1. Modeling Material/Species Transport Reacting Flows - Lecture 8
Instructor: André Bakker
© André Bakker (2006)

2. Outline
In addition to flow fields, we often need to model additional physics.
The fluid velocities transport a number of properties:
Mass of one or more materials.
Momentum.
Energy.
Proper modeling of material transport is necessary if we want to model mixing or reaction.
Methods to model material transport:
Discrete phase modeling (DPM), aka particle tracking.
Species transport, aka scalar transport.
Multiphase flow modeling, e.g. Eulerian flow models.

3. Multiphase flow  multiple momentum eqns.
Multiphase flow is simultaneous flow of:
Materials with different states or phases (i.e. gas, liquid or solid).
Materials in the same state or phase, but that are immiscible (i.e. liquid-liquid systems such as oil droplets in water).
Each phase has its own velocity field and its own momentum.
It is therefore often necessary to solve multiple sets of momentum equations, one set for each phase.
Interaction between the phases requires the introduction of momentum exchange terms.
Models are often complex, and time consuming to solve.
Will not discuss here.

4. Systems with single set of momentum eqns.
We will discuss material transport in systems that are adequately described by a single set of momentum eqns:
Species or scalar transport.
Particle tracking (DPM).
One fluid flow field is solved.
The rate of transport of the species or particles is derived from that single fluid flow field.
The local concentration of species or particles may affect the flow field itself.

5. Species transport
The species transport equation (constant density, incompressible flow) is given by:
The concentration of the chemical species is c.
The velocity is ui.
D is the diffusion coefficient.
S is a source term.
This equation is solved in discretized
form to calculate the transport and
local species concentrations. cP cE cN cS cW

6. Species transport – the convective term
Convection is transport of material due to the velocity of the fluid.
Flux from one grid cell to the next is area times normal-velocity times concentration.
From cell “p” to “E”: Ae.ue.ce.
Values at cell faces required!
Implication: for best accuracy, use higher order discretization. cP cE cW
Ae,ce,ue

7. Species transport – the diffusive term
Diffusion is transport resulting from concentration gradients.
Diffusion flux from one grid cell to the next is area times the concentration gradient times the diffusion coefficient.
From cell “p” to “E”:
Gradient at interface between cells is easily calculated.
The main difficulty is the calculation of the diffusion coefficient. cP cE cW
Ae,ce,ue

8. The diffusion term - molecular
Molecular diffusion:
As a result of concentration gradients: mass diffusion.
As a result of temperature gradients: thermodiffusion.
Mass diffusion coefficient:
Constant dilute approximation: same constant for all species.
Dilute approximation: different constant for each species.
Multi-component: a separate binary diffusion coefficient Dij for each combination of species “i” into species “j”.
Thermodiffusion: flux is proportional to thermal diffusion coefficient DT and temperature gradients:
Not usually important in industrial chemical reactors.

9. The diffusion term - turbulence
Turbulent diffusion: transport due to the mixing action of the chaotic turbulent velocity fluctuations.
The turbulent diffusion coefficient is calculated from the turbulent viscosity t:
The turbulent Schmidt number Sct is a model constant. Recommended values are:
0.7 if an eddy viscosity turbulence model is used, e.g. k-.
1.0 if the Reynolds stress model (RSM) is used.

10. Species transport – source terms
The source term:
This describes all other effects:
Creation or destruction of species due to chemical reaction.
Any other physical phenomena the user wants to implement.

11. Model setup Model setup:
Specify which species are present in the mixture.
Specify properties of all species.
If N species are present, N-1 equations are solved. The concentration of the Nth species follows from the fact that all mass fractions Yi should sum to unity.

12. Boundary conditions Wall boundary conditions:
Either specified mass fraction, or zero flux.
Inlet boundary conditions:
At inlets, the inlet flux is calculated as:
Need to specify inlet concentration/mass fraction.
The inlet diffusion flux depends on the concentration gradient. Value can not be predicted beforehand. If a fixed mass flow rate is desired, this term should be disabled.
Outlet boundary conditions: specify species mass fraction in case backflow occurs at outlet.

13. Species equation is one-way!
All species have the same convective velocity.
Diffusion usually reduces concentration gradients  mixing.
As a result, the diffusion equation can not usually be used to model separation!
To model separation, multiphase models where the phases have different velocities are necessary.
Exceptions:
Some laminar flow, thermal diffusion dominated cases.
Cases with complex transport models implemented through source terms.

