fluent dpm chp19

Chapter 19. Discrete Phase Models

This chapter describes the Lagrangian discrete phase capabilities avail-able in FLUENT and how to use them.

Information is organized into the following sections:

• Section 19.1: Overview and Limitations of the Discrete Phase Mod-els

• Section 19.2: Trajectory Calculations

• Section 19.3: Heat and Mass Transfer Calculations

• Section 19.4: Spray Models

• Section 19.5: Coupling Between the Discrete and Continuous Phases

• Section 19.6: Overview of Using the Discrete Phase Models

• Section 19.7: Discrete Phase Model Options

• Section 19.8: Unsteady Particle Tracking

• Section 19.9: Setting Initial Conditions for the Discrete Phase

• Section 19.10: Setting Boundary Conditions for the Discrete Phase

• Section 19.11: Setting Material Properties for the Discrete Phase

• Section 19.12: Calculation Procedures for the Discrete Phase

• Section 19.13: Postprocessing for the Discrete Phase

c© Fluent Inc. December 3, 2001 19-1

Discrete Phase Models

19.1 Overview and Limitations of the Discrete PhaseModels

19.1.1 Introduction

In addition to solving transport equations for the continuous phase, FLU-ENT allows you to simulate a discrete second phase in a Lagrangian frameof reference. This second phase consists of spherical particles (which maybe taken to represent droplets or bubbles) dispersed in the continuousphase. FLUENT computes the trajectories of these discrete phase en-tities, as well as heat and mass transfer to/from them. The couplingbetween the phases and its impact on both the discrete phase trajecto-ries and the continuous phase flow can be included.

FLUENT provides the following discrete phase modeling options:

• Calculation of the discrete phase trajectory using a Lagrangianformulation that includes the discrete phase inertia, hydrodynamicdrag, and the force of gravity, for both steady and unsteady flows

• Prediction of the effects of turbulence on the dispersion of particlesdue to turbulent eddies present in the continuous phase

• Heating/cooling of the discrete phase

• Vaporization and boiling of liquid droplets

• Combusting particles, including volatile evolution and char com-bustion to simulate coal combustion

• Optional coupling of the continuous phase flow field prediction tothe discrete phase calculations

• Droplet breakup and coalescence

These modeling capabilities allow FLUENT to simulate a wide rangeof discrete phase problems including particle separation and classifica-tion, spray drying, aerosol dispersion, bubble stirring of liquids, liquidfuel combustion, and coal combustion. The physical equations used forthese discrete phase calculations are described in Sections 19.2–19.5, and

19-2 c© Fluent Inc. December 3, 2001

19.1 Overview and Limitations of the Discrete Phase Models

instructions for setup, solution, and postprocessing are provided in Sec-tions 19.6–19.13.

19.1.2 Particles in Turbulent Flows

The dispersion of particles due to turbulence in the fluid phase can bepredicted using the stochastic tracking model or the particle cloud model(see Section 19.2.2). The stochastic tracking (random walk) model in-cludes the effect of instantaneous turbulent velocity fluctuations on theparticle trajectories through the use of stochastic methods (see Sec-tion 19.2.2). The particle cloud model tracks the statistical evolutionof a cloud of particles about a mean trajectory (see Section 19.2.2). Theconcentration of particles within the cloud is represented by a Gaus-sian probability density function (PDF) about the mean trajectory. Inboth models, the particles have no direct impact on the generation ordissipation of turbulence in the continuous phase.

19.1.3 Limitations

Limitation on the Particle Volume Fraction

The discrete phase formulation used by FLUENT contains the assumptionthat the second phase is sufficiently dilute that particle-particle interac-tions and the effects of the particle volume fraction on the gas phase arenegligible. In practice, these issues imply that the discrete phase must bepresent at a fairly low volume fraction, usually less than 10–12%. Notethat the mass loading of the discrete phase may greatly exceed 10–12%:you may solve problems in which the mass flow of the discrete phaseequals or exceeds that of the continuous phase. See Chapters 18 and 20for information about when you might want to use one of the generalmultiphase models instead of the discrete phase model.

Limitation on Modeling Continuous Suspensions of Particles

The steady-particle Lagrangian discrete phase model described in thischapter is suited for flows in which particle streams are injected into acontinuous phase flow with a well-defined entrance and exit condition.



The Lagrangian model does not effectively model flows in which par-ticles 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 continuous suspensions of particles. See Chap-ters 18 and 20 for information about when you might want to use one ofthe general multiphase models instead of the discrete phase models.

Limitations on Using the Discrete Phase Model with OtherFLUENT Models

The following restrictions exist on the use of other models with the dis-crete phase model:

• Streamwise periodic flow (either specified mass flow rate or spec-ified pressure drop) cannot be modeled when the discrete phasemodel is used.

• Adaptive time stepping cannot be used with the discrete phasemodel.

• Only non-reacting particles can be included when the premixedcombustion model is used.

• When multiple reference frames are used in conjunction with thediscrete phase model, the display of particle tracks will not, by de-fault, be meaningful. Similarly, coupled discrete-phase calculationsare not meaningful.

An alternative approach for particle tracking and coupled discrete-phase calculations with multiple reference frames is to track parti-cles based on absolute velocity instead of relative velocity. To makethis change, use the define/models/dpm/tracking/track-in-absolute-frame text command. Note, however, that tracking par-ticles based on absolute velocity may result in incorrect particle-wall interaction.

The particle injection velocities (specified in the Set Injection Prop-erties panel) are defined relative to the frame of reference in whichthe particles are tracked. By default, the injection velocities are


19.1 Overview and Limitations of the Discrete Phase Models

specified relative to the local reference frame. If you enable thetrack-in-absolute-frame option, the injection velocities are spec-ified relative to the absolute frame.

19.1.4 Overview of Discrete Phase Modeling Procedures

You can include a discrete phase in your FLUENT model by definingthe initial position, velocity, size, and temperature of individual parti-cles. These initial conditions, along with your inputs defining the phys-ical properties of the discrete phase, are used to initiate trajectory andheat/mass transfer calculations. The trajectory and heat/mass transfercalculations are based on the force balance on the particle and on theconvective/radiative heat and mass transfer from the particle, using thelocal continuous phase conditions as the particle moves through the flow.The predicted trajectories and the associated heat and mass transfer canbe viewed graphically and/or alphanumerically.

