the normal

Particle R

There are sevedomains, or onrelease frequenspecific releasein the tangent-normal coordinate system. The velocity components ns of the incident particle velocity, energy or any other quantity. otal momentum of the particle is not necessarily conserved with this

ndary conditions for the particles striking the wall can be sed on either an expression or a probability. The Probability option ndary condition with a certain probability. The Evaluation

ion only applies the boundary condition if the expression evaluates onzero.

ARTICLES

icles can be introduced into the modeling domain when a primary the wall. It is possible to specify the Number of secondary particles rticle as well as their initial velocity. The initial velocity can either d, Isotropic hemisphere, which releases the secondary particles with d and hemispherical velocity direction with the north pole directed direction away from the wall, or Reflection of primary particle.

elease Mechanisms

ral ways to release particles, either from a grid, by selecting specific specific boundaries. In each case the number of particles and the cy can be specified. There are also specific options available for the features, described below.

RELEASE FROM A GRIDParticles can be released from a grid which may either be regular or graded. In addition to the initial coordinates of the particles the initial velocity and times to release the particles may be specified. There are also three predefined initial velocity distributions aparticles with aIsotropic ConsIsotropic Hemdistribution fu

RELEASE FROMParticles can bemodel geometIntroduction | 9

vailable to release Maxwellian, tant Speed, or ispherical nction.

A DOMAIN released from a domain according to the mesh associated with the

ry or according to an arbitrary expression.

10 | Introductio

When the particle release is Mesh based, there is an option to specify the Refinement factor. The higher this factor, the more particles are released from within each mesh element. When the particle release is Density based, the release of particles is weighted according to aexpression. Thflexible optionparticles becaufunction can bexpression, a raor even the soldifferential equ

INLETThe inlet featuoptions for theoption to relean

user defined is is the most for releasing se the weighting e an analytic ndom function,

ution to a partial ation.

re allows for particles to be released from boundaries. Both release Release from domain feature are available, as well as an additional se particles uniformly from a selected boundary.

Modeling Tools

The Particle Tracing Module provides a wide range of specialized modeling tools to assist in extracting specific quantities of interest.

SPECIAL VARIABLESThe particle tracing user interfaces define a number of special variables, some of which can only be used during results processing. These variables can be found in the Particle sta

An example ofwith the MathThe following Particle ind

the total nuwhich can cparticle.

Particle releuseful to beinitial releasvalue, starti

The releasesecondary pparticles we

The time at

There are also only be evalua Total numb

known in ademission.Modeling Tools | 11

tistics plot group during results processing, as shown below:

variables available from the particle statistics menu and available ematical Particle Tracing interface. variables are defined:ex. Each particle is assigned a unique index starting from 1 up to mber of particles. This expression can be passed into a function, reate, for example, random forces that are unique for each

ase feature. If there are multiple release features in a model, it is able to visualize how the particles mix together based on their e position. The Particle release feature variable takes a numeric ng at 1, which is unique to each release feature. time of a given particle. This works for both primary and articles and thus allows for extraction of the time at which re released into the modeling domain. which a particle stopped at a boundary.

variables that are only available during results processing and can ted using the Global Evaluation node under Derived Values.er of particles. The total number of particles released may not be vance, especially in models that include secondary particle

12 | Modeling T

Total number of particles in selection. If a selection has been applied to the Particle data set, the number of particles in that selection can be evaluated.

Transmission probability. Often the transmission probability is the main quantity of interest in a particle tracing model. The Transmission probability variable is available on domains, boundaries, or combinations of both.

MONTE CARLO MODELINGIt is possible to add forces on particles which are stochastic. In the Particle Tracing for Fluid Flowinclude randommay be necessastatistical averarandom numbBrownian Forcrandom numbbe set in a paraAll three particallows for genediscretely chanCharged ParticCollision Forcand backgrounshould occur, tan elastic collisof a charged pools

interface, the Drag Force and Brownian Force features potentially contributions. When these features are included in a model, it

ry to solve the problem multiple times and take some kind of ge of the results. Each time the model is solved, a new set of ers, which are used to compute the force, should be generated. The e feature has a parameter called Additional input argument to er generator. This value can be set by a parameter, which can then metric sweep.le tracing interfaces contain a velocity reinitialization feature. This ral purpose Monte Carlo modeling, since the velocity vector can be ged at each timestep according to some logical expression. The le Tracing interface exploits this functionality in the Elastic

e feature. The collision probability between the charged particles d gas are determined by the collision frequency. If a collision he charged particle has its velocity vector reinitialized according to ion with isotropic scattering. This provides an accurate description article interacting with a background gas.

