Particle Tracking in the Circle of Willis Pim van Ooij A ... · Particle Tracking in the Circle of...

Particle Tracking in the Circle of Willis

Pim van OoijA report of work carried out atthe Centre of Bioengineering

University of CanterburyTU EindhovenBMTE05.40

June 2005

1

Summary

An ischemic stroke is most commonly caused by a blood clot that blocks anartery in the brain, resulting in oxygen deficit in brain tissue and accompanyingbrain damage. To examine the route a blood clot travels in the Circle of Willis,the ring-like structure of blood vessels that distribute the blood flow to thecerebral mass, the influence of outlet diameter, bifurcation angle and mass fluxof five different geometries on particle trajectory is studied, with the intentionto predict particle trajectories in the Circle of Willis. The finite volume packageFluent is used, which supports two particle models: the Discrete Phase Model(DPM) and the Macroscopic Particle Model (MPM). The difference betweenthe models is the degree of interaction between the fluid and the particles. Theresults show a larger effect of outlet diameter and bifurcation angle on theparticle trajectory for MPM than for DPM. For DPM, there is a linear relationbetween mass flux and density of particles. The higher the mass flux, the higherthe density of particles in the flow. Since momentum transfer between fluid andparticles for MPM is larger, the relation between mass flux and amount ofparticles is not as obvious. A simulation using DPM of particle tracks in theCircle of Willis produces predicted results. Unfortunately it was not possible toverify the prediction of particle tracks in the Circle of Willis for MPM.

Contents

1 Introduction 3

2 Theory and Methodology 42.1 Emboli . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.2 Circle of Willis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.3 Geometry model . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.3.1 Geometries . . . . . . . . . . . . . . . . . . . . . . . . . . 62.3.2 Circle of Willis . . . . . . . . . . . . . . . . . . . . . . . . 8

2.4 Fluids model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.4.1 Geometries . . . . . . . . . . . . . . . . . . . . . . . . . . 92.4.2 Circle of Willis . . . . . . . . . . . . . . . . . . . . . . . . 10

2.5 Particle model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.5.1 Discrete Phase Model . . . . . . . . . . . . . . . . . . . . 112.5.2 Macroscopic Particle Model . . . . . . . . . . . . . . . . . 13

3 Results 163.1 Geometries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

3.1.1 General . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163.1.2 Outlet diameter . . . . . . . . . . . . . . . . . . . . . . . 183.1.3 Bifurcation angle . . . . . . . . . . . . . . . . . . . . . . . 233.1.4 Mass flux . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3.2 Circle of Willis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333.2.1 General . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333.2.2 Outlet diameter . . . . . . . . . . . . . . . . . . . . . . . 333.2.3 Bifurcation angle . . . . . . . . . . . . . . . . . . . . . . . 343.2.4 Mass flux . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

4 Discussion 364.1 Geometries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

4.1.1 Outlet diameter . . . . . . . . . . . . . . . . . . . . . . . 364.1.2 Bifurcation angle . . . . . . . . . . . . . . . . . . . . . . . 374.1.3 Mass flux . . . . . . . . . . . . . . . . . . . . . . . . . . . 384.1.4 Differences between DPM and MPM . . . . . . . . . . . . 41

4.2 Circle of Willis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

5 Conclusion 43

6 Recommendations 44

1

CONTENTS CONTENTS

A Discrete Phase Model Manual 46A.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46A.2 Limitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47A.3 Overview of Discrete Phase Modeling Procedures . . . . . . . . . 48A.4 Equations of Motion of Particles . . . . . . . . . . . . . . . . . . 49A.5 Coupling Between the Discrete and Continuous Phases . . . . . . 52

B Macroscopic Particle Model Manual 54

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Chapter 1

Introduction

The cardiovascular disease called stroke is the third largest cause of death, af-ter heart diseases and all forms of cancer. There are two types of strokes: theischemic stroke and the hemorrhagic stroke. Although the latter type of strokehas a much higher fatality rate than the former, this project will focus on theformer type of stroke which causes 70-80 percent of all strokes. An ischemicstroke occurs when a blood vessel to the brain is blocked by an abnormal bloodclot, called embolus. Due to the following deprivation of oxygen, brain tissuedies which causes irreparable brain damage or death.The embolus most often blocks an artery in the Circle of Willis, a ring-like struc-ture of blood vessels found beneath the hypothalamus at the base of the brain,which main function is to distribute oxygen-rich blood to the cerebral mass.To know more about the route a blood clot travels in the brain, simulations ofparticle trajectories in the Circle of Willis are needed, which is the main goal ofthis project.Five different geometries are used to study the influence of artery diameter,bifurcation angles and mass flux on the particle trajectories. Mass flux is variedby prescribing different boundary conditions on the outlets of the geometries.With this information the particle tracks in the Circle of Willis are predicted.This prediction is then verified by a simulation of particle tracks on the Circleof Willis.The null hypothesis of this project is that there is no relationship between thegeometry of the Circle of Willis and the distribution of the particles. How-ever, it is expected that more mass flux through an artery causes more particleflow through this artery. Also, for equal pressure gradients, a large diameter ofan artery is supposed to cause more mass flux through this particular artery,bringing more particles with it, then through an artery with a smaller diameter.Furthermore, it is assumed that a smaller bifurcation angle causes more massflux through the adjacent artery, causing more particle flow. By comparing theresulting particle distribution on the outlets with the particle distribution onthe inlet of the geometries, it is possible to predict particle tracks in the Circleof Willis, so that more knowledge is obtained about the trajectory of blood clotsin the brain.

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Chapter 2

Theory and Methodology

2.1 Emboli

Bloods clots are created in so-called thromboembolic conditions, which are usu-ally caused twofold [1]. First, a roughened surface, caused by arteriosclerosis,infection or trauma, is likely to initiate the clotting process. Secondly, clottingoccurs in blood that flows very slowly through blood vessels because small quan-tities of thrombin and other procoagulants are always being formed. These aregenerally removed from the blood by the macrophage system, mainly the Kupf-fer cells of the liver. The concentration of procoagulants rises often high enoughto initiate clotting, but when blood flows rapidly, they are mixed with largequantities of blood and are removed during passage through the liver. When anabnormal blood clot develops in a blood vessel, it is called a thrombus. Con-tinued flow of blood past the clot is likely to break it away from its attachmentresulting in a clot that flows along with the blood. These free flowing clots areknown as emboli, which do not stop flowing until they come to a narrow pointin the circulatory system. Thus, emboli originated in the venous system and inthe right side of the heart flow into the vessels of the lung and cause pulmonaryarterial embolism. Emboli originated in large arteries or in the left side of theheart will plug smaller arteries of arterioles in the brain, kidneys or elsewhere.

2.2 Circle of Willis

The Circle of Willis is a ring-like structure of blood vessels in the brain, displayedin figure 2.1 and distributes oxygenated blood throughout the cerebral mass.

It is estimated that among the general population, 50% have a completeCircle of Willis [2]. Variations include underdeveloped of even absent bloodvessels. This can present a health risk, mostly for ischaemic stroke, while anindividual with one of these variations may suffer no ill effects. Examples ofthese variations are the fetal P1, where the P1 section of the Posterior CerebralArtery (PCA) is underdeveloped, and the missing A1, where the A1 sectionof the Anterior Cerebral Artery is missing, as displayed in figure 2.2. In thesefigures are displayed the abbreviations of the names of the arteries in the Circleof Willis. These will be briefly discussed.

In figure 2.2, R stands for right and L for left. The Internal Carotid Arteries

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Theory and Methodology 2.2 Circle of Willis

Figure 2.1: The Circle of Willis, frontal

(a) Fetal P1 (b) Missing A1

Figure 2.2: Variations in the Circle of Willis

(ICA), the Vertebral Arteries (VA) and the Basilar Artery (BA) are the vesselsthat transport blood into the Circle of Willis and are called afferent arteries.The Middle Cerebral Artery (MCA), Posterior Cerebral Artery (PCA) and theAnterior Cerebral Artery (ACA) are the vessels that transport blood from theCircle of Willis and are called efferent arteries. The ICA is connected withthe PCA via the Posterior Communicating Artery (PCoA) and the Left ACA isconnected with the Right ACA via the Anterior Communicating Artery (ACoA).

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Theory and Methodology 2.3 Geometry model

2.3 Geometry model

2.3.1 Geometries

To study the influence of outlet diameter, bifurcation angle and mass flux onparticle trajectories, five different geometries are used. The geometry meshesare created by the meshing software package GAMBIT and comprise of approx-imately 300,000 tetrahedral volumes. The different geometries all consist of oneinlet and two outlets. The diameter of the inlet is 5 millimeters. From inlet tooutlet the geometries measure approximately 45 millimeters and from outlet tooutlet the geometries measure approximately 30 millimeters, see figure 2.3.1.