14. Mixing mechanism Laminar mixing. CFD simulation. Six elements.
Each element splits, stretches and folds the fluid parcels.
Every two elements the fluid is moved inside-out.

15. Mixing quantification
Species concentration in sample points at different axial locations.
Coefficient of variance:
Kenics mixer
Six elements
Re=10
88 evenly spaced sample points in each axial plane.

16. Locations of sample plane points
Surfaces | Plane
File | Write | Profile

17. Particle tracking
Solve one set of momentum equations for the fluid flow.
In an Eulerian reference frame, i.e. on the grid locations.
Simulate a second, discrete phase consisting of individual particles.
Known as discrete phase modeling (DPM).
In a Lagrangian frame of reference, i.e. following the particles.
Trajectories are calculated, as well as particle heat and mass transfer.
Particles may affect fluid flow field. This is done by introducing source terms in the fluid flow equations.
Particle trajectories in a cyclone

18. DPM theory
Trajectory is calculated by integrating the particle force balance equation:
drag force is
a function of the
relative velocity
Gravity force
Additional forces:
Pressure gradient
Thermophoretic
Rotating reference frame
Brownian motion
Saffman lift
Other (user defined)
typical continuous phase control volume
mass, momentum and heat exchange
The discrete phase particle trajectories are computed by integrating the force balance equation. This equation has terms for the drag force and the gravity force and there are also numerous additional forces that can be included. Typically for combustion applications the drag and gravity forces are the main influences on particle trajectories.
The particle may exchange mass, momentum and heat with the gas phase and that occurs within the control volumes that are crossed by the particle trajectory.
Saffman's lift force = lift due to shear
Thermophoretic Force: Small particles suspended in a gas that has a temperature gradient experience a force in the direction opposite to that of the gradient. This phenomenon is known as thermophoresis.

19. Coupling between phases
One-way coupling:
Fluid phase influences particulate phase via drag and turbulence.
Particulate phase has no influence on the gas phase.
Two-way coupling:
Particulate phase influences fluid phase via source terms of mass, momentum, and energy.
Examples include:
Inert particle heating and cooling.
Droplet evaporation.
Droplet boiling.
Devolatilization.
Surface combustion.
In combustion systems, typically we have two-way coupling between the discrete phase and the continuous phase. The fluid phase influences the particles via drag, turbulence and momentum transfer. The particulate influences the fluid phase through source terms. We may have a mass transfer, a momentum transfer, as well as an energy transfer between the phases. So there’s definitely a strong coupling in combustion applications between the discrete phase and the continuous phases.
Some of the examples of this coupling may include particles that are heated or cooled; droplet evaporation; droplet boiling; devolatilization and surface combustion. These last two are generally features of coal combustion applications.

20. Discrete phase model
Trajectories of particles/droplets are computed in a Lagrangian frame.
Exchange (couple) heat, mass, and momentum with Eulerian frame gas phase.
Discrete phase volume fraction should preferably be less than 10%.
Mass loading can be large (+100%).
No particle-particle interaction or break up.
Turbulent dispersion modeled by:
Stochastic tracking.
Particle cloud model.
Model particle separation, spray drying, liquid fuel or coal combustion, etc.
continuous phase flow field calculation
particle trajectory calculation
update continuous phase source terms
Let’s start with some background on the discrete phase model or DPM model. The trajectories of particles or droplets are computed in a Lagrangian frame and we have the capabilities of coupling these particles to the gas phase in the Eulerian frame. We can also exchange mass, momentum and energy between the two phases.
There are some specific requirements that you should keep in mind if you’re using the discrete phase model. It is recommended that you keep a volume fraction that’s less then 10%. The reason for that is because the discrete phase model doesn’t take into account any particle-particle interactions. So particles don’t know about one another. And if you were at a volume fraction much higher then 10%, then the influence of particles interaction may become a significant factor that could effect the accuracy of your results.
On the other hand, the mass loading can be very large, in excess of 100%. So that’s not an issue. It’s the volume loading or the volume fraction that is the issue.
The DPM model accounts for the effect of turbulence on the particle trajectories. This turbulent dispersion can be modeled via DPM either with the stochastic tracking model or with the particle cloud model. And we’ll talk about these two different approaches soon.
Typical applications for DPM might be for particle separation such as coal classifiers or cyclones. DPM is also applied to spray drying applications, and of course, liquid fuel or coal combustion problems.