Brownian motion fright, t = 10 s, low

AUXILIARY DAuxiliary deperesidence timeAuxiliary Depeadditional ordThe residence particle status time and the pdomain). The Modeling Tools | 13

or particles all starting at a single point. They diffuse outwards. Top left t = 0 s, top er left t = 30 s and lower right t = 100 s.

EPENDENT VARIABLES AND RESIDENCE TIMEndent variables can be used to help keep track of things like , particle trajectory length, integrated shear rate etc. When an ndent Variable feature is added to the physics interface an

inary differential equation (ODE) is solved for each particle.time can also be computed in a different way, by setting the Store data property to On. This creates variables for the particle release article stop time (the time when a particle left the modeling residence time is then simply the difference between the two.

14 | Modeling T

Results Processing Tools

PARTICLE DATA SETThe Particle data set is automatically created when solving a model containing one of the Particle Tracing Module interfaces, provided that the Generate default plots option is selected in the Study. Selections may be added to the particle data set which allows, for example, the number or fraction of particles in a given domain or on

PARTICLE TRAA Particle Trajcreated automsolving for a pinterface is comoptions to set trajectories to or Tubes. It is render the partcomet tails, in location of theselected outpumodels where ttimes is small, option is availasmoother lookplots.ools

a given boundary to be computed during results processing.

JECTORY PLOTSectories plot is atically when a Study article tracing

puted. There are the particle be rendered as Lines also possible to icles as points or which case only the particles at the t time is shown. For he number of output an interpolation ble that will create ing particle trajectory

POINCAR MAPS AND PHASE PORTRAITSPoincar maps are available in both 2D and 3D . This plot type is useful to visualize the particle trajectories in a plot that represents the position of the particles in a section that is usually transversal to the particle trajectories. The Poincar map represents the particle trajectories in a space dimension that is one dimension lower than the original particle space..

Phase portraitsPortrait plot toof a phase portvelocity on theModeling Tools | 15

are available as a 2D plot type under More Plots. Use a Phase visualize large data sets of particle trajectories. The traditional use rait is to plot the particle position on the x-axis and the particle y-axis. Each dot in the xy-plane represents a particle.

16 | Modeling T

PARTICLE EVALUATIONInformation about expressions and variables along particle trajectories can be written to the Results Table using the Particle Evaluation option under Derived values . Once the data has been written to the results table it can be manipulated anaccordingly. Tto only write daa fraction or a sparticles.

OPERATIONS It is possible toMaximum operations to bvelocity, maxim

HISTOGRAMSStatistical inforwith a histogranumber of binsvisualization o

FILTERSVisualizing theconsume a lot is possible to fools

d plotted here are options ta to the table for pecific number of

ON PARTICLE DATA set the source data set for the Integration , Average , and Minimum to be a Particle data set. This allows e performed on the particle data set, to compute average particle um energy, and so on.

mation about the behavior of the particles is often best visualized m. The histogram sorts the value of a variable into a specified . The most obvious application of the histogram is that it allows for f the velocity and energy distribution function of a set of particles.

trajectory of systems with a very large number of particles can of computer resources and often particles obscure one another. It ilter the type of particle and the number of particles which should

be rendered. To do this, right click the Particle Trajectories plot type and choose Filter.

The Particle type can be set to render Primary particles, Secondary particles, All, or a Logical expression.If particles arehigh, the numto render optioNumber of paModeling Tools | 17

obscuring one another or the burden on the graphics card is very ber of particles rendered can be reduced by changing the Particles n. A Fraction of the total number of particles to render or the total

rticles to render can be set.

18 | The Model

The Model Library

To open any Particle Tracing Module model, select View>Model Library from the main menu in COMSOL Multiphysics. In the Model Library window that opens, expand the Particle Tracing Module folder and browse or search the contents of the folders. Click Open Model and PDF to open both the model in COMSOL Multiphysics and a PDF to read background theory including the step-by-step instructions to build it.Due to the factmodules, somethe Model Libadditional part

AC/DC MOD Magnetic L Multipactor Quadrupole Quadrupole

CFD MODUL Micromixer Thermopho

ACOUSTICS M Acoustic Le

PLASMA MOD Ion energy

The Model Libmodels and to( ) to updat Library

that the Particle Tracing Module enhances the capabilities of other of the models utilizing the particle tracing interfaces are placed in rary folders for other products. The following modules have icle tracing example models:

ULE

ens

Mass Filter Mass Spectrometer

E

resis

ODULE

vitator

ULE

Distribution Function

rary is updated on a regular basis by COMSOL in order to add new improve existing models. Choose View>Model Library Update e the model library.