Figure 2.3: Measures of geometry 1

The diameters of the outlets vary, as well as the bifurcation angles of theoutlets. The geometries are displayed in figure 2.4. The different features of thegeometries are summarized in table 2.1. For bifurcations of vessels in the humanbody, a physiological relation exist between outlet diameter and bifurcationangle and is given by Zamir [3]:

cos θ1 =(1 + α3)4/3 + 1 − α4

2(1 + α3)2/3(2.1)

cos θ2 =(1 + α3)4/3 + α4 − 1

2α2(1 + α3)2/3(2.2)

withα =

a2

a1(2.3)

where θ1 and θ2 are the bifurcation angles, a1 is the diameter of outlet 1and a2 the diameter of outlet 2. The diameters of the geometries are randomlychosen. These equations are used to calculate the bifurcation angles of geometry5. Then geometry 4 is taken as the opposite of geometry 5. The properties ofgeometry 5 are calculated by equations 2.1 and 2.2 and can therefore actuallybe found in the human body, while geometries 1, 2, 3 and 4 are hypotheticalgeometries.In figure 2.4 the right outlets of the geometries are outlets 1, the left ones outlets2.

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(a) Geometry 1 (b) Geometry 2

(c) Geometry 3 (d) Geometry 4

(e) Geometry 5

Figure 2.4: Geometries used for particle tracking

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Table 2.1: Properties of the five different geometriesGeometry D outlet 1 D outlet 2 Angle outlet 1 Angle outlet 2

mm mm o o

1 3 3 40 402 1.5 3 40 403 2.5 2.5 30 504 2 3 21.26 54.685 3 2 21.26 54.68

2.3.2 Circle of Willis

The 3D mesh of the Circle of Willis comprises of approximately 1 million tetra-hedral volumes. The mesh is displayed in figure 2.5. The diameters of thevarious arterial segments of the Circle of Willis were obtained from a popula-tion study of retrospective MRA scans [2]. The measurements and standarddeviations are displayed in table 2.2.

Figure 2.5: Mesh of the Circle of Willis

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Theory and Methodology 2.4 Fluids model

Table 2.2: Circle of Willis MeasurementsArtery Diameter Std Dev

mm mmACA - A1 Anterior Cerebral Artery - A1 2.33 0.22ACA - A2 Anterior Cerebral Artery - A2 2.40 0.31MCA Middle Cerebral Artery 2.86 0.17PCA - P1 Posterior Cerebral Artery - P1 2.13 0.25PCA - P2 Posterior Cerebral Artery - P2 2.10 0.21ACoA Anterior Communicating Artery 1.47 0.17PCoA Posterior Communicating Artery 1.45 0.31BA - B1 Basilar Artery - B1 3.17 0.51BA - B2 Basilar Artery - B2 3.29 0.44ICA Internal Carotid Artery 4.72 0.26

Since the bifurcation angles in the Circle of Willis are not measured, a roughestimation of 90o is made for the bifurcation between the ICA and the PCoA,and between the ICA and the ACA. There is no angle between ICA and MCA.This estimation is done by rotating the Circle of Willis in Fluent and study itsfeatures by eye.

2.4 Fluids model

2.4.1 Geometries

The blood flow through the geometries is modeled as unsteady, incompressibleand viscous. This means that the governing transport equations to be solvedby Fluent are the continuity equation and the momentum equation:∫

A

ρu · dA = 0 (2.4)

∫V

δuδt

+∫

A

ρuu · dA = −∫

A

pIdA +∫

A

η(ε)ε · dA +∫

V

FdV (2.5)

where u is the velocity vector, ρ is the blood density, with a value of 1410kg/m3, p is the pressure, η is the fluid viscosity and F is the body forces vectoron the fluid. V is a closed volume and A is the edge of a closed volume. ε is thestrain rate tensor and is represented as:

ε =12(∇u + ∇uT ) (2.6)

In this project blood is simulated as a non-Newtonian fluid and thereforethe Carreau-Yasuda model for the viscosity is implemented:

η − η∞η0 − η∞

= (1 + (λγ)a)n−1

a (2.7)

where η∞ is the infinite shear viscosity, set at 0.0022 Pa · s and η0 is thezero shear viscosity, taken as 0.022 Pa · s. λ is 0.11 s, a is taken as 0.644 and n

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Theory and Methodology 2.5 Particle model

as 0.392 [2]. γ is the strain rate magnitude, derived from the second invariantof the strain rate tensor, which for an incompressible fluid becomes:

γ = 2√

εij εij (2.8)

These equation properties are taken from Moore [2]. To study the influenceof different mass fluxes on the particle trajectories, for each geometry, threedifferent pressure profiles are prescribed on the outlets. In the first situation,the pressure on outlet 1 is the same as the pressure on outlet 2: 98 mm Hg. Inthe second situation, a pressure of 95 mm Hg is prescribed on outlet 1, whilethe pressure on outlet 2 remains 98 mm Hg. Finally, the pressure on outlet 1returns to 98 mm Hg, while the pressure on outlet 2 is set to 95 mm Hg. Thepressure on the inlet remains constant at 100 mm Hg at all times. These valuesare an estimation of pressure differences in arteries of equal size in the humanbody. The boundary conditions on the outlets of the geometries are summarizedin tabel 2.3. In this report, they will be referred to as boundary condition 1,boundary condition 2 and boundary condition 3.

Table 2.3: Boundary conditions on the geometriesBoundary outlet Pressurecondition mm Hg

1 1 982 98

2 1 952 98

3 1 982 95

A time step of 0.01 seconds is used in all the simulations. The Reynoldsnumber of the flows are defined on the inlets and are calculated by:

Re =4m

πdη∞(2.9)

with m the mass flux through the inlet and d the diameter of the inlet.

2.4.2 Circle of Willis

All the equations mentioned in subsection 2.4.1 apply to the Circle of Willisas well. The boundary conditions on the inlet and outlets of the geometriesare chosen as described in the previous subsection, thus they differ from theboundary conditions on the inlets and outlets of the Circle of Willis, since theseare chosen as the mean values of the pressures in systole and diastole of arteries(afferent arteries) and veins (efferent arteries). These are summarized in table2.4 and are equal for left and right arteries.

2.5 Particle model

Two models to simulate particle tracks are implemented in Fluent: the DiscretePhase Model (DPM) and the Macroscopic Particle Model (MPM). In this section

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Table 2.4: Boundary conditions on the Circle of WillisArtery Pressure

mm HgVA 100ICA 100ACA 4MCA 4PCA 4

both models will be described. Unfortunately, it is not yet possible to applyMPM to the Circle of Willis, since the MPM algorithms are not fully developedyet.To study particle tracks in geometries, first of all, particles need to be injectedin a certain prescribed distribution. The distribution of the particles in theinjection is the same for DPM and MPM at the inlet of the geometries as forDPM in the Circle of Willis. In the latter, the particles are injected in theRICA, see figure 2.2. The injection is displayed in figure 2.6. The distributionof the injection is chosen after the fact that emboli develop attached to vesselsand thus are transported at the sides of a flow. No initial velocity is chosen forthe particles. The particles used both in DPM and MPM, and in the geometriesand the Circle of Willis, are of the material anthracite, with a density of 1550kg/m3 and a diameter of 0.5 mm. The value of the density is chosen afterthe assumption that a particle has a higher density than a liquid of the samematerial. The value of the diameter chosen so that multiple particles in aninjection can be simulated.

