dpm fluent

Chapter 22. Modeling Discrete Phase

This chapter describes the Lagrangian discrete phase capabilities available in FLUENTand how to use them.

Information is organized into the following sections:

• Section 22.1: Introduction

• Section 22.2: Particle Motion Theory

• Section 22.3: Multicomponent Particle Theory

• Section 22.4: Wall-Film Model Theory

• Section 22.5: Particle Erosion and Accretion Theory

• Section 22.6: Dynamic Drag Model Theory

• Section 22.7: Spray Model Theory

• Section 22.8: Atomizer Model Theory

• Section 22.9: One-Way and Two-Way Coupling

• Section 22.10: Discrete Phase Model (DPM) Boundary Conditions

• Section 22.11: Steps for Using the Discrete Phase Models

• Section 22.12: Setting Initial Conditions for the Discrete Phase

• Section 22.13: Setting Boundary Conditions for the Discrete Phase

• Section 22.14: Setting Material Properties for the Discrete Phase

• Section 22.15: Solution Strategies for the Discrete Phase

• Section 22.16: Postprocessing for the Discrete Phase

c© Fluent Inc. September 29, 2006 22-1

Modeling Discrete Phase

22.1 Introduction

In addition to solving transport equations for the continuous phase, FLUENT allowsyou to simulate a discrete second phase in a Lagrangian frame of reference. This secondphase consists of spherical particles (which may be taken to represent droplets or bubbles)dispersed in the continuous phase. FLUENT computes the trajectories of these discretephase entities, as well as heat and mass transfer to/from them. The coupling between thephases and its impact on both the discrete phase trajectories and the continuous phaseflow can be included.

FLUENT provides the following discrete phase modeling options:

• calculation of the discrete phase trajectory using a Lagrangian formulation thatincludes the discrete phase inertia, hydrodynamic drag, and the force of gravity,for both steady and unsteady flows

• prediction of the effects of turbulence on the dispersion of particles due to turbulenteddies 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 to simulatecoal combustion

• optional coupling of the continuous phase flow field prediction to the discrete phasecalculations

• droplet breakup and coalescence

These modeling capabilities allow FLUENT to simulate a wide range of discrete phaseproblems including particle separation and classification, spray drying, aerosol dispersion,bubble stirring of liquids, liquid fuel combustion, and coal combustion. The physicalequations used for these discrete phase calculations are described in Sections 22.2–22.9.1,and instructions for setup, solution, and postprocessing are provided in Sections 22.11–22.16.

22.1.1 Overview

Advances in computational fluid mechanics have provided the basis for further insight intothe dynamics of multiphase flows. Currently there are two approaches for the numericalcalculation of multiphase flows: the Euler-Lagrange approach (discussed below) and theEuler-Euler approach (discussed in Section 23.2.1: Approaches to Multiphase Modeling).

22-2 c© Fluent Inc. September 29, 2006

22.1 Introduction

The Euler-Lagrange Approach

The Lagrangian discrete phase model in FLUENT (described in Chapter 22: ModelingDiscrete Phase) follows the Euler-Lagrange approach. The fluid phase is treated as acontinuum by solving the time-averaged Navier-Stokes equations, while the dispersedphase is solved by tracking a large number of particles, bubbles, or droplets through thecalculated flow field. The dispersed phase can exchange momentum, mass, and energywith the fluid phase.

A fundamental assumption made in this model is that the dispersed second phase occupiesa low volume fraction, even though high mass loading (mparticles ≥ mfluid) is acceptable.The particle or droplet trajectories are computed individually at specified intervals duringthe fluid phase calculation. This makes the model appropriate for the modeling of spraydryers, coal and liquid fuel combustion, and some particle-laden flows, but inappropriatefor the modeling of liquid-liquid mixtures, fluidized beds, or any application where thevolume fraction of the second phase is not negligible.

22.1.2 Limitations

Limitation on the Particle Volume Fraction

The discrete phase formulation used by FLUENT contains the assumption that the secondphase is sufficiently dilute that particle-particle interactions and the effects of the particlevolume fraction on the gas phase are negligible. In practice, these issues imply that thediscrete phase must be present at a fairly low volume fraction, usually less than 10–12%.Note that the mass loading of the discrete phase may greatly exceed 10–12%: you maysolve problems in which the mass flow of the discrete phase equals or exceeds that of thecontinuous phase. See Chapter 23: Modeling Multiphase Flows for information aboutwhen you might want to use one of the general multiphase models instead of the discretephase model.

Limitation on Modeling Continuous Suspensions of Particles

The steady-particle Lagrangian discrete phase model described in this chapter is suitedfor flows in which particle streams are injected into a continuous phase flow with a well-defined entrance and exit condition. The Lagrangian model does not effectively modelflows in which particles are suspended indefinitely in the continuum, as occurs in solidsuspensions within closed systems such as stirred tanks, mixing vessels, or fluidized beds.The unsteady-particle discrete phase model, however, is capable of modeling continuoussuspensions of particles. See Chapter 23: Modeling Multiphase Flows for informationabout when you might want to use one of the general multiphase models instead of thediscrete phase models.



Limitations on Using the Discrete Phase Model with Other FLUENT Models

The following restrictions exist on the use of other models with the discrete phase model:

• When tracking particles in parallel, the DPM model cannot be used with any ofthe multiphase flow models (VOF, mixture, or Eulerian – see Chapter 23: ModelingMultiphase Flows) if the shared memory option is enabled (Section 22.11.9: ParallelProcessing for the Discrete Phase Model). (Note that using the message passingoption, when running in parallel, enables the compatibility of all multiphase flowmodels with the DPM model.)

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

• Only nonreacting particles can be included when the premixed combustion modelis used.

• Surface injections will be moved with the grid when a sliding mesh or a movingor deforming mesh is being used, however only those surfaces associated with aboundary will be recalculated. Injections from cut plane surfaces will not be movedwith the mesh and will be deleted when remeshing occurs.

• The cloud model is not available for unsteady particle tracking, or in parallel, whenusing the message passing option for the particles.

• The wall-film model is only valid for liquid materials. If a nonliquid particle inter-acts with a wall-film boundary, the boundary condition will default to the reflectboundary condition.

• When multiple reference frames are used in conjunction with the discrete phasemodel, the display of particle tracks will not, by default, be meaningful. 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 particles based on absolute velocityinstead of relative velocity. To make this change, use the define/models/dpm/

options/track-in-absolute-frame text command. Note that the results maystrongly depend on the location of walls inside the multiple reference frame.

The particle injection velocities (specified in the Set Injection Properties panel) aredefined relative to the frame of reference in which the particles are tracked. Bydefault, the injection velocities are specified relative to the local reference frame.If you enable the track-in-absolute-frame option, the injection velocities arespecified relative to the absolute frame.

• Relative particle tracking cannot be used in combination with sliding and movingdeforming meshes. If sliding and/or deforming meshes are used with the DPM


22.2 Particle Motion Theory

model, the particles will always be tracked in the absolute frame. Switching to therelative frame is not permitted.


22.2.1 Equations of Motion for Particles

Particle Force Balance

FLUENT predicts the trajectory of a discrete phase particle (or droplet or bubble) byintegrating the force balance on the particle, which is written in a Lagrangian referenceframe. This force balance equates the particle inertia with the forces acting on theparticle, and can be written (for the x direction in Cartesian coordinates) as

dupdt

= FD(u− up) +gx(ρp − ρ)

ρp+ Fx (22.2-1)

where Fx is an additional acceleration (force/unit particle mass) term, FD(u− up) is thedrag force per unit particle mass and

FD =18µ

ρpd2p

CDRe

24(22.2-2)

Here, u is the fluid phase velocity, up is the particle velocity, µ is the molecular viscosityof the fluid, ρ is the fluid density, ρp is the density of the particle, and dp is the particlediameter. Re is the relative Reynolds number, which is defined as

Re ≡ ρdp |up − u|µ

(22.2-3)

Inclusion of the Gravity Term

While Equation 22.2-1 includes a force of gravity on the particle, it is important to notethat in FLUENT the default gravitational acceleration is zero. If you want to includethe gravitational force, you must remember to define the magnitude and direction of thegravity vector in the Operating Conditions panel.



Other Forces

Equation 22.2-1 incorporates additional forces (Fx) in the particle force balance that canbe important under special circumstances. The first of these is the “virtual mass” force,the force required to accelerate the fluid surrounding the particle. This force can bewritten as

Fx =1

2

ρ

ρp

d

dt(u− up) (22.2-4)

and is important when ρ > ρp. An additional force arises due to the pressure gradient inthe fluid:

Fx =

(ρ

ρp

)upi

∂u

∂xi(22.2-5)

Laws for Drag Coefficients

The drag coefficient, CD, can be taken from either

CD = a1 +a2

Re+

a3

Re2 (22.2-6)

where a1, a2, and a3 are constants that apply to smooth spherical particles over severalranges of Re given by Morsi and Alexander [252], or

CD =24

Resph

(1 + b1Resph

b2)

+b3Resph

b4 + Resph

(22.2-7)

where

b1 = exp(2.3288− 6.4581φ+ 2.4486φ2)

b2 = 0.0964 + 0.5565φ

b3 = exp(4.905− 13.8944φ+ 18.4222φ2 − 10.2599φ3)

b4 = exp(1.4681 + 12.2584φ− 20.7322φ2 + 15.8855φ3) (22.2-8)

which is taken from Haider and Levenspiel [131]. The shape factor, φ, is defined as

φ =s

S(22.2-9)

where s is the surface area of a sphere having the same volume as the particle, and Sis the actual surface area of the particle. The Reynolds number Resph is computed withthe diameter of a sphere having the same volume.

i The shape factor cannot exceed a value of 1.



For sub-micron particles, a form of Stokes’ drag law is available [273]. In this case, FDis defined as

FD =18µ

dp2ρpCc

(22.2-10)

The factor Cc is the Cunningham correction to Stokes’ drag law, which you can computefrom

Cc = 1 +2λ

dp(1.257 + 0.4e−(1.1dp/2λ)) (22.2-11)

where λ is the molecular mean free path.

A high-Mach-number drag law is also available. This drag law is similar to the sphericallaw (Equation 22.2-6) with corrections [61] to account for a particle Mach number greaterthan 0.4 at a particle Reynolds number greater than 20.

For unsteady models involving discrete phase droplet breakup, a dynamic drag law optionis also available. See Section 22.6: Dynamic Drag Model Theory for a description of thislaw.

Instructions for selecting the drag law are provided in Section 22.11.4: Alternate DragLaws.

Forces in Rotating Reference Frames

The additional force term, Fx, in Equation 22.2-1 also includes forces on particles thatarise due to rotation of the reference frame. These forces arise when you are modelingflows in rotating frames of reference (see Section 10.2: Flow in a Rotating ReferenceFrame). For rotation defined about the z axis, for example, the forces on the particles inthe Cartesian x and y directions can be written as

(1− ρ

ρp

)Ω2x+ 2Ω

(uy,p −

ρ

ρpuy

)(22.2-12)

where uy,p and uy are the particle and fluid velocities in the Cartesian y direction, and

(1− ρ

ρp

)Ω2y − 2Ω

(ux,p −

ρ

ρpux

)(22.2-13)

where ux,p and ux are the particle and fluid velocities in the Cartesian x direction.



Thermophoretic Force

Small particles suspended in a gas that has a temperature gradient experience a forcein the direction opposite to that of the gradient. This phenomenon is known as ther-mophoresis. FLUENT can optionally include a thermophoretic effect on particles in theadditional acceleration (force/unit mass) term, Fx, in Equation 22.2-1:

Fx = −DT,p1

mpT

∂T

∂x(22.2-14)

where DT,p is the thermophoretic coefficient. You can define the coefficient to be constant,polynomial, or a user-defined function, or you can use the form suggested by Talbot [366]:

Fx = − 6πdpµ2Cs(K + CtKn)

ρ(1 + 3CmKn)(1 + 2K + 2CtKn)

1

mpT

∂T

∂x(22.2-15)

where: Kn = Knudsen number = 2 λ/dpλ = mean free path of the fluidK = k/kpk = fluid thermal conductivity based on translational

energy only = (15/4) µRkp = particle thermal conductivityCS = 1.17Ct = 2.18Cm = 1.14mp = particle massT = local fluid temperatureµ = fluid viscosity

This expression assumes that the particle is a sphere and that the fluid is an ideal gas.



Brownian Force

For sub-micron particles, the effects of Brownian motion can be optionally included in theadditional force term. The components of the Brownian force are modeled as a Gaussianwhite noise process with spectral intensity Sn,ij given by [204]

Sn,ij = S0δij (22.2-16)

where δij is the Kronecker delta function, and

S0 =216νkBT

π2ρd5p

(ρpρ

)2Cc

(22.2-17)

T is the absolute temperature of the fluid, ν is the kinematic viscosity, and kB is theBoltzmann constant. Amplitudes of the Brownian force components are of the form

Fbi = ζi

√πSo∆t

(22.2-18)

where ζi are zero-mean, unit-variance-independent Gaussian random numbers. The am-plitudes of the Brownian force components are evaluated at each time step. The energyequation must be enabled in order for the Brownian force to take effect. Brownian forceis intended only for nonturbulent models.

Saffman’s Lift Force

The Saffman’s lift force, or lift due to shear, can also be included in the additional forceterm as an option. The lift force used is from Li and Ahmadi [204] and is a generalizationof the expression provided by Saffman [312]:

~F =2Kν1/2ρdij

ρpdp(dlkdkl)1/4(~v − ~vp) (22.2-19)

where K = 2.594 and dij is the deformation tensor. This form of the lift force is intendedfor small particle Reynolds numbers. Also, the particle Reynolds number based on theparticle-fluid velocity difference must be smaller than the square root of the particleReynolds number based on the shear field. Since this restriction is valid for submicronparticles, it is recommended to use this option only for submicron particles.



22.2.2 Turbulent Dispersion of Particles

The dispersion of particles due to turbulence in the fluid phase can be predicted usingthe stochastic tracking model or the particle cloud model (see Section 22.2.2: TurbulentDispersion of Particles). The stochastic tracking (random walk) model includes the effectof instantaneous turbulent velocity fluctuations on the particle trajectories through theuse of stochastic methods (see Section 22.2.2: Stochastic Tracking). The particle cloudmodel tracks the statistical evolution of a cloud of particles about a mean trajectory(see Section 22.2.2: Particle Cloud Tracking). The concentration of particles within thecloud is represented by a Gaussian probability density function (PDF) about the meantrajectory. For stochastic tracking a model is available to account for the generation ordissipation of turbulence in the continuous phase (see Section 22.9.1: Coupling Betweenthe Discrete and Continuous Phases).

i Turbulent dispersion of particles cannot be included if the Spalart-Allmarasturbulence model is used.

Stochastic Tracking

When the flow is turbulent, FLUENT will predict the trajectories of particles using themean fluid phase velocity, u, in the trajectory equations (Equation 22.2-1). Optionally,you can include the instantaneous value of the fluctuating gas flow velocity,

u = u+ u′ (22.2-20)

to predict the dispersion of the particles due to turbulence.

In the stochastic tracking approach, FLUENT predicts the turbulent dispersion of particlesby integrating the trajectory equations for individual particles, using the instantaneousfluid velocity, u + u

′(t), along the particle path during the integration. By computing

the trajectory in this manner for a sufficient number of representative particles (termedthe “number of tries”), the random effects of turbulence on the particle dispersion maybe accounted for.

FLUENT uses a stochastic method (random walk model) to determine the instantaneousgas velocity. In the discrete random walk (DRW) model, the fluctuating velocity com-ponents are discrete piecewise constant functions of time. Their random value is keptconstant over an interval of time given by the characteristic lifetime of the eddies.

The DRW model may give nonphysical results in strongly nonhomogeneous diffusion-dominated flows, where small particles should become uniformly distributed. Instead,the DRW will show a tendency for such particles to concentrate in low-turbulence regionsof the flow.



The Integral Time

Prediction of particle dispersion makes use of the concept of the integral time scale, T ,which describes the time spent in turbulent motion along the particle path, ds:

T =∫ ∞

0

up′(t)up

′(t+ s)

up′2

ds (22.2-21)

The integral time is proportional to the particle dispersion rate, as larger values indicatemore turbulent motion in the flow. It can be shown that the particle diffusivity is givenby ui′uj ′T .

For small “tracer” particles that move with the fluid (zero drift velocity), the integral timebecomes the fluid Lagrangian integral time, TL. This time scale can be approximated as

TL = CLk

ε(22.2-22)

where CL is to be determined as it is not well known. By matching the diffusivity oftracer particles, ui′uj ′TL, to the scalar diffusion rate predicted by the turbulence model,νt/σ, one can obtain

TL ≈ 0.15k

ε(22.2-23)

for the k-ε model and its variants, and

TL ≈ 0.30k

ε(22.2-24)

when the Reynolds stress model (RSM) is used [74]. For the k-ω models, substituteω = ε/k into Equation 22.2-22. The LES model uses the equivalent LES time scales.

The Discrete Random Walk Model

In the discrete random walk (DRW) model, or “eddy lifetime” model, the interaction ofa particle with a succession of discrete stylized fluid phase turbulent eddies is simulated.Each eddy is characterized by

• a Gaussian distributed random velocity fluctuation, u′, v′, and w′

• a time scale, τe



The values of u′, v′, and w′ that prevail during the lifetime of the turbulent eddy aresampled by assuming that they obey a Gaussian probability distribution, so that

u′

= ζ

√u′2 (22.2-25)

where ζ is a normally distributed random number, and the remainder of the right-handside is the local RMS value of the velocity fluctuations. Since the kinetic energy ofturbulence is known at each point in the flow, these values of the RMS fluctuatingcomponents can be defined (assuming isotropy) as

√u′2 =

√v′2 =

√w′2 =

√2k/3 (22.2-26)

for the k-ε model, the k-ω model, and their variants. When the RSM is used, nonisotropyof the stresses is included in the derivation of the velocity fluctuations:

u′ = ζ

√u′2 (22.2-27)

v′ = ζ

√v′2 (22.2-28)

w′ = ζ

√w′2 (22.2-29)

when viewed in a reference frame in which the second moment of the turbulence is diag-onal [412]. For the LES model, the velocity fluctuations are equivalent in all directions.See Section 12.9.4: Inlet Boundary Conditions for the LES Model for details.

The characteristic lifetime of the eddy is defined either as a constant:

τe = 2TL (22.2-30)

where TL is given by Equation 22.2-22 in general (Equation 22.2-23 by default), or as arandom variation about TL:

τe = −TL log(r) (22.2-31)

where r is a uniform random number between 0 and 1 and TL is given by Equation 22.2-23.The option of random calculation of τe yields a more realistic description of the correlationfunction.



The particle eddy crossing time is defined as

tcross = −τ ln

[1−

(Le

τ |u− up|

)](22.2-32)

where τ is the particle relaxation time, Le is the eddy length scale, and |u − up| is themagnitude of the relative velocity.

The particle is assumed to interact with the fluid phase eddy over the smaller of theeddy lifetime and the eddy crossing time. When this time is reached, a new value of theinstantaneous velocity is obtained by applying a new value of ζ in Equation 22.2-25.

Using the DRW Model

The only inputs required for the DRW model are the value for the integral time-scaleconstant, CL (see Equations 22.2-22 and 22.2-30) and the choice of the method used forthe prediction of the eddy lifetime. You can choose to use either a constant value or arandom value by selecting the appropriate option in the Set Injection Properties panel foreach injection, as described in Section 22.12.5: Stochastic Tracking.

i Turbulent dispersion of particles cannot be included if the Spalart-Allmarasturbulence model is used.

Stochastic Staggering of Particles

In order to obtain a better representation of an injector, the particles can be staggeredeither spatially or temporally. When particles are staggered spatially, FLUENT randomlysamples from the region in which the spray is specified (e.g., the sheet thickness in thepressure-swirl atomizer) so that as the calculation progresses, trajectories will originatefrom the entire region. This allows the entire geometry specified in the atomizer to besampled while specifying fewer streams in the input panel, thus decreasing computationalexpense.

When injecting particles in a transient calculation using relatively large time steps inrelation to the spray event, the particles can clump together in discrete bunches. Theclumps do not look physically realistic, though FLUENT calculates the trajectory foreach particle as it passes through a cell and the coupling to the gas phase is properlyaccounted for. To obtain a statistically smoother representation of the spray, the particlescan be staggered in time. During the first time step, the particle is tracked for a randompercentage of its initial step. This results in a sample of the initial volume swept out bythe particle during the first time step and a smoother, more uniform spatial distributionat longer time intervals.

The menu for staggering is available in the text user interface, under

define/models/dpm/options/particle-staggering.



The “staggering factor” in the TUI is a constant which multiplies the random sample.The staggering factor controls the percentage of the initial time step that will be sampled.For example, if the staggering factor is 0.5, then the parcels in the injection will be trackedbetween half and all of their full initial time step. If the staggering factor is 0.1, then theparcels will be tracked between ninety percent and all of their initial time step. If thestaggering factor is set to 0.9, the parcels will be tracked between ten percent and all oftheir initial time step. This allows the user to control the amount of smoothing betweeninjections.

The default values for the options in the TUI are no temporal staggering and a temporalstaggering factor of 1.0. The temporal staggering factor is inactive until the flag fortemporal staggering is turned on.

Particle Cloud Tracking

Particle dispersion due to turbulent fluctuations can also be modeled with the particlecloud model [27, 28, 158, 216]. The turbulent dispersion of particles about a meantrajectory is calculated using statistical methods. The concentration of particles aboutthe mean trajectory is represented by a Gaussian probability density function (PDF)whose variance is based on the degree of particle dispersion due to turbulent fluctuations.The mean trajectory is obtained by solving the ensemble-averaged equations of motionfor all particles represented by the cloud (see Section 22.2.2: Particle Cloud Tracking).

The cloud enters the domain either as a point source or with an initial diameter. Thecloud expands due to turbulent dispersion as it is transported through the domain untilit exits. As mentioned before, the distribution of particles in the cloud is defined by aprobability density function (PDF) based on the position in the cloud relative to the cloudcenter. The value of the PDF represents the probability of finding particles representedby that cloud with residence time t at location xi in the flow field. The average particlenumber density can be obtained by weighting the total flow rate of particles representedby that cloud, m, as

〈n(xi)〉 = mP (xi, t) (22.2-33)

The PDFs for particle position are assumed to be multivariate Gaussian. These arecompletely described by their mean, µi, and variance, σi

2, and are of the form

P (xi, t) =1

(2π)3/23∏i=1

σi

e−s/2 (22.2-34)



where

s =3∑i=1

(xi − µiσi

)2

(22.2-35)

The mean of the PDF, or the center of the cloud, at a given time represents the mostlikely location of the particles in the cloud. The mean location is obtained by integratinga particle velocity as defined by an equation of motion for the cloud of particles:

µi(t) ≡ 〈xi(t)〉 =∫ t

0〈Vi(t1)〉dt1 + 〈xi(0)〉 (22.2-36)

The equations of motion are constructed using an ensemble average.

The radius of the particle cloud is based on the variance of the PDF. The variance, σ2i (t),

of the PDF can be expressed in terms of two particle turbulence statistical quantities:

σ2i (t) = 2

∫ t

0〈u′2p,i(t2)〉

∫ t2

0Rp,ii(t2, t1)dt1dt2 (22.2-37)

where 〈u′2p,i〉 are the mean square velocity fluctuations, and Rp,ij(t2, t1) is the particlevelocity correlation function:

Rp,ij(t2, t1) =〈u′p,i(t2)u′p,j(t1)〉[〈u′2p,i(t2)u′2p,j(t2)〉

]1/2 (22.2-38)

By using the substitution τ = |t2 − t1|, and the fact that

Rp,ij(t2, t1) = Rp,ij(t4, t3) (22.2-39)

whenever |t2 − t1| = |t4 − t3|, we can write

σ2i (t) = 2

∫ t

0〈u′2p,i(t2)〉

∫ t2

0Rp,ii(τ)dτdt2 (22.2-40)

Note that cross correlations in the definition of the variance (Rp,ij, i 6= j) have beenneglected.



The form of the particle velocity correlation function used determines the particle disper-sion in the cloud model. FLUENT uses a correlation function first proposed by Wang [388],and used by Jain [158]. When the gravity vector is aligned with the z-coordinate direc-tion, Rij takes the form:

Rp,11 =u′2

θe−(τ/τa) StT

(B − 0.5mTγ

St2TB

2 + 1

θ

)

+u′2

θe−(τB/T )

(−1 +

mTSt2TγB

θ+ 0.5mTγ

τ

T

)(22.2-41)

Rp,22 = Rp,11 (22.2-42)

Rp,33 =u′2StTB

θe−(τ/τa) − u′2

θe−(τB/T ) (22.2-43)

where B =√

1 +m2Tγ

2 and τa is the aerodynamic response time of the particle:

τa =ρpd

2p

18µ(22.2-44)

and

T =mTTmEm

(22.2-45)

TfE =C3/4µ k3/2

ε(23k)1/2

(22.2-46)

γ =τag

u′(22.2-47)

St =τaTmE

(22.2-48)

StT =τaT

(22.2-49)

θ = St2T (1 +m2

Tγ2)− 1 (22.2-50)

m =u

u′(22.2-51)

TmE = TfEu

u′(22.2-52)

mT = m

[1− G(m)

(1 + St)0.4(1+0.01St)

](22.2-53)

G(m) =2√π

∫ ∞0

e−y2dy(

1 + m2

π(√π erf(y)y − 1 + e−y2)

)5/2(22.2-54)


22.3 Multicomponent Particle Theory

Using this correlation function, the variance is integrated over the life of the cloud. Atany given time, the cloud radius is set to three standard deviations in the coordinatedirections. The cloud radius is limited to three standard deviations since at least 99.2%of the area under a Gaussian PDF is accounted for at this distance. Once the cellswithin the cloud are established, the fluid properties are ensemble-averaged for the meantrajectory, and the mean path is integrated in time. This is done with a weighting factordefined as

W (xi, t) ≡

∫Vcell

P (xi, t)dV∫Vcloud

P (xi, t)dV(22.2-55)

If coupled calculations are performed, sources are distributed to the cells in the cloudbased on the same weighting factors.

Using the Cloud Model

The only inputs required for the cloud model are the values of the minimum and maximumcloud diameters. The cloud model is enabled in the Set Injection Properties panel for eachinjection, as described in Section 22.12.5: Cloud Tracking.

i The cloud model is not available for unsteady particle tracking, or in par-allel, when using the message passing option for the particles.

22.3 Multicomponent Particle Theory

A number of industrially important processes, such as distillation, absorption and ex-traction, bring into contact two phases which are not at equilibrium. The rate at which aspecie is transferred from one phase to the other depends on the departure of the systemfrom equilibrium. The quantitative treatment of these rate processes requires knowledgeof the equilibrium states of the system. Apart from these cases, vapor-liquid equilibrium(VLE) relationships in multicomponent systems are needed for the solution of many otherclasses of engineering problems, such as the computation of evaporation rates in spraycombustion applications.

In FLUENT the rate of vaporization of a single component droplet is computed fromEquation 22.9-20, where Ci,s is the equilibrium concentration of the droplet species inthe gas phase, and is computed in Equation 22.9-21 as:

Ci,s = psat/RTp (22.3-1)

where Tp is the droplet temperature, and psat is the saturation pressure of the dropletspecies at Tp.



For the general case where N components are evaporating from a droplet (distillation),the evaporation rate of each species is again given by Equation 22.9-20; however, psat inEquation 22.3-1 must be replaced by pi, the partial pressure of species i, to calculate theconcentration of i at the droplet surface.

The partial pressure of species i can be obtained from the general expression for vaporliquid equilibrium [342],

φiyip = γixiφsat,ipsat,iexp

[ViL(p− psat,i)

RT

](22.3-2)

which is obtained by equating the fugacity of the liquid and vapor mixtures. Here, φi, isthe fugacity coefficient for species i in the mixture, and accounts for nonideality in thegas; φsat,i is the fugacity coefficient for pure i at the saturation pressure; γi is the activitycoefficient for species i in the mixture, and accounts for nonideality in the liquid phase; pis the absolute pressure; T is the temperature; R is the universal gas constant; Vi

L is themolar volume of the liquid; psat,i is the saturation pressure of species i ; and xi and yi arethe equilibrium compositions of species i in the liquid and gas phases, respectively. Theexponential term is the Poynting correction factor and accounts for compressibility effectswithin the liquid. Except at high pressures, the Poynting factor is usually negligible.

Under low pressure conditions where the gas phase may be assumed to be ideal, φi ≈ 1and φsat,i ≈ 1. Furthermore, if the liquid is also assumed to be ideal, γi ≈ 1 andEquation 22.3-2 reduces to Raoult’s law,

yip = xipsat,i (22.3-3)

Raoult’s law is the default vapor-liquid equilibrium expression used in the FLUENT mul-ticomponent droplet model. However, there is a UDF hook available for user-definedvapor-liquid equilibrium models.

While Raoult’s law represents the simplest form of the VLE equation, keep in mind thatit is of limited use, as the assumptions made for its derivation are usually unrealistic.The most critical assumption is that the liquid phase is an ideal solution. This is notlikely to be valid, unless the system is made up of species of similar molecular sizes andchemical nature, such as in the case of benzene and toluene, or n-heptane and n-hexane.

When Raoult’s law is applicable, the vaporization rate of each species from a multicompo-nent droplet can be computed from Equation 22.9-20, with the equilibrium concentrationof species i in the gas phase Ci,s computed as:

Ci,s = xipsat,i/RTp (22.3-4)

where Tp is the droplet temperature, xi is the mole fraction of species i in the droplet,and psat,i is the saturation pressure of species i at Tp.


22.4 Wall-Film Model Theory


22.4.1 Introduction

Spray-wall interaction is an important part of the mixture formation process in port fuelinjected (PFI) engines. A fuel spray impinges on a surface, usually at the intake port nearthe intake valve, as well as at the intake valve itself, where it splashes and subsequentlyevaporates. The evaporated mixture is entrained into the cylinder of the engine, whereit is mixed with the fresh charge and any residual gas in the cylinder. The mixture thatis compressed and burned, finally exits through the exhaust port. The process repeatsitself between 200 and 8000 times per second, depending on the engine.

Several cycles worth of fuel remain in the intake tract due to film formation on thewalls. This in turn makes the film important in hydrocarbon emissions for PFI engines.Additionally, film can form inside combustion chambers of direct injection (DI) typesof engines. In a direct injection engine, fuel is injected directly into the combustionchamber, where the spray can impinge upon the piston if the injection event is early orlate in the cycle. The modeling of the wall-film inside a DI engine, especially in dieselengines, is compounded by the presence of carbon deposits on the surfaces of the engine.This carbon deposit absorbs the liquid film as it impinges upon it. It is believed thatthe carbon deposits adsorb the fuel later in the cycle, however this phenomena is verycomplex and is not well understood.

DPM particles are used to model the wall-film. The wall-film model in FLUENT allowsa single component liquid drop to impinge upon a boundary surface and form a thinfilm. The model can be broken down into four major subtopics: interaction during theinitial impact with a wall boundary, subsequent tracking on surfaces, calculation of filmvariables, and coupling to the gas phase. Figure 22.4.1 schematically shows the basicmechanisms considered for the wall-film model.

The main assumptions and restrictions for the wall-film model are as follows:

• The layer is thin, less than 500 microns in thickness. This limitation is due to theassumption of a linear velocity profile in the film.

• The temperature in the film particles change relatively slowly so that an analyticalintegration scheme can be utilized.

• Film particles are assumed to be in direct contact with the wall surface and theheat transfer from the wall to the film takes place by conduction.

• The film temperature never exceeds the boiling temperature for the liquid.

• The simulation is transient.

• The wall-film model is not available with the Workpile Algorithm shared memoryoption in parallel processing.



Evaporation

Convectiveheat transfer

Film separation

Major Physical Phenomena

Conduction

ImpingingFuel Drops

Splashing

and sheet breakupShear Forces

Figure 22.4.1: Mechanisms of Splashing, Momentum, Heat and Mass Trans-fer for the Wall-Film

If you wish to model a spray impacting a very hot wall, the wall-jet model may bemore appropriate as the assumption in the wall-jet impingement model is that there isa vapor layer underneath the drops which keeps them from making direct contact withthe boundary surface. This may be a more accurate assumption for in-cylinder dieselcomputations at typical operating conditions.

22.4.2 Interaction During Impact with a Boundary

The wall interaction is based on the work of Stanton [353] and O’Rourke [271], wherethe regimes are calculated for a drop-wall interaction based on local information. Thefour regimes, stick, rebound, spread, and splash are based on the impact energy and walltemperature. The following chart is helpful in showing the cutoffs.



-

6

Tb Tw

E

Stick

Spread

Rebound

Splash

Figure 22.4.2: Simplified Decision Chart for Wall Interaction Criterion.

Below the boiling temperature of the liquid, the impinging droplet can either stick, spreador splash, while above the boiling temperature, the particle can either rebound or splash.

The criteria by which the regimes are partitioned are based on the impact energy andthe boiling temperature of the liquid. The impact energy is defined by

E2 =ρV 2

r D

σ

(1

min (h0/D, 1) + δbl/D

)(22.4-1)

where ρ is the liquid density, Vr is the relative velocity of the particle in the frame of thewall (i.e. V 2

r = (Vp−Vw)2), D is the diameter of the droplet, and σ is the surface tensionof the liquid. Here, δbl is a boundary layer thickness, defined by

δbl =D√Re

(22.4-2)

where the Reynolds number is defined as Re = ρVrD/µ. By defining the energy as inEquation 22.4-1, the presence of the film on the wall suppresses the splash, but does notgive unphysical results when the film height approaches zero.

The sticking regime is applied when the dimensionless energy E is less than 16, and theparticle velocity is set equal to the wall velocity. In the spreading regime, the initial di-rection and velocity of the particle are set using the wall-jet model, where the probabilityof the drop having a particular direction along the surface is given by an analogy of aninviscid liquid jet with an empirically defined radial dependence for the momentum flux.

If the wall temperature is above the boiling temperature of the liquid, impingement eventsbelow a critical impact energy (E) results in the particles rebounding from the wall. Forthe rebound regime, the particle rebounds with the following coefficient of restitution:

e = 0.993− 1.76ΘI + 1.56Θ2I − 0.49Θ3

I (22.4-3)



where ΘI is the impingement angle as measured from the wall surface.

Splashing occurs when the impingement energy is above a critical energy threshold,defined as Ecr = 57.7. The number of splashed droplet parcels is set in the BoundaryConditions panel with a default number of 4, however, the user can select numbers betweenzero and ten. The splashing algorithm follows that described by Stanton [353] and isdetailed in Section 22.4.3: Splashing.

22.4.3 Splashing

If the particle impinging on the surface has a sufficiently high energy, the particle splashesand several new particles are created. The number of particles created by each impactis explicitly set by the user in the DPM tab in the Boundary Conditions panel, as inFigure 22.4.3. The number of splashed parcels may be set to an integer value betweenzero and ten. The properties (diameter, magnitude, and direction) of the splashed parcelsare randomly sampled from the experimentally obtained distribution functions describedin the following sections. Setting the number of splashed parcels to zero turns off thesplashing calculation. Bear in mind that each splashed parcel can be considered a discretesample of the distribution curves and that selecting the number of splashed drops in theBoundary Conditions panel does not limit the number of splashed drops, only the numberof parcels representing those drops.