You can use FLUENT to predict the discrete phase patterns based ona fixed continuous phase flow field (an uncoupled approach), or you caninclude the effect of the discrete phase on the continuum (a coupledapproach). In the coupled approach, the continuous phase flow patternis impacted by the discrete phase (and vice versa), and you can alternatecalculations of the continuous phase and discrete phase equations untila converged coupled solution is achieved. See Section 19.5 for details.

Outline of Steady-State Problem Setup and Solution Procedure

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

1. Solve the continuous-phase flow.

2. Create the discrete-phase injections.

3. Solve the coupled flow, if desired.

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



Outline of Unsteady Problem Setup and Solution Procedure

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

1. Create the discrete-phase injections.

2. Initialize the flow field.

3. Advance the solution in time by taking the desired number of timesteps. Particle positions will be updated as the solution advances intime. If you are solving an uncoupled flow, the particle position willbe updated at the end of each time step. For a coupled calculation,the positions are iterated on within each time step.

19.2 Trajectory Calculations

19.2.1 Equations of Motion for Particles

Particle Force Balance

FLUENT predicts the trajectory of a discrete phase particle (or droplet orbubble) by integrating the force balance on the particle, which is writtenin a Lagrangian reference frame. This force balance equates the particleinertia with the forces acting on the particle, and can be written (for thex direction in Cartesian coordinates) as

dup

dt= FD(u− up) +

gx(ρp − ρ)ρp

+ Fx (19.2-1)

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

FD =18µρpd2

p

CDRe24

(19.2-2)

Here, u is the fluid phase velocity, up is the particle velocity, µ is themolecular viscosity of the fluid, ρ is the fluid density, ρp is the density of



the particle, and dp is the particle diameter. Re is the relative Reynoldsnumber, which is defined as

Re ≡ ρdp |up − u|µ

(19.2-3)

The drag coefficient, CD, can be taken from either

CD = a1 +a2

Re+

a3

Re2 (19.2-4)

where a1, a2, and a3 are constants that apply for smooth spherical par-ticles over several ranges of Re given by Morsi and Alexander [163], or

CD =24Re

(1 + b1Reb2

)+

b3Reb4 + Re

(19.2-5)

where

b1 = exp(2.3288 − 6.4581φ + 2.4486φ2)b2 = 0.0964 + 0.5565φb3 = exp(4.905 − 13.8944φ + 18.4222φ2 − 10.2599φ3)b4 = exp(1.4681 + 12.2584φ − 20.7322φ2 + 15.8855φ3)(19.2-6)

which is taken from Haider and Levenspiel [85]. The shape factor, φ, isdefined as

φ =s

S(19.2-7)

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

For sub-micron particles, a form of Stokes’ drag law is available [170]. Inthis case, FD is defined as

FD =18µ

dp2ρpCc

(19.2-8)



The factor Cc is the Cunningham correction to Stokes’ drag law, whichyou can compute from

Cc = 1 +2λdp

(1.257 + 0.4e−(1.1dp/2λ)) (19.2-9)

where λ is the molecular mean free path.

A high-Mach-number drag law is also available. This drag law is similarto the spherical law (Equation 19.2-4) with corrections [38] to account fora particle Mach number greater than 0.4 or a particle Reynolds numbergreater than 20.

For unsteady models involving discrete phase droplet breakup, a dynamicdrag law option is also available. See Section 19.4.4 for a description ofthis law.

Instructions for selecting the drag law are provided in Section 19.7.7.

Including the Gravity Term

While Equation 19.2-1 includes a force of gravity on the particle, it isimportant to note that in FLUENT the default gravitational accelerationis zero. If you want to include the gravity force, you must remember todefine the magnitude and direction of the gravity vector in the OperatingConditions panel.

Other Forces

Equation 19.2-1 incorporates additional forces (Fx) in the particle forcebalance that can be important under special circumstances. The firstof these is the “virtual mass” force, the force required to accelerate thefluid surrounding the particle. This force can be written as

Fx =12ρ

ρp

d

dt(u− up) (19.2-10)

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



Fx =

(ρ

ρp

)up∂u

∂x(19.2-11)

Forces in Rotating Reference Frames

The additional force term, Fx, in Equation 19.2-1 also includes forces onparticles that arise due to rotation of the reference frame. These forcesarise when you are modeling flows in rotating frames of reference (seeSection 9.2). For rotation defined about the z axis, for example, theforces on the particles in the Cartesian x and y directions can be writtenas

(1 − ρ

ρp

)Ω2x+ 2Ω

(uy,p − ρ

ρpuy

)(19.2-12)

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

(1 − ρ

ρp

)Ω2y − 2Ω

(ux,p − ρ

ρpux

)(19.2-13)

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

Thermophoretic Force

Small particles suspended in a gas that has a temperature gradient ex-perience a force in the direction opposite to that of the gradient. Thisphenomenon is known as thermophoresis. FLUENT can optionally in-clude a thermophoretic force on particles in the additional force term,Fx, in Equation 19.2-1:

Fx = −DT,p1

mpT

∂T

∂x(19.2-14)



where DT,p is the thermophoretic coefficient. You can define the coeffi-cient to be constant, polynomial, or a user-defined function, or you canuse the form suggested by Talbot [237]:

Fx = − 6πdpµ2Cs(K + CtKn)

ρ(1 + 3CmKn)(1 + 2K + 2CtKn)1

mpT

∂T

∂x(19.2-15)

where: Kn = Knudsen number = 2 λ/dp

λ = mean free path of the fluidK = k/kp

k = fluid thermal conductivity based on translationalenergy only = (15/4) µR

kp = 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 fluidis an ideal gas.

Brownian Force

For sub-micron particles, the effects of Brownian motion can optionallybe included in the additional force term. The components of the Brow-nian force are modeled as a Gaussian white noise process with spectralintensity Sn,ij given by [135]

Sn,ij = S0δij (19.2-16)

where δij is the Kronecker delta function, and

S0 =216νσT

π2ρd5p

(ρp

ρ

)2Cc

(19.2-17)



T is the absolute temperature of the fluid, ν is the kinematic viscosity,and σ is the Stefan-Boltzmann constant. Amplitudes of the Brownianforce components are of the form

Fbi= ζi

√πSo

∆t(19.2-18)

where ζi are zero-mean, unit-variance-independent Gaussian random num-bers. The amplitudes of the Brownian force components are evaluatedat each time step. The energy equation must be enabled in order forthe Brownian force to take effect. Brownian force is intended only fornon-turbulent models.