21. Particle types
Particle types are inert, droplet and combusting particle.
Particle Type
Description
Inert.
Inert/heating or cooling
Droplet (e.g. oil).
Heating/evaporation/boiling.
Requires modeling of heat transfer and species.
Combusting (e.g. coal).
Heating.
Evolution of volatiles/swelling.
Heterogeneous surface reaction.
There are three different particle types that you can choose from when you’re using DPM: inert particles, droplets and combusting particles. The inert particles have associated with them models for describing the heating and cooling or heat transfer between the particle phase and the continuous phase.
A droplet particle type allows you to model the heating, as well as evaporation and boiling. Energy as well as the mass transfer between the two phases is modeled.
A combusting particle, typically a coal particle, allows modeling the heating, as well as the evolution of volatiles or the devolatilization or the release of volatile fuel gases from the coal particles. It also allows to model heterogeneous surface reactions mainly the char combustion that occurs on the coal particles.
There are several devolatilization and char burnout models that we’ll refer to you in a later slide. But what is most important to remember about DPM and its different submodels, is that if you’re modeling coal combustion or liquid phase combustion, DPM provides you with the ability to describe the trajectory as well as the evolution of gaseous fuel from the discrete phase. However, we still need to combine these physical models with a gas phase combustion model, such as the finite rate model or the PDF model, to complete the set of equations that describes your system. So these kinds of combustion applications have an additional complexity, in that they require not only a gas phase combustion model, but they also require additional models for tracking and modeling the evolution of the gas phase fuel from the discrete phase.

22. Heat and mass transfer to a droplet
Temperature Tb Tv
When using FLUENT's discrete phase modeling capability, reacting particles or droplets can be modeled and their impact on the continuous phase can be examined. Several heat and mass transfer relationships, termed "laws", are available in FLUENT and various physical models employed in these laws. This plot illustrates the laws that are used to describe the different heat and mass transfer steps during the evaporation of a liquid droplet as a function of time after injection.
When the particle is first injected it is subjected to heating. Once the vaporization temperature specified in the material database is reached for the droplet, vaporization begins. The mass transfer rate for vaporization is based on a specific relationship. When the boiling point is reach for the droplet, the boiling law is used to describe the mass transfer from the droplet to the continuous phase.
Tinjection
Inert heating law
Vaporization law
Boiling law
Particle time

23. Particle-wall interaction
Particle boundary conditions at walls, inlets, and outlets:
For particle reflection, a restitution coefficient e is specified:
volatile fraction flashes to vapor
Escape
Reflect
Trap
There are a number of different devolatilization and char burnout models under the DPM umbrella. For devolatilization, we have the default constant rate model that is oftentimes used initially to foster the reaction. There is also a single kinetics rate and a two competing rates model. We also have a number of different char burnout models that are available. The diffusion-limited rate model is the default model where the surface reaction proceeds at a rate determined by the diffusion of the oxidant at the surface. This model ignores kinetics. There is also a kinetics/diffusion-limited rate where the diffusion and the kinetics rates are weighted to yield a global char combustion rate. Finally we have a model termed intrinsic model for char burnout that differs from the previous model in dealing with the kinetics rate. It also has the option of changing the diameter or the density of the particle.
There are a number of different ways that you can inject particles into your domain. You can have a single injection. You can have what’s termed a group injection where you have a number of individual injections that vary, for example, in their initial velocities to give you a fan. There’s the option in three dimensions to do a cone type spray. And you can also choose to inject from a surface and provide injections from the cell centers of a particular surface.
Particles that are traveling through your domain will inevitably find a wall and there’s a number of different ways you can model the interaction of particles with walls. The particle can escape, reflect on the wall or it can be trapped. When the particle strikes the wall in the context of the trapped option, whatever volatile fraction is contained in the particle, it will be instantaneously converted to vapor.

24. Particle fates
“Escaped” trajectories are those that terminate at a flow boundary for which the “escape” condition is set.
“Incomplete” trajectories are those that were terminated when the maximum allowed number of time steps was exceeded.
“Trapped” trajectories are those that terminate at a flow boundary where the “trap” condition has been set.
“Evaporated” trajectories include those trajectories along which the particles were evaporated within the domain.
“Aborted” trajectories are those that fail to complete due to numerical/round-off reasons. If there are many aborted particles, try to redo the calculation with a modified length scale and/or different initial conditions.