Therefore, for each splashed parcel, a different diameter is obtained by sampling a cu-mulative probability distribution function (CPDF), which is obtained from a Weibulldistribution function and fitted to the data from Mundo, et al. [254]. The equation is

pdf

(d

D

)= 2

d

D2exp

−( dD

)2 (22.4-4)

and it represents the probability of finding drops of diameter di in a sample of splasheddrops. This distribution is similar to the Nakamura-Tanasawa distribution function usedby O’Rourke [271], but with the peak of the distribution function being D = dmax/

√2.

To ensure that the distribution functions produce physical results with an increasingWeber number, the following expression for dmax from O’Rourke [271] is used. The peakof the splashed diameter distribution is

dmax/d0 = MAX

(E2crit

E2,

6.4

We, 0.06

)(22.4-5)

where the expression for energy is given by Equation 22.4-1. Low Weber number impactsare described by the second term in Equation 22.4-5 and the peak of the minimumsplashed diameter distribution is never less than 0.06 for very high energy impacts in anyof the experiments analyzed by O’Rourke [271]. The Weber number in Equation 22.4-5is defined using the relative velocity and drop diameter:

We =ρV 2

r D

σ(22.4-6)



Figure 22.4.3: The Discrete Phase Model Panel and the Film Model Parameters



The cumulative probability distribution function (CPDF) is needed so that a diametercan be sampled from the experimental data. The CPDF is obtained from integratingEquation 22.4-4 to obtain

cpdf

(d

D

)= 1− exp

−( dD

)2 (22.4-7)

which is bounded by zero and one. An expression for the diameter (which is a functionof D, the impingement Weber number We, and the impingement energy) is obtained byinverting Equation 22.4-7 and sampling the CPDF between zero and one. The expressionfor the diameter of the ith splashed parcel is therefore given by,

di = D√− ln (1− ci)

where ci is the ith random sample. Once the diameter of the splashed drop has beendetermined, the probability of finding that drop in a given sample is determined byevaluating Equation 22.4-4 at the given diameter. The number of drops per parcel canbe expressed as a function of the total number of splashed drops:

Ni = Ntotpdfi (22.4-8)

where the pdfi is for the ith sample. The values of pdfi are then normalized so that theirsum is one. Both the number per parcel (Ni) and the total number of splashed drops(Ntot) is unknown, but an expression for Ntot can be obtained from the conservation ofmass if the total splashed mass is known.

The amount of mass splashed from the surface is a quadratic function of the splashingenergy, obtained from the experimental data from Mundo [254]. The splashed massfraction ys is given by

ys =

1.8x10−4(E2 − E2

crit) , E2crit < E2 < 7500

0.70 , E2 > 7500

The authors (O’Rourke et al. [271]) noted that nearly all of the impacts for typical dieselsprays are well above the upper bound and so the splashing event nearly always ejects70% of the mass of the impinging drop. To obtain an expression for the total number ofdrops, we note that overall conservation of mass requires that the sum of the total massof the splashed parcel(s) must equal the splashed mass fraction, or

ρπ

6Ntot

Nparcels∑n=1

(pdfnd

3n

)= ysm0 (22.4-9)

where m0 is the total mass of the impinging parcel. The expression for the total numberof splashed drops is

Ntot =ysm0

ρπ6

∑Nparcelsn=1 (pdfnd3

n)



The number of splashed drops per parcel is then determined by Equation 22.4-8 with thevalues of pdfi given by Equation 22.4-4.

To calculate the velocity with which the splashed drops leave the surface, additionalcorrelations are sampled for the normal component of the velocity. A Weibull function,fit to the data from Mundo [254], is used as the PDF for the normal component. Theprobability density is given by

pdf(VniVnd

)=

bvΘv

(Vni/Vnd

Θv

)bv−1 exp

−(Vni/VndΘv

)bv (22.4-10)

where

bv =

2.1, ΘI ≤ 50

1.10 + 0.02ΘI , ΘI > 50(22.4-11)

andΘv = 0.158e0.017ΘI (22.4-12)

where ΘI is the angle at which the parcel impacts the surface, or the impingement angle.The tangential component of the velocity is obtained from the expression for the reflectionangle Θs:

Θs = 65.4 + 0.226Θl (22.4-13)

combined with

Vti =Vni

tan(Θs)(22.4-14)

Finally, an energy balance is performed for the new parcels so that the sum of the kineticand surface energies of the new drops does not exceed that of the old drops. The energybalance is given by

1

2

Nparcel∑i=1

(miV

2i

)+ πσ

Nparcel∑i=1

(Nid

2i

)=

1

2mdV

2d

(miV

2i

)+ πσ

(mdd

2i

)− Ecrit

where Ecrit is the threshold energy for splashing to occur. To ensure conservation ofenergy, the following correction factor is computed:

K =12mdV

2d (miV

2i ) + πσ (mdd

2i )− Ecrit − πσ

∑Nparceli=1 (Nid

2i )

12

∑Nparceli=1 (miV 2

i ). (22.4-15)

This correction factor is needed due to the relatively small number of sampled points forthe velocity of the splashed drops (see Stanton [354] for more detail). The componentsof the splashed parcel are multiplied by the square root of K in Equation 22.4-15 so thatenergy will be conserved. The normal and tangential velocity components of the splashedparcels are therefore given by

V ′ni =√KVni and V ′ti =

√KVti



FLUENT will limit the velocity of the splashed parcels so that they do not exceed theimpact velocity of the original parcel. It is important to note that splashing events areinherently transient, so the splashing submodel is only available with unsteady trackingin FLUENT. Splashing can also cause large increases in source terms in the cells in whichit occurs, which can cause difficulty in convergence between time steps. Thus, it may benecessary to use a smaller time step during the simulation when splashing is enabled.

22.4.4 Separation Criteria

The film can separate from the wall when the stress at an edge of the film exceedsthe adhesion forces holding the film to the wall. These forces are complex and highlydependent on local surface conditions. An order of magnitude analysis of a film roundinga sharp corner shows that the stresses at the edge of a film are proportional to the anglewhich the film negotiates. In FLUENT, you can specify the maximum angle that thefilm can negotiate using a special text command. For more information, contact yourFLUENT support engineer.

22.4.5 Conservation Equations for Wall-Film Particles

Conservation equations for momentum, mass, and energy for individual parcels in thewall-film are described below. The particle-based approach for thin films was first for-mulated by O’Rourke [270] and most of the following derivation is based closely on thatwork.

Momentum

The equation for the momentum of a parcel on the film is

ρhd~updt

+ h(∇spf )α = τg~tg + τw~tw + ~P imp,α − Mimp,α~up + ~F n,α + ρh(~g − ~aw) (22.4-16)

where α denotes the current face on which the particle resides, h denotes the current filmheight at the particle location, ∇s is the gradient operator restricted to the surface, andpf is the pressure on the surface of the film. On the right-hand side of Equation 22.4-16,τg denotes the magnitude of the shear stress of the gas flow on the surface of the film, ~tgis the unit vector in the direction of the relative motion of the gas and the film surface,µl is the liquid viscosity, and τw is the magnitude of the stress that the wall exerts onthe film. Similarly to the expression for ~tg, ~tw is the unit vector in the direction of therelative motion of the film and the wall. The remaining expressions on the right-hand

side of Equation 22.4-16 are ~P imp,α which denotes the impingement pressure on the film

surface, ~M imp,α is the impingement momentum source, and ~F n,α is the force necessaryto keep the film on the surface, as determined by

~up · nα = 0. (22.4-17)



Here, ρh(~g − ~aw) is the body force term. Note that the body force term can be verysignificant, despite the small values of film thickness due to the very high accelera-tion rates seen in simulations with moving boundaries. The requirement representedby Equation 22.4-17 is explicitly enforced at each time step in FLUENT for all particlesrepresenting the wall-film.

The term h(∇spf )α is the surface gradient of the pressure on the face, pf . This pressure,pf , is the sum of the fluid pressure and the impingement pressure from the drops on theface, given by

pf = Pcell − ~P imp,α · n+ Mimp,α~up · nwhere the impingement mass Mimp,α is given by

Mimp,α =∫ ∫ ∫

ρlVp~v · nf(~xs, ~v, r, Td, t)drd~vdTd (22.4-18)

and the impingement pressure is given by

~P imp,α =∫ ∫ ∫

ρlVp~v~v · nf(~xs, ~v, r, Td, t)drd~vdTd (22.4-19)

where Vp is the volume of the drop. An approximation of the impingement mass inEquation 22.4-18 is given by

Mimp,α =

(Ns∑n=0

ρVp

)/Aα∆t, (22.4-20)

and the corresponding expression of the impingement pressure in Equation 22.4-19 isgiven by

~P imp,α =

Ni∑n=0

ρVp(~un+1p − ~unp )

/Aα∆t. (22.4-21)

The summation in Equation 22.4-20 is over all the drops which actually stick to the faceα during the time step (Ns). The summation in Equation 22.4-21 is over all the particleswhich impinge upon the face during the same interval (Ni).

The expression for the stress that the gas exerts on the surface of the wall-film, τg, inEquation 22.4-16 is given by

τg = Cf (~ug − 2~up)2 = CfV

2relg

where Cf is the skin friction coefficient and ~ug is the gas velocity evaluated at the filmheight above the wall. The assumption made in evaluating the skin friction coefficientis that the wall shear stress from the gas is constant over the thickness of the film andthe boundary layer above the film (in the normal direction from the face). The stress istangent to the wall in the direction of the difference between the wall-film velocity andthe gas velocity, so the unit vector in the direction of the velocity difference along thesurface is

tg =~Vrelg − (~Vrelg · n)n

|~Vrelg − (~Vrelg · n)n|



where n is the normal face . The expression for the stress that the wall exerts on thefilm, τw, in Equation 22.4-16 is given by

τw = −µlh|2~up − ~uw| = −

µlh|~Vrelw |

where µl is the liquid viscosity and ~uw is the velocity of the wall. Here, τw acts in thedirection of the velocity difference between the wall and the film, as given by

tw =~Vrelw − (~Vrelw · n)n

|~Vrelw − (~Vrelw · n)n|.

Note that the tangential unit vectors, tg and tw, are independent and can point in com-pletely different directions.

Since FLUENT solves a particle position equation of the form

d~updt

= α− β~up,

Equation 22.4-16 must be rearranged. The film particle acceleration is then given by

d~updt

=

Cf |Vrelg |ρh

tg −(∇spf )α

ρ+µl| ~uw|ρh2

tw +~Pimpρh

+ ~g

−(2Cf |Vrelg |ρh

+2µlρh2

+Mimp

ρh

)~up.

(22.4-22)

The terms for Mimp and ~Pimp are used from the previous time step and the differentialequations for the particle motion are solved with the existing integration routines.

Mass Transfer from the Film

The film vaporization law is applied when the film particle is above the vaporizationtemperature Tvap. A wall particle has the temperature limited by the boiling temperatureTbp and does not have a specific boiling law associated with the physics of film boiling.

The vaporization rate of the film is governed by gradient diffusion from the surfaceexposed to the gas phase. The gradient of vapor concentration between the film surfaceand the gas phase is

Ni = Bf (Ci,s − Ci,∞) (22.4-23)

where Ni is the molar flux of vapor (with units of kgmol/m2-s), Bf is the mass transfercoefficient (in m/s), and Ci,s and Ci,∞ are the vapor concentrations on the film surfaceand in the bulk gas, respectively. The units of vapor concentration are kgmol/m3.

The vapor concentration at the surface is evaluated using the saturated vapor pressureat the film surface temperature and the bulk gas concentration is obtained from the flowfield solution. The vaporization rate is sensitive to the saturated vapor pressure, similarto droplet vaporization.



The mass transfer coefficient is obtained using a Nusselt correlation for the heat transfercoefficient and replacing the Prandtl number with the Schmidt number. The equation is

Nux =Bfx

kf=

0.332Re1/2

x Sc1/3 Rex < 2500, 0.6 < Sc < 500.0296Re4/5

x Sc1/3 Rex > 2500, 0.6 < Sc < 60(22.4-24)

where the Reynolds number is based on a representative length derived from the facearea. The temperature for the film surface is equal to the gas temperature, but is limitedby the boiling temperature of the liquid. The particle properties are evaluated at thesurface temperature when used in correlation 22.4-24.

For multicomponent vaporization, the Schmidt number based on the diffusivity of eachspecies is used to calculate the correlation in equation 22.4-24 for each component.

The mass of the particle is decreased by

mp(t+ ∆t) = mp(t)−NiApMw,i∆t (22.4-25)

where Mw,i is the molecular weight of the gas phase species to which the vapor from theliquid is added. The diameter of the film particle is decreased to account for the massloss in the individual parcel. This keeps the number of drops in the parcel constant andacts only as a place holder. When the parcel detaches from the boundary, the diameteris set to the height of the film and the number in the parcel is adjusted so that the overallmass of the parcel is conserved.

Energy Transfer from the Film

To obtain an equation for the temperature in the film, energy flux from the gas side aswell as energy flux from the wall side must be considered. The assumed temperatureprofile in the liquid is bilinear, with the surface temperature Ts being the maximumtemperature of the gas at the film height. Furthermore, the boiling point of the liquidand the wall temperature will be the maximum of the wall face temperature Tw, andwill be the same boiling temperature as the liquid. An energy balance on a film particleyields

d

dtmpCpTp = Qcond +Qconv (22.4-26)

where Qcond is the conduction from the wall, given by

Qcond =κAph

(Tw − Tp)

where κ is the thermal conductivity of the liquid and h is the film height at the locationof the particle, as seen in Figure 22.4.4. The convection from the top surface, Qconv isgiven by

Qconv = hfAp(Tg − Tp)where hf is the film heat transfer coefficient given by Equation 22.4-24 and Ap is thearea represented by a film particle, taken to be a mass weighted percentage of the face



area, Af . Contributions from the impingement terms are neglected in this formulation,as well as contributions from the gradients of the mean temperature on the edges of thefilm.

Ts

Tw

Tp2h

Figure 22.4.4: Assumption of a Bilinear Temperature Profile in the Film

Assuming that the temperature changes slowly for each particle in the film, the equationfor the change in temperature of a non vaporizing particle can be written as

mpCpdTpdt

= Ap

(−[hf +

κ

h

]Tp + hTg +

κ

hTw

)(22.4-27)

As the particle trajectory is computed, FLUENT integrates Equation 22.4-27 to obtainthe particle temperature at the next time value, yielding

Tp(t+ ∆t) = αp + [Tp(t)− αp]e−βp∆t (22.4-28)

where ∆t is the integration time step and αp and βp are given by

αp =hfTg + κ

hTw

hf + κh

(22.4-29)

and

βp =Ap(hf + κ

h)

mpCp(22.4-30)

When the particle changes its mass during vaporization, an additional term is added toEquation 22.4-27 to account for the enthalpy of vaporization, which is given by

mpCpdTpdt

= Ap

(−[hf +

κ

h

]Tp + hTg +

κ

hTw

)+ mphfg (22.4-31)


22.5 Particle Erosion and Accretion Theory

where hfg is the latent heat of vaporization (with units of J/kg) and the expression mp

is the rate of evaporation in kg/s. This alters the expression for αp in Equation 22.4-29so that

αp =hfTg + κ

hTw + mphfg/Aphf + κ

h

(22.4-32)

When the wall-film model is active, the heat flux from the wall to the liquid film issubtracted from the heat flux from the wall to the gas phase. Additionally, enthalpyfrom vaporization of the liquid from the wall is subtracted from the cell to which thevapor mass goes. Since film boiling is modeled by limiting the liquid phase temperatureto the boiling point of the material, energy in excess of that absorbed by the liquid willbe put into the gas phase. When the thermal boundary conditions on the wall are setto a constant heat flux, the local temperature of the wall face is used as the thermalboundary condition for the wall-film particles.

22.5 Particle Erosion and Accretion Theory

Particle erosion and accretion rates can be monitored at wall boundaries. The erosionrate is defined as

Rerosion =Nparticles∑p=1

mpC(dp)f(α)vb(v)

Aface

(22.5-1)

where C(dp) is a function of particle diameter, α is the impact angle of the particle pathwith the wall face, f(α) is a function of impact angle, v is the relative particle velocity,b(v) is a function of relative particle velocity, and Aface is the area of the cell face at thewall. Default values are C = 1.8× 10−9, f = 1, and b = 0.

Since C, f , and b are defined as boundary conditions at a wall, rather than propertiesof a material, the default values are not updated to reflect the material being used. Youwill need to specify appropriate values at all walls. Values of these functions for sanderoding both carbon steel and aluminum are given by Edwards et al. [94].

The erosion rate as calculated above is displayed in units of removed material/(area-time), i.e., mass flux, and can therefore be changed accordingly to the defined unitsin FLUENT. The functions C and f have to be specified in consistent units to build adimensionless group with the relative particle velocity and its exponent. To computean erosion rate in terms of length/time (mm/year, for example) you can either define acustom field function to divide the erosion rate by the density of the wall material orinclude this division in the units for C and/or f . Note that the units given by FLUENTwhen displaying the erosion rate are no longer valid in the latter case.

A variety of erosion models [105, 234, 93, 262, 137, 313] containing model constants [137,93] and angle functions can be easily implemented into FLUENT. The equations describing



some of the erosion models can be modified to appear in the form of the general equationdescribing the erosion rate, Equation 22.5-1. For example, the Tulsa Angle DependentModel [93] described by Equation 22.5-2

ER = 1559B−0.59Fsv1.73f(α) (22.5-2)

can be rewritten in the form of Equation 22.5-1 with the following substitutions:

v1.73 = vb(v)

1559B−0.59Fs = C(dp)

where ER is the erosion rate, B is the Brinell hardness, and Fs is a particle shapecoefficient.

User-defined functions can be used to describe erosion models of any form. For morecomplex models, such as those models with varying function angles, f(α), the defaultErosion Model in the wall Boundary Conditions panel cannot be used. Hence, a user-definedfunction should be used instead. For information on how to apply user-defined functionsfor DPM erosion models, refer to the DEFINE DPM EROSION macro in the separate UDFManual , or contact your support engineer for further assistance.

Note that the particle erosion and accretion rates can be displayed only when coupledcalculations are enabled.

The accretion rate is defined as

Raccretion =Nparticles∑p=1

mp

Aface

(22.5-3)

22.6 Dynamic Drag Model Theory

Accurate determination of droplet drag coefficients is crucial for accurate spray modeling.FLUENT provides a method that determines the droplet drag coefficient dynamically,accounting for variations in the droplet shape.

The dynamic drag model is applicable in almost any circumstance. It is compatible withboth the TAB and wave models for droplet breakup. When the collision model is turnedon, collisions reset the distortion and distortion velocities of the colliding droplets.

Many droplet drag models assume the droplet remains spherical throughout the domain.With this assumption, the drag of a spherical object is determined by the following [217]:

Cd,sphere =

0.424 Re > 1000

24Re

(1 + 1

6Re2/3

)Re ≤ 1000

(22.6-1)


22.7 Spray Model Theory

However, as an initially spherical droplet moves through a gas, its shape is distortedsignificantly when the Weber number is large. In the extreme case, the droplet shapewill approach that of a disk. The drag of a disk, however, is significantly higher than thatof a sphere. Since the droplet drag coefficient is highly dependent upon the droplet shape,a drag model that assumes the droplet is spherical is unsatisfactory. The dynamic dragmodel accounts for the effects of droplet distortion, linearly varying the drag betweenthat of a sphere (Equation 22.6-1) and a value of 1.54 corresponding to a disk [217]. Thedrag coefficient is given by

Cd = Cd,sphere(1 + 2.632y) (22.6-2)

where y is the droplet distortion, as determined by the solution of

d2y

dt2=CFCb

ρgρl

u2

r2− Ckσ

ρlr3y − Cdµl

ρlr2

dy

dt(22.6-3)

In the limit of no distortion (y = 0), the drag coefficient of a sphere will be obtained,while at maximum distortion (y = 1) the drag coefficient corresponding to a disk will beobtained.

Note that Equation 22.6-3 is obtained from the TAB model for spray breakup, describedin Section 22.7.2: Taylor Analogy Breakup (TAB) Model, but the dynamic drag modelcan be used with either of the breakup models.


In addition to the simple injection types described in Section 22.12.1: Injection Types,FLUENT also provides more complex injection types for sprays. For most types of injec-tions, you will need to provide the initial diameter, position, and velocity of the particles.For sprays, however, there are models available to predict the droplet size and velocitydistributions. Models are also available for droplet breakup and collision, as well as adynamically varying drag coefficient which accounts for variation in droplet shape. Thesemodels for realistic spray simulations are described in this section.

Information is organized into the following subsections:

• Section 22.7.1: Droplet Collision Model

• Section 22.7.2: Droplet Breakup Models



22.7.1 Droplet Collision Model

Introduction

When your simulation includes tracking of droplets, FLUENT provides an option forestimating the number of droplet collisions and their outcomes in a computationallyefficient manner. The difficulty in any collision calculation is that for N droplets, eachdroplet has N−1 possible collision partners. Thus, the number of possible collision pairsis approximately 1

2N2. (The factor of 1

2appears because droplet A colliding with droplet

B is identical to droplet B colliding with droplet A. This symmetry reduces the numberof possible collision events by half.)

An important consideration is that the collision algorithm must calculate 12N2 possible

collision events at every time step. Since a spray can consist of several million droplets,the computational cost of a collision calculation from first principles is prohibitive. Thismotivates the concept of parcels. Parcels are statistical representations of a numberof individual droplets. For example, if FLUENT tracks a set of parcels, each of whichrepresents 1000 droplets, the cost of the collision calculation is reduced by a factor of106. Because the cost of the collision calculation still scales with the square of N , thereduction of cost is significant; however, the effort to calculate the possible intersectionof so many parcel trajectories would still be prohibitively expensive.

The algorithm of O’Rourke [268] efficiently reduces the computational cost of the spraycalculation. Rather than using geometry to see if parcel paths intersect, O’Rourke’smethod is a stochastic estimate of collisions. O’Rourke also makes the assumption thattwo parcels may collide only if they are located in the same continuous-phase cell. Thesetwo assumptions are valid only when the continuous-phase cell size is small comparedto the size of the spray. For these conditions, the method of O’Rourke is second-orderaccurate at estimating the chance of collisions. The concept of parcels together withthe algorithm of O’Rourke makes the calculation of collision possible for practical sprayproblems.

Once it is decided that two parcels of droplets collide, the algorithm further determinesthe type of collision. Only coalescence and bouncing outcomes are considered. Theprobability of each outcome is calculated from the collisional Weber number (Wec) anda fit to experimental observations. Here,

Wec =ρU2

relD

σ(22.7-1)

where Urel is the relative velocity between two parcels and D is the arithmetic meandiameter of the two parcels. The state of the two colliding parcels is modified based onthe outcome of the collision.



Use and Limitations

The collision model assumes that the frequency of collisions is much less than the par-ticle time step. If the particle time step is too large, then the results may be time-step-dependent. You should adjust the particle length scale accordingly. Additionally,the model is most applicable for low-Weber-number collisions where collisions result inbouncing and coalescence. Above a Weber number of about 100, the outcome of collisioncould be shattering.

Sometimes the collision model can cause grid-dependent artifacts to appear in the spray.This is a result of the assumption that droplets can collide only within the same cell.These tend to be visible when the source of injection is at a mesh vertex. The coalescenceof droplets tends to cause the spray to pull away from cell boundaries. In two dimensions,a finer mesh and more computational droplets can be used to reduce these effects. Inthree dimensions, best results are achieved when the spray is modeled using a polar meshwith the spray at the center.

If the collision model is used in a transient simulation, multiple DPM iterations per timestep cannot be specified in the Number of Continuous Phase Iterations per DPM Iterationfield in the Discrete Phase Model panel. In such cases, only one DPM iteration per timestep will be calculated.

Theory

As noted above, O’Rourke’s algorithm assumes that two droplets may collide only if theyare in the same continuous-phase cell. This assumption can prevent droplets that arequite close to each other, but not in the same cell, from colliding, although the effect ofthis error is lessened by allowing some droplets that are farther apart to collide. Theoverall accuracy of the scheme is second-order in space.

Probability of Collision

The probability of collision of two droplets is derived from the point of view of the largerdroplet, called the collector droplet and identified below with the number 1. The smallerdroplet is identified in the following derivation with the number 2. The calculation is inthe frame of reference of the larger droplet so that the velocity of the collector droplet iszero. Only the relative distance between the collector and the smaller droplet is importantin this derivation. If the smaller droplet is on a collision course with the collector, thecenters will pass within a distance of r1 + r2. More precisely, if the smaller droplet centerpasses within a flat circle centered around the collector of area π(r1 + r2)2 perpendicularto the trajectory of the smaller droplet, a collision will take place. This disk can be usedto define the collision volume, which is the area of the aforementioned disk multiplied bythe distance traveled by the smaller droplet in one time step, namely π(r1 + r2)2vrel∆t.

The algorithm of O’Rourke uses the concept of a collision volume to calculate the prob-ability of collision. Rather than calculating whether or not the position of the smaller



droplet center is within the collision volume, the algorithm calculates the probabilityof the smaller droplet being within the collision volume. It is known that the smallerdroplet is somewhere within the continuous-phase cell of volume V . If there is a uniformprobability of the droplet being anywhere within the cell, then the chance of the dropletbeing within the collision volume is the ratio of the two volumes. Thus, the probabilityof the collector colliding with the smaller droplet is

P1 =π(r1 + r2)2vrel∆t

V(22.7-2)

Equation 22.7-2 can be generalized for parcels, where there are n1 and n2 droplets inthe collector and smaller droplet parcels, respectively. The collector undergoes a meanexpected number of collisions given by

n =n2π(r1 + r2)2vrel∆t

V(22.7-3)

The actual number of collisions that the collector experiences is not generally the meanexpected number of collisions. The probability distribution of the number of collisionsfollows a Poisson distribution, according to O’Rourke, which is given by

P (n) = e−nnn

n!(22.7-4)

where n is the number of collisions between a collector and other droplets.

Collision Outcomes

Once it is determined that two parcels collide, the outcome of the collision must bedetermined. In general, the outcome tends to be coalescence if the droplets collide head-on, and bouncing if the collision is more oblique. In the reference frame being used here,the probability of coalescence can be related to the offset of the collector droplet centerand the trajectory of the smaller droplet. The critical offset is a function of the collisionalWeber number and the relative radii of the collector and the smaller droplet.

The critical offset is calculated by O’Rourke using the expression

bcrit = (r1 + r2)

√√√√min

(1.0,

2.4f

We

)(22.7-5)

where f is a function of r1/r2, defined as

f(r1

r2

)=(r1

r2

)3

− 2.4(r1

r2

)2

+ 2.7(r1

r2

)(22.7-6)



The value of the actual collision parameter, b, is (r1 + r2)√Y , where Y is a random

number between 0 and 1. The calculated value of b is compared to bcrit, and if b < bcrit,the result of the collision is coalescence. Equation 22.7-4 gives the number of smallerdroplets that coalesce with the collector. The properties of the coalesced droplets arefound from the basic conservation laws.

In the case of a grazing collision, the new velocities are calculated based on conservationof momentum and kinetic energy. It is assumed that some fraction of the kinetic energyof the droplets is lost to viscous dissipation and angular momentum generation. Thisfraction is related to b, the collision offset parameter. Using assumed forms for the energyloss, O’Rourke derived the following expression for the new velocity:

v′1 =m1v1 +m2v2 +m2(v1 − v2)

m1 +m2

(b− bcrit

r1 + r2 − bcrit

)(22.7-7)

This relation is used for each of the components of velocity. No other droplet propertiesare altered in grazing collisions.

22.7.2 Droplet Breakup Models

FLUENT offers two droplet breakup models: the Taylor analogy breakup (TAB) modeland the wave model. The TAB model is recommended for low-Weber-number injectionsand is well suited for low-speed sprays into a standard atmosphere. For Weber numbersgreater than 100, the wave model is more applicable. The wave model is popular for usein high-speed fuel-injection applications. Details for each model are provided below.

Taylor Analogy Breakup (TAB) Model

Introduction

The Taylor analogy breakup (TAB) model is a classic method for calculating dropletbreakup, which is applicable to many engineering sprays. This method is based uponTaylor’s analogy [368] between an oscillating and distorting droplet and a spring masssystem. Table 22.7.1 illustrates the analogous components.

Table 22.7.1: Comparison of a Spring-Mass System to a Distorting Droplet

Spring-Mass System Distorting and Oscillating Dropletrestoring force of spring surface tension forcesexternal force droplet drag forcedamping force droplet viscosity forces



The resulting TAB model equation set, which governs the oscillating and distortingdroplet, can be solved to determine the droplet oscillation and distortion at any giventime. As described in detail below, when the droplet oscillations grow to a critical valuethe “parent” droplet will break up into a number of smaller “child” droplets. As a dropletis distorted from a spherical shape, the drag coefficient changes. A drag model that incor-porates the distorting droplet effects is available in FLUENT. See Section 22.6: DynamicDrag Model Theory for details.

Use and Limitations

The TAB model is best for low-Weber-number sprays. Extremely high-Weber-numbersprays result in shattering of droplets, which is not described well by the spring-massanalogy.

Droplet Distortion

The equation governing a damped, forced oscillator is [269]

F − kx− ddxdt

= md2x

dt2(22.7-8)

where x is the displacement of the droplet equator from its spherical (undisturbed) po-sition. The coefficients of this equation are taken from Taylor’s analogy:

F

m= CF

ρgu2

ρlr(22.7-9)

k

m= Ck

σ

ρlr3(22.7-10)

d

m= Cd

µlρlr2

(22.7-11)

where ρl and ρg are the discrete phase and continuous phase densities, u is the relativevelocity of the droplet, r is the undisturbed droplet radius, σ is the droplet surfacetension, and µl is the droplet viscosity. The dimensionless constants CF , Ck, and Cd willbe defined later.

The droplet is assumed to break up if the distortion grows to a critical ratio of the dropletradius. This breakup requirement is given as

x > Cbr (22.7-12)

where Cb is a constant equal to 0.5 if breakup is assumed to occur when the distortionis equal to the droplet radius, i.e., the north and south poles of the droplet meet at



the droplet center. This implicitly assumes that the droplet is undergoing only one(fundamental) oscillation mode. Equation 22.7-8 is nondimensionalized by setting y =x/(Cbr) and substituting the relationships in Equations 22.7-9–22.7-11:

d2y

dt2=CFCb

ρgρl

u2

r2− Ckσ

ρlr3y − Cdµl

ρlr2

dy

dt(22.7-13)

where breakup now occurs for y > 1. For under-damped droplets, the equation governingy can easily be determined from Equation 22.7-13 if the relative velocity is assumed tobe constant:

y(t) = Wec + e−(t/td)

[(y0 −Wec) cos(ωt) +

1

ω

(dy0

dt+y0 −Wec

td

)sin(ωt)

](22.7-14)

where

We =ρgu

2r

σ(22.7-15)

Wec =CFCkCb

We (22.7-16)

y0 = y(0) (22.7-17)

dy0

dt=

dy

dt(0) (22.7-18)

1

td=

Cd2

µlρlr2

(22.7-19)

ω2 = Ckσ

ρlr3− 1

t2d(22.7-20)

In Equation 22.7-14, u is the relative velocity between the droplet and the gas phaseand We is the droplet Weber number, a dimensionless parameter defined as the ratioof aerodynamic forces to surface tension forces. The droplet oscillation frequency isrepresented by ω. The default value of y0 is 0, based upon the work of Liu et al. [217].The constants have been chosen to match experiments and theory [189]:

Ck = 8

Cd = 5

CF =1

3

If Equation 22.7-14 is solved for all droplets, those with y > 1 are assumed to break up.The size and velocity of the new child droplets must be determined.



Size of Child Droplets

The size of the child droplets is determined by equating the energy of the parent dropletto the combined energy of the child droplets. The energy of the parent droplet is [269]

Eparent = 4πr2σ +Kπ

5ρlr

5

(dydt

)2

+ ω2y2

(22.7-21)

where K is the ratio of the total energy in distortion and oscillation to the energy in thefundamental mode, of the order (10

3). The child droplets are assumed to be nondistorted

and nonoscillating. Thus, the energy of the child droplets can be shown to be

Echild = 4πr2σr

r32

+π

6ρlr

5

(dy

dt

)2

(22.7-22)

where r32 is the Sauter mean radius of the droplet size distribution. r32 can be found byequating the energy of the parent and child droplets (i.e., Equations 22.7-21 and 22.7-22),setting y = 1, and ω2 = 8σ/ρlr

3:

r32 =r

1 + 8Ky2

20+ ρlr3(dy/dt)2

σ

(6K−5

120

) (22.7-23)

Once the size of the child droplets is determined, the number of child droplets can easilybe determined by mass conservation.

Velocity of Child Droplets

The TAB model allows for a velocity component normal to the parent droplet velocityto be imposed upon the child droplets. When breakup occurs, the equator of the parentdroplet is traveling at a velocity of dx/dt = Cbr(dy/dt). Therefore, the child dropletswill have a velocity normal to the parent droplet velocity given by

vnormal = CvCbrdy

dt(22.7-24)

where Cv is a constant of order (1).



Droplet Breakup

To model droplet breakup, the TAB model first determines the amplitude for an un-damped oscillation (td ≈ ∞) for each droplet at time step n using the following:

A =

√√√√(yn −Wec)2 +

((dy/dt)n

ω

)2

(22.7-25)

According to Equation 22.7-25, breakup is possible only if the following condition issatisfied:

Wec + A > 1 (22.7-26)

This is the limiting case, as damping will only reduce the chance of breakup. If adroplet fails the above criterion, breakup does not occur. The only additional calculationsrequired then, are to update y using a discretized form of Equation 22.7-14 and itsderivative, which are both based on work done by O’Rourke and Amsden [269]:

yn+1 = Wec + e−(∆t/td)

(yn −Wec) cos(ωt) +

1

ω

[(dy

dt

)n+yn −Wec

td

]sin(ωt)

(22.7-27)

(dy

dt

)n+1

=Wec − yn+1

td+

ωe−(∆t/td)

1

ω

[(dy

dt

)n+yn −Wec

td

]cos(ω∆t)− (yn −Wec) sin(ω∆t)

(22.7-28)

All of the constants in these expressions are assumed to be constant throughout the timestep.