Saffman’s Lift Force

The Saffman’s lift force, or lift due to shear, can also be included inthe additional force term as an option. The lift force used is from Liand Ahmadi [135] and is a generalization of the expression provided bySaffman [196]:

~F =2Kν1/2ρdij

ρpdp(dlkdkl)1/4(~v − ~vp) (19.2-19)

where K = 2.594 and dij is the deformation tensor. This form of the liftforce is intended for small particle Reynolds numbers. Also, the particleReynolds number based on the particle-fluid velocity difference must besmaller than the square root of the particle Reynolds number based onthe shear field. Since this restriction is valid for submicron particles, itis recommended to use this option only for submicron particles.

Stochastic Particle Tracking in Turbulent Flow

When the flow is turbulent, FLUENT will predict the trajectories of par-ticles using the mean fluid phase velocity, u, in the trajectory equations(Equation 19.2-1). Optionally, you can include the instantaneous valueof the fluctuating gas flow velocity,



u = u+ u′ (19.2-20)

to predict the dispersion of the particles due to turbulence. FLUENTuses a stochastic method (random walk model) to determine the instan-taneous gas velocity, as detailed in Section 19.2.2.

Particle Cloud Tracking in Turbulent Flow

Particle dispersion due to turbulent fluctuations can also be modeledwith the particle cloud model [14, 15, 99, 141]. The turbulent disper-sion of particles about a mean trajectory is calculated using statisticalmethods. The concentration of particles about the mean trajectory isrepresented by a Gaussian probability density function (PDF) whosevariance is based on the degree of particle dispersion due to turbulentfluctuations. The mean trajectory is obtained by solving the ensemble-averaged equations of motion for all particles represented by the cloud(see Section 19.2.2).

Integration of the Trajectory Equations

The trajectory equations, and any auxiliary equations describing heator mass transfer to/from the particle, are solved by stepwise integrationover discrete time steps. Integration in time of Equation 19.2-1 yieldsthe velocity of the particle at each point along the trajectory, with thetrajectory itself predicted by

dx

dt= up (19.2-21)

Equations similar to 19.2-1 and 19.2-21 are solved in each coordinatedirection to predict the trajectories of the discrete phase.

Assuming that the term containing the body force remains constant overeach small time interval, and linearizing any other forces acting on theparticle, the trajectory equation can be rewritten in simplified form as

dup

dt=

1τp

(u− up) (19.2-22)



where τp is the particle relaxation time. FLUENT uses a trapezoidalscheme for integrating Equation 19.2-22:

un+1p − un

p

∆t=

1τ(u∗ − un+1

p ) + . . . (19.2-23)

where n represents the iteration number and

u∗ =12(un + un+1) (19.2-24)

un+1 = un + ∆tunp · ∇un (19.2-25)

Equations 19.2-21 and 19.2-22 are solved simultaneously to determinethe velocity and position of the particle at any given time. For rotatingreference frames, the integration is carried out in the rotating framewith the extra terms described above (Equations 19.2-12 and 19.2-13)to account for system rotation. In all cases, care must be taken thatthe time step used for integration is sufficiently small that the trajectoryintegration is accurate in time. (See Section 19.12.)

Droplet Size Distributions

For liquid sprays, a convenient representation of the droplet size distri-bution is the Rosin-Rammler expression. The complete range of sizes isdivided into an adequate number of discrete intervals; each representedby a mean diameter for which trajectory calculations are performed. Ifthe size distribution is of the Rosin-Rammler type, the mass fraction ofdroplets of diameter greater than d is given by

Yd = e−(d/d)n(19.2-26)

where d is the size constant and n is the size distribution parameter. Useof the Rosin-Rammler size distribution is detailed in Section 19.9.7.



Discrete Phase Boundary Conditions

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

• Reflection via an elastic or inelastic collision.

• Escape through the boundary. (The particle is lost from the cal-culation at the point where it impacts the boundary.)

• Trap at the wall. Non-volatile material is lost from the calculationat the point of impact with the boundary; volatile material presentin the particle or droplet is released to the vapor phase at this point.

• Passing through an internal boundary zone, such as radiator orporous jump.

You also have the option of implementing a user-defined function tomodel the particle path. See the separate UDF Manual for informationabout user-defined functions.

These boundary condition options are described in detail in Section 19.10.

19.2.2 Turbulent Dispersion of Particles

Turbulent dispersion of particles can be modeled using either a stochas-tic discrete-particle approach or a “cloud” representation of a group ofparticles about a mean trajectory. In addition, these approaches canbe combined to model a set of “clouds” about a mean trajectory thatincludes the effects of turbulent fluctuations in the gas phase velocities.

Turbulent dispersion of particles cannot be included if the Spalart-Allmaras!turbulence model is used.

Stochastic Tracking

In the stochastic tracking approach, FLUENT predicts the turbulent dis-persion of particles by integrating the trajectory equations for individualparticles, using the instantaneous fluid velocity, u+u

′(t), along the par-

ticle path during the integration. By computing the trajectory in this



manner for a sufficient number of representative particles (termed the“number of tries”), the random effects of turbulence on the particle dis-persion may be accounted for. In FLUENT, the Discrete Random Walk(DRW) model is used. In this model, the fluctuating velocity compo-nents are discrete piecewise constant functions of time. Their randomvalue is kept constant over an interval of time given by the characteristiclifetime of the eddies.

The DRW model may give non-physical results in strongly inhomoge-neous diffusion-dominated flows, where small particles should becomeuniformly distributed. Instead, the DRW will show a tendency for suchparticles to concentrate in low-turbulence regions of the flow.

The Integral Time

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

T =∫ ∞

0

up′(t)up

′(t+ s)

up′2

ds (19.2-27)

The integral time is proportional to the particle dispersion rate, as largervalues indicate more turbulent motion in the flow. It can be shown thatthe particle diffusivity is given by ui

′uj′T .

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

TL = CLk

ε(19.2-28)

where CL is to be determined and is not well known. By matchingthe diffusivity of tracer particles, ui

′uj′TL, to the scalar diffusion rate

predicted by the turbulence model, νt/σ, one can obtain

TL ≈ 0.15k

ε(19.2-29)



for the k-ε model and its variants, and

TL ≈ 0.30k

ε(19.2-30)

when the Reynolds stress model (RSM) is used [48]. For the k-ω models,substitute ω = ε/k into Equation 19.2-28. The LES model uses theequivalent LES time scales.