25. Turbulent dispersion of particles
Dispersion of particles due to turbulent fluctuations in the flow can be modeled using either:
Stochastic tracking (discrete random walk).
Particle cloud model.
Turbulent dispersion is important because:
Physically more realistic (at an added computational expense).
Enhances stability by smoothing source terms and eliminating local spikes in coupling to the gas phase.
We mentioned earlier that there are two different models for turbulent dispersion of particles: the stochastic tracking model and the particle cloud model. So, let’s talk about the reasoning behind the need for these different models. Dispersion of particles in a turbulent flow is due to turbulent fluctuations. This turbulent dispersion is important to include in your model for two reasons. The first reason is to provide a degree of physical realism at, of course, an added computational expense. But it’s physically realistic to include the turbulent dispersion in your flow problem. Secondly, from a numerical standpoint, including turbulent dispersion enhances stability by smoothing out source terms and eliminating local spikes in the coupling between the discrete phase and the gas phase. And let’s talk more about this second issue.
The DPM model has associated with it a finite number of injections for any mass loading that you want to introduce into your system. So each individual injection doesn’t represent an individual particle but an individual mass loading, divided equally amongst N number of tracks. Let’s assume you ignore turbulent dispersion in your calculation. And let’s consider a given individual injection that has associated with it some finite mass loading. If you inject that into the system, then you’re assuming that every single particle that’s represented by that finite injection or finite loading follows the exact same path.
Now, first of all, we know that isn’t physically realistic. In a turbulent flow if we were to send out ten particles from the same position with the same initial conditions, we would expect that they would take different paths. But if we don’t include turbulent dispersion, we don’t get that. A second issue is, when your entire mass loading follows an identical trajectory, they you necessarily focus or concentrate your source terms of mass, momentum and energy, along a single path. And numerically, that’s difficult to deal with. So turbulent dispersion is important because it gives you a level of physical realism and because it allows you to smooth out or distribute these sources in a more realistic manner which is also easier for the solver to deal with.
So instead of having all the mass follow in a single path, you may have 100 injections or 100 tracks where each 1/100th of the mass follows a slightly different path that allows you to distribute the source terms.

26. Turbulence: discrete random walk tracking
Each injection is tracked repeatedly in order to generate a statistically meaningful sampling.
Mass flow rates and exchange source terms for each injection are divided equally among the multiple stochastic tracks.
Turbulent fluctuations in the flow field are represented by defining an instantaneous fluid velocity:
where is derived from the local turbulence parameters:
and is a normally distributed random number.
In Fluent 5 there are two methods for modeling the turbulent dispersion of particles. The discrete random walk tracking or stochastic approach in modeling particle trajectories. And the particle cloud tracking
The discrete random walk tracking approach works by tracking each particle injection individually through the domain.
Each injection is repeatedly tracked in order to generate a statistically meaningful sampling. The users specifies the number of attempts or tries. The turbulent fluctuations in the continuous phase flow field can be represented. The turbulent fluid velocity can be decomposed into a mean and a fluctuating component. This turbulent fluctuation is derived from the local turbulence quantities, the local turbulent kinetic energy, where the constant z (zeta) is a normally distributed randomly generated number.
The extent of that fluctuation is reflected by the amount of turbulence and local turbulence.
Mass flow rates and exchange terms for each injection are divided into your multiple tracks. This is important if you're modeling particle combustion where you have a highly coupled flow field between the dispersed phase and the continuous phase and there is exchange of heat mass and momentum. It is important to introduce a large number of stochastic tracks thereby minimizing the source contribution from each track.

27. Stochastic tracking – static mixer
Stochastic tracking turned off.
One track per injection point.
Uses steady state velocities only and ignores effect of turbulence.
Stochastic tracking turned on.
Ten tracks per injection point.
Adds random turbulent dispersion to each track.
Tracks that start in the same point are all different.
Particle residence time (s)

28. Turbulence: cloud tracking
Uses statistical methods to trace the turbulent dispersion of particles about a mean trajectory.
Calculate mean trajectory 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 probability density function.
The cloud tracking is the second method for modeling turbulent dispersion of the particle trajectories
The cloud tracking uses a statistical method to trace the dispersion of a particle about a mean trajectory. The cloud tracking calculates the mean trajectory from the ensemble average of the equations of motion for the particles represented in the cloud.
The distribution of particles in this cloud is represented by a Gaussian probability density function.