If the criterion of Equation 22.7-26 is met, then breakup is possible. The breakup time,tbu, must be determined to see if breakup occurs within the time step ∆t. The value oftbu is set to the time required for oscillations to grow sufficiently large that the magnitudeof the droplet distortion, y, is equal to unity. The breakup time is determined under theassumption that the droplet oscillation is undamped for its first period. The breakup timeis therefore the smallest root greater than tn of an undamped version of Equation 22.7-14:

Wec + A cos[ω(t− tn) + φ] = 1 (22.7-29)



where

cosφ =yn −Wec

A(22.7-30)

and

sinφ = −(dy/dt)n

Aω(22.7-31)

If tbu > tn+1 , then breakup will not occur during the current time step, and y and(dy/dt) are updated by Equations 22.7-27 and 22.7-28. The breakup calculation thencontinues with the next droplet. Conversely, if tn < tbu < tn+1, then breakup will occurand the child droplet radii are determined by Equation 22.7-23. The number of childdroplets, N , is determined by mass conservation:

Nn+1 = Nn(rn

rn+1

)3

(22.7-32)

It is assumed that the child droplets are neither distorted nor oscillating; i.e., y =(dy/dt) = 0. The child droplets are represented by a number of child parcels whichare created from the original parcel. These child parcels are distributed equally alongthe equator of the parent droplet in a plane normal to the parent relative velocity vector.The diameter of each of the child parcels is sampled from a Rosin Rammler distributionbased on the Sauter mean radius (Equation 22.7-23) and a spread parameter of 3.5.

A velocity component normal to the relative velocity vector, with magnitude computedby Equation 22.7-24, is imposed upon the child droplets. It is decomposed at the equatorinto components pointing radially outward.

i A large number of child parcels ensures a smooth distribution of particlediameters and source terms which is needed when simulating, for example,evaporating sprays.



Wave Breakup Model

Introduction

An alternative to the TAB model that is appropriate for high-Weber-number flows is thewave breakup model of Reitz [298], which considers the breakup of the droplets to beinduced by the relative velocity between the gas and liquid phases. The model assumesthat the time of breakup and the resulting droplet size are related to the fastest-growingKelvin-Helmholtz instability, derived from the jet stability analysis described below. Thewavelength and growth rate of this instability are used to predict details of the newly-formed droplets.

Use and Limitations

The wave model is appropriate for high-speed injections, where the Kelvin-Helmholtzinstability is believed to dominate droplet breakup (We > 100). Because this breakupmodel can increase the number of computational parcels, you may wish to inject a modestnumber of droplets initially.

Jet Stability Analysis

The jet stability analysis described in detail by Reitz and Bracco [300] is presented brieflyhere. The analysis considers the stability of a cylindrical, viscous, liquid jet of radiusa issuing from a circular orifice at a velocity v into a stagnant, incompressible, inviscidgas of density ρ2. The liquid has a density, ρ1, and viscosity, µ1, and a cylindricalpolar coordinate system is used which moves with the jet. An arbitrary infinitesimalaxisymmetric surface displacement of the form

η = η0eikz+ωt (22.7-33)

is imposed on the initially steady motion and it is thus desired to find the dispersionrelation ω = ω(k) which relates the real part of the growth rate, ω, to its wave number,k = 2π/λ.

In order to determine the dispersion relation, the linearized equations for the hydrody-namics of the liquid are solved assuming wave solutions of the form

φ1 = C1I0(kr)eikz+ωt (22.7-34)

ψ1 = C2I1(Lr)eikz+ωt (22.7-35)

where φ1 and ψ1 are the velocity potential and stream function, respectively, C1 andC2 are integration constants, I0 and I1 are modified Bessel functions of the first kind,L2 = k2 + ω/ν1, and ν1 is the liquid kinematic viscosity [298]. The liquid pressure is



obtained from the inviscid part of the liquid equations. In addition, the inviscid gasequations can be solved to obtain the fluctuating gas pressure at r = a:

− p21 = −ρ2(U − iωk)2kηK0(ka)

K1(ka)(22.7-36)

where K0 and K1 are modified Bessel functions of the second kind and u is the relativevelocity between the liquid and the gas. The linearized boundary conditions are

v1 =∂η

∂t(22.7-37)

∂u1

∂r= −∂v1

∂z(22.7-38)

and

− p1 + 2µ1 −σ

a2

(η + a2∂

2η

∂z2

)+ p2 = 0 (22.7-39)

which are mathematical statements of the liquid kinematic free surface condition, conti-nuity of shear stress, and continuity of normal stress, respectively. Note that u1 is theaxial perturbation liquid velocity, v1 is the radial perturbation liquid velocity, and σ isthe surface tension. Also note that Equation 22.7-38 was obtained under the assumptionthat v2 = 0.

As described by Reitz [298], Equations 22.7-37 and 22.7-38 can be used to eliminatethe integration constants C1 and C2 in Equations 22.7-34 and 22.7-35. Thus, whenthe pressure and velocity solutions are substituted into Equation 22.7-39, the desireddispersion relation is obtained:

ω2 + 2ν1k2ω

[I ′1(ka)

I0(ka)− 2kL

k2 + L2

I1(ka)

I0(ka)

I ′1(La)

I1(La)

]=

σk

ρ1a2(1− k2a2)

(L2 − a2

L2 + a2

)I1(ka)

I0(ka)+ρ2

ρ1

(U − iω

k

)2(L2 − a2

L2 + a2

)I1(ka)

I0(ka)

K0(ka)

K1(ka)(22.7-40)



As shown by Reitz [298], Equation 22.7-40 predicts that a maximum growth rate (ormost unstable wave) exists for a given set of flow conditions. Curve fits of numericalsolutions to Equation 22.7-40 were generated for the maximum growth rate, Ω, and thecorresponding wavelength, Λ, and are given by Reitz [298]:

Λ

a= 9.02

(1 + 0.45Oh0.5)(1 + 0.4Ta0.7)

(1 + 0.87We1.672 )0.6

(22.7-41)

Ω

(ρ1a

3

σ

)=

(0.34 + 0.38We1.52 )

(1 + Oh)(1 + 1.4Ta0.6)(22.7-42)

where Oh =√

We1/Re1 is the Ohnesorge number and Ta = Oh√

We2 is the Taylornumber. Furthermore, We1 = ρ1U

2a/σ and We2 = ρ2U2a/σ are the liquid and gas

Weber numbers, respectively, and Re1 = Ua/ν1 is the Reynolds number.

Droplet Breakup

In the wave model, breakup of droplet parcels is calculated by assuming that the radiusof the newly-formed droplets is proportional to the wavelength of the fastest-growingunstable surface wave on the parent droplet. In other words,

r = B0Λ (22.7-43)

where B0 is a model constant set equal to 0.61 based on the work of Reitz [298]. Fur-thermore, the rate of change of droplet radius in the parent parcel is given by

da

dt= −(a− r)

τ, r ≤ a (22.7-44)

where the breakup time, τ , is given by

τ =3.726B1a

ΛΩ(22.7-45)

and Λ and Ω are obtained from Equations 22.7-41 and 22.7-42, respectively. The breakuptime constant, B1, is set to a value of 1.73 as recommended by Liu et al. [217]. Values ofB1 can range between 1 and 60, depending on the injector characterization.

In the wave model, mass is accumulated from the parent drop at a rate given by Equa-tion 22.7-45 until the shed mass is equal to 5% of the initial parcel mass. At this time, anew parcel is created with a radius given by Equation 22.7-43. The new parcel is giventhe same properties as the parent parcel (i.e., temperature, material, position, etc.) withthe exception of radius and velocity. The new parcel is given a component of velocity



randomly selected in the plane orthogonal to the direction vector of the parent parcel,and the momentum of the parent parcel is adjusted so that momentum is conserved. Thevelocity magnitude of the new parcel is the same as the parent parcel.

You must also specify the model constants which determine how the gas phase interactswith the liquid droplets. For example, the breakup time constant B1 is the constantmultiplying the time scale which determines how quickly the parcel will loose mass.Therefore, a larger number means that it takes longer for the particle to loose a givenamount. A larger number for B1 in the context of interaction with the gas phase wouldmean that the interaction with the subgrid is less intense. B0 is the constant for the dropsize and is generally taken to be 0.61.

22.8 Atomizer Model Theory

All of the atomization models use physical atomizer parameters, such as orifice diameterand mass flow rate, to calculate initial droplet size, velocity, and position.

For realistic atomizer simulations, the droplets must be randomly distributed, both spa-tially through a dispersion angle and in their time of release. For other types of injectionsin FLUENT (nonatomizer), all of the droplets are released along fixed trajectories at thebeginning of the time step. The atomizer models use stochastic trajectory selection andstaggering to attain a random distribution. Further information on staggering can befound in section Section 22.2.2: Stochastic Staggering of Particles.

Stochastic trajectory selection is the random dispersion of initial droplet directions. Allof the atomizer models provide an initial dispersion angle, and the stochastic trajectoryselection picks an initial direction within this angle. This approach improves the accuracyof the results for spray-dominated flows. The droplets will be more evenly spread amongthe computational cells near the atomizer, which improves the coupling to the gas phaseby spreading drag more smoothly over the cells near the injection. Source terms inthe energy and species conservation equations are also more evenly distributed amongneighboring cells, improving solution convergence.

Five atomizer models are available in FLUENT to predict the spray characteristics fromknowledge of global parameters such as nozzle type and liquid flow rate:

• plain-orifice atomizer

• pressure-swirl atomizer

• flat-fan atomizer

• air-blast/air-assisted atomizer

• effervescent/flashing atomizer



You can choose them as injection types and define the associated parameters in the SetInjection Properties panel, as described in Section 22.12.1: Injection Types. Details aboutthe atomizer models are provided below.

22.8.1 The Plain-Orifice Atomizer Model

The plain-orifice is the most common type of atomizer and the most simply made. How-ever there is nothing simple about the physics of the internal nozzle flow and the exter-nal atomization. In the plain-orifice atomizer model in FLUENT, the liquid is acceleratedthrough a nozzle, forms a liquid jet and then breaks up to form droplets. This apparentlysimple process is dauntingly complex. The plain orifice may operate in three differentregimes: single-phase, cavitating and flipped [347]. The transition between regimes isabrupt, producing dramatically different sprays. The internal regime determines thevelocity at the orifice exit, as well as the initial droplet size and the angle of dropletdispersion. Diagrams of each case are shown in Figures 22.8.1, 22.8.2, and 22.8.3.

downstreamgas

liquid jet

orifice walls

d

L

p1

p2

r

Figure 22.8.1: Single-Phase Nozzle Flow (Liquid Completely Fills the Ori-fice)



vapor

vapor

downstreamgas

liquid jet

orifice walls

Figure 22.8.2: Cavitating Nozzle Flow (Vapor Pockets Form Just after theInlet Corners)

downstreamgas

liquid jet

orifice walls

Figure 22.8.3: Flipped Nozzle Flow (Downstream Gas Surrounds the LiquidJet Inside the Nozzle)



Internal Nozzle State

To accurately predict the spray characteristics, the plain-orifice model in FLUENT mustidentify the correct state of the internal nozzle flow because the nozzle state has a tremen-dous effect on the external spray. Unfortunately, there is no established theory for deter-mining the nozzle state. One must rely on empirical models obtained from experimentaldata. FLUENT uses several dimensionless parameters to determine the internal flowregime for the plain-orifice atomizer model. These parameters and the decision-makingprocess are summarized below.

A list of the parameters that control internal nozzle flow is given in Table 22.8.1. Theseparameters may be combined to form nondimensional characteristic lengths such as r/dand L/d, as well as nondimensional groups like the Reynolds number based on hydraulic“head” (Reh) and the cavitation parameter (K).

Table 22.8.1: List of Governing Parameters for Internal Nozzle Flow

nozzle diameter dnozzle length Lradius of curvature of the inlet corner rupstream pressure p1

downstream pressure p2

viscosity µliquid density ρlvapor pressure pv

Reh =dρlµ

√2(p1 − p2)

ρl(22.8-1)

K =p1 − pvp1 − p2

(22.8-2)

The liquid flow often contracts in the nozzle, as can be seen in Figures 22.8.2 and 22.8.3.Nurick [265] found it helpful to use a coefficient of contraction (Cc) to represent thereduction in the cross-sectional area of the liquid jet. The coefficient of contraction isdefined as the area of the stream of contracting liquid over the total cross-sectional areaof the nozzle. FLUENT uses Nurick’s fit for the coefficient of contraction:

Cc =1√

1C2

ct− 11.4r

d

(22.8-3)



Here, Cct is a theoretical constant equal to 0.611, which comes from potential flow analysisof flipped nozzles.

Coefficient of Discharge

Another important parameter for describing the performance of nozzles is the coefficientof discharge (Cd). The coefficient of discharge is the ratio of the mass flow rate throughthe nozzle to the theoretical maximum mass flow rate:

Cd =meff

A√

2ρl(p1 − p2)(22.8-4)

where meff is the effective mass flow rate of the nozzle, defined by

meff =2πm

∆φ(22.8-5)

Here, m is the mass flow rate specified in the user interface, and ∆φ is the differencebetween the azimuthal stop angle and the azimuthal start angle

∆φ = φstop − φstart (22.8-6)

as input by the user (see Section 22.12.1: Point Properties for Plain-Orifice AtomizerInjections). Note that the mass flow rate input by the user should be for the appropriatestart and stop angles, in other words the correct mass flow rate for the sector beingmodeled. Note also that for ∆φ of 2π, the effective mass flow rate is identical to themass flow rate in the interface.

The cavitation number (K in Equation 22.8-2) is an essential parameter for predictingthe inception of cavitation. The inception of cavitation is known to occur at a valueof Kincep ≈ 1.9 for short, sharp-edged nozzles. However, to include the effects of inletrounding and viscosity, an empirical relationship is used:

Kincep = 1.9(

1− r

d

)2

− 1000

Reh(22.8-7)

Similarly, a critical value of K where flip occurs is given by

Kcrit = 1 +1(

1 + L4d

) (1 + 2000

Reh

)e70r/d

(22.8-8)

If r/d is greater than 0.05, then flip is deemed impossible and Kcrit is set to 1.0.



The cavitation number, K, is compared to the values of Kincep and Kcrit to identify thenozzle state. The decision tree is shown in Figure 22.8.4. Depending on the state of thenozzle, a unique closure is chosen for the above equations.

For a single-phase nozzle (K > Kincep, K ≥ Kcrit) [206], the coefficient of discharge isgiven by

Cd =1

1Cdu

+ 20 (1+2.25L/d)Reh

(22.8-9)

where Cdu is the ultimate discharge coefficient, and is defined as

Cdu = 0.827− 0.0085L

d(22.8-10)

For a cavitating nozzle (Kcrit ≤ K ≤ Kincep) [265] the coefficient of discharge is deter-mined from

Cd = Cc√K (22.8-11)

For a flipped nozzle (K < Kcrit) [265],

Cd = Cct = 0.611 (22.8-12)

incep

crit

incep

crit crit crit

K > KK ≤ K

K < K K ≥ K K < K K ≥ K

flipped cavitating flipped single phase

Figure 22.8.4: Decision Tree for the State of the Cavitating Nozzle

All of the nozzle flow equations are solved iteratively, along with the appropriate relation-ship for coefficient of discharge as given by the nozzle state. The nozzle state may changeas the upstream or downstream pressures change. Once the nozzle state is determined,the exit velocity is calculated, and appropriate correlations for spray angle and initialdroplet size distribution are determined.



Exit Velocity

For a single-phase nozzle, the estimate of exit velocity (u) comes from the conservationof mass and the assumption of a uniform exit velocity:

u =meff

ρlA(22.8-13)

For the cavitating nozzle, Schmidt and Corradini [322] have shown that the uniform exitvelocity is not accurate. Instead, they derived an expression for a higher velocity over areduced area:

u =2Ccp1 − p2 + (1− 2Cc)pv

Cc√

2ρl(p1 − pv)(22.8-14)

This analytical relation is used for cavitating nozzles in FLUENT. For the case of flippednozzles, the exit velocity is found from the conservation of mass and the value of thereduced flow area:

u =meff

ρlCctA(22.8-15)

Spray Angle

The correlation for the spray angle (θ) comes from the work of Ranz [294]:

θ

2=

tan−1

[4πCA

√ρgρl

√3

6

]single phase, cavitating

0.01 flipped

(22.8-16)

The spray angle for both single-phase and cavitating nozzles depends on the ratio of thegas and liquid densities and also the parameter CA. For flipped nozzles, the spray anglehas a constant value.

The parameter CA, which you must specify, is thought to be a constant for a given nozzlegeometry. The larger the value, the narrower the spray. Reitz [299] suggests the followingcorrelation for CA:

CA = 3 +L

3.6d(22.8-17)



The spray angle is sensitive to the internal flow regime of the nozzle. Hence, you maywish to choose smaller values of CA for cavitating nozzles than for single-phase nozzles.Typical values range from 4.0 to 6.0. The spray angle for flipped nozzles is a small,arbitrary value that represents the lack of any turbulence or initial disturbance from thenozzle.

Droplet Diameter Distribution

One of the basic characteristics of an injection is the distribution of drop size. Foran atomizer, the droplet diameter distribution is closely related to the nozzle state.FLUENT’s spray models use a two-parameter Rosin-Rammler distribution, characterizedby the most probable droplet size and a spread parameter. The most probable dropletsize, d0 is obtained in FLUENT from the Sauter mean diameter, d32 [199]. For moreinformation about the Rosin-Rammler size distribution, see Section 22.12.1: Using theRosin-Rammler Diameter Distribution Method.

For single-phase nozzle flows, the correlation of Wu et al. [406] is used to calculate d32

and relate the initial drop size to the estimated turbulence quantities of the liquid jet:

d32 = 133.0λWe−0.74, (22.8-18)

where λ = d/8, λ is the radial integral length scale at the jet exit based upon fully-developed turbulent pipe flow, and We is the Weber number, defined as

We ≡ ρlu2λ

σ. (22.8-19)

Here, σ is the droplet surface tension. For a more detailed discussion of droplet surfacetension and the Weber number, see Section 22.7.2: Droplet Breakup Models. For moreinformation about mean particle diameters, see Section 22.16.8: Summary Reporting ofCurrent Particles.

For cavitating nozzles, FLUENT uses a slight modification of Equation 22.8-18. Theinitial jet diameter used in Wu’s correlation, d, is calculated from the effective area ofthe cavitating orifice exit, and thus represents the effective diameter of the exiting liquidjet, deff . For an explanation of effective area of cavitating nozzles, please see Schmidtand Corradini [322].

The length scale for a cavitating nozzle is λ = deff/8, where

deff =

√4meff

πρlu. (22.8-20)



For the case of the flipped nozzle, the initial droplet diameter is set to the diameter ofthe liquid jet:

d0 = d√Cct (22.8-21)

where d0 is defined as the most probable diameter.

The second parameter required to specify the droplet size distribution is the spreadparameter, s. The values for the spread parameter are chosen from past modeling expe-rience and from a review of experimental observations. Table 22.8.2 lists the values of sfor the three nozzle states. The larger the value of the spread parameter, the narrowerthe droplet size distribution.

Table 22.8.2: Values of Spread Parameter for Different Nozzle States

State Spread Parametersingle phase 3.5cavitating 1.5

flipped ∞

Since the correlations of Wu et al. provide the Sauter mean diameter, d32, these areconverted to the most probable diameter, d0. Lefebvre [199] gives the most generalrelationship between the Sauter mean diameter and most probable diameter for a Rosin-Rammler distribution. The simplified version for s=3.5 is as follows:

d0 = 1.2726d32

(1− 1

s

)1/s

(22.8-22)

At this point, the droplet size, velocity, and spray angle have been determined and theinitialization of the injections is complete.



22.8.2 The Pressure-Swirl Atomizer Model

Another important type of atomizer is the pressure-swirl atomizer, sometimes referred toby the gas-turbine community as a simplex atomizer. This type of atomizer acceleratesthe liquid through nozzles known as swirl ports into a central swirl chamber. The swirlingliquid pushes against the walls of the swirl chamber and develops a hollow air core. Itthen emerges from the orifice as a thinning sheet, which is unstable, breaking up intoligaments and droplets. The pressure-swirl atomizer is very widely used for liquid-fuelcombustion in gas turbines, oil furnaces, and direct-injection spark-ignited automobileengines. The transition from internal injector flow to fully-developed spray can be dividedinto three steps: film formation, sheet breakup, and atomization. A sketch of how thisprocess is thought to occur is shown in Figure 22.8.5.

dispersion angle

half angle

Figure 22.8.5: Theoretical Progression from the Internal Atomizer Flow tothe External Spray

The interaction between the air and the sheet is not well understood. It is generally ac-cepted that an aerodynamic instability causes the sheet to break up. The mathematicalanalysis below assumes that Kelvin-Helmholtz waves grow on the sheet and eventuallybreak the liquid into ligaments. It is then assumed that the ligaments break up intodroplets due to varicose instability. Once the liquid droplets are formed, the spray evo-lution is determined by drag, collision, coalescence, and secondary breakup.



The pressure-swirl atomizer model used in FLUENT is called the Linearized InstabilitySheet Atomization (LISA) model of Schmidt et al. [324]. The LISA model is divided intotwo stages:

1. film formation

2. sheet breakup and atomization

Both parts of the model are described below.

Film Formation

The centrifugal motion of the liquid within the injector creates an air core surroundedby a liquid film. The thickness of this film, t, is related to the mass flow rate by

meff = πρut(dinj − t) (22.8-23)

where dinj is the injector exit diameter, and meff is the effective mass flow rate, which isdefined by equation Equation 22.8-5 . The other unknown in Equation 22.8-23 is u, theaxial component of velocity at the injector exit. This quantity depends on internal detailsof the injector and is difficult to calculate from first principles. Instead, the approachof Han et al. [133] is used. The total velocity is assumed to be related to the injectorpressure by

U = kv

√2∆p

ρl(22.8-24)

where kv is the velocity coefficient. Lefebvre [199] has noted that kv is a function of theinjector design and injection pressure. If the swirl ports are treated as nozzles and if it isassumed that the dominant portion of the pressure drop occurs at those ports, kv is theexpression for the discharge coefficient (Cd). For single-phase nozzles with sharp inletcorners and L/d ratios of 4, a typical Cd value is 0.78 or less [206]. If the nozzles arecavitating, the value of Cd may be as low as 0.61. Hence, 0.78 should be a practical upperbound for kv. Reducing kv by 10% to 0.7 approximates the effect of other momentumlosses on the discharge coefficient.

Physical limits on kv require that it be less than unity from conservation of energy, yetbe large enough to permit sufficient mass flow. To guarantee that the size of the air coreis non-negative, the following expression is used for kv:

kv = max

[0.7,

4meff

d20ρl cos θ

√ρl

2∆p

](22.8-25)



Assuming that ∆p is known, Equation 22.8-24 can be used to find U . Once U is deter-mined, u is found from

u = U cos θ (22.8-26)

where θ is the spray angle, which is assumed to be known. At this point, the thicknessand axial component of the liquid film are known at the injector exit. The tangentialcomponent of velocity (w = U sin θ) is assumed to be equal to the radial velocity compo-nent of the liquid sheet downstream of the nozzle exit. The axial component of velocityis assumed to remain constant.

Sheet Breakup and Atomization

The pressure-swirl atomizer model includes the effects of the surrounding gas, liquidviscosity, and surface tension on the breakup of the liquid sheet. Details of the theoreticaldevelopment of the model are given in Senecal et al. [325] and are only briefly presentedhere. For a more robust implementation, the gas-phase velocity is neglected in calculatingthe relative liquid-gas velocity and is instead set by the user. This avoids having theinjector parameters depend too heavily on the usually under-resolved gas-phase velocityfield very near the injection location.

The model assumes that a two-dimensional, viscous, incompressible liquid sheet of thick-ness 2h moves with velocity U through a quiescent, inviscid, incompressible gas medium.The liquid and gas have densities of ρl and ρg, respectively, and the viscosity of theliquid is µl. A coordinate system is used that moves with the sheet, and a spectrum ofinfinitesimal wavy disturbances of the form

η = η0eikx+ωt (22.8-27)

is imposed on the initially steady motion. The spectrum of disturbances results in fluc-tuating velocities and pressures for both the liquid and the gas. In Equation 22.8-27, η0

is the initial wave amplitude, k = 2π/λ is the wave number, and ω = ωr + iωi is thecomplex growth rate. The most unstable disturbance has the largest value of ωr, denotedhere by Ω, and is assumed to be responsible for sheet breakup. Thus, it is desired toobtain a dispersion relation ω = ω(k) from which the most unstable disturbance can becalculated as a function of wave number.

Squire [352], Li and Tankin [205], and Hagerty and Shea [130] have shown that twosolutions, or modes, exist that satisfy the governing equations subject to the boundaryconditions at the upper and lower interfaces. The first solution, called the sinuous mode,has waves at the upper and lower interfaces in phase. The second solution is called thevaricose mode which has the waves at the upper and lower interfaces π radians out ofphase. It has been shown by numerous authors (e.g., Senecal et. al. [325]) that the



sinuous mode dominates the growth of varicose waves for low velocities and low gas-to-liquid density ratios. In addition, it can be shown that the sinuous and varicose modesbecome indistinguishable for high-velocity flows. As a result, the atomization model inFLUENT is based upon the growth of sinuous waves on the liquid sheet.

As derived in Senecal et al. [325], the dispersion relation for the sinuous mode is givenby

ω2[tanh(kh) +Q] + [4νlk2 tanh(kh) + 2iQkU ] +

4νlk4 tanh(kh)− 4ν2

l k3` tanh(`h)−QU2k2 +

σk3

ρl= 0 (22.8-28)

where Q = ρg/ρl and `2 = k2 + ω/νl.

Above a critical Weber number of Weg = 27/16 (based on the liquid velocity, gas density,and sheet half-thickness), the fastest-growing waves are short. For Weg < 27/16, thewavelengths are long compared to the sheet thickness. The speed of modern high pressurefuel injection systems is high enough to ensure that the film Weber number is well abovethis critical limit.

An order-of-magnitude analysis using typical values shows that the terms of second orderin viscosity can be neglected in comparison to the other terms in Equation 22.8-28. Usingthis assumption, Equation 22.8-28 reduces to

ωr =1

tanh(kh) +Q

−2νlk2 tanh(kh) +

√√√√4ν2l k

4 tanh2(kh)−Q2U2k2 − [tanh(kh) +Q]

[−QU2k2 +

σk3

ρl

] (22.8-29)

For waves that are long compared with the sheet thickness, a mechanism of sheet disin-tegration proposed by Dombrowski and Johns [82] is adopted. For long waves, ligamentsare assumed to form from the sheet breakup process once the unstable waves reach acritical amplitude. If the surface disturbance has reached a value of ηb at breakup, abreakup time, τ , can be evaluated:

ηb = η0eΩτ ⇒ 1

Ωln

(ηbη0

)(22.8-30)

where Ω, the maximum growth rate, is found by numerically maximizing Equation 22.8-29as a function of k. The maximum is found using a binary search that checks the sign ofthe derivative. The sheet breaks up and ligaments will be formed at a length given by



Lb = Uτ =U

Ωln

(ηbη0

)(22.8-31)

where the quantity ln( ηbη0

) is an empirical sheet constant that you must specify. The

default value of 12 was obtained theoretically by Weber [391] for liquid jets. Dombrowskiand Hooper [81] showed that a value of 12 for the sheet constant agreed favorably withexperimental sheet breakup lengths over a range of Weber numbers from 2 to 200.

The diameter of the ligaments formed at the point of breakup can be obtained from amass balance. If it is assumed that the ligaments are formed from tears in the sheet twiceper wavelength, the resulting diameter is given by

dL =

√8h

Ks

(22.8-32)

where Ks is the wave number corresponding to the maximum growth rate, Ω. Theligament diameter depends on the sheet thickness, which is a function of the breakuplength. The film thickness is calculated from the breakup length and the radial distancefrom the center line to the mid-line of the sheet at the atomizer exit, r0:

hend =r0h0

r0 + Lb sin(θ2

) (22.8-33)

This mechanism is not used for waves that are short compared to the sheet thickness.For short waves, the ligament diameter is assumed to be linearly proportional to thewavelength that breaks up the sheet,

dL =2πCLKs

(22.8-34)

where CL, or the ligament constant, is equal to 0.5 by default.

In either the long-wave or the short-wave case, the breakup from ligaments to dropletsis assumed to behave according to Weber’s [391] analysis for capillary instability.

d0 = 1.88dL(1 + 3Oh)1/6 (22.8-35)

Here, Oh is the Ohnesorge number which is a combination of the Reynolds numberand the Weber number (see Section 22.7.2: Jet Stability Analysis for more details aboutOh). Once d0 has been determined from Equation 22.8-35, it is assumed that this dropletdiameter is the most probable droplet size of a Rosin-Rammler distribution with a spreadparameter of 3.5 and a default dispersion angle of 6 (which can be modified in theGUI). The choice of spread parameter and dispersion angle is based on past modeling



experience [323]. It is important to note that the spray cone angle must be specified bythe user when using this model.

22.8.3 The Air-Blast/Air-Assist Atomizer Model

In order to accelerate the breakup of liquid sheets from an atomizer, an additional airstream is often directed through the atomizer. The liquid is formed into a sheet by a noz-zle, and air is then directed against the sheet to promote atomization. This technique iscalled air-assisted atomization or air-blast atomization, depending on the quantity of airand its velocity. The addition of the external air stream past the sheet produces smallerdroplets than without the air. Though the exact mechanism for this enhanced perfor-mance is not completely understood, it is thought that the assisting air may acceleratethe sheet instability. The air may also help disperse the droplets, preventing collisionsbetween them. Air-assisted atomization is used in many of the same applications aspressure-swirl atomization, where especially fine atomization is required.

FLUENT’s air-blast atomization model is a variation of the pressure-swirl model. Oneimportant difference between them is that the sheet thickness is set directly in the air-blast atomizer model. This input is necessary because of the variety of sheet formationmechanisms used in air-blast atomizers. Hence the air-blast atomizer model does notcontain the sheet formation equations that were included in the pressure-swirl atomizermodel (Equations 22.8-23–22.8-26). You will also specify the maximum relative velocitythat is produced by the sheet and air. Though this quantity could be calculated, specify-ing a value relieves you from the necessity of finely resolving the atomizer internal flow.This feature is convenient for simulations in large domains, where the atomizer is verysmall by comparison.

An additional difference is that the air-blast atomizer model assumes that the sheetbreakup is always due to short waves. This assumption is a consequence of the greatersheet thickness commonly found in air-blast atomizers. Hence the ligament diameter isassumed to be linearly proportional to the wavelength of the fastest-growing wave on thesheet, and is calculated from Equation 22.8-34.

Other inputs are similar to the pressure-swirl model – the user must provide the massflow rate and spray angle. The angle in the case of the air-blast atomizer is the initialtrajectory of the film as it leaves the end of the orifice. The value of the angle is negativeif the initial film trajectory is inward, towards the centerline. Specification of the innerand outer diameters of the film at the atomizer exit are also required, in addition to thedispersion angle whose default value is 6 (which can be modified in the GUI).

Since the air-blast atomizer model does not include internal gas flows, the user must createthe air streams surrounding the injector as boundary conditions within the FLUENTsimulation. These streams are ordinary continuous-phase flows and require no specialtreatment.



22.8.4 The Flat-Fan Atomizer Model

The flat-fan atomizer is very similar to the pressure-swirl atomizer, but it makes a flatsheet and does not use swirl. The liquid emerges from a wide, thin orifice as a flat liquidsheet that breaks up into droplets. The primary atomization process is thought to besimilar to the pressure-swirl atomizer. Some researchers believe that flat-fan atomization,because of jet impingement, is very similar to the atomization of a flat sheet. The flat-fanmodel could serve doubly for this application.

The flat-fan atomizer is available only for 3D models. An image of the three-dimensionalflat fan is shown in Figure 22.8.6. The model assumes that the fan originates from avirtual origin. You need to provide the location of this origin, which is the intersectionof the lines that mark the sides of the fan as well as the location of the center pointof the arc from which the fan originates. FLUENT will find the vector that points fromthe origin to the center point in order to determine the direction of the injection. Youalso need to provide the half-angle of the fan arc, the width of the orifice (in the normaldirection) and the mass flow rate of the liquid to use the flat-fan atomizer model.

dispersion angle

half angle

dispersion angle

Figure 22.8.6: Flat Fan Viewed from Above and from the Side



The breakup of the flat fan is calculated very much like the breakup of the sheet in thepressure-swirl atomizer. The sheet breaks up into ligaments which then form individualdroplets. The only difference is that for short waves, the flat fan sheet is assumed to formligaments at half-wavelength intervals. Hence the ligament diameter for short waves isgiven by

dL =

√16h

Ks

(22.8-36)

In this case, dL in Equation 22.8-36 is taken to be the most probable diameter, with aRosin-Rammler spread parameter of 3.5 and a default dispersion angle of 6. This anglecan be set in the Set Injection Properties panel. In all other respects, the flat-fan atomizermodel is like the sheet breakup portion of the pressure-swirl atomizer.

22.8.5 The Effervescent Atomizer Model

Effervescent atomization is the injection of liquid infused with a super-heated (withrespect to downstream conditions) liquid or propellant. As the volatile liquid exits thenozzle, it rapidly changes phase. This phase change quickly breaks up the stream intosmall droplets with a wide dispersion angle. The model also applies to cases where avery hot liquid is discharged.

Since the physics of effervescence is not well understood, the model must rely on roughempirical fits. The photographs of Reitz and Bracco [299] provide some insights. Thesephotographs show a dense liquid core to the spray, surrounded by a wide shroud of smallerdroplets.

The initial velocity of the droplets is computed from conservation of mass, assuming theexiting jet has a cross-sectional area that is Cct times the nozzle area, where Cct is a fixedconstant, equal to 0.611 as specified in Equations 22.8-3 and 22.8-12.

u =meff

ρlCctA(22.8-37)

The maximum droplet diameter is set to the effective diameter of the exiting jet:

dmax = d√Cct (22.8-38)


22.9 One-Way and Two-Way Coupling

The droplet size is then sampled from a Rosin-Rammler distribution with a spread pa-rameter of 4.0. (See Section 22.12.1: Using the Rosin-Rammler Diameter DistributionMethod for details on the Rosin-Rammler distribution.) The most probable droplet sizedepends on the angle, θ, between the droplet’s stochastic trajectory and the injectiondirection:

d0 = dmaxe−(θ/Θs)

2

(22.8-39)

The dispersion angle multiplier, Θs, is computed from the quality, x, and the specifiedvalue for the dispersion constant, Ceff :

x =mvapor

(mvapor + mliquid)(22.8-40)

Θs =x

Ceff

(22.8-41)

This technique creates a spray with large droplets in the central core and a shroud ofsmaller surrounding droplets. The droplet temperature is initialized to 0.99 times thesaturation temperature, such that the temperature of the droplet is close to boiling. Tocomplete the model, the flashing vapor must also be included in the calculation. Thisvapor is part of the continuous phase and not part of the discrete phase model. Youmust create an inlet at the point of injection when you specify boundary conditions forthe continuous phase. When the effervescent atomizer model is selected, you will need tospecify the nozzle diameter, mass flow rate, mixture quality, saturation temperature ofthe volatile substance, spray half-angle and dispersion constant in addition to specifyingthe position and direction of the injector.