The Discrete Random Walk Model

In the Discrete Random Walk (DRW) model, or “eddy lifetime” model,the interaction of a particle with a succession of discrete stylized fluidphase 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 turbu-lent eddy are sampled by assuming that they obey a Gaussian probabilitydistribution, so that

u′= ζ

√u′2 (19.2-31)

where ζ is a normally distributed random number, and the remainder ofthe right-hand side is the local RMS value of the velocity fluctuations.Since the kinetic energy of turbulence is known at each point in theflow, these values of the RMS fluctuating components can be obtained(assuming isotropy) as

√u′2 =

√v′2 =

√w′2 =

√2k/3 (19.2-32)

for the k-ε model, the k-ω model, and their variants. When the RSMis used, non-isotropy of the stresses is included in the derivation of thevelocity fluctuations:



u′ = ζ

√u′2 (19.2-33)

v′ = ζ

√v′2 (19.2-34)

w′ = ζ

√w′2 (19.2-35)

when viewed in a reference frame in which the second moment of theturbulence is diagonal [274]. For the LES model, the velocity fluctuationsare equivalent in all directions. See Section 10.7.3 for details.

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

τe = 2TL (19.2-36)

where TL is given by Equation 19.2-28 in general (Equation 19.2-29 bydefault), or as a random variation about TL:

τe = −TL log(r) (19.2-37)

where r is a uniform random number between 0 and 1 and TL is given byEquation 19.2-29. The option of random calculation of τe yields a morerealistic description of the correlation function.

The particle eddy crossing time is defined as

tcross = −τ ln

[1 −

(Le

τ |u− up|

)](19.2-38)

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

The particle is assumed to interact with the fluid phase eddy over thesmaller of the eddy lifetime and the eddy crossing time. When thistime is reached, a new value of the instantaneous velocity is obtained byapplying a new value of ζ in Equation 19.2-31.



Using the DRW Model

The only inputs required for the DRW model are the value for the integraltime-scale constant, CL (see Equations 19.2-28 and 19.2-36) and thechoice of the method used for the prediction of the eddy lifetime. You canchoose to use either a constant value or a random value by selecting theappropriate option in the Set Injection Properties panel for each injection,as described in Section 19.9.15.

Turbulent dispersion of particles cannot be included if the Spalart-Allmaras!turbulence model is used.

Particle Cloud Tracking

The particle cloud model is based on the stochastic transport of particlesmodel developed by Litchford and Jeng [141], Baxter and Smith [15], andJain [99]. The approach uses statistical methods to trace the turbulentdispersion of particles about a mean trajectory. The mean trajectory iscalculated from the ensemble average of the equations of motion for theparticles represented by the cloud. The cloud enters the domain eitheras a point source or with an initial diameter. The cloud expands dueto turbulent dispersion as it is transported through the domain until itexits. The distribution of particles in the cloud is defined by a probabilitydensity function (PDF) based on the position in the cloud relative to thecloud center. The value of the PDF represents the probability of findingparticles represented by that cloud with residence time t at location xi

in the flow field. The average particle number density can be obtainedby weighting the total flow rate of particles represented by that cloud,m, as

〈n(xi)〉 = mP (xi, t) (19.2-39)

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

2, andare of the form



P (xi, t) =1

(8π)3/23∏

i=1

σi

e−(s2/2) (19.2-40)

where

s =3∑

i=1

xi − µi

σi(19.2-41)

The mean of the PDF, or the center of the cloud, at a given time rep-resents the most likely location of the particles in the cloud. The meanlocation is obtained by integrating a particle velocity as defined by anequation of motion for the cloud of particles:

µi(t) ≡ 〈xi(t)〉 =∫ t

0〈Vi(t1)〉dt1 + 〈xi(0)〉 (19.2-42)

The equations of motion are constructed using an ensemble average.

The radius of the particle cloud is based on the variance of the PDF. Thevariance, σ2

i (t), of the PDF can be expressed in terms of two particleturbulence statistical quantities:

σ2i (t) = 2

∫ t

0〈u′2p,i(t2)〉

∫ t2

0Rp,ii(t2, t1)dt1dt2 (19.2-43)

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

is the particle velocity correlation function:

Rp,ij(t2, t1) =〈u′p,i(t2)u

′p,j(t1)〉[

〈u′2p,i(t2)u′2p,j(t2)〉

]1/2(19.2-44)

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

Rp,ij(t2, t1) = Rp,ij(t4, t3) (19.2-45)



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

σ2i (t) = 2

∫ t

0〈u′2p,i(t2)〉

∫ t2

0Rp,ii(τ)dτdt2 (19.2-46)

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

The form of the particle velocity correlation function used determinesthe particle dispersion in the cloud model. FLUENT uses a correlationfunction first proposed by Wang [254], and used by Jain [99]. When thegravity vector is aligned with the z-coordinate direction, Rij takes theform:

Rp,11 =u′2pθe−(τ/τa) StT

(B − 0.5mT γ

St2TB2 + 1θ

)

+u′2

θe−(τB/T )

(−1 +

mT St2TγBθ

+ 0.5mT γτ

T

)

(19.2-47)Rp,22 = Rp,11 (19.2-48)

Rp,33 =u′2StTB

θe−(τ/τa) − u′2

θe−(τB/T ) (19.2-49)

where B =√

1 +m2Tγ

2 and τa is the aerodynamic response time of theparticle:

τa =ρpd

2p

18µ(19.2-50)

and

T =mTTmE

m(19.2-51)

TfE =C

3/4µ k3/2

ε(23k)

1/2(19.2-52)



γ =τag

u′(19.2-53)

St =τaTmE

(19.2-54)

StT =τaT

(19.2-55)

θ = St2T (1 +m2Tγ

2) − 1 (19.2-56)

m =u

u′(19.2-57)

TmE = TfEu

u′(19.2-58)

mT = m

[1 − G(m)

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

](19.2-59)

G(m) =2√π

∫ ∞

0

e−y2dy(

1 + m2

π

(√π erf(y)y − 1 + e−y2))5/2

(19.2-60)

Using this correlation function, the variance is integrated over the life ofthe cloud. At any given time, the cloud radius is set to three standarddeviations in the coordinate directions. The cloud radius is limited tothree standard deviations since at least 99.2% of the area under a Gaus-sian PDF is accounted for at this distance. Once the cells within thecloud are established, the fluid properties are ensemble-averaged for themean trajectory, and the mean path is integrated in time. This is donewith a weighting factor defined as

W (xi, t) ≡

∫Vcell

P (xi, t)dV∫Vcloud

P (xi, t)dV(19.2-61)

If coupled calculations are performed, sources are distributed to the cellsin the cloud based on the same weighting factors.