Figure 2.6: Injected particle distribution in all geometries

2.5.1 Discrete Phase Model

In addition to solving the transport equations for the continuous phase, Fluentis able to simulate a discrete second phase that consists of spherical particlesdispersed in the continuous phase. To calculate the discrete phase trajectories,Fluent uses a Lagrangian formulation that includes the discrete phase inertiaand hydrodynamic drag [4]. This formulation is only suited for a continuousphase flow with a well-defined entrance and exit. It contains the assumptionthat the second phase is sufficiently dilute, so that particle-particle interactionsare negligible. This implies that the volume fraction of the discrete phase mustbe sufficiently low, usually less than 10-12%. At a total particle volume ofapproximately 5 mm3 and a fluid volume around 500 mm3, this is the case forthe geometries. Since the total volume of the Circle of Willis is larger than the

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volume of a geometry, the volume fraction of the discrete phase of the Circle ofWillis is lower than the volume fraction of a geometry.The trajectory of a discrete phase particle is calculated by use of integrationof the force balance on the particle, which is written in a Lagrangian referenceframe. This force balance equates the particle inertia with the forces acting onthe particle, and can be written as:

dup

dt= FD(u − up) +

gx(ρp − ρ)ρp

+ Fx (2.10)

Since gravity is omitted, and the additional forces, Fx only play a role whenρ > ρp [4], equation A.1 reduces to:

dup

dt= FD(u − up) (2.11)

with FD(u − up) the drag force per unit particle mass and FD defined as

FD =18µ

ρpd2p

CDRe

24(2.12)

with Reynolds number Re defined as

Re =ρdp | up − u |

µ(2.13)

and drag coefficient CD as

CD = a1 +a2

Re+

a3

Re2(2.14)

where u is the continuous phase (fluid) velocity, up the particle velocity, µthe kinematic viscosity of the fluid, ρ the density of the fluid, ρp the density ofthe particle, dp the particle diameter and a1, a2 and a3 constants that applyfor smooth spherical particles over several ranges of Re given by Morsi andAlexander [5].The trajectory equations are solved by stepwise integration over discrete timesteps. Integration in time of equation 2.11 yields the velocity of the particle ateach point along the trajectory, with the trajectory itself predicted by:

dx

dt= up (2.15)

With y and z in place of x for every coordinate direction. Equation 2.11 canbe rewritten in simplified form as:

dup

dt=

1τp

(u − up) (2.16)

where τp is the particle relaxation time. Then a trapezoidal scheme is usedfor integrating 2.16:

un+1p − un

p

∆t=

1τp

(u∗ − un+1p ) + ... (2.17)

where n represents the iteration number and

12


u∗ =12(un + un+1) (2.18)

un+1p = un

p + ∆tunp · ∇un

p (2.19)

These final two equations are solved simultaneously to determine the veloc-ity and position of the particle at any given time.In these simulations the coupled approach is used, which means that the con-tinuous flow pattern is impacted by the discrete phase and vice versa. Thecalculations of the continuous and discrete phase are alternated until a con-verged solution is achieved. The momentum transfer from the continuous phaseto the discrete phase is computed in Fluent by examining the change in mo-mentum of a particle as it passes through each control volume in the model, asillustrated in figure 2.7.

Figure 2.7: Momentum transfer between the discrete and the continuous phases

The momentum transfer from the continuous phase to the discrete phaseis computed by examining the change in momentum of a particle as it passesthrough each control volume in the Fluent model:

F =∑

(18µCDRe

ρpd2p24

(up − u) + Fother)mp∆t (2.20)

where Fother are other interaction forces, in this case 0, mp is the mass flowrate of the particles and ∆t is the time step.When a particle strikes a wall, it slides along the wall depending on the particleproperties and the impact angle [4].

2.5.2 Macroscopic Particle Model

The Macroscopic Particle Model takes into account:

• The blockage and the momentum transfer of the fluid by particles.

13


• Evaluation of the drag force and the torque experienced by the particles.

• Particle-particle as well as particle-wall collision.

In MPM, the particles are treated in a Lagrangian frame of reference as well.Rigid body velocity of a particle is imposed on the fluid cells that are touchedby the particle, as displayed in figure 2.8 [6].

Figure 2.8: Particle velocity of a particle imposed on touched cells

Firstly, this means that the particles add momentum to the fluid. Themomentum is integrated and the particle drag and particle torque vectors arecalculated for each particle. From these vectors the new position, velocity andangular velocity of the particles are calculated. Secondly, this means that mo-mentum can also diffuse from touched cells to the particles. This momentumrepresents the hydrodynamic forces on the particles.Particle-wall collision works in the following manner. First, the boundary facesthat are intersected by the particle are identified. Secondly, the particle velocityonto the normal and tangential vector of the wall is projected. Finally, the resti-tution coefficient is applied to calculate the outgoing velocity of the particle.Particle-particle collision is determined in a similar manner. At first, the parti-cles which are going to collide are detected. The line-of-action of the collisionis found next, which identifies the normal direction. Then, incoming particlevelocities are projected onto the line-of-action to get the normal and tangentialcomponents. Finally, the coefficient of restitution and conservation of momen-tum to the normal components of the incoming velocities are applied to obtainthe final velocities of the particles. The coefficients of restitution in the sim-ulations are set to 0.8. For MPM, it is necessary to define the mass and themoment of inertia of the particles. The mass of the particles is set to 1e-5 kg,the moment of inertia to 1e-3 kg/m2, sufficiently small to neglect. Furthermore,particle-particle attraction force and particle-wall attraction force are set to 1.For simulations of flow and pressure on the Circle of Willis, Fluent needs alibrary containing user-defined functions that describes the behavior of theseproperties. However, the MPM code for Fluent is also written in a library ofuser-defined functions and unfortunately, for this version of Fluent, it was not

14


possible to use multiple libraries for the same simulation. Nor was it possibleto combine both the libraries, since the user-defined functions of MPM are notknown, as it is a part of the Fluent program. In future versions of Fluent andMPM, it will be possible to use multiple libraries and the simulation using MPMon the Circle of Willis can be done.

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Chapter 3

Results

3.1 Geometries

3.1.1 General

On each geometry, three different boundary condition situations are set. Since5 geometries are used, 15 simulations are done for DPM and 15 simulations forMPM. A total of 30 simulations.The DPM results are obtained using Fluent. It is possible in DPM to trap theparticles on the outlets, so that the positions of the particles on the outletsare easy to determine. For MPM, trapping particles on outlets is not possible.The positions of the particles in MPM are saved in files, the final positions areisolated, and displayed using Matlab. Therefore, the results of DPM and MPMlook slightly different. Unfortunately, it was not possible in Matlab to displaythe particles at real size. An example of a result is given in figure 3.1.

16

Results 3.1 Geometries

(a) DPM, Geometry 1, equal boundary conditions

(b) MPM, Geometry 1, equal boundary conditions

Figure 3.1: Example of DPM results and MPM results

17


From these results, it is easy to determine the amount of particles on outlet1 and how many on outlet 2. This data in combination with the geometry andboundary conditions is used to evaluate particle trajectories in various circum-stances.In tables 3.1 and 3.2 duration of the simulations is summarized.

Table 3.1: Time for all particles to be trapped on the outlets for DPM in seconds

Geometry 1 2 3 4 5boundaryconditions98 outlet 198 outlet 2 0.64 0.52 1.11 0.49 0.4995 outlet 198 outlet 2 0.41 0.49 0.42 0.49 0.4098 outlet 195 outlet 2 0.40 0.27 0.43 0.41 0.69

Table 3.2: Time for all particles to exit the outlets for MPM in seconds

Geometry 1 2 3 4 5boundaryconditions98 outlet 198 outlet 2 0.60 0.71 0.68 0.69 0.6495 outlet 198 outlet 2 0.54 0.67 0.56 0.55 0.5998 outlet 195 outlet 2 0.50 0.85 0.53 0.58 0.57

First, the influence of diameter of the outlets of the geometries and bifurca-tion angle on the particle trajectories is evaluated. Then, the influence of massflux on the particle trajectories is evaluated.

3.1.2 Outlet diameter

To evaluate the influence of outlet diameter on the distribution of the particleson the outlets, first, a comparison is made between geometry 1 and geometry2, since the diameter of outlet 1 of geometry 2 is half the size of outlet 1 ofgeometry 1, while outlets 2 of both geometries remain the same size. Secondly,a comparison is made between geometries 4 and 5, where geometry 5 is theinverse is of geometry 4, see figure 2.4 and table 2.1. The results for DPM aredisplayed in figure 3.3, the results for MPM are displayed in figure 3.5

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(a) Percentage of particles against different mass fluxes on geometry 1 where the diametersof outlets 1 and 2 both are 3 mm.

(b) Percentage of particles against different mass fluxes on geometry 2 where the diameterof outlet 1 is 1.5 mm and the diameter of outlet 2 is 3 mm.

Figure 3.2: Percentage of particles against the diameters of geometries 1 and 2for DPM

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(a) Percentage of particles against different mass fluxes on geometry 4 where the diameterof outlet 1 is 2 mm and the diameter of outlet 2 is 3 mm.

(b) Percentage of particles against different mass fluxes on geometry 5 where the diameterof outlet 1 is 3 mm and the diameter of outlet 2 is 2 mm.

Figure 3.3: Percentage of particles against the diameters of geometries 4 and 5for DPM.

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(a) Percentage of particles against different mass fluxes on geometry 1 where the diametersof outlets 1 and 2 both are 3 mm.

(b) Percentage of particles against different mass fluxes on geometry 2 where the diameterof outlet 1 is 1.5 mm and the diameter of outlet 2 is 3 mm.

Figure 3.4: Percentage of particles against the diameters of geometries 1 and 2for MPM.

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(a) Percentage of particles against different mass fluxes on geometry 4 where the diameterof outlet 1 is 2 mm and the diameter of outlet 2 is 3 mm.

(b) Percentage of particles against different mass fluxes on geometry 5 where the diameterof outlet 1 is 3 mm and the diameter of outlet 2 is 2 mm.