You can use FLUENT to predict the discrete phase patterns based on a fixed continu-ous phase flow field (an uncoupled approach or ”one-way coupling”), or you can includethe effect of the discrete phase on the continuum (a coupled approach or ”two-way cou-pling”). In the coupled approach, the continuous phase flow pattern is impacted by thediscrete phase (and vice versa), and you can alternate calculations of the continuousphase and discrete phase equations until a converged coupled solution is achieved. SeeSection 22.9.1: Coupling Between the Discrete and Continuous Phases for details.

Using FLUENT’s discrete phase modeling capability, reacting particles or droplets can bemodeled and their impact on the continuous phase can be examined. Several heat andmass transfer relationships, termed “laws”, are available in FLUENT and the physicalmodels employed in these laws are described in this section.



22.9.1 Coupling Between the Discrete and Continuous Phases

As the trajectory of a particle is computed, FLUENT keeps track of the heat, mass, andmomentum gained or lost by the particle stream that follows that trajectory and thesequantities can be incorporated in the subsequent continuous phase calculations. Thus,while the continuous phase always impacts the discrete phase, you can also incorporatethe effect of the discrete phase trajectories on the continuum. This two-way couplingis accomplished by alternately solving the discrete and continuous phase equations untilthe solutions in both phases have stopped changing. This interphase exchange of heat,mass, and momentum from the particle to the continuous phase is depicted qualitativelyin Figure 22.9.1.

mass-exchangeheat-exchangemomentum-exchange

typical

particle

trajectory

typical continuous

phase control volume

Figure 22.9.1: Heat, Mass, and Momentum Transfer Between the Discreteand Continuous Phases

Momentum Exchange

The momentum transfer from the continuous phase to the discrete phase is computed inFLUENT by examining the change in momentum of a particle as it passes through eachcontrol volume in the FLUENT model. This momentum change is computed as

F =∑(

18µCDRe

ρpd2p24

(up − u) + Fother

)mp∆t (22.9-1)



whereµ = viscosity of the fluidρp = density of the particledp = diameter of the particleRe = relative Reynolds numberup = 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 continuous phase momen-tum balance in any subsequent calculations of the continuous phase flow field and canbe reported by FLUENT as described in Section 22.16: Postprocessing for the DiscretePhase.

Heat Exchange

The heat transfer from the continuous phase to the discrete phase is computed in FLUENTby examining the change in thermal energy of a particle as it passes through each controlvolume in the FLUENT model. In the absence of a chemical reaction (i.e., for all particlelaws except Law 5) the heat exchange is computed as

Q = (mpin −mpout)[−Hlatref +Hpyrol]−mpout

∫ Tpout

Tref

cppdT +mpin

∫ Tpin

Tref

cppdT (22.9-2)

wherempin = mass of the particle on cell entry (kg)mpout = mass of the particle on cell exit (kg)cpp = heat capacity of the particle (J/kg-K)

Hpyrol = heat of pyrolysis as volatiles are evolved (J/kg)Tpin = temperature of the particle on cell entry (K)Tpout = temperature of the particle on cell exit (K)Tref = reference temperature for enthalpy (K)Hlatref = latent heat at reference conditions (J/kg)

The latent heat at the reference conditions Hlatref for droplet particles is computed asthe difference of the liquid and gas standard formation enthalpies, and can be related tothe latent heat at the boiling point as follows:

Hlatref = Hlat −∫ Tbp

Tref

cpgdT +∫ Tbp

Tref

cppdT (22.9-3)



wherecpg = heat capacity of gas product species (J/kg-K)

Tbp = boiling point temperature (K)Hlat = latent heat at the boiling point temperature (J/kg)

For the volatile part of the combusting particles, some constraints are applied to ensurethat the enthalpy source terms do not depend on the particle history. The formulationshould be consistent with the mixing of two gas streams, one consisting of the fluid andthe other consisting of the volatiles. Hence Hlatref is derived by applying a correction toHlat, which accounts for different heat capacities in the particle and gaseous phase:

Hlatref = Hlat −∫ Tp,init

Tref

cpgdT +∫ Tp,init

Tref

cppdT (22.9-4)

whereTp,init = particle initial temperature (K)

Mass Exchange

The mass transfer from the discrete phase to the continuous phase is computed in FLU-ENT by examining the change in mass of a particle as it passes through each controlvolume in the FLUENT model. The mass change is computed simply as

M =∆mp

mp,0

mp,0 (22.9-5)

This mass exchange appears as a source of mass in the continuous phase continuityequation and as a source of a chemical species defined by you. The mass sources areincluded in any subsequent calculations of the continuous phase flow field and are reportedby FLUENT as described in Section 22.16: Postprocessing for the Discrete Phase.

Under-Relaxation of the Interphase Exchange Terms

Note that the interphase exchange of momentum, heat, and mass is under-relaxed duringthe calculation, so that

Fnew = Fold + α(Fcalculated − Fold) (22.9-6)

Qnew = Qold + α(Qcalculated −Qold) (22.9-7)

Mnew = Mold + α(Mcalculated −Mold) (22.9-8)



where α is the under-relaxation factor for particles/droplets that you can set in theSolution Controls panel. The default value for α is 0.5. This value may be reduced toimprove the stability of coupled calculations. Note that the value of α does not influencethe predictions obtained in the final converged solution.

Two options exist when updating the new particle source terms Fnew, Qnew andMnew. Thefirst option is to compute the new source terms and the particle source terms, Fcalculated,Qcalculated and Mcalculated, at the same time. The second option is to update the newsource terms, Fnew, Qnew and Mnew, every flow iteration, while the particle source terms,Fcalculated, Qcalculated and Mcalculated, are calculated every Discrete Phase Model iteration.The latter option is recommended for transient flows, where the particles are updatedonce per flow time step.

Interphase Exchange During Stochastic Tracking

When stochastic tracking is performed, the interphase exchange terms, computed viaEquations 22.9-1 to 22.9-8, are computed for each stochastic trajectory with the particlemass flow rate, mp0, divided by the number of stochastic tracks computed. This impliesthat an equal mass flow of particles follows each stochastic trajectory.

Interphase Exchange During Cloud Tracking

When the particle cloud model is used, the interphase exchange terms are computed viaEquations 22.9-1 to 22.9-8 based on ensemble-averaged flow properties in the particlecloud. The exchange terms are then distributed to all the cells in the cloud based on theweighting factor defined in Equation 22.2-55.

22.9.2 Particle Types in FLUENT

The laws that you activate depend upon the particle type that you select. In the SetInjection Properties panel you will specify the Particle Type, and FLUENT will use a givenset of heat and mass transfer laws for the chosen type. All particle types have predefinedsequences of physical laws as shown in the table below:

Particle Type Description Laws ActivatedInert inert/heating or cooling 1, 6Droplet heating/evaporation/boiling 1, 2, 3, 6Combusting heating;

evolution of volatiles/swelling;heterogeneous surface reaction

1, 4, 5, 6

Multicomponent multicomponent droplets/particles 7



In addition to the above laws, you can define your own laws using a user-defined function.See the separate UDF Manual for information about user-defined functions.

You can also extend combusting particles to include an evaporating/boiling material byselecting Wet Combustion in the Set Injection Properties panel.

FLUENT’s physical laws (Laws 1 through 6), which describe the heat and mass transferconditions listed in this table, are explained in detail in the sections that follow.

Inert Heating or Cooling (Law 1/Law 6)

The inert heating or cooling laws (Laws 1 and 6) are applied when the particle tempera-ture is less than the vaporization temperature that you define, Tvap, and after the volatilefraction, fv,0, of a particle has been consumed. These conditions may be written as

Law 1:Tp < Tvap (22.9-9)

Law 6:mp ≤ (1− fv,0)mp,0 (22.9-10)

where Tp is the particle temperature, mp,0 is the initial mass of the particle, and mp isits current mass.

Law 1 is applied until the temperature of the particle/droplet reaches the vaporiza-tion temperature. At this point a noninert particle/droplet may proceed to obey oneof the mass-transfer laws (2, 3, 4, and/or 5), returning to Law 6 when the volatileportion of the particle/droplet has been consumed. (Note that the vaporization temper-ature, Tvap, is an arbitrary modeling constant used to define the onset of the vaporiza-tion/boiling/volatilization laws.)

When using Law 1 or Law 6, FLUENT uses a simple heat balance to relate the parti-cle temperature, Tp(t), to the convective heat transfer and the absorption/emission ofradiation at the particle surface:

mpcpdTpdt

= hAp(T∞ − Tp) + εpApσ(θ4R − T 4

p ) (22.9-11)

wheremp = mass of the particle (kg)cp = heat capacity of the particle (J/kg-K)Ap = surface area of the particle (m2)T∞ = local temperature of the continuous phase (K)h = convective heat transfer coefficient (W/m2-K)εp = particle emissivity (dimensionless)σ = Stefan-Boltzmann constant (5.67 x 10−8 W/m2-K4)θR = radiation temperature, ( G

4σ)1/4



Equation 22.9-11 assumes that there is negligible internal resistance to heat transfer, i.e.,the particle is at uniform temperature throughout.

G is the incident radiation in W/m2:

G =∫

Ω=4πIdΩ (22.9-12)

where I is the radiation intensity and Ω is the solid angle.

Radiation heat transfer to the particle is included only if you have enabled the P-1 ordiscrete ordinates radiation model and you have activated radiation heat transfer toparticles using the Particle Radiation Interaction option in the Discrete Phase Model panel.

Equation 22.9-11 is integrated in time using an approximate, linearized form that assumesthat the particle temperature changes slowly from one time value to the next:

mpcpdTpdt

= Ap−[h+ εpσT

3p

]Tp +

[hT∞ + εpσθ

4R

](22.9-13)

As the particle trajectory is computed, FLUENT integrates Equation 22.9-13 to obtainthe particle temperature at the next time value, yielding


where ∆t is the integration time step and

αp =hT∞ + εpσθ

4R

h+ εpσT 3p (t)

(22.9-15)

and

βp =Ap(h+ εpσT

3p (t))

mpcp(22.9-16)

FLUENT can also solve Equation 22.9-13 in conjunction with the equivalent mass transferequation using a stiff coupled solver. See Section 22.11.7: Including Coupled Heat-MassSolution Effects on the Particles for details.

The heat transfer coefficient, h, is evaluated using the correlation of Ranz and Mar-shall [295, 296]:

Nu =hdpk∞

= 2.0 + 0.6Re1/2d Pr1/3 (22.9-17)



where

dp = particle diameter (m)k∞ = thermal conductivity of the continuous phase (W/m-K)Red = Reynolds number based on the particle diameter and

the relative velocity (Equation 22.2-3)Pr = Prandtl number of the continuous phase (cpµ/k∞)

Finally, the heat lost or gained by the particle as it traverses each computational cellappears as a source or sink of heat in subsequent calculations of the continuous phaseenergy equation. During Laws 1 and 6, particles/droplets do not exchange mass with thecontinuous phase and do not participate in any chemical reaction.

Droplet Vaporization (Law 2)

Law 2 is applied to predict the vaporization from a discrete phase droplet. Law 2 isinitiated when the temperature of the droplet reaches the vaporization temperature,Tvap, and continues until the droplet reaches the boiling point, Tbp, or until the droplet’svolatile fraction is completely consumed:

Tp < Tbp (22.9-18)

mp > (1− fv,0)mp,0 (22.9-19)

The onset of the vaporization law is determined by the setting of Tvap, a modeling pa-rameter that has no physical significance. Note that once vaporization is initiated (bythe droplet reaching this threshold temperature), it will continue to vaporize even if thedroplet temperature falls below Tvap. Vaporization will be halted only if the droplettemperature falls below the dew point. In such cases, the droplet will remain in Law2 but no evaporation will be predicted. When the boiling point is reached, the dropletvaporization is predicted by a boiling rate, Law 3, as described in a section that follows.

Mass Transfer During Law 2

During Law 2, the rate of vaporization is governed by gradient diffusion, with the fluxof droplet vapor into the gas phase related to the gradient of the vapor concentrationbetween the droplet surface and the bulk gas:

Ni = kc(Ci,s − Ci,∞) (22.9-20)



where

Ni = molar flux of vapor (kgmol/m2-s)kc = mass transfer coefficient (m/s)Ci,s = vapor concentration at the droplet surface (kgmol/m3)Ci,∞ = vapor concentration in the bulk gas (kgmol/m3)

Note that FLUENT’s vaporization law assumes that Ni is positive (evaporation). Ifconditions exist in which Ni is negative (i.e., the droplet temperature falls below the dewpoint and condensation conditions exist), FLUENT treats the droplet as inert (Ni = 0.0).

The concentration of vapor at the droplet surface is evaluated by assuming that thepartial pressure of vapor at the interface is equal to the saturated vapor pressure, psat,at the particle droplet temperature, Tp:

Ci,s =psat(Tp)

RTp(22.9-21)

where R is the universal gas constant.

The concentration of vapor in the bulk gas is known from solution of the transport equa-tion for species i or from the PDF look-up table for nonpremixed or partially premixedcombustion calculations:

Ci,∞ = Xip

RT∞(22.9-22)

where Xi is the local bulk mole fraction of species i, p is the local absolute pressure,and T∞ is the local bulk temperature in the gas. The mass transfer coefficient in Equa-tion 22.9-20 is calculated from the Sherwood number correlation [295, 296]:

ShAB =kcdpDi,m

= 2.0 + 0.6Re1/2d Sc1/3 (22.9-23)

where Di,m = diffusion coefficient of vapor in the bulk (m2/s)Sc = the Schmidt number, µ

ρDi,m

dp = particle (droplet) diameter (m)

The vapor flux given by Equation 22.9-20 becomes a source of species i in the gas phasespecies transport equation, (see Section 22.14: Setting Material Properties for the DiscretePhase) or from the PDF look-up table for nonpremixed combustion calculations.

The mass of the droplet is reduced according to

mp(t+ ∆t) = mp(t)−NiApMw,i∆t (22.9-24)



where Mw,i = molecular weight of species i (kg/kgmol)mp = mass of the droplet (kg)Ap = surface area of the droplet (m2)

FLUENT can also solve Equation 22.9-24 in conjunction with the equivalent heat transferequation using a stiff coupled solver. See Section 22.11.7: Including Coupled Heat-MassSolution Effects on the Particles for details.

Defining the Vapor Pressure and Diffusion Coefficient

You must define the vapor pressure as a polynomial or piecewise linear function of tem-perature (psat(T )) during the problem definition. Note that the vapor pressure definitionis critical, as psat is used to obtain the driving force for the evaporation process (Equa-tions 22.9-20 and 22.9-21). You should provide accurate vapor pressure values for tem-peratures over the entire range of possible droplet temperatures in your problem. Vaporpressure data can be obtained from a physics or engineering handbook (e.g., [278]).

You must also input the diffusion coefficient, Di,m, during the setup of the discrete phasematerial properties. Note that the diffusion coefficient inputs that you supply for thecontinuous phase are not used in the discrete phase model.

Heat Transfer to the Droplet

Finally, the droplet temperature is updated according to a heat balance that relates thesensible heat change in the droplet to the convective and latent heat transfer betweenthe droplet and the continuous phase:

mpcpdTpdt

= hAp(T∞ − Tp) +dmp

dthfg + Apεpσ(θR

4 − Tp4) (22.9-25)

where cp = droplet heat capacity (J/kg-K)Tp = droplet temperature (K)h = convective heat transfer coefficient (W/m2-K)T∞ = temperature of continuous phase (K)dmpdt

= rate of evaporation (kg/s)hfg = latent heat (J/kg)εp = particle emissivity (dimensionless)σ = Stefan-Boltzmann constant (5.67 x 10−8 W/m2-K4)θR = radiation temperature, ( I

4σ)1/4, where I is the radiation intensity


The heat transferred to or from the gas phase becomes a source/sink of energy duringsubsequent calculations of the continuous phase energy equation.



Droplet Boiling (Law 3)

Law 3 is applied to predict the convective boiling of a discrete phase droplet when thetemperature of the droplet has reached the boiling temperature, Tbp, and while the massof the droplet exceeds the nonvolatile fraction, (1− fv,0):

Tp ≥ Tbp (22.9-26)

and

mp > (1− fv,0)mp,0 (22.9-27)

When the droplet temperature reaches the boiling point, a boiling rate equation is ap-plied [187]:

d(dp)

dt=

4k∞ρpcp,∞dp

(1 + 0.23√

Red) ln

[1 +

cp,∞(T∞ − Tp)hfg

](22.9-28)

where cp,∞ = heat capacity of the gas (J/kg-K)ρp = droplet density (kg/m3)k∞ = thermal conductivity of the gas (W/m-K)

Equation 22.9-28 was derived assuming steady flow at constant pressure. Note that themodel requires T∞ > Tbp in order for boiling to occur and that the droplet remains atfixed temperature (Tbp) throughout the boiling law.

When radiation heat transfer is active, FLUENT uses a slight modification of Equa-tion 22.9-28, derived by starting from Equation 22.9-25 and assuming that the droplettemperature is constant. This yields

− dmp

dthfg = hAp(T∞ − Tp) + Apεpσ(θR

4 − Tp4) (22.9-29)

or

− d(dp)

dt=

2

ρphfg

[k∞Nu

dp(T∞ − Tp) + εpσ(θ4

R − T 4p )

](22.9-30)

Using Equation 22.9-17 for the Nusselt number correlation and replacing the Prandtlnumber term with an empirical constant, Equation 22.9-30 becomes

− d(dp)

dt=

2

ρphfg

[2k∞[1 + 0.23

√Red]

dp(T∞ − Tp) + εpσ(θ4

R − T 4p )

](22.9-31)



In the absence of radiation, this result matches that of Equation 22.9-28 in the limit thatthe argument of the logarithm is close to unity. FLUENT uses Equation 22.9-31 whenradiation is active in your model and Equation 22.9-28 when radiation is not active.Radiation heat transfer to the particle is included only if you have enabled the P-1 ordiscrete ordinates radiation model and you have activated radiation heat transfer toparticles using the Particle Radiation Interaction option in the Discrete Phase Model panel.

The droplet is assumed to stay at constant temperature while the boiling rate is applied.Once the boiling law is entered it is applied for the duration of the particle trajectory.The energy required for vaporization appears as a (negative) source term in the energyequation for the gas phase. The evaporated liquid enters the gas phase as species i, asdefined by your input for the destination species (see Section 22.14: Setting MaterialProperties for the Discrete Phase).

Devolatilization (Law 4)

The devolatilization law is applied to a combusting particle when the temperature of theparticle reaches the vaporization temperature, Tvap, and remains in effect while the massof the particle, mp, exceeds the mass of the nonvolatiles in the particle:

Tp ≥ Tvap and Tp ≥ Tbp (22.9-32)

and

mp > (1− fv,0)(1− fw,0)mp,0 (22.9-33)

where fw,0 is the mass fraction of the evaporating/boiling material if Wet Combustion isselected (otherwise, fw,0 = 0). As implied by Equation 22.9-32, the boiling point, Tbp,and the vaporization temperature, Tvap, should be set equal to each other when Law 4is to be used. When wet combustion is active, Tbp and Tvap refer to the boiling andevaporation temperatures for the combusting material only.

FLUENT provides a choice of four devolatilization models:

• the constant rate model (the default model)

• the single kinetic rate model

• the two competing rates model (the Kobayashi model)

• the chemical percolation devolatilization (CPD) model

Each of these models is described, in turn, below.



Choosing the Devolatilization Model

You will choose the devolatilization model when you are setting physical properties for thecombusting-particle material in the Materials panel, as described in Section 22.14.2: De-scription of the Properties. By default, the constant rate model (Equation 22.9-34) willbe used.

The Constant Rate Devolatilization Model

The constant rate devolatilization law dictates that volatiles are released at a constantrate [26]:

− 1

fv,0(1− fw,0)mp,0

dmp

dt= A0 (22.9-34)

where mp = particle mass (kg)fv,0 = fraction of volatiles initially present in the particlemp,0 = initial particle mass (kg)A0 = rate constant (s−1)

The rate constant A0 is defined as part of your modeling inputs, with a default value of 12s−1 derived from the work of Pillai [283] on coal combustion. Proper use of the constantdevolatilization rate requires that the vaporization temperature, which controls the onsetof devolatilization, be set appropriately. Values in the literature show this temperatureto be about 600 K [26].

The volatile fraction of the particle enters the gas phase as the devolatilizing species i,defined by you (see Section 22.14: Setting Material Properties for the Discrete Phase).Once in the gas phase, the volatiles may react according to the inputs governing the gasphase chemistry.

The Single Kinetic Rate Model

The single kinetic rate devolatilization model assumes that the rate of devolatilization isfirst-order dependent on the amount of volatiles remaining in the particle [16]:

− dmp

dt= k[mp − (1− fv,0)(1− fw,0)mp,0] (22.9-35)

where mp = particle mass (kg)fv,0 = mass fraction of volatiles initially present in the particlefw,0 = mass fraction of evaporating/boiling material (if wet combustion

is modeled)mp,0 = initial particle mass (kg)k = kinetic rate (s−1)



Note that fv,0, the fraction of volatiles in the particle, should be defined using a valueslightly in excess of that determined by proximate analysis. The kinetic rate, k, is definedby input of an Arrhenius type pre-exponential factor and an activation energy:

k = A1e−(E/RT ) (22.9-36)

FLUENT uses default rate constants, A1 and E, as given in [16].

Equation 22.9-35 has the approximate analytical solution:

mp(t+ ∆t) = (1− fv,0)(1− fw,0)mp,0 + [mp(t)− (1− fv,0)(1− fw,0)mp,0]e−k∆t (22.9-37)

which is obtained by assuming that the particle temperature varies only slightly betweendiscrete time integration steps.

FLUENT can also solve Equation 22.9-37 in conjunction with the equivalent heat transferequation using a stiff coupled solver. See Section 22.11.7: Including Coupled Heat-MassSolution Effects on the Particles for details.

The Two Competing Rates (Kobayashi) Model

FLUENT also provides the kinetic devolatilization rate expressions of the form proposedby Kobayashi [184]:

R1 = A1e−(E1/RTp) (22.9-38)

R2 = A2e−(E2/RTp) (22.9-39)

where R1 and R2 are competing rates that may control the devolatilization over differenttemperature ranges. The two kinetic rates are weighted to yield an expression for thedevolatilization as

mv(t)

(1− fw,0)mp,0 −ma

=∫ t

0(α1R1 + α2R2) exp

(−∫ t

0(R1 +R2) dt

)dt (22.9-40)

where mv(t) = volatile yield up to time tmp,0 = initial particle mass at injectionα1, α2 = yield factorsma = ash content in the particle



The Kobayashi model requires input of the kinetic rate parameters, A1, E1, A2, andE2, and the yields of the two competing reactions, α1 and α2. FLUENT uses defaultvalues for the yield factors of 0.3 for the first (slow) reaction and 1.0 for the second(fast) reaction. It is recommended in the literature [184] that α1 be set to the fractionof volatiles determined by proximate analysis, since this rate represents devolatilizationat low temperature. The second yield parameter, α2, should be set close to unity, whichis the yield of volatiles at very high temperature.

By default, Equation 22.9-40 is integrated in time analytically, assuming the particletemperature to be constant over the discrete time integration step. FLUENT can alsosolve Equation 22.9-40 in conjunction with the equivalent heat transfer equation using astiff coupled solver. See Section 22.11.7: Including Coupled Heat-Mass Solution Effectson the Particles for details.

The CPD Model

In contrast to the coal devolatilization models presented above, which are based on em-pirical rate relationships, the chemical percolation devolatilization (CPD) model char-acterizes the devolatilization behavior of rapidly heated coal based on the physical andchemical transformations of the coal structure [108, 109, 126].

General Description

During coal pyrolysis, the labile bonds between the aromatic clusters in the coal structurelattice are cleaved, resulting in two general classes of fragments. One set of fragmentshas a low molecular weight (and correspondingly high vapor pressure) and escapes fromthe coal particle as a light gas. The other set of fragments consists of tar gas precursorsthat have a relatively high molecular weight (and correspondingly low vapor pressure)and tend to remain in the coal for a long period of time during typical devolatilizationconditions. During this time, reattachment with the coal lattice (which is referred to ascrosslinking) can occur. The high molecular weight compounds plus the residual latticeare referred to as metaplast. The softening behavior of a coal particle is determinedby the quantity and nature of the metaplast generated during devolatilization. Theportion of the lattice structure that remains after devolatilization is comprised of charand mineral-compound-based ash.

The CPD model characterizes the chemical and physical processes by considering thecoal structure as a simplified lattice or network of chemical bridges that link the aromaticclusters. Modeling the cleavage of the bridges and the generation of light gas, char, andtar precursors is then considered to be analogous to the chemical reaction scheme shownin Figure 22.9.2.

The variable £ represents the original population of labile bridges in the coal lattice.Upon heating, these bridges become the set of reactive bridges, £∗. For the reactivebridges, two competing paths are available. In one path, the bridges react to form side



Figure 22.9.2: Coal Bridge

chains, δ. The side chains may detach from the aromatic clusters to form light gas, g1.As bridges between neighboring aromatic clusters are cleaved, a certain fraction of thecoal becomes detached from the coal lattice. These detached aromatic clusters are theheavy-molecular-weight tar precursors that form the metaplast. The metaplast vaporizesto form coal tar. While waiting for vaporization, the metaplast can also reattach to thecoal lattice matrix (crosslinking). In the other path, the bridges react and become a charbridge, c, with the release of an associated light gas product, g2. The total population ofbridges in the coal lattice matrix can be represented by the variable p, where p = £ + c.

Reaction Rates

Given this set of variables that characterizes the coal lattice structure during devolatiliza-tion, the following set of reaction rate expressions can be defined for each, starting withthe assumption that the reactive bridges are destroyed at the same rate at which theyare created (∂£∗

∂t= 0):

d£

dt= −kb£ (22.9-41)

dc

dt= kb

£

ρ+ 1(22.9-42)

dδ

dt=

[2ρkb

£

ρ+ 1

]− kgδ (22.9-43)

dg1

dt= kgδ (22.9-44)

dg2

dt= 2

dc

dt(22.9-45)



where the rate constants for bridge breaking and gas release steps, kb and kg, are expressedin Arrhenius form with a distributed activation energy:

k = Ae−(E±Eσ)/RT (22.9-46)

where A, E, and Eσ are, respectively, the pre-exponential factor, the activation energy,and the distributed variation in the activation energy, R is the universal gas constant,and T is the temperature. The ratio of rate constants, ρ = kδ/kc, is set to 0.9 in thismodel based on experimental data.

Mass Conservation

The following mass conservation relationships are imposed:

g = g1 + g2 (22.9-47)

g1 = 2f − σ (22.9-48)

g2 = 2(c− c0) (22.9-49)

where f is the fraction of broken bridges (f = 1 − p). The initial conditions for thissystem are given by the following:

c(0) = c0 (22.9-50)

£(0) = £0 = p0 − c0 (22.9-51)

δ(0) = 2f0 = 2(1− c0 −£0) (22.9-52)

g(0) = g1(0) = g2(0) = 0 (22.9-53)

where c0 is the initial fraction of char bridges, p0 is the initial fraction of bridges in thecoal lattice, and £0 is the initial fraction of labile bridges in the coal lattice.



Fractional Change in the Coal Mass

Given the set of reaction equations for the coal structure parameters, it is necessary torelate these quantities to changes in coal mass and the related release of volatile products.To accomplish this, the fractional change in the coal mass as a function of time is dividedinto three parts: light gas (fgas), tar precursor fragments (ffrag), and char (fchar). This isaccomplished by using the following relationships, which are obtained using percolationlattice statistics:

fgas(t) =r(g1 + g2)(σ + 1)

4 + 2r(1− c0)(σ + 1)(22.9-54)

ffrag(t) =2

2 + r(1− c0)(σ + 1)[ΦF (p) + rΩK(p)] (22.9-55)

fchar(t) = 1− fgas(t)− ffrag(t) (22.9-56)

The variables Φ, Ω, F (p), and K(p) are the statistical relationships related to the cleavingof bridges based on the percolation lattice statistics, and are given by the followingequations:

Φ = 1 + r

[£

p+

(σ − 1)δ

4(1− p)

](22.9-57)

Ω =δ

2(1− p)− £

p(22.9-58)

F (p) =

(p′

p

)σ+1σ−1

(22.9-59)

K(p) =[1−

(σ + 1

2

)p′] (

p′

p

)σ+1σ−1

(22.9-60)

r is the ratio of bridge mass to site mass, mb/ma, where

mb = 2Mw,δ (22.9-61)

ma = Mw,1 − (σ + 1)Mw,δ (22.9-62)



where Mw,δ and Mw,1 are the side chain and cluster molecular weights respectively. σ+ 1is the lattice coordination number, which is determined from solid-state nuclear magneticesonance (NMR) measurements related to coal structure parameters, and p′ is the rootof the following equation in p (the total number of bridges in the coal lattice matrix):

p′(1− p′)σ−1 = p(1− p)σ−1 (22.9-63)

In accounting for mass in the metaplast (tar precursor fragments), the part that vaporizesis treated in a manner similar to flash vaporization, where it is assumed that the finitefragments undergo vapor/liquid phase equilibration on a time scale that is rapid withrespect to the bridge reactions. As an estimate of the vapor/liquid that is present atany time, a vapor pressure correlation based on a simple form of Raoult’s Law is used.The vapor pressure treatment is largely responsible for predicting pressure-dependentdevolatilization yields. For the part of the metaplast that reattaches to the coal lattice,a cross-linking rate expression given by the following equation is used:

dmcross

dt= mfragAcrosse

−(Ecross/RT ) (22.9-64)

where mcross is the amount of mass reattaching to the matrix, mfrag is the amount ofmass in the tar precursor fragments (metaplast), and Across and Ecross are rate expressionconstants.

CPD Inputs

Given the set of equations and corresponding rate constants introduced for the CPDmodel, the number of constants that must be defined to use the model is a primaryconcern. For the relationships defined previously, it can be shown that the followingparameters are coal independent [108]:

• Ab, Eb, Eσb, Ag, Eg, and Eσg for the rate constants kb and kg

• Across, Ecross, and ρ

These constants are included in the submodel formulation and are not input or modifiedduring problem setup.



There are an additional five parameters that are coal-specific and must be specified duringthe problem setup:

• initial fraction of bridges in the coal lattice, p0

• initial fraction of char bridges, c0

• lattice coordination number, σ + 1

• cluster molecular weight, Mw,1

• side chain molecular weight, Mw,δ

The first four of these are coal structure quantities that are obtained from NMR exper-imental data. The last quantity, representing the char bridges that either exist in theparent coal or are formed very early in the devolatilization process, is estimated basedon the coal rank. These quantities are entered in the Materials panel as described inSection 22.14.2: Description of the Properties. Values for the coal-dependent parametersfor a variety of coals are listed in Table 22.9.1.

Table 22.9.1: Chemical Structure Parameters for 13C NMR for 13 Coals

Coal Type σ + 1 p0 Mw,1 Mw,δ c0

Zap (AR) 3.9 .63 277 40 .20Wyodak (AR) 5.6 .55 410 42 .14Utah (AR) 5.1 .49 359 36 0Ill6 (AR) 5.0 .63 316 27 0Pitt8 (AR) 4.5 .62 294 24 0Stockton (AR) 4.8 .69 275 20 0Freeport (AR) 5.3 .67 302 17 0Pocahontas (AR) 4.4 .74 299 14 .20Blue (Sandia) 5.0 .42 410 47 .15Rose (AFR) 5.8 .57 459 48 .101443 (lignite, ACERC) 4.8 .59 297 36 .201488 (subbituminous, ACERC) 4.7 .54 310 37 .151468 (anthracite, ACERC) 4.7 .89 656 12 .25

AR refers to eight types of coal from the Argonne premium sample bank [346, 386]. Sandia refers tothe coal examined at Sandia National Laboratories [107]. AFR refers to coal examined at AdvancedFuel Research. ACERC refers to three types of coal examined at the Advanced Combustion EngineeringResearch Center.



Particle Swelling During Devolatilization

The particle diameter changes during devolatilization according to the swelling coefficient,Csw, which is defined by you and applied in the following relationship:

dpdp,0

= 1 + (Csw − 1)(1− fw,0)mp,0 −mp

fv,0(1− fw,0)mp,0

(22.9-65)

where dp,0 = particle diameter at the start of devolatilizationdp = current particle diameter

The term (1−fw,0)mp,0−mpfv,0(1−fw,0)mp,0

is the ratio of the mass that has been devolatilized to the total

volatile mass of the particle. This quantity approaches a value of 1.0 as the devolatiliza-tion law is applied. When the swelling coefficient is equal to 1.0, the particle diameterstays constant. When the swelling coefficient is equal to 2.0, the final particle diam-eter doubles when all of the volatile component has vaporized, and when the swellingcoefficient is equal to 0.5 the final particle diameter is half of its initial diameter.

Heat Transfer to the Particle During Devolatilization

Heat transfer to the particle during the devolatilization process includes contributionsfrom convection, radiation (if active), and the heat consumed during devolatilization:

mpcpdTpdt

= hAp(T∞ − Tp) +dmp

dthfg + Apεpσ(θR

4 − Tp4) (22.9-66)


By default, Equation 22.9-66 is solved analytically, by assuming that the temperatureand mass of the particle do not change significantly between time steps:


where

αp =hApT∞ + dmp

dthfg + ApεpσθR

4

hAp + εpApσTp3 (22.9-68)

and

βp =Ap(h+ εpσTp

3)

mpcp(22.9-69)



FLUENT can also solve Equation 22.9-66 in conjunction with the equivalent mass transferequation using a stiff coupled solver. See Section 22.11.7: Including Coupled Heat-MassSolution Effects on the Particles for details.

Surface Combustion (Law 5)

After the volatile component of the particle is completely evolved, a surface reactionbegins which consumes the combustible fraction, fcomb, of the particle. Law 5 is thusactive (for a combusting particle) after the volatiles are evolved:

mp < (1− fv,0)(1− fw,0)mp,0 (22.9-70)

and until the combustible fraction is consumed:

mp > (1− fv,0 − fcomb)(1− fw,0)mp,0 (22.9-71)

When the combustible fraction, fcomb, has been consumed in Law 5, the combustingparticle may contain residual “ash” that reverts to the inert heating law, Law 6 (describedpreviously).

With the exception of the multiple surface reactions model, the surface combustion lawconsumes the reactive content of the particle as governed by the stoichiometric require-ment, Sb, of the surface “burnout” reaction:

char(s) + Sbox(g) −→ products(g) (22.9-72)

where Sb is defined in terms of mass of oxidant per mass of char, and the oxidant andproduct species are defined in the Set Injection Properties panel.

FLUENT provides a choice of four heterogeneous surface reaction rate models for com-busting particles:

• the diffusion-limited rate model (the default model)

• the kinetics/diffusion-limited rate model

• the intrinsic model

• the multiple surface reactions model

Each of these models is described in detail below. You will choose the surface combustionmodel when you are setting physical properties for the combusting-particle material inthe Materials panel, as described in Section 22.14.2: Description of the Properties. Bydefault, the diffusion-limited rate model will be used.