Using the Cloud Model

The only inputs required for the cloud model are the values of the min-imum and maximum cloud diameters. The cloud model is enabled in



the Set Injection Properties panel for each injection, as described in Sec-tion 19.9.15.

The cloud model is not available for unsteady tracking.!

19.2.3 Particle Erosion and Accretion

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

Rerosion =Nparticles∑

p=1

mpC(dp)f(α)vb(v)

Aface(19.2-62)

where C(dp) is a function of particle diameter, α is the impact angle ofthe particle path with the wall face, f(α) is a function of impact angle, vis the relative particle velocity, and b(v) is a function of relative particlevelocity. Default values are C = 1, f = 1, and b = 0.

Since C, f , and b are defined as boundary conditions at a wall, ratherthan properties of a material, the default values are not updated to reflectthe material being used. You will need to specify appropriate values atall walls. Values of these functions for sand eroding both carbon steeland aluminum are given by Edwards et al. [60].

Note that the erosion rate as calculated above is displayed as dimension-less (that is, no units are listed) to provide some flexibility. The functionsC and f can be defined so that they account for the wall material density,resulting in erosion-rate units of length/time (mm/year, for example).When the default values for C and f are used, the erosion-rate units aremass of material removed/(area-time).

Note that the particle erosion and accretion rates can be displayed onlywhen coupled calculations are enabled.

The accretion rate is defined as

Raccretion =Nparticles∑

p=1

mp

Aface(19.2-63)


19.3 Heat and Mass Transfer Calculations


Using FLUENT’s discrete phase modeling capability, reacting particlesor droplets can be modeled and their impact on the continuous phasecan be examined. Several heat and mass transfer relationships, termed“laws”, are available in FLUENT and the physical models employed inthese laws are described in this section.

19.3.1 Particle Types in FLUENT

Which laws are to be active depends upon the particle type that youselect. In the Set Injection Properties panel you will specify the ParticleType, and FLUENT will use a given set of heat and mass transfer lawsfor the chosen type. All particle types have pre-defined sequences ofphysical 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

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

You can also extend combusting particles to include an evaporating/boilingmaterial by selecting Wet Combustion in the Set Injection Properties panel.

FLUENT’s physical laws (Laws 1 through 6), which describe the heatand mass transfer conditions listed in this table, are explained in detailin Sections 19.3.2–19.3.6.

19.3.2 Law 1/Law 6: Inert Heating or Cooling

The inert heating or cooling laws (Laws 1 and 6) are applied while theparticle temperature is less than the vaporization temperature that you



define, Tvap, and after the volatile fraction, fv,0, of a particle has beenconsumed. These conditions may be written as

Law 1:Tp < Tvap (19.3-1)

Law 6:mp ≤ (1 − fv,0)mp,0 (19.3-2)

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

Law 1 is applied until the temperature of the particle/droplet reachesthe vaporization temperature. At this point a non-inert particle/dropletmay proceed to obey one of the mass-transfer laws (2, 3, 4, and/or 5),returning to Law 6 when the volatile portion of the particle/droplethas been consumed. (Note that the vaporization temperature, Tvap,is thus an arbitrary modeling constant used to define the onset of thevaporization/boiling/volatilization laws.)

When using Law 1 or Law 6, FLUENT uses a simple heat balance torelate the particle temperature, Tp(t), to the convective heat transferand the absorption/emission of radiation at the particle surface:

mpcpdTp

dt= hAp(T∞ − Tp) + εpApσ(θ4

R − T 4p ) (19.3-3)

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 19.3-3 assumes that there is negligible internal resistance toheat transfer, i.e., the particle is at uniform temperature throughout.



G is the incident radiation in W/m2:

G =∫Ω=4π

IdΩ (19.3-4)

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

Radiation heat transfer to the particle is included only if you have en-abled the P-1 or discrete ordinates radiation model and you have ac-tivated radiation heat transfer to particles using the Particle RadiationInteraction option in the Discrete Phase Model panel.

Equation 19.3-3 is integrated in time using an approximate, linearizedform that assumes that the particle temperature changes slowly fromone time value to the next:

mpcpdTp

dt= Ap

−[h+ εpσT

3p

]Tp +

[hT∞ + εpσθ

4R

](19.3-5)

As the particle trajectory is computed, FLUENT integrates Equation 19.3-5to obtain the particle temperature at the next time value, yielding

Tp(t+ ∆t) = αp + [Tp(t) − αp]e−βp∆t (19.3-6)

where ∆t is the integration time step and

αp =hT∞ + εpσθ

4R

h+ εpσT 3p (t)

(19.3-7)

and

βp =Ap(h+ εpσT

3p (t))

mpcp(19.3-8)

FLUENT can also solve Equation 19.3-5 in conjunction with the equiva-lent mass transfer equation using a stiff coupled solver. See Section 19.7.3for details.



The heat transfer coefficient, h, is evaluated using the correlation of Ranzand Marshall [185, 186]:

Nu =hdp

k∞= 2.0 + 0.6Re1/2

d Pr1/3 (19.3-9)

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 19.2-3)Pr = Prandtl number of the continuous phase (cpµ/k∞)

Finally, the heat lost or gained by the particle as it traverses each compu-tational cell appears as a source or sink of heat in subsequent calculationsof the continuous phase energy equation. During Laws 1 and 6, parti-cles/droplets do not exchange mass with the continuous phase and donot participate in any chemical reaction.

19.3.3 Law 2: Droplet Vaporization

Law 2 is applied to predict the vaporization from a discrete phase droplet.Law 2 is initiated when the temperature of the droplet reaches the va-porization temperature, Tvap, and continues until the droplet reaches theboiling point, Tbp, or until the droplet’s volatile fraction is completelyconsumed:

Tp < Tbp (19.3-10)

mp > (1 − fv,0)mp,0 (19.3-11)

The onset of the vaporization law is determined by the setting of Tvap,a temperature that has no other physical significance. Note that oncevaporization is initiated (by the droplet reaching this threshold temper-ature), it will continue even if the droplet temperature falls below Tvap.



Vaporization will be halted only if the droplet temperature falls belowthe dew point. In such cases, the droplet will remain in Law 2 but noevaporation will be predicted. When the boiling point is reached, thedroplet vaporization is predicted by a boiling rate, Law 3, as describedin Section 19.3.4.

Mass Transfer During Law 2

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

Ni = kc(Ci,s − Ci,∞) (19.3-12)

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 (evap-oration). If conditions exist in which Ni is negative (i.e., the droplettemperature falls below the dew point and condensation conditions ex-ist), FLUENT treats the droplet as inert (Ni = 0.0).