Figure 3.5: Percentage of particles against the diameters of geometries 4 and 5for MPM.

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3.1.3 Bifurcation angle

To evaluate the influence of the bifurcation angles of the geometries, the resultsof geometry 1 are compared with the results of geometry 3. This is because bothgeometries have equal diameters on the outlets, while the bifurcation anglesdiffer. There is, however, a difference between the diameters of the outlets ofgeometry 1 and geometry 3, but this is not considered as of influence on theparticle distribution, since equal diameter is thought to cause equal distributionof particles. Furthermore, a comparison is made between the outlets with thesame diameter of geometry 4 and 5. This means that outlet 1 of geometry 4is compared with outlet 2 of geometry 5 and the opposite. See figure 2.4 andtable 2.1 for details.

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(a) Percentage of particles against different mass fluxes on geometry 1 where the bifurcationangles of outlets 1 and 2 both are 40o.

(b) Percentage of particles against different mass fluxes on geometry 3 where the bifurcationangle of outlet 1 is 30o and where the bifurcation angle of outlet 2 is 50o.

Figure 3.6: Percentage of particles against the bifurcation angles of geometries1 and 3 for DPM.

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(a) Percentage of particles against different mass fluxes on geometry 4 where the bifurcationangles of outlet 1 is 21.26o and where the bifurcation angle of outlet 2 is 54.68o.

(b) Percentage of particles against different mass fluxes on geometry 4 where the bifurcationangles of outlet 1 is 21.26o and where the bifurcation angle of outlet 2 is 54.68o.

Figure 3.7: Percentage of particles against the bifurcation angles of geometries4 and 5 for DPM.

25


(a) Percentage of particles against different mass fluxes on geometry 1 where the bifurcationangles of outlets 1 and 2 both are 40o.

(b) Percentage of particles against different mass fluxes on geometry 3 where the bifurcationangles of outlet 1 is 30o and where the bifurcation angle of outlet 2 is 50o.

Figure 3.8: Percentage of particles against the bifurcation angles of geometries1 and 3 for MPM.

26


(a) Percentage of particles against different mass fluxes on geometry 4 where the bifurcationangles of outlet 1 is 21.26o and where the bifurcation angle of outlet 2 is 54.68o.

(b) Percentage of particles against different mass fluxes on geometry 5 where the bifurcationangles of outlet 1 is 21.26o and where the bifurcation angle of outlet 2 is 54.68o.

Figure 3.9: Percentage of particles against the bifurcation angles of geometries4 and 5 for MPM.

27


3.1.4 Mass flux

Different boundary conditions on the outlets will cause different mass fluxesthrough the artery. For every geometry, three different pressure situations isapplied, see table 2.3. To examine the influence of mass flux on particle trajec-tory, the percentage of particles on the outlets is plotted against the mass fluxfor every geometry. This means that each plot contains six points: the percent-age of particles on outlet 1 for three different mass fluxes, and the percentage ofparticles on outlet 2 for three different mass fluxes. First, the results for DPMare displayed. The results of DPM can be found in figure 3.10, the results ofMPM in figure 3.11. In figure 3.12(a) and figure 3.12(b) all percentages of theparticles on the outlets of all simulations are plotted against every mass flux onthe outlets. The mass fluxes on the inlets are used to calculate the Reynoldsnumbers of the different flows. These are displayed in table 3.3 for DPM and intable 3.4 for MPM.



28


(e) Geometry 5

Figure 3.10: Percentage of particles on the outlets as a function of mass flux inDPM

29




(e) Geometry 5

Figure 3.11: Percentage of particles on the outlets as a function of mass flux inMPM

30


(a) Percentage of particles as a function of mass flux of every simulation in DPM

(b) Percentage of particles as a function of mass flux of every simulation in MPM

Figure 3.12: The percentage of particles as a function of mass flux for everysimulation in DPM and MPM

31


Table 3.3: Reynolds numbers for different boundary conditions of the geometriesbased on the inlet for DPM

Geometry 1 2 3 4 5boundaryconditions98 outlet 198 outlet 2 407 242 284 272 28295 outlet 198 outlet 2 574 274 384 345 49398 outlet 195 outlet 2 572 443 380 476 368

Table 3.4: Reynolds numbers for different boundary conditions of the geometriesbased on the inlet for MPM

Geometry 1 2 3 4 5boundaryconditions98 outlet 198 outlet 2 442 258 278 307 30595 outlet 198 outlet 2 699 305 423 396 56098 outlet 195 outlet 2 678 526 427 566 397

32

Results 3.2 Circle of Willis


3.2.1 General

In figure 3.13 the end state of the the results of DPM on the Circle of Willis isdisplayed. The whole simulation can be found in .mpeg file called sequence-1or ParticlesCoW. The development of each particle from injection to end statecan be studied in this movie. From the figure only the final distribution on theoutlets of the Circle of Willis can be derived. Since some particles are trappedin the same positions on the MCA-outlet, it’s not clear to see that there are6 particles trapped on the the MCA-outlet. Two particles are trapped on theACA, and one on the PCA.

Figure 3.13: Result of particle trajectory on the Circle of Willis using DPM


In figure 3.14 the percentage of the particles on the outlets is plotted. The bloodflow through the PCA is an addition of the blood flow through the BA and the

33


blood flow through the PCoA. Since the particle trajectory of the particle foundon the PCA contains the PCoA and not the BA, the contribution of blood flowfrom the BA is not considered. Therefore, instead of displaying the percentage ofparticles on the PCA, the percentage of particles through the PCoA is displayed.The diameters of the displayed outlets can be found in table 2.2 in subsection2.1.2.

Figure 3.14: Percentage of particles as a function of diameter outlet of the Circleof Willis using DPM


For the results of bifurcation angle, figure 3.14 can be used. The angles betweenthe arteries are mentioned in subsection 2.1.2.

34


3.2.4 Mass flux

Figure 3.15: Percentage of particles as a function of mass flux through theoutlets of the Circle of Willis using DPM

The Reynolds number defined on the inlet of the right ICA is 769. Thesimulation time is 1.09 seconds.

35

Chapter 4

Discussion

4.1 Geometries


DPM

The diameter of outlet 1 of geometry 2 is two times as small as the diameterof outlet 1 of geometry 1. In comparison of figure 3.2(a) and 3.2(b) it showsthat, for all three boundary conditions, the percentage of particles on outlet 1of geometry 2 is significantly less than the percentage of particles on outlet 1of geometry 1. Since there is no difference in bifurcation angle, this indicatesthat a smaller diameter of an outlet is of significant influence on the particledistribution on the outlets for DPM. This is supported by comparing figure3.3(a) with figure 3.3(b) where outlet 1 of geometry 4 traps a significant smallerpercentage of particles than outlet 1 of geometry 5 for all pressures submittedon the outlets.It is obvious that if a smaller diameter of an outlet causes a smaller percentage ofparticles on this outlet, a larger diameter of an outlet causes a larger percentageof particles on this outlet. This can also be found by observing figure 3.3.

MPM

When comparing figure 3.4(a) with 3.4(b), it is seen that for geometry 2, thediameter outlet of outlet 1 has a large effect on the particle distribution, sincethe percentages of particles are significantly lower than the percentages on outlet1 of geometry 1. Furthermore, when figure 3.5(a) is compared with 3.5(b), itcan be seen that for the three boundary conditions, the percentage of particleson outlet 1 of geometry 5 is significantly higher than the percentage of particleson outlet 1 of geometry 4. The hypothesis that a smaller diameter of an outletcauses a smaller percentage of particles on the outlet is thus supported by theresults of MPM.

36

Discussion 4.1 Geometries


DPM

Note that figure 3.6(a) is exactly the same as figure 3.6(b). The same amount ofparticles is found on the outlets of geometry 3 as on the outlets of geometry 1,while the bifurcation angles of the geometries differ. This result does not supportthe hypothesis that a larger bifurcation angle causes a smaller percentage ofparticles through the artery.The comparison between figure 3.7(a) and figure 3.7(b) is as follows. Outlet 1 ofgeometry 4 needs to be compared with outlet 2 of geometry 5, since these outletsboth have a diameter of 2 mm, while the bifurcation angles differ. Outlet 2 ofgeometry 4 needs to be compared with outlet 1 of geometry 5, since these outletshave a diameter of 3 mm. This means that boundary condition 2 of geometry4 needs to be compared with the boundary condition 3 of geometry 5 and theopposite. Now, it can be seen that the results for boundary condition 1 for bothgeometries and boundary condition 3 for geometry 4 and boundary condition2 for geometry 5 are equal. The only indication that a different bifurcationangle might be of influence on the distribution of particles on the outlets canbe found in the comparison between boundary condition 2 of geometry 4 andboundary condition 3 of geometry 5. Here it can be seen that more particles arefound on outlet 1 of geometry 5, where the bifurcation angle is 21.26o, than onoutlet 2 of geometry 4, where the bifurcation angle is 54.68o, and less particleson outlet 2 of geometry 5, where the bifurcation angle is 54.68o, than on outlet1 of geometry 4, where the bifurcation angle is 21.26o. It is from these results,however, not clear that, for DPM, a smaller bifurcation angle causes a largerpercentage of particles on the outlet.