The Diffusion-Limited Surface Reaction Rate Model

The diffusion-limited surface reaction rate model which is the default model in FLUENT,assumes that the surface reaction proceeds at a rate determined by the diffusion of thegaseous oxidant to the surface of the particle:

dmp

dt= −4πdpDi,m

YoxT∞ρ

Sb(Tp + T∞)(22.9-73)

where Di,m = diffusion coefficient for oxidant in the bulk (m2/s)Yox = local mass fraction of oxidant in the gasρ = gas density (kg/m3)Sb = stoichiometry of Equation 22.9-72

Equation 22.9-73 is derived from the model of Baum and Street [26] with the kinetic con-tribution to the surface reaction rate ignored. The diffusion-limited rate model assumesthat the diameter of the particles does not change. Since the mass of the particles isdecreasing, the effective density decreases, and the char particles become more porous.

The Kinetic/Diffusion Surface Reaction Rate Model

The kinetic/diffusion-limited rate model assumes that the surface reaction rate is deter-mined either by kinetics or by a diffusion rate. FLUENT uses the model of Baum andStreet [26] and Field [104], in which a diffusion rate coefficient

D0 = C1[(Tp + T∞)/2]0.75

dp(22.9-74)

and a kinetic rate

R = C2e−(E/RTp) (22.9-75)

are weighted to yield a char combustion rate of

dmp

dt= −Appox

D0RD0 +R

(22.9-76)



where Ap is the surface area of the droplet (πd2p), pox is the partial pressure of oxidant

species in the gas surrounding the combusting particle, and the kinetic rate, R, incorpo-rates the effects of chemical reaction on the internal surface of the char particle (intrinsicreaction) and pore diffusion. In FLUENT, Equation 22.9-76 is recast in terms of theoxidant mass fraction, Yox, as

dmp

dt= −Ap

ρRT∞Yox

Mw,ox

D0RD0 +R

(22.9-77)

The particle size is assumed to remain constant in this model while the density is allowedto decrease.

When this model is enabled, the rate constants used in Equations 22.9-74 and 22.9-75 areentered in the Materials panel, as described in Section 22.14: Setting Material Propertiesfor the Discrete Phase.

The Intrinsic Model

The intrinsic model in FLUENT is based on Smith’s model [341], assuming the order ofreaction is equal to unity. Like the kinetic/diffusion model, the intrinsic model assumesthat the surface reaction rate includes the effects of both bulk diffusion and chemicalreaction (see Equation 22.9-77). The intrinsic model uses Equation 22.9-74 to computethe diffusion rate coefficient, D0, but the chemical rate, R, is explicitly expressed in termsof the intrinsic chemical and pore diffusion rates:

R = ηdp6ρpAgki (22.9-78)

η is the effectiveness factor, or the ratio of the actual combustion rate to the rate attain-able if no pore diffusion resistance existed [197]:

η =3

φ2(φ cothφ− 1) (22.9-79)

where φ is the Thiele modulus:

φ =dp2

[SbρpAgkipox

Deρox

]1/2

(22.9-80)



ρox is the density of oxidant in the bulk gas (kg/m3) and De is the effective diffusioncoefficient in the particle pores. Assuming that the pore size distribution is unimodaland the bulk and Knudsen diffusion proceed in parallel, De is given by

De =θ

τ 2

[1

DKn

+1

D0

]−1

(22.9-81)

where D0 is the bulk molecular diffusion coefficient and θ is the porosity of the charparticle:

θ = 1− ρpρt

(22.9-82)

ρp and ρt are, respectively, the apparent and true densities of the pyrolysis char.

τ (in Equation 22.9-81) is the tortuosity of the pores. The default value for τ in FLUENTis√

2, which corresponds to an average intersecting angle between the pores and theexternal surface of 45 [197].

DKn is the Knudsen diffusion coefficient:

DKn = 97.0rp

√Tp

Mw,ox

(22.9-83)

where Tp is the particle temperature and rp is the mean pore radius of the char parti-cle, which can be measured by mercury porosimetry. Note that macropores (rp > 150A) dominate in low-rank chars while micropores (rp < 10 A) dominate in high-rankchars [197].

Ag (in Equations 22.9-78 and 22.9-80) is the specific internal surface area of the charparticle, which is assumed in this model to remain constant during char combustion.Internal surface area data for various pyrolysis chars can be found in [340]. The meanvalue of the internal surface area during char combustion is higher than that of thepyrolysis char [197]. For example, an estimated mean value for bituminous chars is 300m2/g [52].

ki (in Equations 22.9-78 and 22.9-80) is the intrinsic reactivity, which is of Arrheniusform:

ki = Aie−(Ei/RTp) (22.9-84)

where the pre-exponential factor Ai and the activation energy Ei can be measured foreach char. In the absence of such measurements, the default values provided by FLUENT(which are taken from a least squares fit of data of a wide range of porous carbons,including chars [340]) can be used.



To allow a more adequate description of the char particle size (and hence density) vari-ation during combustion, you can specify the burning mode α, relating the char particlediameter to the fractional degree of burnout U (where U = 1−mp/mp,0) by [339]

dpdp,0

= (1− U)α (22.9-85)

where mp is the char particle mass and the subscript zero refers to initial conditions (i.e.,at the start of char combustion). Note that 0 ≤ α ≤ 1/3 where the limiting values 0 and1/3 correspond, respectively, to a constant size with decreasing density (zone 1) and adecreasing size with constant density (zone 3) during burnout. In zone 2, an intermediatevalue of α = 0.25, corresponding to a decrease of both size and density, has been foundto work well for a variety of chars [339].

When this model is enabled, the rate constants used in Equations 22.9-74, 22.9-78,22.9-80, 22.9-81, 22.9-83, 22.9-84, and 22.9-85 are entered in the Materials panel, asdescribed in Section 22.14: Setting Material Properties for the Discrete Phase.

The Multiple Surface Reactions Model

Modeling multiple particle surface reactions follows a pattern similar to the wall surfacereaction models, where the surface species is now a “particle surface species”. For themixture material defined in the Species Model panel, the particle surface species can bedepleted or produced by the stoichiometry of the particle surface reaction (defined in theReactions panel). The particle surface species constitutes the reactive char mass of theparticle, hence, if a particle surface species is depleted, the reactive “char” content of theparticle is consumed, and in turn, when a surface species is produced, it is added to theparticle “char” mass. Any number of particle surface species and any number of particlesurface reactions can be defined for any given combusting particle.

Multiple injections can be accommodated, and combusting particles reacting accordingto the multiple surface reactions model can coexist in the calculation, with combustingparticles following other char combustion laws. The model is based on oxidation studiesof char particles, but it is also applicable to gas-solid reactions in general, not only tochar oxidation reactions.

See Section 14.3: Particle Surface Reactions for information about particle surface reac-tions.



Limitations

Note the following limitations of the multiple surface reactions model:

• The model is not available together with the unsteady tracking option.

• The model is available only with the species transport model for volumetric reac-tions, and not with the nonpremixed, premixed, or partially premixed combustionmodels.

Heat and Mass Transfer During Char Combustion

The surface reaction consumes the oxidant species in the gas phase; i.e., it supplies a(negative) source term during the computation of the transport equation for this species.Similarly, the surface reaction is a source of species in the gas phase: the product ofthe heterogeneous surface reaction appears in the gas phase as a user-selected chemicalspecies. The surface reaction also consumes or produces energy, in an amount determinedby the heat of reaction defined by you.

The particle heat balance during surface reaction is

mpcpdTpdt

= hAp(T∞ − Tp)− fhdmp

dtHreac + Apεpσ(θR

4 − Tp4) (22.9-86)

where Hreac is the heat released by the surface reaction. Note that only a portion (1−fh)of the energy produced by the surface reaction appears as a heat source in the gas-phase energy equation: the particle absorbs a fraction fh of this heat directly. For coalcombustion, it is recommended that fh be set at 1.0 if the char burnout product is COand 0.3 if the char burnout product is CO2 [38].


By default, Equation 22.9-86 is solved analytically, by assuming that the temperatureand mass of the particle do not change significantly between time steps. FLUENT canalso solve Equation 22.9-86 in conjunction with the equivalent mass transfer equationusing a stiff coupled solver. See Section 22.11.7: Including Coupled Heat-Mass SolutionEffects on the Particles for details.



Multicomponent Particle Definition (Law 7)

Multicomponent particles are described in FLUENT as a mixture of species within droplets/particles.The particle mass m is the sum of the masses of the components

m =∑i

mi (22.9-87)

The density of the particle ρp can be either constant, or volume-averaged:

ρp =

(∑i

mi

mρi

)−1

(22.9-88)

For particles containing more than one component it is difficult to assign the wholeparticle to one process like boiling or heating. Therefore it can be only modeled by a lawintegrating all processes of relevance in one equation. The source terms for temperatureand component mass are the sum of the sources from the partial processes:

mpcp

(dTpdt

)= Apεpσ(θ4

R − T 4p ) + hAp(T∞ − Tp) +

∑i

dmi

dt(hi,p − hi,g) (22.9-89)

(dmi

dt

)= Mw,ikc,i(Ci,s − Ci,∞) (22.9-90)

The equation for the particle temperature T consists of terms for radiation, convectiveheating (Equation 22.9-11) and vaporization. Radiation heat transfer to the particle isincluded only if you have enabled P-1 or Discrete-Ordinates (DO) radiation and you haveactivated radiation heat transfer to the particles using the Particle Radiation Interactionoption in the Discrete Phase Model panel.

The mass of the particle components mi is only influenced by the vaporization (Equa-tion 22.9-20), whereMw,i is the molecular weight of species i. The mass transfer coefficientkc,i of component i is calculated from the Sherwood correlation (Equation 22.9-23). Theconcentration of vapor at the particle surface Ci,s depends on the saturation pressure ofthe component.

Raoult’s Law

The correlation between the vapor concentration of a species Ci,s over the surface andits mole fraction in the condensed phase Xi (under the assumption of an ideal gas law)is described by Raoult’s law:

Ci,s =piRT

=Xip

RT(22.9-91)


22.10 Discrete Phase Model (DPM) Boundary Conditions

You can define your own law other than Raoult’s Law for vapor concentration at theparticle surface using a user-defined function.

See Section 2.5.15: DEFINE DPM VP EQUILIB of the FLUENT UDF Manual for details.

22.10 Discrete Phase Model (DPM) Boundary Conditions

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

• The particle may be reflected via an elastic or inelastic collision.

• The particle may escape through the boundary. The particle is lost from the cal-culation at the point where it impacts the boundary.

• The particle may be trapped at the wall. Nonvolatile material is lost from thecalculation at the point of impact with the boundary; volatile material present inthe particle or droplet is released to the vapor phase at this point.

• The particle may pass through an internal boundary zone, such as radiator orporous jump.

• The particle may slide along the wall, depending on particle properties and impactangle.

You also have the option of implementing a user-defined function to model the particlebehavior when hitting the boundary. See the separate UDF Manual for informationabout user-defined functions.

These boundary condition options are described in detail in Section 22.13: Setting Bound-ary Conditions for the Discrete Phase.

22.11 Steps for Using the Discrete Phase Models

You can include a discrete phase in your FLUENT model by defining the initial position,velocity, size, and temperature of individual particles. These initial conditions, alongwith your inputs defining the physical properties of the discrete phase, are used to initiatetrajectory and heat/mass transfer calculations. The trajectory and heat/mass transfercalculations are based on the force balance on the particle and on the convective/radiativeheat and mass transfer from the particle, using the local continuous phase conditions asthe particle moves through the flow. The predicted trajectories and the associated heatand mass transfer can be viewed graphically and/or alphanumerically.



The procedure for setting up and solving a problem involving a discrete phase is outlinedbelow, and described in detail in Sections 22.11.1–22.16. Only the steps related specifi-cally to discrete phase modeling are shown here. For information about inputs related toother models that you are using in conjunction with the discrete phase models, see theappropriate sections for those models.

1. Enable any of the discrete phase modeling options, if relevant, as described in thissection.

2. Choose a transient or steady treatment of particles as described inSection 22.11.2: Steady/Transient Treatment of Particles.

3. Enable the required physical submodels for the discrete phase model, as describedin Section 22.11.5: Physical Models for the Discrete Phase Model.

4. Set the numerics parameters and solve the problem, as described in Section 22.11.7: Nu-merics of the Discrete Phase Model and Section 22.15: Solution Strategies for theDiscrete Phase.

5. Specify the initial conditions and particle size distributions, as described in Sec-tion 22.12: Setting Initial Conditions for the Discrete Phase.

6. Define the boundary conditions, as described in Section 22.13: Setting BoundaryConditions for the Discrete Phase.

7. Define the material properties, as described in Section 22.14: Setting Material Prop-erties for the Discrete Phase.

8. Initialize the flow field.

9. For transient cases, advance the solution in time by taking the desired number oftime steps. Particle positions will be updated as the solution advances in time. Ifyou are solving an uncoupled flow, the particle position will be updated at the endof each time step. For a coupled calculation, the positions are iterated on or withineach time step.

10. Solve the coupled or uncoupled flow (Section 22.15: Solution Strategies for theDiscrete Phase).

11. Examine the results, as described in Section 22.16: Postprocessing for the DiscretePhase.



22.11.1 Options for Interaction with the Continuous Phase

If the discrete phase interacts (i.e., exchanges mass, momentum, and/or energy) with thecontinuous phase, you should enable the Interaction with the Continuous Phase option.An input for the Number of Continuous Phase Iterations per DPM Iteration will appear,which allows you to control the frequency at which the particles are tracked and theDPM sources are updated.

For steady-state simulations, increasing the Number of Continuous Phase Iterations perDPM Iteration will increase stability but require more iterations to converge. Figure 22.11.1shows how the source term, S, when applied to the flow equations, changes with the num-ber of updates for varying under-relaxation factors. In Figure 22.11.1, S∞ is the finalsource term for which a value is reached after a certain number of updates and S0 is theinitial source term at the start of the computation. The value of S0 is typically zero atthe beginning of the calculation.

In addition, another option exists which allows you to control the numerical treatmentof the source terms and how they are applied to the continuous phase equations. UpdateDPM Sources Every Flow Iteration is recommended when doing unsteady simulations;at every DPM Iteration, the particle source terms are recalculated. The source termsapplied to the continuous phase equations transition to the new values every flow iterationbased on Equations 22.9-6 to 22.9-8. This process is controlled by the under-relaxationfactor, specified in the Solution Controls panel, see Section 22.15.2: Under-Relaxation ofthe Interphase Exchange Terms.

Figure 22.11.1 can be applied to this option as well. Keep in mind that the DPM sourceterms are updated every continuous flow iteration.

22.11.2 Steady/Transient Treatment of Particles

The Discrete Phase Model utilizes a Lagrangian approach to derive the equations forthe underlying physics which are solved transiently. Transient numerical procedures inthe Discrete Phase Model can be applied to resolve steady flow simulations as well astransient flows.

In the Discrete Phase Model panel you have the option of choosing whether you wantto treat the particles in an unsteady or a steady fashion. This option can be chosenindependent of the settings for the solver. Thus, you can perform steady state trajectorysimulations even when selecting a transient solver for numerical reasons. You can alsospecify unsteady particle tracking when solving the steady continuous phase equations.This can be used to improve numerical stability for very large particle source terms orsimply for postprocessing purposes. Whenever you enable a breakup or collision modelto simulate sprays, the Unsteady Particle Tracking will be switched on automatically.



Figure 22.11.1: Effect of Number of Source Term Updates on Source TermApplied to Flow Equations



When Unsteady Particle Tracking is enabled, several new options appear. If steady stateequations are solved for the continuous phase, you simply enter the Particle Time StepSize and the Number of Time Steps, thus tracking particles every time a DPM iterationis conducted. When you increase the Number of Time Steps, the droplets penetrate thedomain faster.

i Note that you must enter the start and stop times for each injection.

When solving unsteady equations for the continuous phase, you must decide whether youwant to use Fluid Flow Time Step to inject the particles, or whether you prefer a ParticleTime Step Size independent of the Fluid Flow Time Step. With the latter option, youcan use the Discrete Phase Model in combination with changes in the time step for thecontinuous equations, as it is done when using adaptive flow time stepping.

If you do not use Fluid Flow Time Step, you will need to decide when to inject the particlesfor a new time step. You can either Inject Particles at Particle Time step or at the flow timestep. In any case, the particles will always be tracked in such a way that they coincidewith the flow time of the continuous flow solver.

You can use a user-defined function (DEFINE DPM TIMESTEP) to change the time step forDPM particle tracking. The time step can be prescribed for special applications wherea certain time step is needed. See Section 2.5.14: DEFINE DPM TIMESTEP of the FLUENTUDF Manual for details on changing the time step size for DPM particle tracking.

i When the density-based explicit solver is used with the explicit unsteadyformulation, the particles are advanced once per time step and are calcu-lated at the start of the time step (before the flow is updated).

Additional inputs are required for each injection in the Set Injection Properties panel.For Unsteady Particle Tracking, the injection Start Time and Stop Time must be specifiedunder Point Properties. Injections with start and stop times set to zero will be injectedonly at the start of the calculation (t = 0). Changing injection settings during a transientsimulation will not affect particles currently released in the domain. At any point duringa simulation, you can clear particles that are currently in the domain by clicking on theClear Particles button in the Discrete Phase Model panel.



For unsteady simulations (with the implicit solvers), several methods can be chosen tocontrol when the particles are advanced.

• If the Number of Continuous Phase Iterations per DPM Iteration is less than thenumber of iterations required to converge the continuous phase between time steps,then sub-iterations are done. Here, particles are tracked to their new positionsduring a time step and DPM sources are updated; particles are then returned totheir original state at the beginning of the time step. At the end of the timestep, particles are advanced to their new positions based on the continuous-phasesolution.

• If the Number of Continuous Phase Iterations per DPM Iteration is larger than thenumber of iterations specified to converge the continuous phase between time steps,the particles are advanced at the beginning of the time step to compute the particlesource terms.

• When you specify a value of Zero as the Number of Continuous Phase Iterations perDPM Iteration, the particles are advanced at the end of the time step. For thisoption, it may be better if the particle source terms are not reset at the begin-ning of the time step. This can be done with the TUI command define/models

/dpm/interaction/reset-sources-at-timestep?.

In all the above cases, you must provide a sufficient number of particle source termupdates to better control when the particles are advanced, see Figure 22.11.1.

i In steady-state discrete phase modeling, particles do not interact with eachother and are tracked one at a time in the domain.

i If the collision model is used, you will not be able to set the Number of Con-tinuous Phase Iterations per DPM Iteration. Refer to Section 22.7.1: DropletCollision Model for details about this limitation.

22.11.3 Parameter Tracking for the Discrete Phase Model

You will use two parameters to control the time integration of the particle trajectoryequations:

• the length scale/step length factor

This factor is used to set the time step for integration within each control volume.

• the maximum number of time steps

This factor is used to abort trajectory calculations when the particle never exitsthe flow domain.



Each of these parameters is set in the Discrete Phase Model panel (Figure 22.11.2) underTracking Parameters in the Tracking tab.

Define −→ Models −→Discrete Phase...

Figure 22.11.2: The Discrete Phase Model Panel and the Tracking Parameters



Max. Number Of Steps is the maximum number of time steps used to compute a singleparticle trajectory via integration of Equations 22.2-1 and 22.15-1. When the max-imum number of steps is exceeded, FLUENT abandons the trajectory calculationfor the current particle injection and reports the trajectory fate as “incomplete”.The limit on the number of integration time steps eliminates the possibility of aparticle being caught in a recirculating region of the continuous phase flow fieldand being tracked infinitely. Note that you may easily create problems in whichthe default value of 500 time steps is insufficient for completion of the trajectorycalculation. In this case, when trajectories are reported as incomplete within thedomain and the particles are not recirculating indefinitely, you can increase themaximum number of steps (up to a limit of 109).

Length Scale controls the integration time step size used to integrate the equationsof motion for the particle. The integration time step is computed by FLUENTbased on a specified length scale L, and the velocity of the particle (up) and of thecontinuous phase (uc):

∆t =L

up + uc(22.11-1)

where L is the Length Scale that you define. As defined by Equation 22.11-1, L isproportional to the integration time step and is equivalent to the distance that theparticle will travel before its motion equations are solved again and its trajectoryis updated. A smaller value for the Length Scale increases the accuracy of thetrajectory and heat/mass transfer calculations for the discrete phase.

(Note that particle positions are always computed when particles enter/leave a cell;even if you specify a very large length scale, the time step used for integration willbe such that the cell is traversed in one step.)

Length Scale will appear in the Discrete Phase Model panel when the Specify LengthScale option is on.

Step Length Factor also controls the time step size used to integrate the equations ofmotion for the particle. It differs from the Length Scale in that it allows FLUENT tocompute the time step in terms of the number of time steps required for a particleto traverse a computational cell. To set this parameter instead of the Length Scale,turn off the Specify Length Scale option.

The integration time step is computed by FLUENT based on a characteristic timethat is related to an estimate of the time required for the particle to traverse thecurrent continuous phase control volume. If this estimated transit time is definedas ∆t∗, FLUENT chooses a time step ∆t as

∆t =∆t∗

λ(22.11-2)



where λ is the Step Length Factor. As defined by Equation 22.11-2, λ is inverselyproportional to the integration time step and is roughly equivalent to the numberof time steps required to traverse the current continuous phase control volume. Alarger value for the Step Length Factor decreases the discrete phase integration timestep. The default value for the Step Length Factor is 5. Step Length Factor willappear in the Discrete Phase Model panel when the Specify Length Scale option isoff (the default setting).

One simple rule of thumb to follow when setting the parameters above is that if you wantthe particles to advance through a domain consisting of N grid cells into the main flowdirection, the Step Length Factor times N should be approximately equal to the Max.Number Of Steps.

22.11.4 Alternate Drag Laws

There are five drag laws for the particles that can be selected in the Drag Law drop-downlist under Drag Parameters.

The spherical, nonspherical, Stokes-Cunningham, and high-Mach-number laws described inSection 22.2.1: Particle Force Balance are always available, and the dynamic-drag lawdescribed in Section 22.6: Dynamic Drag Model Theory is available only when one ofthe droplet breakup models is used in conjunction with unsteady tracking. See Sec-tion 22.11.6: Modeling Spray Breakup for information about enabling the droplet breakupmodels.

If the spherical law, the high-Mach-number law, or the dynamic-drag law is selected, nofurther inputs are required. If the nonspherical law is selected, the particle Shape Factor(φ in Equation 22.2-9) must be specified. The shape factor value cannot exceed 1. Forthe Stokes-Cunningham law, the Cunningham Correction factor (Cc in Equation 22.2-11)must be specified.



22.11.5 Physical Models for the Discrete Phase Model

This section provides instructions for using the optional discrete phase models available inFLUENT. All of them can be turned on in the Discrete Phase Model panel (Figure 22.11.3).


Figure 22.11.3: The Discrete Phase Model Panel and the Physical Models



Including Radiation Heat Transfer Effects on the Particles

If you want to include the effect of radiation heat transfer to the particles (Equa-tion 13.3-13), you must turn on the Particle Radiation Interaction option under the PhysicalModels tab, in the Discrete Phase Model panel (Figure 22.11.3). You will also need todefine additional properties for the particle materials (emissivity and scattering factor),as described in Section 22.14.2: Description of the Properties. This option is availableonly when the P-1 or discrete ordinates radiation model is used.

Including Thermophoretic Force Effects on the Particles

If you want to include the effect of the thermophoretic force on the particle trajectories(Equation 22.2-14), turn on the Thermophoretic Force option under the Physical Modelstab, in the Discrete Phase Model panel. You will also need to define the thermophoreticcoefficient for the particle material, as described in Section 22.14.2: Description of theProperties.

Including Brownian Motion Effects on the Particles

For sub-micron particles in laminar flow, you may want to include the effects of Brow-nian motion (described in Section 22.2.1: Brownian Force) on the particle trajectories.To do so, turn on the Brownian Motion option under the Physical Models tab. WhenBrownian motion effects are included, it is recommended that you also select the Stokes-Cunningham drag law in the Drag Law drop-down list under Drag Parameters, and specifythe Cunningham Correction (Cc in Equation 22.2-11).

Including Saffman Lift Force Effects on the Particles

For sub-micron particles, you can also model the lift due to shear (the Saffman lift force,described in Section 22.2.1: Saffman’s Lift Force) in the particle trajectory. To do this,turn on the Saffman Lift Force option under the Physical Models tab, in the Discrete PhaseModel panel.

Monitoring Erosion/Accretion of Particles at Walls

Particle erosion and accretion rates can be monitored at wall boundaries. These ratecalculations can be enabled in the Discrete Phase Model panel when the discrete phaseis coupled with the continuous phase (i.e., when Interaction with Continuous Phase is se-lected). Turning on the Erosion/Accretion option will cause the erosion and accretion ratesto be calculated at wall boundary faces when particle tracks are updated. You will alsoneed to set the Impact Angle Function (f(α) in Equation 22.5-1), Diameter Function (C(dp)in Equation 22.5-1), and Velocity Exponent Function (b(v) in Equation 22.5-1) in the Wallboundary conditions panel for each wall zone (as described in Section 22.13.1: DiscretePhase Boundary Condition Types).



Including the Effect of Particles on Turbulent Quantities

Particles can damp or produce turbulent eddies ([268]. In FLUENT, the work done by theturbulent eddies on the particles is subtracted from the turbulent kinetic energy usingthe formulation described in [98] and [9].

If you want to consider these effects in the chosen turbulence model, you can turn thison using Two-Way Turbulence Coupling, under the Physical Models tab.

Tracking in a Reference Frame

Particle tracking is related to a coordinate system. With Track in Absolute Frame en-abled, you can choose to track the particles in the absolute reference frame. All particlecoordinates and velocities are then computed in this frame. The forces due to frictionwith the continuous phase are transformed to this frame automatically.

In rotating flows it might be appropriate for numerical reasons to track the particles inthe relative reference frame. If several reference frames exist in one simulation, then theparticle velocities are transformed to each reference frame when they enter the fluid zoneassociated with this reference frame.

22.11.6 Options for Spray Modeling

When you enable particle tracking, the Physical Models tab in Discrete Phase Model panelwill expand to show options related to spray modeling.

Modeling Spray Breakup

To enable the modeling of spray breakup, select the Droplet Breakup option under SprayModel and then select the desired model (TAB or Wave). A detailed description of thesemodels can be found in Section 22.7.2: Droplet Breakup Models.

For the TAB model, you will need to specify a value for y0 (the initial distortion attime equal to zero in Equation 22.7-14) in the y0 field. The default value (y0 = 0) isrecommended. You will also set the number of Breakup Parcels (under Breakup Constants),to split the droplet into several child parcels, as described in Section 22.7.2: Velocity ofChild Droplets.

For the wave model, you will need to specify values for B0 and B1, where B0 is theconstant B0 in Equation 22.7-43 and B1 is the constant B1 in Equation 22.7-45. You willgenerally not need to modify the value of B0, as the default value 0.61 is acceptable fornearly all cases. A value of 1.73 is recommended for B1.

For steady-state simulations, you will also need to specify an appropriate Particle TimeStep Size and the Number of Time Steps which will control the spray density. See Sec-tion 22.11.1: Options for Interaction with the Continuous Phase for more information.



Note that you may want to use the dynamic drag law when you use one of the spraybreakup models. See Section 22.11.4: Alternate Drag Laws for information about choos-ing the drag law.

Modeling Droplet Collisions

To include the effect of droplet collisions, as described in Section 22.7.1: Droplet CollisionModel, select the Droplet Collision option under Spray Models. There are no further inputsfor this model.

22.11.7 Numerics of the Discrete Phase Model

The underlying physics of the Discrete Phase Model is described by ordinary differentialequations (ODE) as opposed to the continuous flow which is expressed in the form ofpartial differential equations (PDE). Therefore, the Discrete Phase Model uses its ownnumerical mechanisms and discretization schemes, which are completely different fromother numerics used in FLUENT.

The Numerics tab gives you control over the numerical schemes for particle tracking aswell as solutions of heat and mass equations (Figure 22.11.4).

Numerics for Tracking of the Particles

To solve equations of motion for the particles, the following numerical schemes are avail-able:

implicit uses an implicit Euler integration of Equation 22.2-1 which is unconditionallystable for all particle relaxation times.

trapezoidal uses a semi-implicit trapezoidal integration.

analytic uses an analytical integration of Equation 22.2-1 where the forces are heldconstant during the integration.

runge-kutta facilitates a 5th order Runge Kutta scheme derived by Cash and Karp [49].



Figure 22.11.4: The Discrete Phase Model Panel and the Numerics



You can either choose a single tracking scheme, or switch between higher order and lowerorder tracking schemes using an automated selection based on the accuracy to be achievedand the stability range of each scheme. In addition, you can control how accurately theequations need to be solved.

Accuracy Control enables the solution of equations of motion within a specified toler-ance. This is done by computing the error of the integration step and reducing theintegration step if the error is too large. If the error is within the given tolerance,the integration step will also be increased in the next steps.

Tolerance is the maximum relative error which has to be achieved by the trackingprocedure. Based on the numerical scheme, different methods are used to estimatethe relative error. The implemented Runge-Kutta scheme uses an embedded errorcontrol mechanism. The error of the other schemes is computed by comparing theresult of the integration step with the outcome of a two step procedure with halfthe step size.

Max. Refinements is the maximum number of step size refinements in one single inte-gration step. If this number is exceeded the integration will be conducted with thelast refined integration step size.

Automated Tracking Scheme Selection provides a mechanism to switch in an automatedfashion between numerically stable lower order schemes and higher order schemes,which are stable only in a limited range. In situations where the particle is far fromhydrodynamic equilibrium, an accurate solution can be achieved very quickly witha higher order scheme, since these schemes need less step refinements for a certaintolerance. When the particle reaches hydrodynamic equilibrium, the higher orderschemes become inefficient since their step length is limited to a stable range. Inthis case, the mechanism switches to a stable lower order scheme and facilitateslarger integration steps.

i This mechanism is only available when Accuracy Control is enabled.

Higher Order Scheme can be chosen from the group consisting of trapezoidal and runge-kutta scheme.

Lower Order Scheme consists of implicit and the exponential analytic integration scheme.

Tracking Scheme is selectable only if Automated Tracking Scheme Selection is switchedoff. You can choose any of the tracking schemes. You also can combine each of thetracking schemes with Accuracy Control.



Including Coupled Heat-Mass Solution Effects on the Particles

By default, the solution of the particle heat and mass equations are solved in a segregatedmanner. If you enable the Coupled Heat-Mass Solution option, FLUENT will solve thispair of equations using a stiff, coupled ODE solver with error tolerance control. Theincreased accuracy, however, comes at the expense of increased computational expense.

22.11.8 User-Defined Functions

User-defined functions can be used to customize the discrete phase model to includeadditional body forces, modify interphase exchange terms (sources), calculate or integratescalar values along the particle trajectory, and incorporate nonstandard erosion ratedefinitions. See the separate UDF Manual for information about user-defined functions.

In the Discrete Phase Model panel, under User-Defined Functions in the UDF tab, thereare drop-down lists labeled Body Force, Scalar Update, Source, Spray Collide Function,and DPM Time Step (Figure 22.11.5). If Erosion/Accretion is enabled under the PhysicalModels tab, there will be an additional drop-down list labeled Erosion/Accretion. Theselists will show available user-defined functions that can be selected to customize thediscrete phase model.

In addition, you can specify a Number of Scalars which are allocated to each particle andcan be used to store information when implementing your own particle models.

22.11.9 Parallel Processing for the Discrete Phase Model

FLUENT offers two modes of parallel processing for the discrete phase model: the SharedMemory and the Message Passing options under the Parallel tab, in the Discrete PhaseModel panel. The Shared Memory option is suitable for computations where the machinerunning the FLUENT host process is an adequately large, shared memory, multiprocessormachine. The Message Passing option is turned on by default and is suitable for genericdistributed memory cluster computing.

i When tracking particles in parallel, the DPM model cannot be used withany of the multiphase flow models (VOF, mixture, or Eulerian) if the SharedMemory option is enabled. (Note that using the Message Passing option,when running in parallel, enables the compatibility of all multiphase flowmodels with the DPM model.)

The Shared Memory option is implemented using POSIX Threads (pthreads) based on ashared memory model. Once the Shared Memory option is enabled, you can then selectalong with it the Workpile Algorithm and specify the Number of Threads. By default, theNumber of Threads is equal to the number of compute nodes specified for the parallelcomputation. You can modify this value based on the computational requirements of theparticle calculations. If, for example, the particle calculations require more computation



Figure 22.11.5: The Discrete Phase Model Panel and the UDFs



than the flow calculation, you can increase the Number of Threads (up to the number ofavailable processors) to improve performance. When using the Shared Memory option, theparticle calculations are entirely managed by the FLUENT host process. You must makesure that the machine executing the host process has enough memory to accommodatethe entire grid.

i Note that the Shared Memory option is not available for Windows 2000.

The Message Passing option enables cluster computing and also works on shared memorymachines. With this option enabled, the compute node processes perform the particlework on their local partitions. Particle migration to other compute nodes is implementedusing message passing primitives. There are no special requirements for the host machine.Note that this model is not available if the Cloud Model option is turned on under theTurbulent Dispersion tab of the Set Injection Properties panel. When running FLUENT inparallel, by default, pathline displays are computed in serial on the host node. Pathlinedisplays may be computed in parallel on distributed memory systems if the MessagePassing parallel option is selected in the Discrete Phase Model panel.

You may seamlessly switch between the Shared Memory option and the Message Passingoption at any time during the FLUENT session.

In addition to performing general parallel processing of the Discrete Phase Model, youhave the option of implementing DPM-specific user-defined functions in parallel FLUENT.See Section 7.4: Parallelization of Discrete Phase Model (DPM) UDFs of the separateUDF Manual for details on parallelization of DPM UDFs.


22.12 Setting Initial Conditions for the Discrete Phase


For liquid sprays, a convenient representation of the droplet size distribution is the Rosin-Rammler expression. The complete range of sizes is divided into an adequate number ofdiscrete intervals; each represented by a mean diameter for which trajectory calculationsare performed. If the size distribution is of the Rosin-Rammler type, the mass fractionof droplets of diameter greater than d is given by

Yd = e−(d/d)n (22.12-1)

where d is the size constant and n is the size distribution parameter. Use of the Rosin-Rammler size distribution is detailed in Section 22.12.1: Using the Rosin-Rammler Di-ameter Distribution Method.

The primary inputs that you must provide for the discrete phase calculations in FLUENTare the initial conditions that define the starting positions, velocities, and other param-eters for each particle stream. These initial conditions provide the starting values for allof the dependent discrete phase variables that describe the instantaneous conditions ofan individual particle, and include the following:

• position (x, y, z coordinates) of the particle

• velocities (u, v, w) of the particle

Velocity magnitudes and spray cone angle can also be used (in 3D) to define theinitial velocities (see Section 22.12.1: Point Properties for Cone Injections). Formoving reference frames, relative velocities should be specified.

• diameter of the particle, dp

• temperature of the particle, Tp

• mass flow rate of the particle stream that will follow the trajectory of the individualparticle/droplet, mp (required only for coupled calculations)

• additional parameters if one of the atomizer models described in Section 22.8: At-omizer Model Theory is used for the injection

i When an atomizer model is selected, you will not input initial diameter,velocity, and position quantities for the particles due to the complexitiesof sheet and ligament breakup. Instead of initial conditions, the quantitiesyou will input for the atomizer models are global parameters.