The MPH-files in the COMSOL model libraries can have two formatsFull MPH-files or Compact MPH-files

Full MPH-fmodels appewith the texcursor at th

Compact Mmeshes andMPH-files hto study theto downloadmodels throappear withthe Model LMPH-file isincludes a MThe Model Library | 19

iles include all meshes and solutions. In the Model Library these ar with the icon. If the MPH-files size exceeds 25MB, a tip t Large file and the file size appears when you position the e models node in the Model Library tree.PH-files include all settings for the model but have no built solution data to save space on the DVD (a few compact ave no solutions for other reasons). You can open these models settings and to mesh and re-solve the models. It is also possible the full versionswith meshes and solutionsof most of these ugh Model Library Update. In the Model Library these models the icon. If you position the cursor at a compact model in ibrary window, a No solutions stored message appears. If a full

available for download, the corresponding nodes context menu odel Library Update item.

20 | Computing

Computing Particle Trajectories through a Laminar Static Mixer

This section takes you through the modeling stages of computing particle trajectories based on a computed velocity field. In static mixers, also called motionless or in-line mixers, a fluid is pumped through a pipe containing stationary blades. This mixing technique is particularly well suited for laminar flow mixing because it generates only small pressure losses in this flow regime. This example studieperformance bthe mixer..

Model Wiz

1 Go to the M2 In the Add p

.3 Click the Ad4 Click Next 5 Find the Stu6 Click Finish Particle Trajectories through a Laminar Static Mixer

s the flow in a twisted-blade static mixer. It evaluates the mixing y calculating the trajectory of suspended particles through

ard

odel Wizard window. Click Next .hysics tree, select Fluid Flow>Single-Phase Flow>Laminar Flow

d Selected button..

dies subsection. In the tree, select Preset Studies>Stationary . .

Global Definit ions and Definit ions

1 In the Model Builder window, right-click Global Definitions and choose Parameters .

2 In the table, enter the following settings:

Geometry

The mixer geo1 In the Mode

choose Imp2 In the Impo3 Click the Br4 Browse to th

laminar_mixvaries based drive, the filFiles\COMS

5 Click the Imwindow as sComputing Particle Trajectories through a Laminar Static Mixer | 21

metry is quite complicated so start by importing it from a file.l Builder window, under Model 1 right-click Geometry 1 and

ort .rt settings window, locate the Import section.owse button.e models Model Library folder and double-click the file er_particle.mphbin. The location of the files used in this exercise on your installation. For example, if the installation is on your hard e path might be similar to C:\Program OL43b\models\.port button. You should see the geometry appear in the Graphics hown below.

22 | Computing

Materials

1 In the ModeMaterial

2 In the Mate3 In the table, Particle Trajectories through a Laminar Static Mixer

l Builder window, under Model 1 right-click Materials and choose .

rial settings window, locate the Material Contents section. enter the following settings:

Physics in

Now add an ex1 In the Mode2 Right-click L3 From the m4 Select Boun5 In the Inlet 6 In the U0 edThe boundarynecessary to gemodules all haComputing Particle Trajectories through a Laminar Static Mixer | 23

terfaces

pression for the inflow velocity which is parabolic.l Builder window, expand the Model 1>Laminar Flow node.aminar Flow and choose Inlet.

ain menu select View>Selection List.dary 23 only then right click to confirm the selection.settings window, locate the Velocity section.it field, type 2*(1-(x^2+z^2)/ra^2)*u_mean. condition which was just added was rather complicated but t a fully developed flow profile. The CFD, Microfluidics and Plasma ve a special Laminar Inflow boundary condition which ensures

24 | Computing

laminar flow. It is not necessary to enter a complicated expression for the velocity profile, just the average velocity of flowrate.7 Click back on the Model Builder tab.8 Right-click Laminar Flow and choose Outlet.9 Click on the Selection List tab and select Boundary 20 only. Right click to

confirm the selection.