MPM

From figure 3.8(a) and figure 3.8(b) it is observed that a higher percentage ofparticles, for all three boundary conditions, is found on outlet 1 of geometry3, where the bifurcation angle is 30o, than on outlet 1 of geometry 1, wherethe bifurcation angle is 40o. Consequently, a lower percentage of particles isfound on outlet 2 of geometry 3, where the bifurcation angle is 50o, than onoutlet 2 of geometry 1, where the bifurcation angle is 40o. Furthermore, figures3.9(a) and 3.9(b) show that a difference in bifurcation angle, for MPM, doescause a difference in particle distribution. There are significantly more particlesfound on outlet 1 of geometry 5 than on outlet 2 of geometry 4, when a pressureof 98 mm Hg is applied on both outlets. This is also the case for the othertwo boundary condition situations, when outlet 1 of geometry 4 is comparedwith outlet 2 of geometry 5 and vice-versa. It can be seen that when boundarycondition 2 is applied on geometry 4, a larger amount of particles is found thanon outlet 2 of geometry 5, when boundary condition 3 is applied. This is also thecase for boundary condition 2 on geometry 5 compared with boundary condition3 on geometry 4. Thus, the smaller the bifurcation angle, the larger the amountof particles. This result indicates that bifurcation angle is of significant influenceon particle trajectories in MPM.

37


4.1.3 Mass flux

DPM

From figure 3.10(a), it can be derived that when boundary condition 1 on geom-etry 1 is applied, this results in a nearly equal mass flux through both outlets.There is a slightly larger mass flux through outlet 2, which causes one particlemore on this outlet than on outlet 1. The small difference in mass flux is prob-ably due to the mesh, which is unstructured and thus not totally symmetrical.From this figure, it can also be derived that a pressure of 95 mm Hg on anoutlet causes a larger mass flux through this outlet and consequently a smallermass flux through the other outlet where a pressure of 98 mm Hg is applied.As a result, a larger amount of particles is found on the outlet where the massflux is higher. This indicates that there exists a linear relation between massflux and particle percentage.Since the Reynolds number is a measure for the mass flux through the inlet ofthe flow, in table 3.3, for geometry 1, it can be seen that the different boundaryconditions have an obvious effect on the total mass flux of the flow. When apressure of 95 mm Hg is applied on one on the outlets, and 98 mm Hg on theother, this causes a higher total mass flux of the flow than when a pressure of98 mm Hg on both outlets is applied.When these Reynolds numbers are compared with the time of the simulationsfound in table 3.1, it can be seen that a higher mass flux through the flow causesa higher particle velocity for geometry 1, since the travel time of the particlesis shorter than the travel time of the particles when a lower Reynolds numberis found.In figure 3.10(b) it can be seen that the mass flux through outlet 1 of geometry2, where the diameter is 1.5 mm, is for every boundary condition less than themass flux through outlet 1 of geometry 1, where the diameter is 3 mm. Thisresults in a small amount of percentages of particles on outlet 1, even 0 whena pressure of 95 mm Hg is applied on outlet 2. The mass flux through outlet2 of geometry 2 does not differ much from the mass flux through outlet 2 ofgeometry 1. This means that the total mass flux of the flow through geometry 2is lower than the total mass flux of the flow through geometry 1. This indicatesthat there exists a linear relation between mass flux and outlet diameter.The above can be verified by observing the Reynolds numbers in table 3.3. Asexpected, since total mass flux is lower, the Reynolds number of the flow throughgeometry 2 is lower than the Reynolds number of the flow through geometry 1.Also, when a pressure of 95 mm Hg is applied to outlet 1 of geometry 2, it has asmaller effect on the mass flux through the flow than when a pressure of 95 mmHg is applied to outlet 2. These results support the hypothesis that a smallerdiameter of an outlet causes less mass flux, and consequently less particles, toflow through this outlet.It can be seen in table 3.1 that the simulation times between the first twoboundary conditions situations are nearly equal. This is as expected, since thecorresponding Reynolds numbers do not differ much. When the Reynolds num-ber increases in boundary condition 3, the time for the particles to be trappedon the outlet decreases.There is, however, a large difference between simulation times of geometry 1and geometry 2. The simulation time of equal pressures of geometry 2 is lower

38


than the simulation time of equal pressures of geometry 1, while the Reynoldsnumber of the flow through geometry 2 is smaller than the Reynolds number ofthe flow through geometry 1. The opposite is expected. It is not known whythis is the case. Apparently, for different geometries, different relations betweenReynolds number and duration of simulation apply.A remarkable result is found in figure 3.10(c). For boundary condition 1, themass flux through outlet 1 is slightly higher than the mass flux through outlet2, as expected by bifurcation angle difference. On outlet 1, however, a loweramount of particles is found than on outlet 2. This indicates that the interac-tion between continuous phase and discrete phase can cause unexpected resultsin DPM. This result can be explained by a momentum transfer of fluid to theparticles, which causes a decrease in mass flux of the flow, while the velocityof the particles increases. Since the difference in mass flux between the outletsis extremely small, 3.2e-5 kg/s, which is 0.12 % of the mass flux through theinlet, it is assumed that in DPM, this unexpected mass flux difference does notoccur often. The results of the other boundary conditions is as expected.When figure 3.10(c) is compared with figure 3.10(a), it is seen that the massfluxes are smaller for geometry 3 than for geometry 1, in all three boundaryconditions. This is not due to the difference in bifurcation angle, as discussed inthe previous section, but to the diameter of 2.5 mm of the outlets of geometry3, instead of 3 mm for the outlets of geometry 1.Although the Reynolds numbers are smaller than the Reynolds numbers of ge-ometry 1, they show a similar pattern. It is possible that the smaller Reynoldsnumber of geometry 3, in boundary condition 3, in comparison with the Reynoldsnumber of boundary condition 2, is caused by the larger bifurcation angle of out-let 2. This can, however, not be known for certain, since the Reynolds numberfor this simulation in geometry 1 is also smaller.In table 3.1 a similar result for geometry 3 as for geometry 1 can be found.The simulation time when equal pressures on both outlets are applied, however,is extremely long. This is due to one particle that interacted with the wall ofthe geometry. It probably lost some of its velocity during this interaction andtherefore it took longer to travel to the outlet. After the interaction the velocityof the particle increases, which means that momentum of the fluid is transferredonto the particle, causing less mass flux on the outlet.In figure 3.10(d) it can be seen that the mass flux through outlet 1, where thediameter is 2 mm and the bifurcation angle 21.26o , for the three circumstances(see table 2.3), is smaller than the mass flux through outlet 2, where the diame-ter is 3 mm and the bifurcation angle is 54.68o. This indicates that the diameterof the outlet is of larger influence on the mass flux than the bifurcation angle,which means, as seen before, that the outlet with the larger diameter traps moreparticles than the outlet with the smaller diameter, although the bifurcation an-gle of the latter is smaller than the former.As a consequence the Reynolds number, in boundary condition 2, is larger thanthe Reynolds number when both pressures are equal, but is smaller than theReynolds number of boundary condition 3, as can be seen in table 3.3.The simulation times for geometry 4 in table 3.1 are not in correspondence withthe Reynolds numbers of geometry 4. If it is assumed that the times of bound-ary condition 1 and boundary condition 3 are correct, than a simulation time ofapproximately 0.45 seconds is expected for the simulation with boundary con-dition 2. There is no suitable explanation for this.

39


As expected, the mass flux through outlet 1 of geometry 5, where the diameteris 3 mm is, for all three boundary conditions, higher than the mass flux throughoutlet 2, where the diameter is 2 mm, as can be seen in figure 3.10(e). Sincethe mass fluxes through geometry 5 are of approximately the same size as themass fluxes through geometry 4, Reynolds numbers are of same size as well, ascan be seen from 3.3. For the simulation time, the same problem arises as forgeometry 4. The simulation time of geometry 5 with boundary condition 3, isexpected to be lower than the time of boundary condition 1, and higher thanthe time of boundary condition 3. Here, however, the long simulation time canbe explained by a particle losing velocity while interacting with the wall.