The concentration of vapor at the droplet surface is evaluated by as-suming that the partial pressure of vapor at the interface is equal to thesaturated vapor pressure, psat, at the particle droplet temperature, Tp:

Ci,s =psat(Tp)RTp

(19.3-13)

where R is the universal gas constant.

The concentration of vapor in the bulk gas is known from solution ofthe transport equation for species i or from the PDF look-up table fornon-premixed or partially premixed combustion calculations:



Ci,∞ = Xipop

RT∞(19.3-14)

where Xi is the local bulk mole fraction of species i, pop is the operatingpressure, and T∞ is the local bulk temperature in the gas.

The mass transfer coefficient in Equation 19.3-12 is calculated from aNusselt correlation [185, 186]:

NuAB =kcdp

Di,m= 2.0 + 0.6Re1/2

d Sc1/3 (19.3-15)

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 19.3-12 becomes a source of species iin the gas phase species transport equation, as specified by you (see Sec-tion 19.11) or from the PDF look-up table for non-premixed combustioncalculations.

The mass of the droplet is reduced according to

mp(t+ ∆t) = mp(t) −NiApMw,i∆t (19.3-16)

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 19.3-16 in conjunction with the equiva-lent heat transfer equation using a stiff coupled solver. See Section 19.7.3for details.

Defining the Vapor Pressure and Diffusion Coefficient

You define the vapor pressure as a polynomial or piecewise linear functionof temperature (psat(T )) during the problem definition. Note that thevapor pressure definition is critical, as psat is used to obtain the driving



force for the evaporation process (Equations 19.3-12 and 19.3-13). Youshould provide accurate vapor pressure values for temperatures over theentire range of possible droplet temperatures in your problem. Vaporpressure data can be obtained from a physics or engineering handbook(e.g., [175]).

You also input the diffusion coefficient, Di,m, during the setup of thediscrete phase material properties. Note that the diffusion coefficientinputs that you supply for the continuous phase are not used in thediscrete phase model.

Heat Transfer to the Droplet

Finally, the droplet temperature is updated according to a heat balancethat relates the sensible heat change in the droplet to the convective andlatent heat transfer between the droplet and the continuous phase:

mpcpdTp

dt= hAp(T∞ − Tp) +

dmp

dthfg +Apεpσ(θR

4 − Tp4) (19.3-17)

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)dmp

dt = 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 theradiation intensity


The heat transferred to or from the gas phase becomes a source/sinkof energy during subsequent calculations of the continuous phase energy



equation.

19.3.4 Law 3: Droplet Boiling

Law 3 is applied to predict the convective boiling of a discrete phasedroplet when the temperature of the droplet has reached the boilingtemperature, Tbp, and while the mass of the droplet exceeds the non-volatile fraction, (1 − fv,0):

Tp ≥ Tbp (19.3-18)

and

mp > (1 − fv,0)mp,0 (19.3-19)

When the droplet temperature reaches the boiling point, a boiling rateequation is applied [120]:

d(dp)dt

=4k∞

ρpcp,∞dp(1 + 0.23

√Red) ln

[1 +

cp,∞(T∞ − Tp)hfg

](19.3-20)

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 19.3-20 has been derived assuming steady flow at constant pres-sure. Note that the model requires T∞ > Tbp in order for boiling to occurand that the droplet remains at fixed temperature (Tbp) throughout theboiling law.

When radiation heat transfer is active, FLUENT uses a slight modifica-tion of Equation 19.3-20, derived by starting from Equation 19.3-17 andassuming that the droplet temperature is constant. This yields

−dmp

dthfg = hAp(T∞ − Tp) +Apεpσ(θR

4 − Tp4) (19.3-21)



or

−d(dp)dt

=2

ρphfg

[k∞Nudp

(T∞ − Tp) + εpσ(θ4R − T 4

p )

](19.3-22)

Using Equation 19.3-9 for the Nusselt number correlation and replacingthe Prandtl number term with an empirical constant, Equation 19.3-22becomes

−d(dp)dt

=2

ρphfg

[2k∞[1 + 0.23

√Red]

dp(T∞ − Tp) + εpσ(θ4

R − T 4p )

]

(19.3-23)

In the absence of radiation, this result matches that of Equation 19.3-20in the limit that the argument of the logarithm is close to unity. FLU-ENT uses Equation 19.3-23 when radiation is active in your model andEquation 19.3-20 when radiation is not active. Radiation heat transferto the particle is included only if you have enabled the P-1 or discreteordinates radiation model and you have activated radiation heat transferto particles using the Particle Radiation Interaction option in the DiscretePhase Model panel.

The droplet is assumed to stay at constant temperature while the boilingrate is applied. Once the boiling law is entered it is applied for theduration of the particle trajectory. The energy required for vaporizationappears as a (negative) source term in the energy equation for the gasphase. The evaporated liquid enters the gas phase as species i, as definedby your input for the destination species (see Section 19.11).

19.3.5 Law 4: Devolatilization

The devolatilization law is applied to a combusting particle when thetemperature of the particle reaches the vaporization temperature, Tvap,and remains in effect while the mass of the particle, mp, exceeds themass of the non-volatiles in the particle:

Tp ≥ Tvap and Tp ≥ Tbp (19.3-24)



and

mp > (1 − fv,0)(1 − fw,0)mp,0 (19.3-25)

where fw,0 is the mass fraction of the evaporating /boiling material ifWet Combustion is selected (otherwise, fw,0 = 0). As implied by Equa-tion 19.3-24, the boiling point Tbp and the vaporization temperature Tvap

should be set equal to each other when Law 4 is to be used. When wetcombustion is active, Tbp and Tvap refer to the boiling and evaporationtemperatures 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 physicalproperties for the combusting-particle material in the Materials panel,as described in Section 19.11.2. By default, the constant rate model(Equation 19.3-26) will be used.

The Constant Rate Devolatilization Model

The constant rate devolatilization law dictates that volatiles are releasedat a constant rate [13]:

− 1fv,0(1 − fw,0)mp,0

dmp

dt= A0 (19.3-26)



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 adefault value of 12 s−1 derived from the work of Pillai [180] on coal com-bustion. Proper use of the constant devolatilization rate requires thatthe vaporization temperature, which controls the onset of devolatiliza-tion, be set appropriately. Values in the literature show this temperatureto be about 600 K [13].

The volatile fraction of the particle enters the gas phase as the devolatiliz-ing species i, defined by you (see Section 19.11). Once in the gas phase,the volatiles may react according to the inputs governing the gas phasechemistry.