29. Stochastic vs. cloud tracking
Stochastic tracking:
Accounts for local variations in flow properties.
Requires a large number of stochastic tries in order to achieve a statistically significant sampling (function of grid density).
Insufficient number of stochastic tries results in convergence problems due to non-smooth particle source term distributions.
Recommended for use in complex geometry.
Cloud tracking:
Local variations in flow properties get averaged inside the particle cloud.
Smooth distributions of particle coupling source terms.
Each diameter size requires its own cloud trajectory calculation.
The differences and similarities between stochastic tracking and the cloud tracking are listed here. The stochastic tracking approach accounts for local variations of flow properties like temperature or concentration. Each particle trajectory that's released from a stochastic tracking point of view traverses the domain independently and counters the fluid flow properties of each cell as it tracks through. You need a large number of tries or stochastic attempts to achieve a stochastically or a statistically significant sampling. And this is also a function of your grid density. If you don't have enough tries you can have convergence problems. You can also have non-smooth particle concentrations and you can get an artificial distribution of your particle dispersion in the turbulent flow field. It is particularly important to introduce a large number of stochastic tries when you're working with coupled DPM calculations. It is also the method of choice if you're going to be working in a complex geometry with a lot of near walls and tight spaces where the particles can flow through.
Cloud tracking on the other hand, averages out local variations in your flow properties, like temperature. As shown in the graphic all the properties of the particles get smeared out over that cloud diameter resulting in a smooth distribution of particle concentrations. One aspect of cloud tracking is that you can only have one particle diameter per cloud. So if you want to model a distribution using the cloud tracking approach, you have to introduce a large number of clouds and likewise with the stochastic tracking. The more injections you release or the more clouds you model, the more computationally expensive your problem becomes.

30. Injection set-up Injections may be defined as:
Single: a particle stream is injected from a single point.
Group: particle streams are injected along a line.
Cone: (3-D) particle streams are injected in a conical pattern.
Surface: particle streams are injected from a surface (one from each face).
File: particle streams injection locations and initial conditions are read in from an external file.
Injections may be defined as:
a single injection where a particle stream is injected from a single point.
A Group injection where particle streams are injected along a line.
A Cone injection for 3-D models only where particle streams are injected in a conical pattern.
A Surface injection where particle streams are injected from a surface with one injection created from each face
Or a File injection where particle streams injection locations and initial conditions are read in from an external file.

31. Injection definition Every injection definition includes:
Particle type (inert, droplet, or combusting particle).
Material (from data base).
Initial conditions (except when read from a file).
Combusting particles and droplets require definition of destination species.
Combusting particles may include an evaporating material.
Turbulent dispersion may be modeled by stochastic tracking.
Every injection definition includes:
particle type (inert, droplet, or combusting particle)
material (from data base)
initial conditions (except when read from a file)
Combusting particles and droplets require the definition of destination species.
Combusting particles may include an evaporating material.
Turbulent dispersion may be modeled by:
stochastic tracking
cloud tracking

32. Solution strategy: particle tracking
Cell should be crossed in a minimum of two or three particle steps. More is better.
Adjust step length to either a small size, or 20 or more steps per cell.
Adjust “Maximum Number of Steps.”
Take care for recirculation zones.
Heat and mass transfer: reduce the step length if particle temperature wildly fluctuates at high vaporization heats.
Some solution strategies for applying the DPM model.
Cell should be crossed in 2 or 3 particle steps. This can be set explicitly by specifying the maximum number of steps or by adjusting the step length
When regions of flow recirculation exist in the domain, the mean particle trajectories can get trapped. Adjusting the maximum number of steps to accommodate long particle residence times may be necessary.
If heat or mass transfer are occurring it is recommended to reduce the step length if particle temperature is wildly fluctuating at high vaporization heats

33. Solution strategy: coupled calculation
Two strategies possible:
Closer coupling between dispersed and continuous flow:
Increase underrelaxation for discrete phase.
Decrease number of continuous phase calculations between trajectory calculations to less than three.
Reduce underrelaxation factors for continuous phase.
Decoupling of dispersed and continuous flow:
Reduce underrelaxation factor for discrete phase.
Increase number of continuous phase calculations between trajectory calculations to more than fifteen.
Smooth out particle source terms.
Increase number of particle trajectories.
Two solution strategies are possible for the coupled calculation:
For the first strategy, Allow for closer coupling between dispersed and continuous flow by:
Increasing the underrelaxation factor for the Discrete Phase
Decrease number of continuous phase calculations between trajectory calculations < 3.
Reduce the underrelaxation factors for the continuous phase.
The second strategy involves decoupling the dispersed and continuous phase flow by reducing the underrelaxation factor for Discrete Phase and Increasing number of continuous phase calculations between trajectory calculations > 15
To smooth out particle source terms as mentioned previously, increase number of particle trajectories or tries.