MPM

Note in figure 3.11(a) that for boundary condition 1, on outlet 1 a higher per-centage of particles is found, while the mass flux is lower than on outlet 2, wherethe mass flux is higher, as is seen before in the previous section. This can onlybe explained by assuming that momentum of the fluid is imposed on the par-ticles that travel towards an outlet, resulting in a lower mass flux through thisoutlet. The mass flux difference of the outlets is 7.0541e-5 kg/s, which is 1.8% of the mass flux through the inlet. This particular result, obviously, causesa decrease in linearity in the graph. The results of boundary conditions 2 and3 are as expected: a higher mass flux causes a higher percentage of particles, alower mass flux causes a lower percentage.In table 3.4 a similar pattern for the Reynolds numbers in MPM is found as inDPM. The Reynolds numbers for boundary conditions 2 and 3 are higher thanthe Reynolds number for boundary condition 1. Furthermore, the former twoReynolds numbers do not differ significantly, as expected.In table 3.2 it can be seen that the pattern for simulation time in MPM is notas obvious as it is in DPM. This is due to the mutual momentum interactionbetween the particles and the flow. Since, for a particular flow, the exact momen-tum transfers are not known, it is not possible to verify this effect on simulationtime. The results presented in figure 3.11(b) show that mass fluxes throughoutlets with small diameters are small for the three boundary conditions, andsubsequently large through the other outlet, as expected and discussed in theprevious section.The Reynolds numbers show the same pattern as for DPM, where boundarycondition 3 causes a larger mass flux through the geometry than boundary con-dition 2.The time of the simulation does not correspond with the Reynolds numbers.Here, mutual momentum interaction is assumed to be the cause as well.In figure 3.11(c) it is shown that for boundary condition 1, a lower mass fluxthrough outlet 1 causes a higher particle percentage, where the opposite is ex-pected. The momentum of the fluid has a large effect on the particles, as de-scribed earlier. The difference between mass fluxes is 9.5e-5 kg/s, which is 3.95% of the mass flux through the inlet. The other boundary conditions producethe expected results.There is a slightly higher Reynolds number of geometry 4 when boundary con-dition 3 is applied, than when boundary condition 2 is applied. The opposite isexpected. It can be explained by the fluid transferring more momentum to theparticles in outlet 2 for boundary condition 3, than fluid transferring momentum

40


to the particles in outlet 1 for boundary condition 2. This causes the Reynoldsnumber for boundary condition 3 to be lower than the Reynolds number forboundary condition 2.Here, the simulation times show a correlation with the Reynolds numbers: alower Reynolds number causes a larger simulation time, and a higher Reynoldsnumber a lower simulation time.The momentum transfer has a large effect on the particles in geometry 4 forboundary condition 2 as well, as can be seen in figure 3.11(d), where the massflux through outlet 1 is lower than the mass flux through outlet 2, but causesa higher amount of particles on the outlet. Apparently, a pressure of 95 mmHg on outlet 1 at first causes a larger mass flux through outlet 1, resulting inmore particle flow towards outlet 1. Then momentum transfer causes the massflux to reduce while the particles continue to travel towards outlet 1. In thissimulation the momentum transfer must have been large, since the mass fluxdifference between the outlets is 72.28 kg/s, which is 21.1 % of the mass fluxthrough the inlet. The other boundary conditions produce the expected results.The Reynolds numbers show the same pattern as for DPM, as expected.The time of the simulations do not corresponds with the Reynolds number.The results presented in 3.11(e) show the expected results, as well as the Reynoldsnumbers and the simulation times.

4.1.4 Differences between DPM and MPM

As seen in the previous section, there is more interaction between the fluid andthe particles in MPM than in DPM. As mentioned in section 2.5, there is adifference in momentum transfer between DPM and MPM. In DPM, the ex-change of momentum between the continuous phase and the discrete phase iscalculated as the particle passes through each control volume in the model. InMPM, momentum, supplied by the particles, is added to the fluid and particlevelocity is imposed on touched cells. By doing this, momentum is transferredto the particle, which causes a lower particle velocity, while the velocity of thefluid subsequently increases. A higher fluid velocity will cause a higher massflux through the geometry and consequently a higher Reynolds number, as canbe found in tables 3.3 and 3.4. However, since the particles lost velocity to theflow, they will take longer to travel to the geometry. This explains the longersimulation time of MPM, as seen in tables 3.1 and 3.2. Interaction betweenparticles and fluid also means that fluid velocity can be imposed on the parti-cles, so that the velocity of the fluid decreases and the velocity of the particlesincreases.Since, as seen before, the influence of bifurcation angle is larger in MPM than inDPM, the difference between effect of outlet diameter of the models can not bedefined. The only difference in figures 3.2(a) and 3.2(b), and 3.4(a) and 3.4(b),is the smaller percentage of particles on outlet 1 of geometry 2 in boundarycondition 2 for MPM than on outlet 1 of geometry 2 in boundary condition 2for DPM. This indicates a slightly larger influence of outlet diameter on particledistribution for MPM.The difference between the influence of bifurcation angle on the particle distri-bution between DPM and MPM can be explained by the fact that, for MPM,the velocity vectors of the fluid have a significant influence on the velocity vec-tors of the particles at every touched cell, as explained earlier. The interaction

41

Discussion 4.2 Circle of Willis

between the continuous and discrete phase for control volumes in DPM is notas large as the interaction between flow and particles in MPM. This means thatwhen the direction of the fluid velocity is affected by a bifurcation angle, thisdirection will have a larger influence on the direction of particles in MPM thanon the direction of DPM. Since both diameter and bifurcation angle contributemore to the particle tracks in MPM, it is considered more accurate than DPM.The main difference between DPM and MPM is that the particles in MPM in-teract more with the fluid, transferring momentum, which causes an increaseor decrease of fluid velocity, a larger or smaller mass flux and a lower or higherparticle velocity. This means that under certain boundary conditions, an unex-pected higher mass flux through an outlet is found and thus a higher amountof particles than on the other outlet. This can also mean that a lower massflux is found, when the flow transfers momentum to the particles, causing alower amount of particles on the outlet. As a result, the results in DPM aremore consistent than the results of MPM, as can be seen in figures 3.12(a) and3.12(b).


Summarizing these outcomes, it is seen that, for DPM, the diameter of theoutlet contributes most to the mass flux found on the outlets and thus to theamount of particles found on the outlets. Bifurcation angle is of small influenceon the mass flux and the amount of particles. With these results it is possible toroughly predict the particle distribution on the Circle of Willis. It is expectedthat in the Circle of Willis, the highest percentage of particles will be found onthe MCA, where the outlets is largest, and the bifurcation angle the smallest,causing the largest mass flux. On the ACA a larger percentage of particles willbe found than on the PCA, since the diameter of the ACA is larger than thediameter of the PcOA as can be verified by table 2.2, and thus causing a largermass flux through the ACA, while there is not much difference between thebifurcation angles.This is exactly what is seen in figures 3.13, 3.14 and 3.15.

For MPM, such a prediction is more difficult to make, since the results ofMPM are not as consistent as the results of DPM. In MPM, both outlet diameterand bifurcation angle are of influence on particle distribution. From this resultit can be derived that the largest amount of injected particles will travel throughthe MCA, since the diameter is largest and there is no bifurcation angle betweenthe ICA and MCA. If is assumed that the bifurcation angles between MCA andPCoA, and MCA and ACA is equal, since the influence of outlet diameter forMPM is higher than for DPM, it is most likely that the higher percentage ofthe particles will be found on the ACA, and the smallest amount of particles onthe PCA, as in DPM.

42

Chapter 5

Conclusion

To predict particle trajectory in the Circle of Willis, 5 geometries are usedto study the effect of outlet diameter, bifurcation angle and mass flux on theparticle distribution, Two different models are used to simulate particle tracking:the Discrete Phase Model (DPM) and the Macroscopic Particle Model (MPM).It has been found that outlet diameter effects the particle distribution greatly inDPM, while bifurcation angle barely influences the distribution. Furthermore,for DPM, a larger diameter of outlet causes a higher mass flux through thisoutlet, while a smaller diameter causes a smaller mass flux. Mass flux is linearlyrelated to the amount of particles. These findings are used for a prediction ofparticle trajectory in the Circle of Willis, and results are as predicted.Results of the simulations performed with MPM differ from the the results ofDPM. Firstly, it is seen that bifurcation angle is of significantly larger influenceon the particle distribution. Secondly, also the outlet diameter has a larger effecton the particle tracks in MPM than in DPM. This difference can be explainedby a difference in interaction between fluid and particles for DPM and MPM. InDPM, momentum transfer of the particles to the fluid, or vice-versa, is calculatedfor every control volume, whereas for MPM, this transfer takes place in everycell that is touched by the particle. As a result, the percentage of particles onthe outlets as a function of mass flux does not show as much linearity as DPM.This means that the prediction of particle trajectory in the Circle of Willis isslightly different. It is expected for MPM, that more particles will be found onthe outlet where diameter is largest and bifurcation smallest, than for DPM.Consequentially, less particles are found on the other outlets. Unfortunately, itwas for MPM not possible to verify this prediction by simulating particle tracksin the Circle of Willis.Nevertheless, the conclusion can be drawn that, since for DPM most particlesare found on the MCA, and that for MPM most particles are expected to befound on the MCA, an embolus, causing a stroke, is most likely to block bloodflow through the MCA.