Mesh

The mesh needthrough the mfine on the mix1 In the Mode

More Opera2 Click the W3 Click on the4 Right-click M5 In the Size s6 From the Ca7 From the Pr8 In the Mode9 In the Size s10From the Pr11Click the Cu12Locate the E

field, type 0.13In the Mode

Operations>14Select Boun15Right-click M16In the Size s17From the Ca18From the Pr Particle Trajectories through a Laminar Static Mixer

s to be quite fine to ensure that the particle motion is accurate odeling domain. In this case, take care to ensure that the mesh is ing blades.l Builder window, under Model 1 right-click Mesh 1 and choose tions>Free Triangular .ireframe Rendering button on the Graphics toolbar. Selection List tab and select Boundaries 5, 1618, and 5355 only.

odel 1>Mesh 1>Free Triangular 1 and choose Size .ettings window, locate the Element Size section.librate for list, choose Fluid dynamics.edefined list, choose Extremely fine.l Builder window, under Model 1>Mesh 1 click Size.ettings window, locate the Element Size section.edefined list, choose Extremely fine.stom button.lement Size Parameters section. In the Resolution of curvature edit 15.l Builder window, right-click Mesh 1 and choose More Free Triangular .dary 23 only.

odel 1>Mesh 1>Free Triangular 2 and choose Size .ettings window, locate the Element Size section.librate for list, choose Fluid dynamics.edefined list, choose Extra fine.

19In the Model Builder window, right-click Mesh 1 and choose Free Tetrahedral .

20In the Settings window, click Build All .

Study

1 In the Mode

Model Wiz

Now that the fstudy to comp1 In the Mode2 Go to the M3 In the Add p4 Click Add S5 Click Next 6 Find the Stu

Physics>Tim7 Click Finish

Physics in

1 In the Mode2 Right-click M3 Select Doma4 In the Drag5 From the u 6 From the Computing Particle Trajectories through a Laminar Static Mixer | 25

l Builder window, right-click Study 1 and choose Compute .

ard

luid velocity has been computed, add a physics interface and a new ute the particle trajectories.l Builder window, right-click Model 1 and choose Add Physics.odel Wizard window.hysics tree, select Fluid Flow>Particle Tracing for Fluid Flow .

elected ..

dies subsection. In the tree, select Preset Studies for Selected e Dependent.

.

terfaces

l Builder window, expand the Particle Tracing for Fluid Flow node.odel 1>Particle Tracing for Fluid Flow and choose Drag Force.

in 1 only. Force settings window, locate the Drag Force section.list, choose Velocity field (spf/fp1).list, choose Dynamic viscosity (spf/fp1).

26 | Computing

The goal is to release particles in a way where they are more dense where the velocity field is higher. To do this you use the Density option for the particles Initial Position.7 In the Model Builder window, right-click Particle Tracing for Fluid Flow and

choose Inlet.8 Select Boundary 23 only.9 In the Inlet settings window, locate the Initial Position section.10From the Initial position list, choose Density.11In the N edi12In the edit13Locate the I(spf/fp1).

14Right-click P15Select Boun16In the Mode

Flow>Partic17In the Partic18In the dp ed

Study 2

1 In the ModeDependent

2 In the TimeSelection sec

3 In the Solvethe green chFlow is not

4 Expand the variables not

5 From the M6 From the St7 Locate the S8 Go to the R Particle Trajectories through a Laminar Static Mixer

t field, type 3000. field, type spf.U.nitial Velocity section. From the u list, choose Velocity field

article Tracing for Fluid Flow and choose Outlet.dary 20 only.l Builder window click on Model 1>Particle Tracing for Fluid

le Properties 1.le Properties settings window, locate the Particle Properties section.it field, type 5E-7[m].

l Builder window, expand the Study 2 node, then click Step 1: Time .

Dependent settings window, locate the Physics and Variables tion.

For column, deactivate the Laminar Flow interface by clicking on eck . It will turn into a red cross indicating that the Laminar going to be solved for in this study.Values of Dependent Variables section. Select the Values of solved for check box.ethod list, choose Solution.udy list, choose Study 1, Stationary.tudy Settings section. Click the Range button .ange dialog box.

9 In the Step edit field, type 0.2.10In the Stop edit field, type 5.11Click the Replace button. The Study settings should look like this:

12In the ModeComputing Particle Trajectories through a Laminar Static Mixer | 27

l Builder window, right-click Study 2 and choose Compute.