These dependent variables are updated according to the equations of motion(Section 22.2: Particle Motion Theory) and according to the heat/mass transfer relations



applied (Section 22.9: One-Way and Two-Way Coupling) as the particle/droplet movesalong its trajectory. You can define any number of different sets of initial conditions fordiscrete phase particles/droplets provided that your computer has sufficient memory.

22.12.1 Injection Types

You will define the initial conditions for a particle/droplet stream by creating an “injec-tion” and assigning properties to it. FLUENT provides 11 types of injections:

• single

• group

• cone (only in 3D)

• solid-cone (only in 3D)

• surface

• plain-orifice atomizer

• pressure-swirl atomizer

• flat-fan-atomizer

• air-blast-atomizer

• effervescent-atomizer

• file

For each nonatomizer injection type, you will specify each of the initial conditions listedin Section 22.12: Setting Initial Conditions for the Discrete Phase, the type of particlethat possesses these initial conditions, and any other relevant parameters for the particletype chosen.

You should create a single injection when you want to specify a single value for each ofthe initial conditions (Figure 22.12.1). Create a group injection (Figure 22.12.2) whenyou want to define a range for one or more of the initial conditions (e.g., a range ofdiameters or a range of initial positions). To define hollow spray cone injections in 3Dproblems, create a cone injection (Figure 22.12.3). To release particles from a surface(either a zone surface or a surface you have defined using the items in the Surface menu),you will create a surface injection. (If you create a surface injection, a particle streamwill be released from each facet of the surface. You can use the Bounded and SamplePoints options in the Plane Surface panel to create injections from a rectangular grid ofparticles in 3D (see Section 27.6: Plane Surfaces for details).



Figure 22.12.1: Particle Injection Defining a Single Particle Stream

Figure 22.12.2: Particle Injection Defining an Initial Spatial Distribution ofthe Particle Streams

Figure 22.12.3: Particle Injection Defining an Initial Spray Distribution ofthe Particle Velocity



Particle initial conditions (position, velocity, diameter, temperature, and mass flow rate)can also be read from an external file if none of the injection types listed above can beused to describe your injection distribution. The file has the following form:

(( x y z u v w diameter temperature mass-flow) name )

with all of the parameters in SI units. All the parentheses are required, but the name isoptional.

The inputs for setting injections are described in detail in Section 22.12.4: Defining In-jection Properties.

Point Properties for Single Injections

For a single injection, you will define the following initial conditions for the particlestream under the Point Properties heading (in the Set Injection Properties panel):

• position

Set the x, y, and z positions of the injected stream along the Cartesian axes of theproblem geometry in the X-, Y-, and Z-Position fields. (Z-Position will appear onlyfor 3D problems.)

• velocity

Set the x, y, and z components of the stream’s initial velocity in the X-, Y-, andZ-Velocity fields. (Z-Velocity will appear only for 3D problems.)

• diameter

Set the initial diameter of the injected particle stream in the Diameter field.

• temperature

Set the initial (absolute) temperature of the injected particle stream in the Tem-perature field.

• mass flow rate

For coupled phase calculations (see Section 22.15: Solution Strategies for the Dis-crete Phase), set the mass of particles per unit time that follows the trajectorydefined by the injection in the Flow Rate field. Note that in axisymmetric problemsthe mass flow rate is defined per 2π radians and in 2D problems per unit meterdepth (regardless of the reference value for length).

• duration of injection

For unsteady particle tracking (see Section 22.11.2: Steady/Transient Treatment ofParticles), set the starting and ending time for the injection in the Start Time andStop Time fields.



Point Properties for Group Injections

For group injections, you will define the properties described in Section 22.12.1: PointProperties for Single Injections for single injections for the First Point and Last Point inthe group. That is, you will define a range of values, φ1 through φN , for each initialcondition φ by setting values for φ1 and φN . FLUENT assigns a value of φ to the ithinjection in the group using a linear variation between the first and last values for φ:

φi = φ1 +φN − φ1

N − 1(i− 1) (22.12-2)

Thus, for example, if your group consists of 5 particle streams and you define a range forthe initial x location from 0.2 to 0.6 meters, the initial x location of each stream is asfollows:

• Stream 1: x = 0.2 meters





i In general, you should supply a range for only one of the initial conditionsin a given group—leaving all other conditions fixed while a single conditionvaries among the stream numbers of the group. Otherwise you may find,for example, that your simultaneous inputs of a spatial distribution anda size distribution have placed the small droplets at the beginning of thespatial range and the large droplets at the end of the spatial range.

Note that you can use a different method for defining the size distribution of the particles,as discussed below.



Using the Rosin-Rammler Diameter Distribution Method

By default, you will define the size distribution of particles by inputting a diameter forthe first and last points and using the linear equation (22.12-2) to vary the diameterof each particle stream in the group. When you want a different mass flow rate foreach particle/droplet size, however, the linear variation may not yield the distributionyou need. Your particle size distribution may be defined most easily by fitting the sizedistribution data to the Rosin-Rammler equation. In this approach, the complete rangeof particle sizes is divided into a set of discrete size ranges, each to be defined by a singlestream that is part of the group. Assume, for example, that the particle size data obeysthe following distribution:

Diameter Range (µm) Mass Fraction in Range0–7070–100100–120120–150150–180180–200

0.050.100.350.300.150.05

The Rosin-Rammler distribution function is based on the assumption that an exponentialrelationship exists between the droplet diameter, d, and the mass fraction of droplets withdiameter greater than d, Yd:

Yd = e−(d/d)n (22.12-3)

FLUENT refers to the quantity d in Equation 22.12-3 as the Mean Diameter and to n asthe Spread Parameter. These parameters are input by you (in the Set Injection Propertiespanel under the First Point heading) to define the Rosin-Rammler size distribution. Tosolve for these parameters, you must fit your particle size data to the Rosin-Rammlerexponential equation. To determine these inputs, first recast the given droplet size datain terms of the Rosin-Rammler format. For the example data provided above, this yieldsthe following pairs of d and Yd:

Diameter, d (µm) Mass Fraction withDiameter Greater than d, Yd

70100120150180200

0.950.850.500.200.05(0.00)



A plot of Yd vs. d is shown in Figure 22.12.4.

1.0

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

50 70 90 110 130 150 170 190 210 230 250

Mas

sFr

acti

on>

d,Y d

Diameter, d ( µm)

Figure 22.12.4: Example of Cumulative Size Distribution of Particles

Next, derive values of d and n such that the data in Figure 22.12.4 fit Equation 22.12-3.The value for d is obtained by noting that this is the value of d at which Yd = e−1 ≈ 0.368.From Figure 22.12.4, you can estimate that this occurs for d ≈ 131 µm. The numericalvalue for n is given by

n =ln(− lnYd)

ln(d/d

)By substituting the given data pairs for Yd and d/d into this equation, you can obtainvalues for n and find an average. Doing so yields an average value of n = 4.52 for theexample data above. The resulting Rosin-Rammler curve fit is compared to the exampledata in Figure 22.12.5. You can input values for d and n, as well as the diameter rangeof the data and the total mass flow rate for the combined individual size ranges, usingthe Set Injection Properties panel.



This technique of fitting the Rosin-Rammler curve to spray data is used when reportingthe Rosin-Rammler diameter and spread parameter in the discrete phase summary panelin Section 22.16.8: Summary Reporting of Current Particles.

1.0

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

50 70 90 110 130 150 170 190 210 230 250

Mas

sFr

acti

on>

d,Y d

Diameter, d ( µm)

Figure 22.12.5: Rosin-Rammler Curve Fit for the Example Particle Size Data

A second Rosin-Rammler distribution is also available based on the natural logarithm ofthe particle diameter. If in your case, the smaller-diameter particles in a Rosin-Rammlerdistribution have higher mass flows in comparison with the larger-diameter particles, youmay want better resolution of the smaller-diameter particle streams, or “bins”. You cantherefore choose to have the diameter increments in the Rosin-Rammler distribution doneuniformly by ln d.

In the standard Rosin-Rammler distribution, a particle injection may have a diameterrange of 1 to 200 µm. In the logarithmic Rosin-Rammler distribution, the same diameterrange would be converted to a range of ln 1 to ln 200, or about 0 to 5.3. In this way, themass flow in one bin would be less-heavily skewed as compared to the other bins.



When a Rosin-Rammler size distribution is being defined for the group of streams, youshould define (in addition to the initial velocity, position, and temperature) the followingparameters, which appear under the heading for the First Point:

• Total Flow Rate

This is the total mass flow rate of the N streams in the group. Note that inaxisymmetric problems this mass flow rate is defined per 2π radians and in 2Dproblems per unit meter depth.

• Min. Diameter

This is the smallest diameter to be considered in the size distribution.

• Max. Diameter

This is the largest diameter to be considered in the size distribution.

• Mean Diameter

This is the size parameter, d, in the Rosin-Rammler equation (22.12-3).

• Spread Parameter

This is the exponential parameter, n, in Equation 22.12-3.

The Stochastic Rosin-Rammler Diameter Distribution Method

For atomizer injections, a Rosin-Rammler distribution is assumed for the particles exitingthe injector. In order to decrease the number of particles necessary to accurately describethe distribution, the diameter distribution function is randomly sampled for each instancewhere new particles are introduced into the domain.

The Rosin-Rammler distribution can be written as

1− Y = exp

[−(D

d

)n](22.12-4)

where Y is the mass fraction smaller than a given diameter D, d is the Rosin-Rammlerdiameter and n is the Rosin-Rammler exponent. This expression can be inverted bytaking logs of both sides and rearranging,

D = d (− ln(1− Y ))1/n . (22.12-5)

Given a mass fraction Y along with parameters d and n, this function will explic-itly provide a diameter, D. Diameters for the atomizer injectors described in Sec-tion 22.12.1: Point Properties for Plain-Orifice Atomizer Injections are obtained by uni-formly sampling Y in equation 22.12-5.



Point Properties for Cone Injections

In 3D problems, you can define a hollow or solid cone of particle streams using the coneor solid-cone injection type, respectively. For both types of cone injections, the inputsare as follows:

• position

Set the coordinates of the origin of the spray cone in the X-, Y-, and Z-Positionfields.

• diameter

Set the diameter of the particles in the stream in the Diameter field.

• temperature

Set the temperature of the streams in the Temperature field.

• axis

Set the x, y, and z components of the vector defining the cone’s axis in the X-Axis,Y-Axis, and Z-Axis fields.

• velocity

Set the velocity magnitude of the particle streams that will be oriented along thespecified spray cone angle in the Velocity Mag. field.

• cone angle

Set the included half-angle, θ, of the hollow spray cone in the Cone Angle field, asshown in Figure 22.12.6.

θ

r origin axis

Figure 22.12.6: Cone Half Angle and Radius



• radius

A nonzero inner radius can be specified to model injectors that do not emanatefrom a single point. Set the radius r (defined as shown in Figure 22.12.6) in theRadius field. The particles will be distributed about the axis with the specifiedradius.

• swirl fraction (hollow cone only)

Set the fraction of the velocity magnitude to go into the swirling component ofthe flow in the Swirl Fraction field. The direction of the swirl component is definedusing the right-hand rule about the axis (a negative value for the swirl fraction canbe used to reverse the swirl direction).

• mass flow rate

For coupled calculations, set the total mass flow rate for the streams in the spraycone in the Total Flow Rate field.

The distribution of the particle streams for the solid cone injection is random, as seen inFigure 22.12.3. Furthermore, duplicating this injection may not necessarily result in thesame distribution, at the same location.

i For transient calculations, the spatial distribution of streams at the ini-tial injection location is recalculated at each time step. Sampling differentpossible trajectories allows a more accurate representation of a solid coneusing fewer computational parcels. For steady state calculations, the tra-jectories are initialized one time and kept the same for subsequent DPMiterations. The trajectories are recalculated when a change in the injectionpanel occurs or when a case and data file are saved. If the residuals and so-lution change when a small change is made to the injection or when a caseand data file are saved, it may mean that there are not enough trajectoriesbeing used to represent the solid cone with sufficient accuracy.

Note that you may want to define multiple spray cones emanating from the same initiallocation in order to specify a size known distribution of the spray or to include a knownrange of cone angles.

Point Properties for Surface Injections

For surface injections, you will define all the properties described in Section 22.12.1: PointProperties for Single Injections for single injections except for the initial position of theparticle streams. The initial positions of the particles will be the location of the datapoints on the specified surface(s). Note that you will set the Total Flow Rate of allparticles released from the surface (required for coupled calculations only). If you want,you can scale the individual mass flow rates of the particles by the ratio of the area of



the face they are released from to the total area of the surface. To scale the mass flowrates, select the Scale Flow Rate By Face Area option under Point Properties.

Note that many surfaces have nonuniform distributions of points. If you want to generatea uniform spatial distribution of particle streams released from a surface in 3D, you cancreate a bounded plane surface with a uniform distribution using the Plane Surface panel,as described in Section 27.6: Plane Surfaces. In 2D, you can create a rake using theLine/Rake Surface panel, as described in Section 27.5: Line and Rake Surfaces.

In addition to the option of scaling the flow rate by the face area, the normal directionof a face can be used for the injection direction. To use the face normal direction for theinjection direction, select the Inject Using Normal Direction option under Point Properties(Figure 22.12.9). Once this option is selected, you only need to specify the velocitymagnitude of the injection, not the individual components of the velocity magnitude.

i Note also that only surface injections from boundary surfaces will be movedwith the grid when a sliding mesh or a moving or deforming mesh is beingused.

A nonuniform size distribution can be used for surface injections, as described below.

Using the Rosin-Rammler Diameter Distribution Method

The Rosin-Rammler size distributions described in Section 22.12.1: Using the Rosin-Rammler Diameter Distribution Method for group injections is also available for surfaceinjections. If you select one of the Rosin-Rammler distributions, you will need to spec-ify the following parameters under Point Properties, in addition to the initial velocity,temperature, and total flow rate:

• Min. Diameter

This is the smallest diameter to be considered in the size distribution.

• Max. Diameter

This is the largest diameter to be considered in the size distribution.

• Mean Diameter

This is the size parameter, d, in the Rosin-Rammler equation (Equation 22.12-3).

• Spread Parameter

This is the exponential parameter, n, in Equation 22.12-3.

• Number of Diameters

This is the number of diameters in each distribution (i.e., the number of differentdiameters in the stream injected from each face of the surface).



FLUENT will inject streams of particles from each face on the surface, with diameters de-fined by the Rosin-Rammler distribution function. The total number of injection streamstracked for the surface injection will be equal to the number of diameters in each distri-bution (Number of Diameters) multiplied by the number of faces on the surface.

Point Properties for Plain-Orifice Atomizer Injections

For a plain-orifice atomizer injection, you will define the following initial conditions underPoint Properties:

• position

Set the x, y, and z positions of the injected stream along the Cartesian axes of theproblem geometry in the X-Position, Y-Position, and Z-Position fields. (Z-Positionwill appear only for 3D problems).

• axis (3D only)

Set the x, y, and z components of the vector defining the axis of the orifice in theX-Axis, Y-Axis, and Z-Axis fields.

• temperature


• mass flow rate

Set the mass flow rate for the streams in the atomizer in the Flow Rate field. Notethat in 3D sectors, the flow rate must be appropriate for the sector defined by theAzimuthal Start Angle and Azimuthal Stop Angle.



• vapor pressure

Set the vapor pressure governing the flow through the internal orifice (pv in Ta-ble 22.8.1) in the Vapor Pressure field.

• diameter

Set the diameter of the orifice in the Injector Inner Diam. field (d in Table 22.8.1).

• orifice length

Set the length of the orifice in the Orifice Length field (L in Table 22.8.1).



• radius of curvature

Set the radius of curvature of the inlet corner in the Corner Radius of Curv. field (rin Table 22.8.1).

• nozzle parameter

Set the constant for the spray angle correlation in the Constant A field (CA inEquation 22.8-17).

• azimuthal angles

For 3D sectors, set the Azimuthal Start Angle and Azimuthal Stop Angle.

See Section 22.8.1: The Plain-Orifice Atomizer Model for details about how these inputsare used.

Point Properties for Pressure-Swirl Atomizer Injections

For a pressure-swirl atomizer injection, you will specify some of the same properties as fora plain-orifice atomizer. In addition to the position, axis (if 3D), temperature, mass flowrate, duration of injection (if unsteady), injector inner diameter, and azimuthal angles(if relevant) described in Section 22.12.1: Point Properties for Plain-Orifice AtomizerInjections, you will need to specify the following parameters under Point Properties:

• spray angle

Set the value of the spray angle of the injected stream in the Spray Half Angle field(θ in Equation 22.8-26).

• pressure

Set the absolute pressure upstream of the injection in the Upstream Pressure field(p1 in Table 22.8.1).

• sheet breakup

Set the value of the empirical constant that determines the length of the ligamentsthat are formed after sheet breakup in the Sheet Constant field (ln( ηb

η0) in Equa-

tion 22.8-31).

• ligament diameter

For short waves, set the proportionality constant that linearly relates the ligamentdiameter, dL, to the wavelength that breaks up the sheet in the Ligament Constantfield (see Equations 22.8-32–22.8-35).

See Section 22.8.2: The Pressure-Swirl Atomizer Model for details about how these inputsare used.



Point Properties for Air-Blast/Air-Assist Atomizer Injections

For an air-blast/air-assist atomizer, you will specify some of the same properties as for aplain-orifice atomizer. In addition to the position, axis (if 3D), temperature, mass flowrate, duration of injection (if unsteady), injector inner diameter, and azimuthal angles(if relevant) described in Section 22.12.1: Point Properties for Plain-Orifice AtomizerInjections, you will need to specify the following parameters under Point Properties:

• outer diameter

Set the outer diameter of the injector in the Injector Outer Diam. field. This valueis used in conjunction with the Injector Inner Diam. to set the thickness of the liquidsheet (t in Equation 22.8-23).

• spray angle

Set the initial trajectory of the film as it leaves the end of the orifice in the SprayHalf Angle field (θ in Equation 22.8-26).

• relative velocity

Set the maximum relative velocity that is produced by the sheet and air in theRelative Velocity field.

• sheet breakup

Set the value of the empirical constant that determines the length of the ligamentsthat are formed after sheet breakup in the Sheet Constant field (ln( ηb

η0) in Equa-

tion 22.8-31).

• ligament diameter

For short waves, set the proportionality constant (CL in Equation 22.8-34) thatlinearly relates the ligament diameter, dL, to the wavelength that breaks up thesheet in the Ligament Constant field.

See Section 22.8.3: The Air-Blast/Air-Assist Atomizer Model for details about how theseinputs are used.



Point Properties for Flat-Fan Atomizer Injections

The flat-fan atomizer model is available only for 3D models. For this type of injection,you will define the following initial conditions under Point Properties:

• arc position

Set the coordinates of the center point of the arc from which the fan originates inthe X-Center, Y-Center, and Z-Center fields (see Figure 22.8.6).

• virtual position

Set the coordinates of the virtual origin of the fan in the X-Virtual Origin, Y-VirtualOrigin, and Z-Virtual Origin fields. This point is the intersection of the lines thatmark the sides of the fan (see Figure 22.8.6).

• normal vector

Set the direction that is normal to the fan in the X-Fan Normal Vector, Y-Fan NormalVector, and Z-Fan Normal Vector fields.

• temperature


• mass flow rate

Set the mass flow rate for the streams in the atomizer in the Flow Rate field.



• spray half angle

Set the initial half angle of the drops as they leave the end of the orifice in theSpray Half Angle field.

• orifice width

Set the width of the orifice (in the normal direction) in the Orifice Width field.

• sheet breakup

Set the value of the empirical constant that determines the length of the ligamentsthat are formed after sheet breakup in the Flat Fan Sheet Constant field (see Equa-tion 22.8-31).

See Section 22.8.4: The Flat-Fan Atomizer Model for details about how these inputs areused.



Point Properties for Effervescent Atomizer Injections

For an effervescent atomizer injection, you will specify some of the same properties asfor a plain-orifice atomizer. In addition to the position, axis (if 3D), temperature, massflow rate (including both flashing and nonflashing components), duration of injection (ifunsteady), vapor pressure, injector inner diameter, and azimuthal angles (if relevant)described in Section 22.12.1: Point Properties for Plain-Orifice Atomizer Injections, youwill need to specify the following parameters under Point Properties:

• mixture quality

Set the mass fraction of the injected mixture that vaporizes in the Mixture Qualityfield (x in Equation 22.8-41).

• saturation temperature

Set the saturation temperature of the volatile substance in the Saturation Temp.field.

• droplet dispersion

Set the parameter that controls the spatial dispersion of the droplet sizes in theDispersion Constant field (Ceff in Equation 22.8-41).

• spray angle

Set the initial trajectory of the film as it leaves the end of the orifice in the MaximumHalf Angle field.

See Section 22.8.5: The Effervescent Atomizer Model for details about how these inputsare used.



22.12.2 Particle Types

When you define a set of initial conditions (as described in Section 22.12.4: DefiningInjection Properties), you will need to specify the type of particle. The particle typesavailable to you depend on the range of physical models that you have defined in theModels family of panels.

• An “inert” particle is a discrete phase element (particle, droplet, or bubble) thatobeys the force balance (Equation 22.2-1) and is subject to heating or cooling viaLaw 1 (Section 22.9.2: Inert Heating or Cooling (Law 1/Law 6)). The inert type isavailable for all FLUENT models.

• A “droplet” particle is a liquid droplet in a continuous-phase gas flow that obeysthe force balance (Equation 22.2-1) and that experiences heating/cooling via Law1 followed by vaporization and boiling via Laws 2 and 3 (Section 22.9.2: DropletVaporization (Law 2) and Section 22.9.2: Droplet Boiling (Law 3)). The droplettype is available when heat transfer is being modeled and at least two chemicalspecies are active or the nonpremixed or partially premixed combustion model isactive. You should use the ideal gas law to define the gas-phase density (in theMaterials panel, as discussed in Section 8.3.5: Density Inputs for the IncompressibleIdeal Gas Law) when you select the droplet type.

• A “combusting” particle is a solid particle that obeys the force balance (Equa-tion 22.2-1) and experiences heating/cooling via Law 1 followed by devolatilizationvia Law 4 (Section 22.9.2: Devolatilization (Law 4)), and a heterogeneous surfacereaction via Law 5 (Section 22.9.2: Surface Combustion (Law 5)). Finally, thenonvolatile portion of a combusting particle is subject to inert heating via Law6. You can also include an evaporating material with the combusting particle byselecting the Wet Combustion option in the Set Injection Properties panel. Thisallows you to include a material that evaporates and boils via Laws 2 and 3 (Sec-tion 22.9.2: Droplet Vaporization (Law 2) and Section 22.9.2: Droplet Boiling (Law3)) before devolatilization of the particle material begins. The combusting typeis available when heat transfer is being modeled and at least three chemical speciesare active or the nonpremixed combustion model is active. You should use the idealgas law to define the gas-phase density (in the Materials panel) when you select thecombusting particle type.

• A ”multicomponent” particle is, as the name implies, a mixture of droplet particles.These particles contain more than one component, which due to its complexity ofassigning a whole particle to one process, have to be modeled by a law that inte-grates all processes of relevance in one equation. Law 7, the multicomponent law(Section 22.9.2: Multicomponent Particle Definition (Law 7)) is used for such sys-tems. You should use the volume weighted mixing law to define the particle mixturedensity (in the Materials panel) when you select the particle-mixture material type.



22.12.3 Creating, Modifying, Copying, Deleting, and Listing Injections

You will use the Injections panel (Figure 22.12.7) to create, copy, delete, and list injections.

Define −→Injections...

Figure 22.12.7: The Injections Panel

(You can also click on the Injections... button in the Discrete Phase Model panel to openthe Injections panel.)

Creating Injections

To create an injection, click on the Create button. A new injection will appear in theInjections list and the Set Injection Properties panel will open automatically to allowyou to set the injection properties (as described in Section 22.12.4: Defining InjectionProperties).

Modifying Injections

To modify an existing injection, select its name in the Injections list and click on the Set...button. The Set Injection Properties panel will open, and you can modify the propertiesas needed.

If you have two or more injections for which you want to set some of the same properties,select their names in the Injections list and click on the Set... button. The Set MultipleInjection Properties panel will open, which will allow you to set the common properties.



For instructions about using this panel, see Section 22.12.7: Defining Properties Commonto More than One Injection.

Copying Injections

To copy an existing injection to a new injection, select the existing injection in theInjections list and click on the Copy button. The Set Injection Properties panel will openwith a new injection that has the same properties as the injection you selected. This isuseful if you want to set another injection with similar properties.

Deleting Injections

You can delete an injection by selecting its name in the Injections list and clicking on theDelete button.

Listing Injections

To list the initial conditions for the particle streams in the selected injection, click onthe List button. FLUENT reports the initial conditions (in SI units) in the console undervarious columns:

• The particle stream number is in the column headed NO.

• The particle type (IN for inert, DR for droplet, or CP for combusting particle) is inthe column headed TYP.

• The x, y, and z positions are in the columns headed (X), (Y), and (Z).

• The x, y, and z velocities are in the columns headed (U), (V), and (W).

• The temperature is in the column headed (T).

• The diameter is in the column headed (DIAM).

• The mass flow rate in the column headed (MFLOW).

Shortcuts for Selecting Injections

FLUENT provides a shortcut for selecting injections with names that match a specifiedpattern. To use this shortcut, enter the pattern under Injection Name Pattern and thenclick Match to select the injections with names that match the specified pattern. Forexample, if you specify drop*, all injections that have names beginning with drop (e.g.,drop-1, droplet) will be selected automatically. If they are all selected already, they willbe deselected. If you specify drop?, all surfaces with names consisting of drop followedby a single character will be selected (or deselected, if they are all selected already).



22.12.4 Defining Injection Properties

Once you have created an injection (using the Injections panel, as described in Sec-tion 22.12.3: Creating, Modifying, Copying, Deleting, and Listing Injections), you willuse the Set Injection Properties panel (Figure 22.12.8) to define the injection properties.(Remember that this panel will open when you create a new injection, or when you selectan existing injection and click on the Set... button in the Injections panel.)

Figure 22.12.8: The Set Injection Properties Panel



The procedure for defining an injection is as follows:

1. If you want to change the name of the injection from its default name, enter anew one in the Injection Name field. This is recommended if you are defining alarge number of injections so you can easily distinguish them. When assigningnames to your injections, keep in mind the selection shortcut described in Sec-tion 22.12.3: Creating, Modifying, Copying, Deleting, and Listing Injections.

2. Choose the type of injection in the Injection Type drop-down list. The eleven choices(single, group, cone, solid-cone, surface, plain-orifice-atomizer, pressure-swirl-atomizer,air-blast-atomizer, flat-fan-atomizer, effervescent-atomizer, and file) are described inSection 22.12.1: Injection Types. Note that if you select any of the atomizer models,you will also need to set the Viscosity and Droplet Surface Tension in the Materialspanel.

i If you are using sliding or moving/deforming meshes in your simulation,you should not use surface injections because they are not compatible withmoving meshes.

3. If you are defining a single injection, go to the next step. For a group, cone, solid-cone, or any of the atomizer injections, set the Number of Particle Streams in thegroup, spray cone, or atomizer.

If you are defining a surface injection (see Figure 22.12.9), choose the surface(s)from which the particles will be released in the Release From Surfaces list. If youare reading the injection from a file, click on the File... button at the bottom of theSet Injection Properties panel and specify the file to be read in the resulting SelectFile dialog box. The parameters in the injection file must be in SI units.



Figure 22.12.9: Setting Surface Injection Properties



4. Select Inert, Droplet, Combusting, or Multicomponent as the Particle Type. Theavailable types are described in Section 22.12.2: Particle Types.

5. Choose the material for the particle(s) in the Material drop-down list. If this isthe first time you have created a particle of this type, you can choose from all ofthe materials of this type defined in the database. If you have already created aparticle of this type, the only available material will be the material you selectedfor that particle. You can define additional materials by copying them from thedatabase or creating them from scratch, as discussed in Section 22.14.2: SettingDiscrete-Phase Physical Properties and described in detail in Section 8.1.2: Usingthe Materials Panel.

6. If you are defining a group or surface injection and you want to change from thedefault linear (for group injections) or uniform (for surface injections) interpola-tion method used to determine the size of the particles, select rosin-rammler orrosin-rammler-logarithmic in the Diameter Distribution drop-down list. The Rosin-Rammler method for determining the range of diameters for a group injectionis described in Section 22.12.1: Using the Rosin-Rammler Diameter DistributionMethod.

7. If you have created a customized particle law using user-defined functions, turnon the Custom option under Laws and specify the appropriate laws as described inSection 22.12.6: Custom Particle Laws.

8. If your particle type is Inert, go to the next step. If you are defining Droplet particles,select the gas phase species created by the vaporization and boiling laws (Laws 2and 3) in the Evaporating Species drop-down list.

If you are defining Combusting particles, select the gas phase species created bythe devolatilization law (Law 4) in the Devolatilizing Species drop-down list, thegas phase species that participates in the surface char combustion reaction (Law5) in the Oxidizing Species list, and the gas phase species created by the surfacechar combustion reaction (Law 5) in the Product Species list. Note that if theCombustion Model for the selected combusting particle material (in the Materialspanel) is the multiple-surface-reaction model, then the Oxidizing Species and ProductSpecies lists will be disabled because the reaction stoichiometry has been definedin the mixture material.

If you are defining Multicomponent particles, law 7 will go into effect. Notice thatthe Components tab will become active when this particle type is selected. Seebelow for information on the Components tab.



9. Click the Point Properties tab (the default), and specify the point properties (posi-tion, velocity, diameter, temperature, and—if appropriate—mass flow rate and anyatomizer-related parameters) as described for each injection type in Sections 22.12.1–22.12.1.

For surface injections, you can enable the Scale Flow Rate by Face Area and youcan choose the injection direction. To use the face normal direction for the injec-tion direction, select the Inject Using Normal Direction option under Point Properties(Figure 22.12.9). Once this option is selected, you only need to specify the ve-locity magnitude of the injection, not the individual components of the velocitymagnitude.

10. If the flow is turbulent and you wish to include the effects of turbulence on the parti-cle dispersion, click the Turbulent Dispersion tab, turn on the Stochastic Model or theCloud Model, and set the related parameters as described in Section 22.12.5: Mod-eling Turbulent Dispersion of Particles.

11. If your combusting particle includes an evaporating material, click the Wet Com-bustion tab, select the Wet Combustion option, and then select the material thatis evaporating/boiling from the particle before devolatilization begins in the Liq-uid Material drop-down list. You should also set the volume fraction of the liquidpresent in the particle by entering the value of the Liquid Fraction. Finally, selectthe gas phase species created by the evaporating and boiling laws in the EvaporatingSpecies drop-down list in the top part of the panel.

12. If you include multicomponent droplets as the material in your discrete phasemodel, a Components tab will become active. In this tab, you will specify theMass Fraction of each of the components. Note that the sum of the Mass fractionsshould add up to unity, otherwise FLUENT will adjust the values such that you havea sum of 1 for the mass fraction, and will prompt you to accept the entry. UnderVaporized Species, select not-vaporizing if the component in the particle does notvaporize. Otherwise, select from the Vaporized Species drop-down list the speciesthat will be vaporized.

To change the components of a multicomponent droplet, copy the droplet materialsfrom the Fluent Database Materials panel, or define the droplet materials in theMaterials panel, then add them to the Selected Species list in the Species panel byclicking the Edit... button (in the Materials panel) next to Mixture Species.

13. If you want to use a user-defined function to initialize the injection properties, clickthe UDF tab to access the UDF inputs. You can select an Initialization functionunder User-Defined Functions to modify injection properties at the time the particlesare injected into the domain. This allows the position and/or properties of theinjection to be set as a function of flow conditions. See the separate UDF Manualfor information about user-defined functions.



22.12.5 Modeling Turbulent Dispersion of Particles

As mentioned in Section 22.12.4: Defining Injection Properties, you can choose for eachinjection stochastic tracking or cloud tracking as the method for modeling turbulentdispersion of particles.

Stochastic Tracking

For turbulent flows, if you choose to use the stochastic tracking technique, you mustenable it and specify the “number of tries”. Stochastic tracking includes the effect of tur-bulent velocity fluctuations on the particle trajectories using the DRW model describedin Section 22.2.2: Stochastic Tracking.

1. Click the Turbulent Dispersion tab in the Set Injection Properties panel.

2. Enable stochastic tracking by turning on the Stochastic Model under StochasticTracking.

3. Specify the Number Of Tries:

• An input of zero tells FLUENT to compute the particle trajectory based on themean continuous phase velocity field (Equation 22.2-1), ignoring the effects ofturbulence on the particle trajectories.

• An input of 1 or greater tells FLUENT to include turbulent velocity fluctua-tions in the particle force balance as in Equation 22.2-20. The trajectory iscomputed more than once if your input exceeds 1: two trajectory calculationsare performed if you input 2, three trajectory calculations are performed ifyou input 3, etc. Each trajectory calculation includes a new stochastic repre-sentation of the turbulent contributions to the trajectory equation.

When a sufficient number of tries is requested, the trajectories computed willinclude a statistical representation of the spread of the particle stream due toturbulence. Note that for unsteady particle tracking, the Number of Tries isset to 1 if Stochastic Tracking is enabled.



If you want the characteristic lifetime of the eddy to be random (Equation 22.2-31),enable the Random Eddy Lifetime option. You will generally not need to change theTime Scale Constant (CL in Equation 22.2-22) from its default value of 0.15, unless youare using the Reynolds Stress turbulence model (RSM), in which case a value of 0.3 isrecommended.

Figure 22.12.10 illustrates a discrete phase trajectory calculation computed via the “mean”tracking (number of tries = 0) and Figure 22.12.11 illustrates the “stochastic” tracking(number of tries > 1) option.

When multiple stochastic trajectory calculations are performed, the momentum and massdefined for the injection are divided evenly among the multiple particle/droplet tracks,and are thus spread out in terms of the interphase momentum, heat, and mass transfercalculations. Including turbulent dispersion in your model can thus have a significantimpact on the effect of the particles on the continuous phase when coupled calculationsare performed.

Particle Traces Colored by Particle Time (s)

3.04e-02

2.84e-02

2.63e-02

2.43e-02

2.23e-02

2.03e-02

1.82e-02

1.62e-02

1.42e-02

1.22e-02

1.01e-02

8.10e-03

6.08e-03

4.05e-03

2.03e-03

0.00e+00

Figure 22.12.10: Mean Trajectory in a Turbulent Flow



Particle Traces Colored by Particle Time (s)

3.00e-02

2.80e-02

2.60e-02

2.40e-02

2.20e-02

2.00e-02

1.80e-02

1.60e-02

1.40e-02

1.20e-02

1.00e-02

8.00e-03

6.00e-03

4.00e-03

2.00e-03

0.00e+00

Figure 22.12.11: Stochastic Trajectories in a Turbulent Flow

Cloud Tracking

For turbulent flows, you can also include the effects of turbulent dispersion on the injec-tion. When cloud tracking is used, the trajectory will be tracked as a cloud of particlesabout a mean trajectory, as described in Section 22.2.2: Particle Cloud Tracking.