The Single Kinetic Rate Model

The single kinetic rate devolatilization model assumes that the rate ofdevolatilization is first-order dependent on the amount of volatiles re-maining in the particle [5]:

−dmp

dt= k[mp − (1 − fv,0)(1 − fw,0)mp,0] (19.3-27)

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 de-fined using a value slightly in excess of that determined by proximateanalysis. The kinetic rate, k, is defined by input of an Arrhenius typepre-exponential factor and an activation energy:



k = A1e−(E/RT ) (19.3-28)

FLUENT uses default rate constants, A1 and E, as given in [5].

Equation 19.3-27 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 (19.3-29)

which is obtained by assuming that the particle temperature varies onlyslightly between discrete time integration steps.

FLUENT can also solve Equation 19.3-29 in conjunction with the equiva-lent heat transfer equation using a stiff coupled solver. See Section 19.7.3for details.

The Two Competing Rates Kobayashi Model

FLUENT also provides the kinetic devolatilization rate expressions of theform proposed by Kobayashi [117]:

R1 = A1e−(E1/RTp) (19.3-30)

R2 = A2e−(E2/RTp) (19.3-31)

where R1 and R2 are competing rates that may control the devolatiliza-tion over different temperature ranges. The two kinetic rates are weightedto yield an expression for the devolatilization as

mv(t)(1 − fw,0)mp,0 −ma

=∫ t

0(α1R1 + α2R2) exp

(−∫ t

0(R1 + R2) dt

)dt

(19.3-32)



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, and E2, and the yields of the two competing reactions, α1 andα2. FLUENT uses default values for the yield factors of 0.3 for the first(slow) reaction and 1.0 for the second (fast) reaction. It is recommendedin the literature [117] that α1 be set to the fraction of volatiles determinedby proximate analysis, since this rate represents devolatilization at lowtemperature. The second yield parameter, α2, should be set close tounity, which is the yield of volatiles at very high temperature.

By default, Equation 19.3-32 is integrated in time analytically, assumingthe particle temperature to be constant over the discrete time integra-tion step. FLUENT can also solve Equation 19.3-32 in conjunction withthe equivalent heat transfer equation using a stiff coupled solver. SeeSection 19.7.3 for details.

The CPD Model

In contrast to the coal devolatilization models presented above, whichare based on empirical rate relationships, the chemical percolation de-volatilization (CPD) model characterizes the devolatilization behavior ofrapidly heated coal based on the physical and chemical transformationsof the coal structure [68, 69, 81].

General Description

During coal pyrolysis, the labile bonds between the aromatic clusters inthe coal structure lattice are cleaved, resulting in two general classes offragments. One set of fragments has a low molecular weight (and corre-spondingly high vapor pressure) and escapes from the coal particle as alight gas. The other set of fragments consists of tar gas precursors thathave a relatively high molecular weight (and correspondingly low vaporpressure) and tend to remain in the coal for a long period of time duringtypical devolatilization conditions. During this time, reattachment withthe coal lattice (which is referred to as crosslinking) can occur. The high



molecular weight compounds plus the residual lattice are referred to asmetaplast. The softening behavior of a coal particle is determined by thequantity and nature of the metaplast generated during devolatilization.The portion of the lattice structure that remains after devolatilization iscomprised of char and mineral-compound-based ash.

The CPD model characterizes the chemical and physical processes byconsidering the coal structure as a simplified lattice or network of chem-ical bridges that link the aromatic clusters. Modeling the cleavage ofthe bridges and the generation of light gas, char, and tar precursors isthen considered to be analogous to the chemical reaction scheme shownin Figure 19.3.1.

£ k b

£ *

2 δ 2g1

c+ 2g2

k c

k δ

k g

Figure 19.3.1: Coal Bridge

The variable £ represents the original population of labile bridges inthe coal lattice. Upon heating, these bridges become the set of reactivebridges, £∗. For the reactive bridges, two competing paths are available.In one path, the bridges react to form side chains, δ. The side chainsmay detach from the aromatic clusters to form light gas, g1. As bridgesbetween neighboring aromatic clusters are cleaved, a certain fraction ofthe coal becomes detached from the coal lattice. These detached aro-matic clusters are the heavy-molecular-weight tar precursors that formthe metaplast. The metaplast vaporizes to form coal tar. While wait-ing for vaporization, the metaplast can also reattach to the coal latticematrix (crosslinking). In the other path, the bridges react and becomea char bridge, c, with the release of an associated light gas product,g2. The total population of bridges 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 structureduring devolatilization, the following set of reaction rate expressionscan be defined for each, starting with the assumption that the reac-tive bridges are destroyed at the same rate at which they are created(∂£∗

∂t = 0):

d£dt

= −kb£ (19.3-33)

dc

dt= kb

£ρ+ 1

(19.3-34)

dδ

dt=

[2ρkb

£ρ+ 1

]− kgδ (19.3-35)

dg1dt

= kgδ (19.3-36)

dg2dt

= 2dc

dt(19.3-37)

where the rate constants for bridge breaking and gas release steps, kb

and kg, are expressed in Arrhenius form with a distributed activationenergy:

k = Ae−(E±Eσ)/RT (19.3-38)

where A, E, and Eσ are, respectively, the pre-exponential factor, the ac-tivation energy, and the distributed variation in the activation energy, Ris the universal gas constant, and T is the temperature. The ratio of rateconstants, ρ = kδ/kc, is set to 0.9 in this model based on experimentaldata.

Mass Conservation

The following mass conservation relationships are imposed:



g = g1 + g2 (19.3-39)g1 = 2f − σ (19.3-40)g2 = 2(c − c0) (19.3-41)

where f is the fraction of broken bridges (f = 1 − p). The initial condi-tions for this system are given by the following:

c(0) = c0 (19.3-42)£(0) = £0 = p0 − c0 (19.3-43)δ(0) = 2f0 = 2(1 − c0 − £0) (19.3-44)g(0) = g1(0) = g2(0) = 0 (19.3-45)

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

Fractional Change in the Coal Mass

Given the set of reaction equations for the coal structure parameters, itis necessary to relate these quantities to changes in coal mass and therelated release of volatile products. To accomplish this, the fractionalchange in the coal mass as a function of time is divided into three parts:light gas (fgas), tar precursor fragments (ffrag), and char (fchar). Thisis accomplished by using the following relationships, which are obtainedusing percolation lattice statistics:

fgas(t) =r(g1 + g2)(σ + 1)