34. Particle tracking in unsteady flows
Each particle advanced in time along with the flow.
For coupled flows using implicit time stepping, sub-iterations for the particle tracking are performed within each time step.
For non-coupled flows or coupled flows with explicit time stepping, particles are advanced at the end of each time step.
Fluent allows you to perform particle tracking for unsteady flows. Typically this won’t be of interest for people that are doing combustion because they generally solve steady state problems but let’s go through the basics of unsteady tracking anyway.
Traditionally, DPM has really only been valid for steady state simulations because we assume that from the time a particle enters the system, to the time it exits, the flow field doesn’t change. But now the particle can change its motion due to changes in the flow field. At every time step, the changes in the continuous phase affect the trajectory of the particle and the interaction with the discrete phase.

35. Sample planes and particle histograms
Track mean particle trajectory as particles pass through sample planes (lines in 2D), properties (position, velocity, etc.) are written to files.
These files can then be read into the histogram plotting tool to plot histograms of residence time and distributions of particle properties.
The particle property mean and standard deviation are also reported.
sample plane
data
The DPM model comes with some interesting features that allow identifying sample planes and creating useful particle histograms.
Trajectory sampling consists of tracking particles and writing their status to a file when they encounter selected surfaces or sample planes. This file can then be read into the histogram-plotting tool. And you can take a look at histograms, for example, of residence time and distribution of particle properties as described here in this coal fired furnace example.

36. Particle locations at outlet (HEV)
Flow following particles.
Y(m)
Y(m)
X(m)
X(m)
Particles from small center inlet
Particles from large outer inlet

37. Residence time distribution
Residence time histograms can be made from particle times at outlet for flow following particles.
For this mixer, volume is l, total volumetric flow rate is l/s, and an average residence time of 1.8 s is expected.
1870 particles
Average 1.37s
Stdv s
2936 particles
Average 1.90s
Stdv. 0.51s
Particles from small center inlet
Particles from large outer inlet

38. Effect of particle properties
Sand particles (0.2mm, 2000 kg/m3).
Flow following particles.
gravity

39. Effect of particle properties
Flow following particles.
Sand particles (in water).
Y(m)
Y(m)
gravity
X(m)
X(m)
Particles from small center inlet
Particles from small center inlet.

40. Massive particle tracking
Massive particle tracking refers to analyses where tens of thousands to millions of particles are tracked to visualize flows or to derive statistics of the flow field.
Examples:
An unbaffled mixing tank with a Rushton turbine.
An unbaffled mixing tank with four A310 impellers.
A static mixer.

41. Lattice-Boltzmann Method
Calculations by Jos Derksen, Delft University, 2003.
Unbaffled stirred tank equipped with a Rushton turbine.
Cross Section Vessel Wall

42. Lattice-Boltzmann Method
Calculations by Jos Derksen, Delft University, 2003.
Unbaffled stirred tank equipped with four impellers.

43. Particle tracking animations

44. Particle tracking accuracy
There are three types of errors: discretization, time integration, and round-off.
Research has shown that in regular laminar flows the error in the particle location increases as t², and in chaotic flows almost exponentially.
Errors tend to align with the direction of the streamlines in most flows.
As a result, even though errors multiply rapidly (e.g. 0.1% error for 20,000 steps is ,000 = 4.8E8), qualitative features of the flow as shown by the deformation of material lines can be properly reproduced. But the length of the material lines may be of by as much as 100%.
Overall, particle tracking, when properly done, is less diffusive than solving for species transport, but numerical diffusion does exist.
Errors tend to align with the direction of the streamlines in regular flows and with the flow manifolds in chaotic flows.

45. Summary There are different main ways to model material transport:
Multiphase flow: multiple momentum equations.
Species flow: one set of momentum equations.
Discrete phase modeling (DPM; particle tracking): one set of momentum equations for the fluid flow. Additional force balance for the individual particles.
Species mixing:
Material distribution, mixing parameters.
Basis for chemical reaction calculations.
DPM:
Heat and mass transfer from particles.
Mixing analysis.
Unlike species, do not necessarily follow fluid flow exactly.