1. Click the Turbulent Dispersion tab in the Set Injection Properties panel.

2. Enable cloud tracking by turning on the Cloud Model under Cloud Tracking.

3. Specify the minimum and maximum cloud diameters. Particles enter the domainwith an initial cloud diameter equal to the Min. Cloud Diameter. The particlecloud’s maximum allowed diameter is specified by the Max. Cloud Diameter.

You may want to restrict the Max. Cloud Diameter to a relevant length scale for theproblem to improve computational efficiency in complex domains where the meantrajectory may become stuck in recirculation regions.



22.12.6 Custom Particle Laws

If the standard FLUENT laws, Laws 1 through 7, do not adequately describe the physicsof your discrete phase model, you can modify them by creating custom laws with user-defined functions. See the separate UDF Manual for information about user-definedfunctions. You can also create custom laws by using a subset of the existing FLUENTlaws (e.g., Laws 1, 2, and 4), or a combination of existing laws and user-defined functions.

Once you have defined and loaded your user-defined function(s), you can create a customlaw by enabling the Custom option under Laws in the Set Injection Properties panel. Thiswill open the Custom Laws panel. In the drop-down list to the left of each of the sixparticle laws, you can select the appropriate particle law for your custom law. Each listcontains the available options that can be chosen (the standard laws plus any user-definedfunctions you have loaded).

Figure 22.12.12: The Custom Laws Panel

There is a seventh drop-down list in the Custom Laws panel labeled Switching. You maywish to have FLUENT vary the laws used depending on conditions in the model. You cancustomize the way FLUENT switches between laws by selecting a user-defined functionfrom this drop-down list.

An example of when you might want to use a custom law might be to replace the stan-dard devolatilization law with a specialized devolatilization law that more accuratelydescribes some unique aspects of your model. After creating and loading a user-definedfunction that details the physics of your devolatilization law, you would visit the CustomLaws panel and replace the standard devolatilization law (Law 2) with your user-definedfunction.



22.12.7 Defining Properties Common to More than One Injection

If you have a number of injections for which you want to set the same properties, FLUENTprovides a shortcut so that you do not need to visit the Set Injection Properties panel foreach injection to make the same changes.

As described in Section 22.12.4: Defining Injection Properties, if you select more thanone injection in the Injections panel, clicking the Set... button will open the Set MultipleInjection Properties panel (Figure 22.12.13) instead of the Set Injection Properties panel.

Figure 22.12.13: The Set Multiple Injection Properties Panel

Depending on the type of injections you have selected (single, group, atomizers, etc.),there will be different categories of properties listed under Injections Setup. The namesof these categories correspond to the headings within the Set Injection Properties panel(e.g., Particle Type and Stochastic Tracking). Only those categories that are appropriatefor all of your selected injections (which are shown in the Injections list) will be listed. Ifall of these injections are of the same type, more categories of properties will be availablefor you to modify. If the injections are of different types, you will have fewer categoriesto select from.



Modifying Properties

To modify a property, perform the following steps:

1. Select the appropriate category in the Injections Setup list. For example, if you wantto set the same flow rate for all of the selected injections, select Point Properties.The panel will expand to show the properties that appear under that heading inthe Set Injection Properties panel.

2. Set the property (or properties) to be modified, as described below.

3. Click Apply. FLUENT will report the change in the console window.

i You must click Apply to save the property settings within each category. If,for example, you want to modify the flow rate and the stochastic trackingparameters, you will need to select Point Properties in the Injections Setuplist, specify the flow rate, and click Apply. You would then repeat theprocess for the stochastic tracking parameters, clicking Apply again whenyou are done.

There are two types of properties that can be modified using the Set Multiple InjectionProperties panel.

The first type involves one of the following actions:

• selecting a value from a drop-down list

• choosing an option using a radio button

The second type involves one of the following actions:

• entering a value in a field

• turning an option on or off

Setting the first type of property works the same way as in the Set Injection Propertiespanel. For example, if you select Particle Type in the Injections Setup list, the panel willexpand to show the portion of the Set Injection Properties panel where you choose theparticle type. You can simply choose the desired type and click Apply.

Setting the second type of property requires an additional step. If you select a categoryin the Injections Setup list that contains this type of property, the expanded portion of thepanel will look like the corresponding part of the Set Injection Properties panel, with theaddition of Modify check buttons (see Figure 22.12.13). To change one of the properties,



first turn on the Modify check button to its left, and then specify the desired status orvalue.

For example, if you would like to enable stochastic tracking, first turn on the Modifycheck button to the left of Stochastic Model. This will make the property active so youcan modify its status. Then, under Property, turn on the Stochastic Model check button.(Be sure to click Apply when you are done setting stochastic tracking parameters.)

If you would like to change the value of Number of Tries, select the Modify check buttonto its left to make it active, and then enter the new value in the field. Make sure youclick Apply when you have finished modifying the stochastic tracking properties.

i The setting for a property that has not been activated with the Modifycheck button is not relevant, because it will not be applied to the selectedinjections when you click Apply. After you turn on Modify for a particularproperty, clicking Apply will modify that property for all of the selectedinjections, so make sure that you have the settings the way that you wantthem before you do this. If you make a mistake, you will have to returnto the Set Injection Properties panel for each injection to fix the incorrectsetting, if it is not possible to do so in the Set Multiple Injection Propertiespanel.

Modifying Properties Common to a Subset of Selected Injections

Note that it is possible to change a property that is relevant for only a subset of theselected injections. For example, if some of the selected injections are using stochastictracking and some are not, enabling the Random Eddy Lifetime option and clicking Applywill turn this option on only for those injections that are using stochastic tracking. Theother injections will be unaffected.

22.13 Setting Boundary Conditions for the Discrete Phase

When a particle reaches a physical boundary (e.g., a wall or inlet boundary) in yourmodel, FLUENT applies a discrete phase boundary condition to determine the fate of thetrajectory at that boundary. The boundary condition, or trajectory fate, can be definedseparately for each zone in your FLUENT model.



22.13.1 Discrete Phase Boundary Condition Types

The available boundary conditions, as noted in Section 22.10: Discrete Phase Model(DPM) Boundary Conditions, include the following:

• “reflect”

The particle rebounds the off the boundary in question with a change in its mo-mentum as defined by the coefficient of restitution. (See Figure 22.13.1.)

coefficientofrestitution

=V2,nV1,n

θ1 θ2

Figure 22.13.1: “Reflect” Boundary Condition for the Discrete Phase

The normal coefficient of restitution defines the amount of momentum in the di-rection normal to the wall that is retained by the particle after the collision withthe boundary [365]:

en =v2,n

v1,n

(22.13-1)

where vn is the particle velocity normal to the wall and the subscripts 1 and 2 referto before and after collision, respectively. Similarly, the tangential coefficient ofrestitution, et, defines the amount of momentum in the direction tangential to thewall that is retained by the particle.

A normal or tangential coefficient of restitution equal to 1.0 implies that the particleretains all of its normal or tangential momentum after the rebound (an elasticcollision). A normal or tangential coefficient of restitution equal to 0.0 implies thatthe particle retains none of its normal or tangential momentum after the rebound.

Nonconstant coefficients of restitution can be specified for wall zones with the“reflect” type boundary condition. The coefficients are set as a function of theimpact angle, θ1, in Figure 22.13.1.

Note that the default setting for both coefficients of restitution is a constant valueof 1.0 (all normal and tangential momentum retained).



• “trap”

The trajectory calculations are terminated and the fate of the particle is recorded as“trapped”. In the case of evaporating droplets, their entire mass instantaneouslypasses into the vapor phase and enters the cell adjacent to the boundary. SeeFigure 22.13.2. In the case of combusting particles, the remaining volatile mass ispassed into the vapor phase.

θ1

volatile fractionflashes to vapor

Figure 22.13.2: “Trap” Boundary Condition for the Discrete Phase

• “escape”

The particle is reported as having “escaped” when it encounters the boundary inquestion. Trajectory calculations are terminated. See Figure 22.13.3.

particle vanishes

Figure 22.13.3: “Escape” Boundary Condition for the Discrete Phase



• “wall-jet”

The direction and velocity of the droplet particles are given by the resulting mo-mentum flux, which is a function of the impingement angle, φ, and Weber number.See Figure 22.13.4.

x

z

φH(Ψ)

y

x

Ψ

side view top view

Figure 22.13.4: “Wall Jet” Boundary Condition for the Discrete Phase

The “wall-jet” type boundary condition assumes an analogy with an inviscid jetimpacting a solid wall. Equation 22.13-2 shows the analytical solution for an ax-isymmetric impingement assuming an empirical function for the sheet height (H)as a function of the angle that the drop leaves the impingement (Ψ).

H(Ψ) = Hπeβ(1−Ψ

π) (22.13-2)

where Hπ is the sheet height at Ψ = π and β is a constant determined from conser-vation of mass and momentum. The probability that a drop leaves the impingementpoint at an angle between Ψ and Ψ + δΨ is given by integrating the expression forH(Ψ)

Ψ = −πβ

ln[1− P (1− e−β)] (22.13-3)

where P is a random number between 0 and 1. The expression for β is given inNaber and Reitz [257] as

sin(φ) =eβ + 1

(eβ − 1)(1 + (πβ)2)

(22.13-4)

The “wall-jet” type boundary condition is appropriate for high-temperature wallswhere no significant liquid film is formed, and in high-Weber-number impacts wherethe spray acts as a jet. The model is not appropriate for regimes where film isimportant (e.g., port fuel injection in SI engines, rainwater runoff, etc.).



• “wall-film”

This boundary condition consists of four regimes: stick, rebound, spread, andsplash, which are based on the impact energy and wall temperature. Detailedinformation on the wall-film model can be found in Section 22.4: Wall-Film ModelTheory.

• “interior”

This boundary condition means that the particles will pass through the internalboundary. This option is available only for internal boundary zones, such as aradiator or a porous jump.

It is also possible to use a user-defined function to compute the behavior of the particlesat a physical boundary. See the separate UDF Manual for information about user-definedfunctions.

Because you can stipulate any of these conditions at flow boundaries, it is possible toincorporate mixed discrete phase boundary conditions in your FLUENT model.

Discrete phase boundary conditions can be set for boundaries in the panels opened fromthe Boundary Conditions panel. When one or more injections have been defined, inputsfor the discrete phase will appear in the panels (e.g., Figure 22.13.5).

Select reflect, trap, escape, wall-jet, or user-defined in the Boundary Cond. Type drop-downlist under Discrete Phase Model Conditions. (In the Walls panel, you will need to click onthe DPM tab to access the Discrete Phase Model Conditions.) If you select user-defined,you can select a user-defined function in the Boundary Cond. Function drop-down list.For internal boundary zones, such as a radiator or a porous jump, you can also choosean interior boundary condition. The interior condition means that the particles will passthrough the internal boundary.

If you select the reflect type at a wall (only), you can define a constant, polynomial,piecewise-linear, or piecewise-polynomial function for the Normal and Tangent coefficientsof restitution under Discrete Phase Reflection Coefficients. See Section 22.13.1: DiscretePhase Boundary Condition Types for details about the boundary condition types andthe coefficients of restitution. The panels for defining the polynomial, piecewise-linear,and piecewise-polynomial functions are the same as those used for defining temperature-dependent properties. See Section 8.2: Defining Properties Using Temperature-DependentFunctions for details.



Figure 22.13.5: Discrete Phase Boundary Conditions in the Wall Panel



Default Discrete Phase Boundary Conditions

FLUENT makes the following assumptions regarding boundary conditions:

• The reflect type is assumed at wall, symmetry, and axis boundaries, with bothcoefficients of restitution equal to 1.0

• The escape type is assumed at all flow boundaries (pressure and velocity inlets,pressure outlets, etc.)

• The interior type is assumed at all internal boundaries (radiator, porous jump, etc.)

The coefficient of restitution can be modified only for wall boundaries.

22.13.2 Setting Particle Erosion and Accretion Parameters

If the Erosion/Accretion option is selected in the Discrete Phase Model panel, the erosionrate expression must be specified at the walls. The erosion rate is defined in Equa-tion 22.5-1 as a product of the mass flux and specified functions for the particle diam-eter, impact angle, and velocity exponent. Under Erosion Model in the Wall panel, youcan define a constant, polynomial, piecewise-linear, or piecewise-polynomial function for theImpact Angle Function, Diameter Function, and Velocity Exponent Function (f(α), C(dp),and b(v) in Equation 22.5-1). See Sections 22.5 and 22.11.5 for a detailed descriptionof these functions and Section 8.2: Defining Properties Using Temperature-DependentFunctions for details about using the panels for defining polynomial, piecewise-linear,and piecewise-polynomial functions.

22.14 Setting Material Properties for the Discrete Phase

In order to apply the physical models described in earlier sections to the prediction ofthe discrete phase trajectories and heat/mass transfer, FLUENT requires many physicalproperty inputs.

22.14.1 Summary of Property Inputs

Tables 22.14.1–22.14.5 summarize which of these property inputs are used for each par-ticle type and in which of the equations for heat and mass transfer each property inputis used. Detailed descriptions of each input are provided in Section 22.14.2: SettingDiscrete-Phase Physical Properties.



Table 22.14.1: Property Inputs for Inert Particles

Property Symboldensity ρp in Eq. 22.2-1specific heat cp in Eq. 22.9-11particle emissivity εp in Eq. 22.9-11particle scattering factor f in Eq. 13.3-13thermophoretic coefficient DT,p in Eq. 22.2-14

Table 22.14.2: Property Inputs for Droplet Particles

Properties Symboldensity ρp in Eq. 22.2-1specific heat cp in Eq. 22.9-25thermal conductivity kp in Eq. 22.2-15viscosity µ in Eq. 22.7-11latent heat hfg in Eq. 22.9-25vaporization temperature Tvap in Eq. 22.9-18boiling point Tbp in Eq. 22.9-18, 22.9-26volatile component fraction fv0 in Eq. 22.9-19, 22.9-27binary diffusivity Di,m in Eq. 22.9-23saturation vapor pressure psat(T ) in Eq. 22.9-21heat of pyrolysis hpyrol in Eq. 22.9-2droplet surface tension σ in Eq. 22.8-19, 22.7-10particle emissivity εp in Eq. 22.9-25, 22.9-31particle scattering factor f in Eq. 13.3-13thermophoretic coefficient DT,p in Eq. 22.2-14



Table 22.14.3: Property Inputs for Combusting Particles (Laws 1–4)

Properties Symboldensity ρp in Eq. 22.2-1specific heat cp in Eq. 22.9-11latent heat hfg in Eq. 22.9-2vaporization temperature Tvap = Tbp in Eq. 22.9-32volatile component fraction fv0 in Eq. 22.9-33swelling coefficient Csw in Eq. 22.9-65burnout stoichiometric ratio Sb in Eq. 22.9-72combustible fraction fcomb in Eq. 22.9-71heat of reaction for burnout Hreac in Eq. 22.9-72 22.9-86fraction of reaction heat given to solid fh in Eq. 22.9-86particle emissivity εp in Eq. 22.9-66, 22.9-86particle scattering factor f in Eq. 13.3-13thermophoretic coefficient DT,p in Eq. 22.2-14devolatilization model– law 4, constant rate– – constant A0 in Eq. 22.9-34– law 4, single rate– – pre-exponential factor A1 in Eq. 22.9-35– – activation energy E in Eq. 22.9-35– law 4, two rates– – pre-exponential factors A1, A2 in Eq. 22.9-38, 22.9-39– – activation energies E1, E2 in Eq. 22.9-38, 22.9-39– – weighting factors α1, α2 in Eq. 22.9-40– law 4, CPD– – initial fraction of bridges in coal lattice p0 in Eq. 22.9-51– – initial fraction of char bridges c0 in Eq. 22.9-50– – lattice coordination number σ + 1 in Eq. 22.9-62– – cluster molecular weight Mw,1 in Eq. 22.9-62– – side chain molecular weight Mw,δ in Eq. 22.9-61



Table 22.14.4: Property Inputs for Combusting Particles (Law 5)

Properties Symbolcombustion model– law 5, diffusion rate– – binary diffusivity Di,m in Eq. 22.9-73– law 5, diffusion/kinetic rate– – mass diffusion limited rate constant C1 in Eq. 22.9-74– – kinetics limited rate pre-exp. factor C2 in Eq. 22.9-75– – kinetics limited rate activ. energy E in Eq. 22.9-75– law 5, intrinsic rate– – mass diffusion limited rate constant C1 in Eq. 22.9-74– – kinetics limited rate pre-exp. factor Ai in Eq. 22.9-84– – kinetics limited rate activ. energy Ei in Eq. 22.9-84– – char porosity θ in Eq. 22.9-81– – mean pore radius rp in Eq. 22.9-83– – specific internal surface area Ag in Eq. 22.9-78, 22.9-80– – tortuosity τ in Eq. 22.9-81– – burning mode α in Eq. 22.9-85– law 5, multiple surface reaction– – binary diffusivity Di,m in Eq. 22.9-73

Table 22.14.5: Property Inputs for Multicomponent Particles (Law 7)

Property Symbolmixture species selected droplets for componentsdensity ρp in Eq. 22.2-1specific heat cp in Eq. 22.9-89thermal conductivity kp in Eq. 22.2-15vapor particle equilibrium Ci,s in Eq. 22.3-4



22.14.2 Setting Discrete-Phase Physical Properties

The Concept of Discrete-Phase Materials

When you create a particle injection and define the initial conditions for the discretephase (as described in Section 22.12: Setting Initial Conditions for the Discrete Phase),you choose a particular material as the particle’s material. All particle streams of thatmaterial will have the same physical properties.

Discrete-phase materials are divided into four categories, corresponding to the four typesof particles available. These material types are inert-particle, droplet-particle, combusting-particle, and multicomponent-particle. Each material type will be added to the MaterialType list in the Materials panel when an injection of that type of particle is defined(in the Set Injection Properties or Set Multiple Injection Properties panel, as described inSection 22.12: Setting Initial Conditions for the Discrete Phase). The first time youcreate an injection of each particle type, you will be able to choose a material from thedatabase, and this will become the default material for that type of particle. That is,if you create another injection of the same type of particle, your selected material willbe used for that injection as well. You may choose to modify the predefined propertiesfor your selected particle material, if you want (as described in Section 8.1.2: ModifyingProperties of an Existing Material). If you need only one set of properties for each typeof particle, you need not define any new materials; you can simply use the same materialfor all particles.

i If you do not find the material you want in the database, you can se-lect a material that is close to the one you wish to use, and then modifythe properties and give the material a new name, as described in Sec-tion 8.1.2: Creating a New Material.

i Note that a discrete-phase material type will not appear in the MaterialType list in the Materials panel until you have defined an injection of thattype of particles. This means, for example, that you cannot define or mod-ify any combusting-particle materials until you have defined a combustingparticle injection (as described in Section 22.12: Setting Initial Conditionsfor the Discrete Phase).

For a particle-mixture material type, you will need to select the species in your mixture.To do this, click the Edit... button next to Mixture Species in the Materials panel. TheSpecies panel will open, where you will include your Selected Species. The selected specieswill now be available in the Set Injection Properties panel, under the Components tab.



Defining Additional Discrete-Phase Materials

In many cases, a single set of physical properties (density, heat capacity, etc.) is appro-priate for each type of discrete phase particle considered in a given model. Sometimes,however, a single model may contain two different types of inert, droplet, combustingparticles, or multicomponent particles (e.g., heavy particles and gaseous bubbles or twodifferent types of evaporating liquid droplets). In such cases, it is necessary to assign adifferent set of properties to the two (or more) different types of particles. This is easilyaccomplished by defining two or more inert, droplet, or combusting particle materialsand using the appropriate one for each particle injection.

You can define additional discrete-phase materials either by copying them from thedatabase or by creating them from scratch. See Section 8.1.2: Using the Materials Panelfor instructions on using the Materials panel to perform these actions.

i Recall that you must define at least one injection (as described in Sec-tion 22.12: Setting Initial Conditions for the Discrete Phase) containingparticles of a certain type before you will be able to define additional ma-terials for that particle type.

Description of the Properties

The properties that appear in the Materials panel vary depending on the particle type (se-lected in the Set Injection Properties or Set Multiple Injection Properties panel, as describedin Sections 22.12.4 and 22.12.7) and the physical models you are using in conjunctionwith the discrete-phase model.

Below, all properties you may need to define for a discrete-phase material are listed. SeeTables 22.14.1–22.14.4 to see which properties are defined for each type of particle.

Density is the density of the particulate phase in units of mass per unit volume of thediscrete phase. This density is the mass density and not the volumetric density.Since certain particles may swell during the trajectory calculations, your input isactually an “initial” density.

Cp is the specific heat, cp, of the particle. The specific heat may be defined as a func-tion of temperature by selecting one of the function types from the drop-down list tothe right of Cp. See Section 8.2: Defining Properties Using Temperature-DependentFunctions for details about temperature-dependent properties. For multicomponentparticles, it can be calculated as a mass-weighted value of the specific heat of thedroplet component.

Thermal Conductivity is the thermal conductivity of the particle. This input is specifiedin units of W/m-K in SI units or Btu/ft-h-F in British units and is treated as aconstant by FLUENT.



Latent Heat is the latent heat of vaporization, hfg, required for phase change from anevaporating liquid droplet (Equation 22.9-25) or for the evolution of volatiles froma combusting particle (Equation 22.9-66). This input is supplied in units of J/kg inSI units or of Btu/lbm in British units and is treated as a constant by FLUENT. Forthe droplet particle, the latent heat value at the boiling point temperature shouldbe used.

Thermophoretic Coefficient is the coefficient DT,p in Equation 22.2-14, and appearswhen the thermophoretic force (which is described in Section 22.2.1: ThermophoreticForce) is included in the trajectory calculation (i.e., when the Thermophoretic Forceoption is enabled in the Discrete Phase Model panel). The default is the expressiondeveloped by Talbot [366] (talbot-diffusion-coeff) and requires no input from you.You can also define the thermophoretic coefficient as a function of temperature byselecting one of the function types from the drop-down list to the right of Ther-mophoretic Coefficient. See Section 8.2: Defining Properties Using Temperature-Dependent Functions for details about temperature-dependent properties.

Vaporization Temperature is the temperature, Tvap, at which the calculation of va-porization from a liquid droplet or devolatilization from a combusting particle isinitiated by FLUENT. Until the particle temperature reaches Tvap, the particle isheated via Law 1, Equation 22.9-11. This temperature input represents a modelingdecision rather than any physical characteristic of the discrete phase.

Boiling Point is the temperature, Tbp, at which the calculation of the boiling rateequation (22.9-28) is initiated by FLUENT. When a droplet particle reaches theboiling point, FLUENT applies Law 3 and assumes that the droplet temperature isconstant at Tbp. The boiling point denotes the temperature at which the particlelaw transitions from the vaporization law to the boiling law.

Volatile Component Fraction (fv0) is the fraction of a droplet particle that may va-porize via Laws 2 and/or 3 (Section 22.9.2: Droplet Vaporization (Law 2)). Forcombusting particles, it is the fraction of volatiles that may be evolved via Law 4(Section 22.9.2: Devolatilization (Law 4)).

Binary Diffusivity is the mass diffusion coefficient, Di,m, used in the vaporization law,Law 2 (Equation 22.9-23). This input is also used to define the mass diffusionof the oxidizing species to the surface of a combusting particle, Di,m, as given inEquation 22.9-73. (Note that the diffusion coefficient inputs that you supply forthe continuous phase are not used for the discrete phase.)

Saturation Vapor Pressure is the saturated vapor pressure, psat, defined as a functionof temperature, which is used in the vaporization law, Law 2 (Equation 22.9-21).The saturated vapor pressure may be defined as a function of temperature byselecting one of the function types from the drop-down list to the right of its name.(See Section 8.2: Defining Properties Using Temperature-Dependent Functions for



details about temperature-dependent properties.) In the case of unrealistic inputs,FLUENT restricts the range of Psat to between 0.0 and the operating pressure.Correct input of a realistic vapor pressure curve is essential for accurate resultsfrom the vaporization model.

Heat of Pyrolysis is the heat of the instantaneous pyrolysis reaction, hpyrol, that theevaporating/boiling species may undergo when released to the continuous phase.This input represents the conversion of the evaporating species to lighter compo-nents during the evaporation process. The heat of pyrolysis should be input as apositive number for exothermic reaction and as a negative number for endother-mic reaction. The default value of zero implies that the heat of pyrolysis is notconsidered. This input is used in Equation 22.9-2.

Swelling Coefficient is the coefficient Csw in Equation 22.9-65, which governs the swellingof the coal particle during the devolatilization law, Law 4 (Section 22.9.2: De-volatilization (Law 4)). A swelling coefficient of unity (the default) implies thatthe coal particle stays at constant diameter during the devolatilization process.

Burnout Stoichiometric Ratio is the stoichiometric requirement, Sb, for the burnoutreaction, Equation 22.9-72, in terms of mass of oxidant per mass of char in theparticle.

Combustible Fraction is the mass fraction of char, fcomb, in the coal particle, i.e., thefraction of the initial combusting particle that will react in the surface reaction,Law 5 (Equation 22.9-71).

Heat of Reaction for Burnout is the heat released by the surface char combustion re-action, Law 5 (Equation 22.9-72). This parameter is input in terms of heat release(e.g., Joules) per unit mass of char consumed in the surface reaction.

React. Heat Fraction Absorbed by Solid is the parameter fh (Equation 22.9-86), whichcontrols the distribution of the heat of reaction between the particle and the con-tinuous phase. The default value of zero implies that the entire heat of reaction isreleased to the continuous phase.

Devolatilization Model defines which version of the devolatilization model, Law 4, isbeing used. If you want to use the default constant rate devolatilization model,Equation 22.9-34, retain the selection of constant in the drop-down list to the rightof Devolatilization Model and input the rate constant A0 in the field below the list.

You can activate one of the optional devolatilization models (the single kinetic rate,two kinetic rates, or CPD model, as described in Section 22.9.2: Devolatilization(Law 4)) by choosing single rate, two-competing-rates, or cpd-model in the drop-downlist.



When the single kinetic rate model (single-rate) is selected, the Single Rate De-volatilization Model panel will appear and you will enter the Pre-exponential Factor,A1, and the Activation Energy, E, to be used in Equation 22.9-36 for the computa-tion of the kinetic rate.

When the two competing rates model (two-competing-rates) is selected, the TwoCompeting Rates Model panel will appear and you will enter, for the First Rateand the Second Rate, the Pre-exponential Factor (A1 in Equation 22.9-38 and A2

in Equation 22.9-39), Activation Energy (E1 in Equation 22.9-38 and E2 in Equa-tion 22.9-39), and Weighting Factor (α1 and α2 in Equation 22.9-40). The constantsyou input are used in Equations 22.9-38 through 22.9-40.

When the CPD model (cpd-model) is selected, the CPD Model panel will appear andyou will enter the Initial Fraction of Bridges in Coal Lattice (p0 in Equation 22.9-51),Initial Fraction of Char Bridges (c0 in Equation 22.9-50), Lattice Coordination Number(σ + 1 in Equation 22.9-62), Cluster Molecular Weight (Mw,1 in Equation 22.9-62),and Side Chain Molecular Weight (Mw,δ in Equation 22.9-61).

Note that the Single Rate Devolatilization Model, Two Competing Rates Model, andCPD Model panels are modal panels, which means that you must tend to themimmediately before continuing the property definitions.

Combustion Model defines which version of the surface char combustion law (Law 5)is being used. If you want to use the default diffusion-limited rate model, retainthe selection of diffusion-limited in the drop-down list to the right of CombustionModel. No additional inputs are necessary, because the binary diffusivity definedabove will be used in Equation 22.9-73.

To use the kinetics/diffusion-limited rate model for the surface combustion model,select kinetics/diffusion-limited in the drop-down list. The Kinetics/Diffusion-LimitedCombustion Model panel will appear and you will enter the Mass Diffusion LimitedRate Constant (C1 in Equation 22.9-74), Kinetics Limited Rate Pre-exponential Fac-tor (C2 in Equation 22.9-75), and Kinetics Limited Rate Activation Energy (E inEquation 22.9-75).

Note that the Kinetics/Diffusion-Limited Combustion Model panel is a modal panel,which means that you must tend to it immediately before continuing the propertydefinitions.

To use the intrinsic model for the surface combustion model, select intrinsic-model inthe drop-down list. The Intrinsic Combustion Model panel will appear and you willenter the Mass Diffusion Limited Rate Constant (C1 in Equation 22.9-74), KineticsLimited Rate Pre-exponential Factor (Ai in Equation 22.9-84), Kinetics Limited RateActivation Energy (Ei in Equation 22.9-84), Char Porosity (θ in Equation 22.9-81),Mean Pore Radius (rp in Equation 22.9-83), Specific Internal Surface Area (Ag inEquations 22.9-78 and 22.9-80), Tortuosity (τ in Equation 22.9-81), and BurningMode, alpha (α in Equation 22.9-85).



Note that the Intrinsic Combustion Model panel is a model panel, which means thatyou must tend to it immediately before continuing the property definitions.

To use the multiple surface reactions model, select multiple-surface-reactions in thedrop-down list. FLUENT will display a dialog box informing you that you will needto open the Reactions panel, where you can review or modify the particle surfacereactions that you specified as described in Section 14.1.2: Overview of User Inputsfor Modeling Species Transport and Reactions.

i If you have not yet defined any particle surface reactions, you must be sureto define them now. See Section 14.3.3: Using the Multiple Surface Reac-tions Model for Discrete-Phase Particle Combustion for more informationabout using the multiple surface reactions model.

You will notice that the Burnout Stoichiometric Ratio and Heat of Reaction forBurnout are no longer available in the Materials panel, as these parameters are nowcomputed from the particle surface reactions you defined in the Reactions panel.

Note that the multiple surface reactions model is available only if the Particle Surfaceoption for Reactions is enabled in the Species Model panel. See Section 14.3.2: UserInputs for Particle Surface Reactions for details.

When the effect of particles on radiation is enabled (for the P-1 or discrete ordinatesradiation model only) in the Discrete Phase Model panel, you will need to define thefollowing additional parameters:

Particle Emissivity is the emissivity of particles in your model, εp, used to computeradiation heat transfer to the particles (Equations 22.9-11, 22.9-25, 22.9-31, 22.9-66,and 22.9-86) when the P-1 or discrete ordinates radiation model is active. Note thatyou must enable radiation to particles, using the Particle Radiation Interaction optionin the Discrete Phase Model panel. Recommended values of particle emissivity are1.0 for coal particles and 0.5 for ash [219].

Particle Scattering Factor is the scattering factor, fp, due to particles in the P-1 ordiscrete ordinates radiation model (Equation 13.3-13). Note that you must enableparticle effects in the radiation model, using the Particle Radiation Interaction optionin the Discrete Phase Model panel. The recommended value of fp for coal combustionmodeling is 0.9 [219]. Note that if the effect of particles on radiation is enabled,scattering in the continuous phase will be ignored in the radiation model.



When an atomizer injection model and/or the droplet breakup or collision model isenabled in the Set Injection Properties panel (atomizers) and/or Discrete Phase Model panel(droplet breakup/collision), you will need to define the following additional parameters:

Viscosity is the droplet viscosity, µl. The viscosity may be defined as a function of tem-perature by selecting one of the function types from the drop-down list to the rightof Viscosity. See Section 8.2: Defining Properties Using Temperature-DependentFunctions for details about temperature-dependent properties. You also have theoption of implementing a user-defined function to model the droplet viscosity. Seethe separate UDF Manual for information about user-defined functions.

Droplet Surface Tension is the droplet surface tension, σ. The surface tension may bedefined as a function of temperature by selecting one of the function types fromthe drop-down list to the right of Droplet Surface Tension. See Section 8.2: DefiningProperties Using Temperature-Dependent Functions for details about temperature-dependent properties. You also have the option of implementing a user-definedfunction to model the droplet surface tension. See the separate UDF Manual forinformation about user-defined functions.

Vapor-Particle-Equilibrium is the selected approach for the calculation of the vaporconcentration of the components at the surface. This can be Raoult’s law (Equa-tion 22.3-4), or a user-defined function that provides this value.

22.15 Solution Strategies for the Discrete Phase

Solution of the discrete phase implies integration in time of the force balance on theparticle (Equation 22.2-1) to yield the particle trajectory. As the particle is movedalong its trajectory, heat and mass transfer between the particle and the continuousphase are also computed via the heat/mass transfer laws (Section 22.9: One-Way andTwo-Way Coupling). The accuracy of the discrete phase calculation thus depends onthe time accuracy of the integration and upon the appropriate coupling between thediscrete and continuous phases when required. Numerical controls are described in Sec-tion 22.11.7: Numerics of the Discrete Phase Model. Coupling and performing trajectorycalculations are described in Section 22.15.2: Performing Trajectory Calculations. Sec-tions 22.15.3 and 22.11.9 provide information about resetting interphase exchange termsand using the parallel solver for a discrete phase calculation.



22.15.1 Integration of Particle Equation of Motion

The trajectory equations, and any auxiliary equations describing heat or mass transferto/from the particle, are solved by stepwise integration over discrete time steps. Integra-tion of time in Equation 22.2-1 yields the velocity of the particle at each point along thetrajectory, with the trajectory itself predicted by

dx

dt= up (22.15-1)

Equations 22.2-1 and 22.15-1 are a set of coupled ordinary differential equations. Equa-tion 22.2-1 can be cast into the following general form

dupdt

=1

τp(u− up) + a (22.15-2)

where the term a includes accelerations due to all other forces except drag force.

This set can be solved for constant u, a and τp by analytical integration. For the particlevelocity at the new location un+1

p we get

un+1p = un + e

−∆tτp

(unp − un

)− aτp

(e−∆tτp − 1

)(22.15-3)

The new location xn+1p can be computed from a similar relationship.

xn+1p = xnp + ∆t(un + aτp) + τp

(1− e−

∆tτp

) (unp − un − aτp

)(22.15-4)

In these equations unp and un represent particle velocities and fluid velocities at the oldlocation. Equations 22.15-3 and 22.15-4 are applied when using the analytic discretizationscheme.

The set of Equations 22.2-1 and 22.15-1 can also be solved using numerical discretizationschemes. When applying the Euler implicit discretization to Equation 22.15-2 we get

un+1p =

unp + ∆t(a+ un

τp)

1 + ∆tτp

(22.15-5)

When applying a trapezoidal discretization to Equation 22.15-2 the variables up and unon the right hand side are taken as averages, while accelerations, a, due to other forcesare held constant. We get

un+1p − unp

∆t=

1

τp(u∗ − u∗p) + an (22.15-6)



The averages u∗p and u∗ are computed from

u∗p =1

2(unp + un+1

p ) (22.15-7)

u∗ =1

2(un + un+1) (22.15-8)

un+1 = un + ∆tunp · ∇un (22.15-9)

The particle velocity at the new location n+ 1 is computed by

un+1p =

unp (1− 12

∆tτp

) + ∆tτp

(un + 1

2∆tunp · ∇un

)+ ∆ta

1 + 12

∆tτp

(22.15-10)

For the implicit and the trapezoidal schemes the new particle location is always computedby a trapezoidal discretization of Equation 22.15-1.

xn+1p = xnp +

1

2∆t

(unp + un+1

p

)(22.15-11)

Equations 22.15-2 and 22.15-1 can also be computed using a Runge-Kutta scheme whichwas published by Cash and Karp [49]. The ordinary differential equations can be con-sidered as vectors, where the left hand side is the derivative ~y′ and the right hand side isan arbitrary function ~f(t, ~y).