28 | Computing

Results

By default you end up in the Particle Trajectories (fpt) node after the model has finished solving. 1 In the Particle Trajectories (fpt) settings window, click to expand the Color

Legend section.2 From the Position list, choose Bottom.3 In the Model Builder window, expand the Particle Trajectories (fpt) node, then

click Particle4 In the Partic

section.5 Find the Lin6 Find the Po7 In the Mode

(fpt)>Particl8 In the Color

upper-right Flow>Shear

9 Click the Plo Particle Trajectories through a Laminar Static Mixer

Trajectories 1.le Trajectories settings window, locate the Coloring and Style

e style subsection. From the Type list, choose Line.int style subsection. From the Type list, choose None.l Builder window, expand the Results>Particle Trajectories e Trajectories 1 node, then click Color Expression 1. Expression settings window, click Replace Expression in the corner of the Expression section. From the menu, choose Laminar rate (spf.sr).t button .

10Click Zoom Extents in the Graphics toolbar.

Now create a nprobability can1 In the Mode2 Right-click P3 Right-click R4 In the Select5 From the G6 Select Boun7 In the Mode

choose Glob8 In the Glob9 From the D10From the Ti11Click Replac

section. Frostatistics>Tr

12Click the Ev0.81 and 0Computing Particle Trajectories through a Laminar Static Mixer | 29

ew Particle data set with a selection at the outlet so the transmission be evaluated.l Builder window, expand the Results>Data Sets node.article 1 and choose Duplicate .esults>Data Sets>Particle 2 and choose Add Selection.

ion settings window, locate the Geometric Entity Selection section.eometric entity level list, choose Boundary.dary 20 only.l Builder window, under Results right-click Derived Values and al Evaluation.

al Evaluation settings window, locate the Data section.ata set list, choose Particle 2.me selection list, choose Last.e Expression in the upper-right corner of the Expression m the menu, choose Particle Tracing for Fluid Flow>Particle ansmission probability (fpt.alpha).aluate button . The transmission probability should be between .83.

30 | Computing

One useful way of visualizing how to particles mix is to use a Poincar plot. The Poincar plot places a colored dot for each particles at the location at which the particle passes through a cut plane (known as a Poincar section). 1 In the Model Builder window, under Results right-click Data Sets and choose

Cut Plane .2 In the Cut Plane settings window, locate the Data section.3 From the Data set list, choose Particle 1.4 Locate the Plane Data section. From the Plane list, choose xz-planes.5 In the y-coo6 Select the A7 In the Dista8 Click the Plo9 In the Mode10In the 3D P11From the D12Click to exp

Bottom.13Right-click R

Map .14In the Poinc15From the Cu16Locate the C17In the assoc18Click the Plo19Right-click R

Expression.20In the Color21In the Expre22Locate the C23In the Mode

.24In the Surfa25From the D26Locate the E27Locate the C Particle Trajectories through a Laminar Static Mixer

rdinates edit field, type 0.006.dditional parallel planes check box.nces edit field, type 0.006 0.016 0.026 0.036 0.042.t button .l Builder window, right-click Results and choose 3D Plot Group.

lot Group settings window, locate the Data section.ata set list, choose Particle 1.and the Color Legend section. From the Position list, choose

esults>3D Plot Group 4 and choose More Plots>Poincar

ar Map settings window, locate the Data section.t plane list, choose Cut Plane 1.oloring and Style section. Select the Radius scale factor check box.

iated edit field, type 6E-5.t button .esults>3D Plot Group 4>Poincar Map 1 and choose Color

Expression settings window, locate the Expression section.ssion edit field, type at(0,qx

28From the Color list, choose Gray.29Click the Go to Default 3D View button on the Graphics toolbar.30Click the Plot button .

In the above fiThe color reprmarked as red initial positioncolor of their ileft in the abovx < 0. As the pBy the end of significant pocComputing Particle Trajectories through a Laminar Static Mixer | 31

gure, the location of the particles at 6 Poincar sections are shown. esents the location of the particle at its initial position. So, particles had an initial position of x < 0 and particles marked as blue had an of x > 0. The at() operator is used to mark the particles with the nitial position. The first Poincar section (the one furthest to the e figure) clearly indicates which particles start with coordinates of articles begin to follow the flow field, they begin to mix together. the mixer, the particles have not mixed completelythere are still kets of only red and only blue particles.

32 | Computing Particle Trajectories through a Laminar Static Mixer

ContentsIntroductionThe ApplicationsPhysics List by Space Dimension and Preset Study TypeBoundary ConditionsParticle Release Mechanisms

Modeling ToolsResults Processing Tools

The Model LibraryComputing Particle Trajectories through a Laminar Static MixerModel WizardGlobal Definitions and DefinitionsGeometryMaterialsPhysics interfacesMeshStudyModel WizardPhysics interfacesStudy 2Results