43

Chapter 6

Recommendations

Further research includes, obviously, applying MPM to the Circle of Willis,using the same injection and boundary conditions as is used in DPM. Othersimulations can be done with the use of boundary conditions that vary in time,simulating pulsatile blood flow. Another interesting simulation is to use MPM tostudy the particle track of one large particle, simulating the particle trajectoryof an emboli, in the Circle of Willis, and the accompanying influence on theflow.

44

References

[1] A. C. Guyton and J. E. Hall. Textbook of medical physiology. W.B. SaundersCompany, Philadelphia, 9th edition, 1996.

[2] S. Moore. T. David. J. G. Chase. J. Arnold and J. Fink. 3D Models ofBlood Flow in the Cerebral Vasculature. Journal of Biomechanics, Articlein Press.

[3] M. Zamir. The Physics of Pulsatile Flow. Springer-Verslag, New York, 1stedition, 2000.

[4] FLUENT 6.1 User’s Guide, Appendix A.

[5] S. A. Morsi and A. J. Alexander. An investigation of particle trajectories intwo-phase flow systems. J. Fluid Mech, 55.

[6] M. Agrawal and B. K. David. Macroscopic Particle Model, Appendix B.

45

Appendix A

Discrete Phase ModelManual

A.1 Introduction

In addition to solving transport equations for the continuous phase, FLUENTallows you to simulate a discrete second phase in a Lagrangian frame of ref-erence. This second phase consists of spherical particles (which may be takento represent droplets of bubbles) dispersed in the continuous phase. FLUENTcomputes the trajectories of these discrete phase entities, as well as heat andmass transfer to/from them. The coupling between the phases and its impacton both the discrete phase trajectories and the continuous phase flow can beincluded.

FLUENT provides the following discrete phase modeling options:

• Calculation of the discrete phase trajectory using a Lagrangian formula-tion that includes the discrete phase inertia, hydrodynamic drag, and theforce of gravity, for both steady and unsteady flows

• Prediction of the effects of turbulence on the dispersion of particles dueto turbulent eddies present in the continuous phase

• Heating/cooling of the discrete phase

• Vaporization and boiling of liquid droplets

• Combusting particles, including volatile evolution and char combustion tosimulate coal combustion

• Optional coupling of the continuous phase flow field prediction to thediscrete phase calculations

• Droplet breakup and coalescence

These modeling capabilities allow FLUENT to simulate a wide range of discretephase problems including particle separation and classification, spray drying,aerosol dispersion, bubble stirring of liquids, liquid fuel combustion, and coal

46

Discrete Phase Model Manual A.2 Limitations

combustion. The physical equations used for discrete phase calculations aredescribed in Sections A.4 and A.5.

A.2 Limitations

Limitation on the Particle Volume Fraction

The discrete phase formulation used by FLUENT contains the assumption thatthe second phase is sufficiently dilute that particle-particle interactions and theeffects of the particle volume fraction on the gas phase are negligible. In practice,these issues imply that the discrete phase must be present at a fairly low volumefraction, usually less than 10-12 %. Note that the mass loading of the discretephase may greatly exceed 10-12 %: you may solve problems in which the massflow of the discrete phase equals or exceeds that of the continuous phase.

Limitation on Modeling Continuous Suspension of Particles

The steady-particle Lagrangian discrete phase model described in this chapteris suited for flows in which particle streams are injected into a continuous phaseflow with a well-defined entrance and exit condition. The Lagrangian model doesnot effectively model flows in which particles are suspended indefinitely in thecontinuum, as occurs in solid suspensions within closed systems such as stirredtanks, mixing vessels, or fluidized beds. The unsteady-particle discrete phasemodel, however, is capable of modeling continuous suspensions of particles.

Limitation on Using the Discrete Phase Model with Other FLUENTModels

The following restrictions on the use of other models with the discrete phasemodel:

• Streamwise periodic flow (either specified mass flow rate or specified pres-sure drop) cannot be modeled when the discrete phase model is used.

• Adaptive time stepping cannot be used with the discrete phase model.

• Only non-reacting particles can be included when the premixed combus-tion model is used

• Surface injections will not be moved with the grid when a sliding mesh ora moving or deforming mesh is being used.

• When multiple reference frames are used in conjunction with the discrete-phase model, the display of particle tracks will not, by default, be mean-ingful. Similarly, coupled discrete-phase calculations are not meaningful.

An alternative approach for particle tracking and coupled discrete-phase calcula-tions with multiple reference frames is to track particle based on absolute veloc-ity instead of relative velocity. To make this change, use the define/models/dpm/tracking/track-in-absolute-frame text command. Note that the re-sults may be strongly depend on the location of walls inside the multiple ref-erence frame. The particle injection velocities (specified in the Set Injection

47

Discrete Phase Model Manual A.3 Overview of Discrete Phase Modeling Procedures

Properties panel) are defined relative to the frame of reference in which theparticles are tracked. By default, the injection velocities are specified relative tothe local reference frame. If you enable the track-in-absolute-frame option,the injection velocities are specified relative to the absolute frame.

A.3 Overview of Discrete Phase Modeling Pro-cedures

You can include a discrete phase in your FLUENT model by defining the initialposition, velocity, size, and temperature of individual particles. These initialconditions, along with your inputs defining the physical properties of the dis-crete phase, are used to initiate trajectory and heat/mass transfer calculations.The trajectory and heat/mass transfer calculations are based on the force bal-ance on the particle and on the convective/radiative heat and mass transferfrom the particle, using the local continuous phase conditions as the particlemoves through the flow. The predicted trajectories and the associated heat andmass transfer can be viewed graphically and/or alphanumerically.

You can use FLUENT to predict the discrete phase patterns based on a fixedcontinuous phase flow field (an uncoupled approach), or you can include the ef-fect of the discrete phase on the continuum (a coupled approach). In the coupledapproach, the continuous phase flow pattern is impacted by the discrete phase(and vice versa), and you can alternate calculations of the continuous phaseand discrete phase equations until a converged coupled solution is achieved. SeeSection A.6 for details.

Outline of Steady-State Problem Setup and Solution Procedure

The general procedure for setting up and solving a steady-state discrete-phaseproblem is outlined below:

1. Solve the continuous-phase flow.

2. Create the discrete-phase injections.

3. Solve the coupled flow, if desired.

4. Track the discrete-phase injections, using plots or reports.

Outline of Unsteady Problem Setup and Solution Procedure

The general procedure for setting up and solving an unsteady discrete-phaseproblem is outlined below:

1. Create the discrete-phase injections.

2. Initialize the flow field.

3. Advance the solution in time by taking the desired number of time steps.

48

Discrete Phase Model Manual A.4 Equations of Motion of Particles

Particle positions will be updated as the solution advances in time. If you aresolving an uncoupled flow, the particle position will be updated at the end ofeach time step. For a coupled calculation, the positions are iterated on withineach time step.

A.4 Equations of Motion of Particles

Particle Force Balance

FLUENT predicts the trajectory of a discrete phase particle (or droplet orbubble) by integrating the force balance on the particle, which is written ina Lagrangian reference frame. This force balance equates the particle inertiawith the forces acting on the particle, and can be written (for the x direction inCartesian coordinates) as

dup

dt= FD(u − up) +

gx(ρp − ρ)ρp

+ Fx (A.1)

where FD(u − up) is the drag force per unit particle mass and

FD =18µ

ρpd2p

CDRe

24(A.2)

Here, u is the fluid phase velocity, up is the particle velocity, µ is the molec-ular velocity of the fluid, ρ is the fluid density, ρp is the density of the particle,and dp is the particle diameter. Re is the relative Reynolds number, which isdefined as

Re =ρdp | up − u |

µ(A.3)

The drag coefficient, CD, can be taken from either

CD = a1 +a2

Re+

a3

Re2(A.4)

where a1, a2 and a3 are constants that apply for smooth spherical particlesover several ranges of Re given by Morsi and Alexander, or

CD =24Re

(1 + b1Reb2) +b3Re

b4 + Re(A.5)

whereb1 = exp(2.3288 − 6.4581φ + 2.4486φ2)b2 = 0.0964 + 0.5565φb3 = exp(4.905 − 13.8944φ + 18.4222φ2 − 10.2599φ3)b4 = exp(1.4681 + 12.2584φ − 20.7322φ2 + 15.8855φ3)

which is taken from Haider and Levenspiel. The shape factor, φ, is definedas

φ =s

S(A.6)

where s is the surface area of a sphere having the same volume as the par-ticle, and S is the actual surface area of the particle.