~y′ = ~f(t, ~y) (22.15-12)

We get

~yn+1 = ~yn + c1~k1 + c2

~k2 + c3~k3 + c4

~k4 + c5~k5 + c6

~k6 (22.15-13)

with

~k1 = ∆t ~f(t, ~yn)

~k2 = ∆t ~f(t+ a2∆t, ~yn + b21~k1)

~k3 = ∆t ~f(t+ a3∆t, ~yn + b31~k1 + b32

~k2)

~k4 = ∆t ~f(t+ a4∆t, ~yn + b41~k1 + b42

~k2 + b43~k3)

~k5 = ∆t ~f(t+ a5∆t, ~yn + b51~k1 + b52

~k2 + b53~k3 + b54

~k4)

~k6 = ∆t ~f(t+ a6∆t, ~yn + b61~k1 + b62

~k2 + b63~k3 + b64

~k4 + b65~k5)



The coefficients a2 . . . a6, b21 . . . b65, and c1 . . . c6 are taken from Cash and Karp [49]

This scheme provides an embedded error control, which is switched off, when no AccuracyControl is enabled.

For rotating reference frames, the integration is carried out in the rotating frame withthe extra terms described in Equations 22.2-12 and 22.2-13, thus accounting for systemrotation. Using the mechanisms available for accuracy control, the trajectory integrationwill be done accurately in time.

The analytic scheme is very efficient. It can become inaccurate for large steps and in situ-ations where the particles are not in hydrodynamic equilibrium with the continuous flow.The numerical schemes implicit and trapezoidal, in combination with Automated TrackingScheme Selection, consider most of the changes in the forces acting on the particles andare chosen as default schemes. The runge-kutta scheme is recommended of nondrag forcechanges along a particle integration step.

The integration step size of the higher-order schemes, trapezoidal and runge-kutta, islimited to a stable range. Therefore it is recommended to use them in combination withAutomated Tracking Scheme Selection.

22.15.2 Performing Trajectory Calculations

The trajectories of your discrete phase injections are computed when you display thetrajectories using graphics or when you perform solution iterations. That is, you can dis-play trajectories without impacting the continuous phase, or you can include their effecton the continuum (termed a coupled calculation). In turbulent flows, trajectories canbe based on mean (time-averaged) continuous phase velocities or they can be impactedby instantaneous velocity fluctuations in the fluid. This section describes the proceduresand commands you use to perform coupled or uncoupled trajectory calculations, with orwithout stochastic tracking or cloud tracking.



Uncoupled Calculations

For the uncoupled calculation, you will perform the following two steps:

1. Solve the continuous phase flow field.

2. Plot (and report) the particle trajectories for discrete phase injections of interest.

In the uncoupled approach, this two-step procedure completes the modeling effort, asillustrated in Figure 22.15.1. The particle trajectories are computed as they are displayed,based on a fixed continuous-phase flow field. Graphical and reporting options are detailedin Section 22.16: Postprocessing for the Discrete Phase.

continuous phase flow field calculation

particle trajectory calculation

Figure 22.15.1: Uncoupled Discrete Phase Calculations

This procedure is adequate when the discrete phase is present at a low mass and mo-mentum loading, in which case the continuous phase is not impacted by the presence ofthe discrete phase.

Coupled Calculations

In a coupled two-phase simulation, FLUENT modifies the two-step procedure above asfollows:

1. Solve the continuous phase flow field (prior to introduction of the discrete phase).

2. Introduce the discrete phase by calculating the particle trajectories for each discretephase injection.

3. Recalculate the continuous phase flow, using the interphase exchange of momentum,heat, and mass determined during the previous particle calculation.

4. Recalculate the discrete phase trajectories in the modified continuous phase flowfield.

5. Repeat the previous two steps until a converged solution is achieved in which boththe continuous phase flow field and the discrete phase particle trajectories are un-changed with each additional calculation.



This coupled calculation procedure is illustrated in Figure 22.15.2. When your FLUENTmodel includes a high mass and/or momentum loading in the discrete phase, the coupledprocedure must be followed in order to include the important impact of the discrete phaseon the continuous phase flow field.

particle trajectory calculation

update continuous phase source terms

continuous phase flow field calculation

Figure 22.15.2: Coupled Discrete Phase Calculations

i When you perform coupled calculations, all defined discrete phase injec-tions will be computed. You cannot calculate a subset of the injectionsyou have defined.

Procedures for a Coupled Two-Phase Flow

If your FLUENT model includes prediction of a coupled two-phase flow, you should beginwith a partially (or fully) converged continuous-phase flow field. You will then createyour injection(s) and set up the coupled calculation.

For each discrete-phase iteration, FLUENT computes the particle/droplet trajectories andupdates the interphase exchange of momentum, heat, and mass in each control volume.These interphase exchange terms then impact the continuous phase when the continuousphase iteration is performed. During the coupled calculation, FLUENT will perform thediscrete phase iteration at specified intervals during the continuous-phase calculation.The coupled calculation continues until the continuous phase flow field no longer changeswith further calculations (i.e., all convergence criteria are satisfied). When convergenceis reached, the discrete phase trajectories no longer change either, since changes in thediscrete phase trajectories would result in changes in the continuous phase flow field.



The steps for setting up the coupled calculation are as follows:

1. Solve the continuous phase flow field.

2. In the Discrete Phase Model panel (Figure 22.11.2), enable the Interaction withContinuous Phase option.

3. Set the frequency with which the particle trajectory calculations are introduced inthe Number Of Continuous Phase Iterations Per DPM Iteration field. If you set thisparameter to 5, for example, a discrete phase iteration will be performed every fifthcontinuous phase iteration. The optimum number of iterations between trajectorycalculations depends upon the physics of your FLUENT model.

i Note that if you set this parameter to 0, FLUENT will not perform anydiscrete phase iterations.

During the coupled calculation (which you initiate using the Iterate panel in the usualmanner) you will see the following information in the FLUENT console as the continuousand discrete phase iterations are performed:

iter continuity x-velocity y-velocity k epsilon energy time/it314 2.5249e-01 2.8657e-01 1.0533e+00 7.6227e-02 2.9771e-02 9.8181e-03 :00:05315 2.7955e-01 2.5867e-01 9.2736e-01 6.4516e-02 2.6545e-02 4.2314e-03 :00:03

DPM Iteration ....number tracked= 9, number escaped= 1, aborted= 0, trapped= 0, evaporated = 8,iDone.316 1.9206e-01 1.1860e-01 6.9573e-01 5.2692e-02 2.3997e-02 2.4532e-03 :00:02317 2.0729e-01 3.2982e-02 8.3036e-01 4.1649e-02 2.2111e-02 2.5369e-01 :00:01318 3.2820e-01 5.5508e-02 6.0900e-01 5.9018e-02 2.6619e-02 4.0394e-02 :00:00

Note that you can perform a discrete phase calculation at any time by using thesolve/dpm-update text command.

Stochastic Tracking in Coupled Calculations

If you include the stochastic prediction of turbulent dispersion in the coupled two-phaseflow calculations, the number of stochastic tries applied each time the discrete phasetrajectories are introduced during coupled calculations will be equal to the Number ofTries specified in the Set Injection Properties panel. Input of this parameter is describedin Section 22.12.5: Stochastic Tracking.

Note that the number of tries should be set to 0 if you want to perform the coupledcalculation based on the mean continuous phase flow field. An input of n ≥ 1 requests n


22.16 Postprocessing for the Discrete Phase

stochastic trajectory calculations for each particle in the injection. Note that when thenumber of stochastic tracks included is small, you may find that the ensemble averageof the trajectories is quite different each time the trajectories are computed. Thesedifferences may, in turn, impact the convergence of your coupled solution. For thisreason, you should include an adequate number of stochastic tracks in order to avoidconvergence troubles in coupled calculations.

Under-Relaxation of the Interphase Exchange Terms

When you are coupling the discrete and continuous phases for steady-state calculations,using the calculation procedures noted above, FLUENT applies under-relaxation to themomentum, heat, and mass transfer terms. This under-relaxation serves to increase thestability of the coupled calculation procedure by letting the impact of the discrete phasechange only gradually:

Enew = Eold + α(Ecalculated − Eold) (22.15-14)

where Enew is the exchange term, Eold is the previous value, Ecalculated is the newlycomputed value, and α is the particle/droplet under-relaxation factor. FLUENT uses adefault value of 0.5 for α. You can modify α by changing the value in the Discrete PhaseSources field under Under-Relaxation Factors in the Solution Controls panel. You may needto decrease α in order to improve the stability of coupled discrete phase calculations.

22.15.3 Resetting the Interphase Exchange Terms

If you have performed coupled calculations, resulting in nonzero interphase sources/sinksof momentum, heat, and/or mass that you do not want to include in subsequent calcu-lations, you can reset these sources to zero.

Solve −→ Initialize −→Reset DPM Sources

When you select the Reset DPM Sources menu item, the sources will immediately be resetto zero without any further confirmation from you.


After you have completed your discrete phase inputs and any coupled two-phase calcula-tions of interest, you can display and store the particle trajectory predictions. FLUENTprovides both graphical and alphanumeric reporting facilities for the discrete phase, in-cluding the following:

• graphical display of the particle trajectories

• summary reports of trajectory fates



• step-by-step reports of the particle position, velocity, temperature, and diameter

• alphanumeric reports and graphical display of the interphase exchange of momen-tum, heat, and mass

• sampling of trajectories at boundaries and lines/planes

• summary reporting of current particles in the domain

• histograms of trajectory data at sample planes

• display of erosion/accretion rates

This section provides detailed descriptions of each of these postprocessing options.

(Note that plotting or reporting trajectories does not change the source terms.)

22.16.1 Displaying of Trajectories

When you have defined discrete phase particle injections, as described in Section 22.12: Set-ting Initial Conditions for the Discrete Phase, you can display the trajectories of thesediscrete particles using the Particle Tracks panel (Figure 22.16.1).

Display −→Particle Tracks...

The procedure for drawing trajectories for particle injections is as follows:

1. Select the particle injection(s) you wish to track in the Release From Injections list.(You can choose to track a specific particle, instead, as described below.)

2. Set the length scale and the maximum number of steps in the Discrete Phase Modelpanel, as described in Section 22.11.7: Numerics of the Discrete Phase Model.


If stochastic and/or cloud tracking is desired, set the related parameters in the SetInjection Properties panel, as described in Section 22.12.5: Stochastic Tracking.

3. Set any of the display options described below.

4. Click on the Display button to draw the trajectories or click on the Pulse buttonto animate the particle positions. The Pulse button will become the Stop ! buttonduring the animation, and you must click on Stop ! to stop the pulsing.

i For unsteady particle tracking simulations, clicking on Display will showonly the current location of the particles. Typically, you should selectpoint in the Style drop-down list when displaying transient particle locationssince individual positions will be displayed. The Pulse button option is notavailable for unsteady tracking.



Figure 22.16.1: The Particle Tracks Panel



Specifying Individual Particles for Display

It is also possible to display the trajectory for an individual particle stream instead offor all the streams in a given injection. To do so, you will first need to determine whichparticle is of interest. Use the Injections panel to list the particle streams in the desiredinjection, as described in Section 22.12.3: Creating, Modifying, Copying, Deleting, andListing Injections.

Define −→Injections...

Note the ID numbers listed in the first column of the listing printed in the FLUENTconsole. Then perform the following steps after step 1 above:

1. Enable the Track Single Particle Stream option in the Particle Tracks panel.

2. In the Stream ID field, specify the ID number of the particle stream for which youwant to plot the trajectory.

Options for Particle Trajectory Plots

The options mentioned above include the following: you can include the grid in thetrajectory display, control the style of the trajectories (including the twisting of ribbon-style trajectories), color them by different scalar fields and control the color scale, andcoarsen trajectory plots. You can also choose node or cell values for display. If you are“pulsing” the trajectories, you can control the pulse mode. Finally, you can generate anXY plot of the particle trajectory data (e.g., residence time) as a function of time or pathlength and save this XY plot data to a file.

Plotting particle trajectories can be very time consuming, therefore, to reduce the plottingtime, a coarsening factor can be used to reduce the number of points that are plotted.Providing a coarsening factor of n, will result in each nth point being plotted for a giventrajectory in any cell. This coarsening factor is specified in the Particle Tracks panel, inthe Coarsen field and is only valid for steady state cases. For example, if the coarseningfactor is set to 2, then FLUENT will plot alternate points.

i Note that if any particle or pathline enters a new cell, this point will alwaysbe plotted.

To reduce plotting time in transient cases, FLUENT has available an option to skipplotting every nth particle in an injection. Selecting this option is also done in theParticle Tracks panel menu by specifying a nonzero integer in the Skip field. For example,if an individual stream is selected and the skip option is set to 1, every other particle willbe plotted. If the entire injection is selected with a skip option of 1, every other particlewill be plotted for all streams in the injection.



These options are controlled in exactly the same way that pathline-plotting options arecontrolled. See Section 28.1.4: Options for Pathline Plots for details about setting thetrajectory plotting options mentioned above.

Note that in addition to coloring the trajectories by continuous phase variables, youcan also color them according to the following discrete phase variables: particle time,particle velocity, particle diameter, particle density, particle mass, particle temperature,particle law number, particle time step, and particle Reynolds number. These variablesare included in the Particle Variables... category of the Color By list. To display theminimum and maximum values in the domain, click the Update Min/Max button.

Graphical Display for Axisymmetric Geometries

For axisymmetric problems in which the particle has a nonzero circumferential velocitycomponent, the trajectory of an individual particle is often a spiral about the centerlineof rotation. FLUENT displays the r and x components of the trajectory (but not the θcomponent) projected in the axisymmetric plane.

22.16.2 Reporting of Trajectory Fates

When you perform trajectory calculations by displaying the trajectories (as describedin Section 22.16.1: Displaying of Trajectories), FLUENT will provide information aboutthe trajectories as they are completed. By default, the number of trajectories with eachpossible fate (escaped, aborted, evaporated, etc.) is reported:

DPM Iteration ....num. tracked = 7, escaped = 4, aborted = 0, trapped = 0, evaporated = 3, incoDone.

You can also track particles through the domain without displaying the trajectories byclicking on the Track button at the bottom of the panel. This allows the listing of reportswithout also displaying the tracks.



Trajectory Fates

The possible fates for a particle trajectory are as follows:

• “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 al-lowed number of time steps—as defined by the Max. Number Of Steps input in theDiscrete Phase Model panel (see Section 22.11.7: Numerics of the Discrete PhaseModel)—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 wereevaporated within the domain.

• “Aborted” trajectories are those that fail to complete due to roundoff reasons. Youmay want to retry the calculation with a modified length scale and/or differentinitial conditions.

• “Shed” trajectories are newly generated particles during the breakup of a largerdroplet. They appear only if a breakup model is enabled.

• “Coalesced” trajectories are removed particles which have coalesced after particle-particle collisions. They appear only if the coalescence model is enabled.

• “Splashed” trajectories are particles which are newly generated when a particletouches a wall-film. Those trajectories appear only if the wall-film model is enabled.

Summary Reports

You can request additional detail about the trajectory fates as the particles exit thedomain, including the mass flow rates through each boundary zone, mass flow rate ofevaporated droplets, and composition of the particles.

1. Follow steps 1 and 2 in Section 22.16.1: Displaying of Trajectories for displayingtrajectories.

2. Select Summary as the Report Type and click Display or Track.

A detailed report similar to the following example will appear in the console window.(You may also choose to write this report to a file by selecting File as the Report to option,clicking on the Write... button (which was originally the Display button), and specifyinga file name for the summary report file in the resulting Select File dialog box.)



DPM Iteration ....

num. tracked = 10, escaped = 8, aborted = 0, trapped = 0, evaporated = 0, inc

Fate Number Elapsed Time (s)Min Max Avg Std Dev

---- ------ ---------- ---------- ---------- ---------- ---Incomplete 2 1.485e+01 2.410e+01 1.947e+01 4.623e+00Escaped - Zone 7 8 4.940e+00 2.196e+01 1.226e+01 4.871e+00

(*)- Mass Transfer Summary -(*)

Fate Mass Flow (kg/s)Initial Final Change

---- ---------- ---------- ----------Incomplete 1.388e-03 1.943e-04 -1.194e-03Escaped - Zone 7 1.502e-03 2.481e-04 -1.254e-03

(*)- Energy Transfer Summary -(*)

Fate Heat Content (W)Initial Final Change

---- ---------- ---------- ----------Incomplete 4.051e+02 3.088e+02 -9.630e+01Escaped - Zone 7 4.383e+02 3.914e+02 -4.696e+01

(*)- Combusting Particles -(*)

Fate Volatile Content (kg/s) Char Content (kg/s)Initial Final %Conv Initial Final

---- ---------- ---------- ------- ---------- ---------- --Incomplete 6.247e-04 0.000e+00 100.00 5.691e-04 0.000e+00 1Escaped - Zone 7 6.758e-04 0.000e+00 100.00 6.158e-04 3.782e-05

Done.

The report groups together particles with each possible fate, and reports the number ofparticles, the time elapsed during trajectories, and the mass and energy transfer. Thisinformation can be very useful for obtaining information such as where particles areescaping from the domain, where particles are colliding with surfaces, and the extent ofheat and mass transfer to/from the particles within the domain. Additional informationis reported for combusting particles.



Elapsed Time

The number of particles with each fate is listed under the Number heading. (Particlesthat escape through different zones or are trapped at different zones are considered tohave different fates, and are therefore listed separately.) The minimum, maximum, andaverage time elapsed during the trajectories of these particles, as well as the standarddeviation about the average time, are listed in the Min, Max, Avg, and Std Dev columns.This information indicates how much time the particle(s) spent in the domain beforethey escaped, aborted, evaporated, or were trapped.

Fate Number Elapsed Time (s)Min Max Avg Std Dev

---- ------ ---------- ---------- ---------- ---------- ---Incomplete 2 1.485e+01 2.410e+01 1.947e+01 4.623e+00Escaped - Zone 7 8 4.940e+00 2.196e+01 1.226e+01 4.871e+00

Also, on the right side of the report are listed the injection name and index of thetrajectories with the minimum and maximum elapsed times. (You may need to use thescroll bar to view this information.)

Elapsed Time (s) Injection, IndexMin Max Avg Std Dev Min Max--- ---------- ---------- ---------- -------------------- ------------------+01 2.410e+01 1.947e+01 4.623e+00 injection-0 1 injection-0 0+00 2.196e+01 1.226e+01 4.871e+00 injection-0 9 injection-0 2

Mass Transfer Summary

For all droplet or combusting particles with each fate, the total initial and final mass flowrates and the change in mass flow rate are reported in the Initial, Final, and Change

columns. With this information, you can determine how much mass was transferred tothe continuous phase from the particles.

(*)- Mass Transfer Summary -(*)

Fate Mass Flow (kg/s)Initial Final Change

---- ---------- ---------- ----------Incomplete 1.388e-03 1.943e-04 -1.194e-03Escaped - Zone 7 1.502e-03 2.481e-04 -1.254e-03



Energy Transfer Summary

For all particles with each fate, the total initial and final heat content and the change inheat content are reported in the Initial, Final, and Change columns. This report tellsyou how much heat was transferred from the continuous phase to the particles.

(*)- Energy Transfer Summary -(*)

Fate Heat Content (W)Initial Final Change

---- ---------- ---------- ----------Incomplete 4.051e+02 3.088e+02 -9.630e+01Escaped - Zone 7 4.383e+02 3.914e+02 -4.696e+01

Combusting Particles

If combusting particles are present, FLUENT will include additional reporting on thevolatiles and char converted. These reports are intended to help you identify the com-position of the combusting particles as they exit the computational domain.

(*)- Combusting Particles -(*)

Fate Volatile Content (kg/s) Char Content (kg/s)Initial Final %Conv Initial Final %Conv

---- ---------- ---------- ------- ---------- ---------- -------Incomplete 6.247e-04 0.000e+00 100.00 5.691e-04 0.000e+00 100.00Escaped - Zone 7 6.758e-04 0.000e+00 100.00 6.158e-04 3.782e-05 93.86

The total volatile content at the start and end of the trajectory is reported in the Initialand Final columns under Volatile Content. The percentage of volatiles that has beendevolatilized is reported in the %Conv column.

The total reactive portion (char) at the start and end of the trajectory is reported in theInitial and Final columns under Char Content. The percentage of char that reactedis reported in the %Conv column.

Combusting Particles with the Multiple Surface Reaction Model

If the multiple surface reaction model is used with combusting particles, FLUENT willinclude additional reporting on the mass of the individual solid species that constitutethe particle mass.



(*)- Multiple Surface Reactions -(*)

Fate Species Species Content (kg/s)Names Initial Final %Conv

---- ------- ---------- ---------- -------Escaped - Zone 6 c<s> 6.080e-02 1.487e-06 100.00Escaped - Zone 6 s<s> 3.200e-03 5.077e-06 99.84Escaped - Zone 6 cao 0.000e+00 1.153e-03 0.00Escaped - Zone 6 caso4 0.000e+00 9.266e-04 0.00Escaped - Zone 6 caco3 8.000e-03 5.260e-03 34.25

The total mass of each solid species in the particles at the start and end of the trajectoryis reported in the Initial and Final columns, respectively. The percentage of eachspecies that is reacted is reported in the %Conv column. Note that for the solid reactionproducts (e.g., if the mass of a solid species has increased in the particle), the conversionis reported to be 0.

22.16.3 Step-by-Step Reporting of Trajectories

At times, you may want to obtain a detailed, step-by-step report of the particle trajec-tory/trajectories. Such reports can be obtained in alphanumeric format. This capabilityallows you to monitor the particle position, velocity, temperature, or diameter as thetrajectory proceeds.

The procedure for generating files containing step-by-step reports is listed below:

1. Follow steps 1 and 2 in Section 22.16.1: Displaying of Trajectories for displayingtrajectories. You may want to track only one particle at a time, using the TrackSingle Particle Stream option.

2. Select Step By Step as the Report Type.

i This option is only available for steady-state cases. For transient cases, seeSection 22.16.4: Reporting of Current Positions for Unsteady Tracking.

3. Select File as the Report to option. (The Display button will become the Write...button.)

4. In the Significant Figures field, enter the number of significant figures to be used inthe step-by-step report.

5. Click on the Write... button and specify a file name for the step-by-step report filein the resulting Select File dialog box.



A detailed report similar to the following example will be saved to the specified file beforethe trajectories are plotted. (You may also choose to print the report in the console bychoosing Console as the Report to option and clicking on Display or Track, but the reportis very long that it is unlikely to be of use to you in that form.)

The step-by-step report lists the particle position and velocity of the particle at selectedtime steps along the trajectory:

Time X-Position Y-Position Z-Velocity X-Velocity Y-Velocity Z-Veloc0.000e+00 1.411e-03 3.200e-03 0.000e+00 2.650e+01 0.000e+00 0.000e3.773e-05 2.411e-03 3.200e-03 0.000e+00 2.648e+01 0.000e+00 0.000e5.403e-05 2.822e-03 3.192e-03 0.000e+00 2.647e+01 0.000e+00 0.000e9.181e-05 3.822e-03 3.192e-03 0.000e+00 2.644e+01 0.000e+00 0.000e1.296e-04 4.821e-03 3.192e-03 0.000e+00 2.642e+01 0.000e+00 0.000e1.608e-04 5.644e-03 3.192e-03 0.000e+00 2.639e+01 0.000e+00 0.000e

. . . . . . .

. . . . . . .

. . . . . . .

Also listed are the diameter, temperature, density, and mass of the particle. (You mayneed to use the scroll bar to view this information.) In addition, the variable you haveselected in the Color By list is also listed. This provides you with a simple way to exportany variable along a particle trajectory onto the console or into a file.

Note that the Coarsen option affects the step-by-step report.

Y-Velocity Z-Velocity Diameter Temperature Density Mass ColorBy0.000e+00 0.000e+00 2.000e-04 3.000e+02 1.30e+03 5.445e-09 0.000e+000.000e+00 0.000e+00 2.000e-04 3.006e+02 1.30e+03 5.445e-09 3.773e-050.000e+00 0.000e+00 2.000e-04 3.009e+02 1.30e+03 5.445e-09 5.403e-050.000e+00 0.000e+00 2.000e-04 3.015e+02 1.30e+03 5.445e-09 9.181e-050.000e+00 0.000e+00 2.000e-04 3.022e+02 1.30e+03 5.445e-09 1.296e-040.000e+00 0.000e+00 2.000e-04 3.027e+02 1.30e+03 5.445e-09 1.608e-04

. . . . . . .

. . . . . . .

. . . . . . .



22.16.4 Reporting of Current Positions for Unsteady Tracking

In transient cases, when using unsteady tracking, you may want to obtain a report of theparticle trajectory/trajectories showing the current positions of the particles. SelectingCurrent Positions under Report Type in the Particle Tracks panel enables the display of thecurrent positions of the particles.

The procedure for generating files containing current position reports is listed below:

1. Follow steps 1 and 2 in Section 22.16.1: Displaying of Trajectories for displayingtrajectories. You may want to track only one particle stream at a time, using theTrack Single Particle Stream option.

2. Select Current Position as the Report Type.

3. Select File as the Report to option. (The Display button will become the Write...button.)

4. In the Significant Figures field, enter the number of significant figures to be used inthe step-by-step report.

5. Click on the Write... button and specify a file name for the current position reportfile in the resulting Select File dialog box.

The current position report lists the positions and velocities of all particles that arecurrently in the domain:

Time X-Position Y-Position Z-Position X-Velocity Y-Velocity Z-Veloc0.000e+00 1.000e-03 3.120e-02 0.000e+00 1.000e+01 5.000e+00 0.000e1.672e-05 1.168e-03 3.128e-02 0.000e+00 1.010e+01 4.988e+00 0.000e3.342e-05 1.337e-03 3.137e-02 0.000e+00 1.019e+01 4.977e+00 0.000e5.010e-05 1.508e-03 3.145e-02 0.000e+00 1.028e+01 4.965e+00 0.000e6.675e-05 1.680e-03 3.153e-02 0.000e+00 1.038e+01 4.954e+00 0.000e8.338e-05 1.854e-03 3.161e-02 0.000e+00 1.047e+01 4.942e+00 0.000e

. . . . . . .

. . . . . . .

. . . . . . .

Also listed are the diameter, temperature, density, mass of the particles, number in parceland the variable selected from the Color By list. (You may need to use the scroll bar toview this information.)



elocity Diameter Temperature Density Mass Number ColorBy000e+00 7.000e-05 3.000e+02 1.300e+03 2.335e-10 3.183e+03 0.000e+00000e+00 7.000e-05 3.009e+02 1.300e+03 2.335e-10 3.183e+03 1.672e-05000e+00 7.000e-05 3.019e+02 1.300e+03 2.335e-10 3.183e+03 3.342e-05000e+00 7.000e-05 3.028e+02 1.300e+03 2.335e-10 3.183e+03 5.010e-05000e+00 7.000e-05 3.037e+02 1.300e+03 2.335e-10 3.183e+03 6.675e-05000e+00 7.000e-05 3.046e+02 1.300e+03 2.335e-10 3.183e+03 8.338e-05

. . . . . . .

. . . . . . .

. . . . . . .

22.16.5 Reporting of Interphase Exchange Terms and Discrete PhaseConcentration

FLUENT reports the magnitudes of the interphase exchange of momentum, heat, andmass in each control volume in your FLUENT model. It can also report the total con-centration of the discrete phase. You can display these variables graphically, by drawingcontours, profiles, etc. They are all contained in the Discrete Phase Model... category ofthe variable selection drop-down list that appears in postprocessing panels:

• DPM Concentration

• DPM Mass Source

• DPM X,Y,Z Momentum Source

• DPM Swirl Momentum Source

• DPM Sensible Enthalpy Source

• DPM Enthalpy Source

• DPM Absorption Coefficient

• DPM Emission

• DPM Scattering

• DPM Burnout

• DPM Evaporation/Devolatilization

• DPM (species) Source

• DPM Erosion



• DPM Accretion

• DPM (species) Concentration

See Chapter 30: Field Function Definitions for definitions of these variables.

Note that these exchange terms are updated and displayed only when coupled calcu-lations are performed. Displaying and reporting particle trajectories (as described inSections 22.16.1 and 22.16.2) will not affect the values of these exchange terms.

22.16.6 Sampling of Trajectories

Particle states (position, velocity, diameter, temperature, and mass flow rate) can bewritten to files at various boundaries and planes (lines in 2D) using the Sample Trajectoriespanel (Figure 22.16.2).

Report −→ Discrete Phase −→Sample...

Figure 22.16.2: The Sample Trajectories Panel



The procedure for generating files containing the particle samples is listed below:

1. Select the injections to be tracked in the Release From Injections list.

2. Select the surfaces at which samples will be written. These can be boundaries fromthe Boundaries list or planes from the Planes list (in 3D) or lines from the Lines list(in 2D).

3. Click on the Compute button. Note that for unsteady particle tracking, the Computebutton will become the Start button (to initiate sampling) or a Stop button (to stopsampling).

Clicking on the Compute button will cause the particles to be tracked and their status tobe written to files when they encounter selected surfaces. The file names will be formedby appending .dpm to the surface name.

For unsteady particle tracking, clicking on the Start button will open the files and writethe file header sections. If the solution is advanced in time by computing some timesteps, the particle trajectories will be updated and the particle states will be written tothe files as they cross the selected planes or boundaries. Clicking on the Stop button willclose the files and end the sampling.

For stochastic tracking, it may be useful to repeat this process multiple times and appendthe results to the same file, while monitoring the sample statistics at each update. Todo this, enable the Append Files option before repeating the calculation (clicking onCompute). Similarly, you can cause erosion and accretion rates to be accumulated forrepeated trajectory calculations by turning on the Accumulate Erosion/Accretion Ratesoption. (See also Section 22.16.9: Postprocessing of Erosion/Accretion Rates.) Theformat and the information written for the sample output can also be controlled througha user-defined function, which can be selected in the Output drop-down list. See theseparate UDF Manual for information about user-defined functions.



22.16.7 Histogram Reporting of Samples

Histograms can be plotted from sample files created in the Sample Trajectories panel(as described in Section 22.16.6: Sampling of Trajectories) using the Trajectory SampleHistograms panel (Figure 22.16.3).

Report −→ Discrete Phase −→Histogram...

Figure 22.16.3: The Trajectory Sample Histograms Panel

The procedure for plotting histograms from data in a sample file is listed below:

1. Select a file to be read by clicking on the Read... button. After you read in thesample file, the boundary name will appear in the Sample list.

2. Select the data sample in the Sample list, and then select the data to be plottedfrom the Fields list.

3. Click on the Plot button at the bottom of the panel to display the histogram.

By default, the percent of particles will be plotted on the y axis. You can plot theactual number of particles by deselecting Percent under Options. The number of “bins”or intervals in the plot can be set in the Divisions field. You can delete samples from thelist with the Delete button and update the Min/Max values with the Compute button.



22.16.8 Summary Reporting of Current Particles

For many mass-transfer and flow processes, it is desirable to know the mean diameterof the particles. A mean diameter, Djk, is calculated from the particle size distributionusing the following general expression [187]:

(Djk)j−k ≡

∫ ∞0Djf(D)dD∫ ∞

0Dkf(D)dD

(22.16-1)

where j and k are integers and f(D) is the distribution function (e.g., Rosin-Rammler).D10, for example, is the average (arithmetic) particle diameter. The Sauter mean diam-eter (SMD), D32, is the diameter of a particle whose ratio of volume to surface area isequal to that of all particles in the computation. A summary of common mean diametersis given in Table 22.16.1.

Table 22.16.1: Common Mean Diameters and Their Fields of Application

j k Orderj + k

Name Field of Application

1 0 1 Mean diameter, D10 Comparisons, evaporation2 0 2 Mean surface diameter, D20 Absorption3 0 3 Mean volume diameter, D30 Hydrology2 1 3 Overall surface diameter, D21 Adsorption3 1 4 Overall volume diameter, D31 Evaporation, molecular diffusion3 2 5 Sauter mean diameter, D32 Combustion, mass transfer, and

efficiency studies4 3 7 De Brouckere diameter, D43 Combustion equilibrium



Summary information (number, mass, average diameter) for particles currently in thecomputational domain can be reported using the Particle Summary panel (Figure 22.16.4).

Report −→ Discrete Phase −→Summary...

Figure 22.16.4: The Particle Summary Panel

The procedure for reporting a summary for particle injections is as follows:

1. Select the particle injection(s) for which you want to generate a summary in theInjections list.

FLUENT provides a shortcut for selecting injections with names that match a spec-ified pattern. To use this shortcut, enter the pattern under Injection Name Patternand then click Match to select the injections with names that match the speci-fied pattern. For example, if you specify drop*, all injections that have namesbeginning with drop (e.g., drop-1, droplet) will be selected automatically. If theyare all selected already, they will be deselected. If you specify drop?, all surfaceswith names consisting of drop followed by a single character will be selected (ordeselected, if they are all selected already).

2. Click Summary to display the injection summary in the console window.



(*)- Summary for Injection: injection-0 -(*)

Total number of parcels : 1862Total number of particles : 1.196710e+05Total mass : 1.128303e-05 (kg)Maximum RMS distance from injector : 7.372527e-01 (m)Maximum particle diameter : 3.072739e-04 (m)Minimum particle diameter : 1.756993e-06 (m)Overall RR Spread Parameter : 1.446806e+00Maximum Error in RR fit : 1.071220e-01Overall RR diameter (D_RR): 9.051303e-05 (m)Overall mean diameter (D_10): 4.663269e-05 (m)Overall mean surface area (D_20): 5.344694e-05 (m)Overall mean volume (D_30): 6.121478e-05 (m)Overall surface diameter (D_21): 6.125692e-05 (m)Overall volume diameter (D_31): 7.013570e-05 (m)Overall Sauter diameter (D_32): 8.030141e-05 (m)Overall De Brouckere diameter (D_43): 1.082971e-04 (m)

22.16.9 Postprocessing of Erosion/Accretion Rates

You can calculate the erosion and accretion rates in a cumulative manner (over a seriesof injections) by using the Sample Trajectories panel. First select an injection in theRelease From Injections list and compute its trajectory. Then turn on the AccumulateErosion/Accretion Rates option, select the next injection (after deselecting the first one),and click Compute again. The rates will accumulate at the surfaces each time you clickCompute.

i Both the erosion rate and the accretion rate are defined at wall face surfacesonly, so they cannot be displayed at node values.


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