49


For sub-micron particles, a form of Stokes’ drag law is available. in this case,FD is defined as

FD =18µ

d2pρpCc

(A.7)

The factor Cc is the Cunningham correction to Stokes’ drag law, which youcan compute from

Cc = 1 +2λ

dp(1.257 + 0.4e−(1.1dp/2λ)) (A.8)

where λ is the molecular mean free path.(rest not relevant, omitted)

Including the Gravity Term

While equation 1 includes a force of gravity on the particle, it is important tonote that in FLUENT the default gravitational acceleration is zero. If you wantto include the gravity force, you must remember to define the magnitude anddirection of the gravity vector in the Operating Conditions panel.

Other Forces

Equation 1 incorporates additional forces (Fx) in the particle force balance thatcan be important under special circumstances. The first of these is the ”virtualmass” force, the force required to accelerate the fluid surrounding the particle.This force can be written as

Fx =12

ρ

ρp

d

dt(u − up) (A.9)

and is important when ρ > ρp. An additional force arises due to the pressuregradient in the fluid:

Fx = (ρ

ρp)up

∂u

∂x(A.10)

(rest not relevant, omitted)

Integration of the Trajectory Equations

The trajectory equations, and any auxiliary equations describing heat of masstransfer to/from the particle, are solved by stepwise integration over discretetime steps. Integration in time of Equation 1 yields the velocity of the particleat each point along the trajectory, with the trajectory itself predicted by

dx

dt= up (A.11)

Equations similar to 1 and 11 are solved in each coordinate direction to pre-dict the trajectories of the discrete phase.

50


Assuming that the term containing the body force remains constant over eachsmall time interval, and linearizing any other forces acting on the particles, thetrajectory equation can be rewritten in simplified form as

dup

dt=

1τp

(u − up) (A.12)

where τp is the particle relaxation time. FLUENT uses a trapezoidal schemefor integrating equation 12:

un+1p − un

p

∆t=

1τp

(u∗ − un+1p ) + ... (A.13)

where n represents the iteration number and

u∗ =12(un + un+1) (A.14)

un+1 = un + ∆tunp · ∇un (A.15)

Equations 14 and 15 are solved simultaneously to determine the velocity andposition of the particle at any given time. For rotating reference frames, theintegration is carried out in the rotating frame with the extra terms describedabove (not relevant) to account for system rotation. In all cases, care mustbe taken that the time step used for integration is sufficiently small that thetrajectory integration is accurate in time.

Discrete Phase Boundary Conditions

When a particle strikes a boundary face, one of several contingencies may arise:

• Reflection via an elastic or inelastic collision.

• Escape through the boundary. (The particle is lost from the calculationat the point where it impacts the boundary.)

• Trap at the wall. non-volatile material is lost from the calculation at thepoint of impact with the boundary; volatile material present in the particleor droplet is released to the vapor phase at this points.

• Passing through an internal boundary zone, such as radiator or porousjump.

• Slide along the wall depending on particle properties and impact angle.

You also have the option of implementing a user-defined function to modelthe particle behavior when hitting the boundary. See the separate UDF Manualfor information about user-defined functions.

51

Discrete Phase Model Manual A.5 Coupling Between the Discrete and Continuous Phases

A.5 Coupling Between the Discrete and Contin-uous Phases

As the trajectory of a particle is computed, FLUENT keeps rack of the heat,mass, and momentum gained or lost by the particle stream that follows thattrajectory and these quantities can be incorporated in the subsequent contin-uous phase calculations. Thus, while the continuous phase always impacts thediscrete phase, you can also incorporate the effect of the discrete phase trajec-tories on the continuum. This two-way coupling is accomplished by alternatelysolving the discrete and continuous phase equations until the solutions in bothphases have stopped changing. This interphase exchange of heat, mass, andmomentum from the particle to the continuous phase is depicted qualitativelyin figure 1.

Figure A.1: Heat, Mass, and Momentum Transfer Between The Discrete andContinuous phases

Momentum Exchange

The momentum transfer from the continuous phase to the discrete phase iscomputed in FLUENT by examining the change in momentum of a particle asit passes through each control volume in the FLUENT model. This momentumexchange is computed by

F =∑

(18µCDRe

ρpd2p24

(up − u) + Fother)mp∆t (A.16)

whereµ = viscosity of bloodρp = density of the particledp = diameter of the particleRe = relative Reynolds number

52

Discrete Phase Model Manual A.5 Coupling Between the Discrete and Continuous Phases

up = velocity of the particleu = velocity of the fluidCD = drag coefficientmp = mass flow rate of the particles∆t = time stepFother = other interaction forces

This momentum exchange appears as a momentum sink in the continuousphase momentum balance in any subsequent calculations of the continuous phaseflow field and can be reported by FLUENT.

53

Appendix B

Macroscopic Particle ModelManual

Overview

• Current ”CFD Particle Tracking Models” assumes that parti-cles are point masses that do not interact.

• Large particles immersed in the fluid flow can not be modeledusing the Fluent’s DPM approach, it requires special treatmentthrough UDFs to take into account:

– The Blockage and the Momentum Transfer to the fluid bythe particles.

– Proper Evaluation of the Drag force and the Torque expe-rienced by the particles.

– Particle-Particle as well as Particle-Wall collision.

• The present ”Macroscopic Particle Model” can be found usefulin many applications, i.e. Pharmaceutical, Chemical, MaterialHandling and Sports Industries.

Technical Approach for Macroscopic Particle Model

• The Particles are treated in a Lagrangian Frame of Reference.

• For each particle, ”Rigid Body Velocity” on the fluid cells isimposed on the cells ”touched” by the particles.

• The imposition of rigid body motion demands that we addmomentum to the fluid. This momentum must be supplied bythe particles.

• The integral of that momentum deficit gives us the ”ParticleDrag” and the ”Particle Torque” for each particle.

54

Macroscopic Particle Model Manual

• New Position, Velocity, and Angular Velocities of the particlesare calculated from the Drag and Torque vectors calculatedabove.

Figure B.1: Particle velocity of a particle imposed on touched cells

Momentum which diffuses from the touched cells represents the actual hy-drodynamic forces on the particle.

• Particle-Wall collision:

– Identify the boundary faces (if any) the particle intersectedduring the last time.

– Project incoming particle velocity onto the wall normal andtangential and apply Coefficient of Restitution to find outgoing velocities.

• Particle-Particle collision:

– Detect which particles are going to collide

– Find the line-of-action of the collision, which will identifythe Normal Direction. Project incoming particle veloci-ties onto the line-of-action to get Normal and Tangentialcomponents.

– Apply Coefficient of Restitution and Conservation of Mo-mentum to the Normal Components of the incoming ve-locities to obtain final velocities.

55


Macroscopic Particle Model User Inputs (Using Graphic User Inter-face Panels)

• List of Walls Boundary zones to assign Static and Dynamicfriction as well as the Restitution Coefficients.

• Dynamic friction as well as the Restitution Coefficients forparticle-particle and particle-wall collisions.

• Body Forces (i.e. Gravity)

• Particle Details (for each particle):

– Diameter, Mass, Momentum of Inertia, Initial Location(x,y and z coordinates), Initial velocities (U, V, W and aswell as the angular velocities)

– Particles Detail can either be read from the file and/or usercan enter (or modify) these inputs in the GUI panel createdfor this specific application through scheme interface.

Fluent Setup for Macroscopic Particle Model

• Perform a Steady State Flow Analysis of Fluent case withoutany Macroscopic Particles.

• Define Macroscopic Model Parameters and Particles Details us-ing special GUI panels. Hook Adjust UDF function to activatethe Macroscopic Particle Model.

• Perform a Transient Simulation with reasonably small time stepsize with steady state solution as the initial solution.

• UDF will automatically create and update iso-surfaces for allthe particles at each time step.

• Fluent’s Animation Capability can be used to plot sphericalparticles (at actual location) at defined time intervals. Theparticle’s iso-surfaces can be superimposed with the particlemass (or any other Fluent’s variables) to create the desiredanimation.

• Particles Locations are also stored in Field View binary datafile format which can be animated in Field View.

Scope for Further Enhancements

• The UDF at present is only the first step towards MacroscopicParticle Model. There are plethora of space for further en-hancements:

56


– The present formulation can be extended from SphericalShaped particles to any desired shapes (ellipsoid, disk...).

– Friction Dynamics implemented at present needs more at-tention.

– The present model may be not practical for Large Numberof Particles.

– Heat Transfer between particles as well as between particleand fluids could be taken into account.

– Multiple/Group Injections can be added in GUI panel.

57

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