4 + 2r(1 − c0)(σ + 1)(19.3-46)

ffrag(t) =2

2 + r(1 − c0)(σ + 1)[ΦF (p) + rΩK(p)] (19.3-47)

fchar(t) = 1 − fgas(t) − ffrag(t) (19.3-48)



The variables Φ, Ω, F (p), and K(p) are the statistical relationships re-lated to the cleaving of bridges based on the percolation lattice statistics,and are given by the following equations:

Φ = 1 + r

[£p

+(σ − 1)δ4(1 − p)

](19.3-49)

Ω =δ

2(1 − p)− £p

(19.3-50)

F (p) =(p′

p

)σ+1σ−1

(19.3-51)

K(p) =[1 −

(σ + 1

2

)p′](

p′

p

) σ+1σ−1

(19.3-52)

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

mb = 2Mw,δ (19.3-53)ma = Mw,1 − (σ + 1)Mw,δ (19.3-54)

whereMw,δ and Mw,1 are the side chain and cluster molecular weights re-spectively. σ+1 is the lattice coordination number, which is determinedfrom solid-state Nuclear Magnetic Resonance (NMR) measurements re-lated to coal structure parameters, and p′ is the root of the followingequation in p (the total number of bridges in the coal lattice matrix):

p′(1 − p′)σ−1 = p(1 − p)σ−1 (19.3-55)

In accounting for mass in the metaplast (tar precursor fragments), thepart that vaporizes is treated in a manner similar to flash vaporization,where it is assumed that the finite fragments undergo vapor/liquid phaseequilibration on a time scale that is rapid with respect to the bridgereactions. As an estimate of the vapor/liquid that is present at any time,a vapor pressure correlation based on a simple form of Raoult’s Law isused. The vapor pressure treatment is largely responsible for predicting



pressure-dependent devolatilization yields. For the part of the metaplastthat reattaches to the coal lattice, a cross-linking rate expression givenby the following equation is used:

dmcross

dt= mfragAcrosse

−(Ecross/RT ) (19.3-56)

wheremcross is the amount of mass reattaching to the matrix, mfrag is theamount of mass in the tar precursor fragments (metaplast), and Across

and Ecross are rate expression constants.

CPD Inputs

Given the set of equations and corresponding rate constants introducedfor the CPD model, the number of constants that must be defined to usethe model is a primary concern. For the relationships defined previously,it can be shown that the following parameters are coal-independent [68]:

• 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 notinput or modified during problem setup.

There are an additional five parameters that are coal-specific and mustbe specified during the 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 obtainedfrom NMR experimental data. The last quantity, representing the charbridges that either exist in the parent coal or are formed very early in thedevolatilization process, is estimated based on the coal rank. These quan-tities are entered in the Materials panel as described in Section 19.11.2.Values for the coal-dependent parameters for a variety of coals are listedin Table 19.3.1.

Table 19.3.1: Chemical Structure Parameters for 13C NMR for 13 Coals

Coal Type σ + 1 p0 Mw,1 Mw,δ c0Zap (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 [224, 251].

Sandia refers to the coal examined at Sandia National Laboratories [67]. AFR refers

to coal examined at Advanced Fuel Research. ACERC refers to three types of coal

examined at the Advanced Combustion Engineering Research Center.

Particle Swelling During Devolatilization

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



dp

dp,0= 1 + (Csw − 1)

(1 − fw,0)mp,0 −mp

fv,0(1 − fw,0)mp,0(19.3-57)

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

The term (1−fw,0)mp,0−mp

fv,0(1−fw,0)mp,0is the ratio of the mass that has been de-

volatilized to the total volatile mass of the particle. This quantity ap-proaches a value of 1.0 as the devolatilization law is applied. When theswelling coefficient is equal to 1.0, the particle diameter stays constant.When the swelling coefficient is equal to 2.0, the final particle diameterdoubles when all of the volatile component has vaporized, and when theswelling coefficient is equal to 0.5 the final particle diameter is half of itsinitial diameter.

Heat Transfer to the Particle During Devolatilization

Heat transfer to the particle during the devolatilization process includescontributions from convection, radiation (if active), and the heat con-sumed during devolatilization:

mpcpdTp

dt= hAp(T∞ − Tp) +

dmp

dthfg +Apεpσ(θR

4 − Tp4) (19.3-58)


By default, Equation 19.3-58 is solved analytically, by assuming that thetemperature and mass of the particle do not change significantly betweentime steps:

Tp(t+ ∆t) = αp + [Tp(t) − αp]e−βpt (19.3-59)

where



αp =hApT∞ + dmp

dt hfg +ApεpσθR4

hAp + εpApσTp3 (19.3-60)

and

βp =Ap(h+ εpσTp

3)mpcp

(19.3-61)

FLUENT can also solve Equation 19.3-58 in conjunction with the equiva-lent mass transfer equation using a stiff coupled solver. See Section 19.7.3for details.

19.3.6 Law 5: Surface Combustion

After the volatile component of the particle is completely evolved, asurface reaction begins, which consumes the combustible fraction, fcomb,of the particle. Law 5 is thus active (for a combusting particle) after thevolatiles are evolved:

mp < (1 − fv,0)(1 − fw,0)mp,0 (19.3-62)

and until the combustible fraction is consumed:

mp > [(1 − fv,0)(1 − fw,0) − fcomb]mp,0 (19.3-63)

When the combustible fraction, fcomb, has been consumed in Law 5, thecombusting particle may contain residual “ash” that reverts to the inertheating law, Law 6 (see Section 19.3.2).

With the exception of the multiple surface reactions model, the surfacecombustion law consumes the reactive content of the particle as governedby the stoichiometric requirement, Sb, of the surface “burnout” reaction:

char(s) + Sbox(g) −→ products(g) (19.3-64)



where Sb is defined in terms of mass of oxidant per mass of char, and theoxidant and product species are defined in the Set Injection Propertiespanel.

FLUENT provides a choice of four heterogeneous surface reaction ratemodels for combusting 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 thesurface combustion model when you are setting physical properties forthe combusting-particle material in the Materials panel, as described inSection 19.11.2. By default, the diffusion-limited rate model will be used.

Diffusion-Limited Surface Reaction Rate Model

The diffusion-limited surface reaction rate model, the default model inFLUENT, assumes that the surface reaction proceeds at a rate determinedby the diffusion of the gaseous oxidant to the surface of the particle:

dmp

dt= −4πdpDi,m

YoxT∞ρg

Sb(Tp + T∞)(19.3-65)

where Di,m = diffusion coefficient for oxidant in the bulk (m2/s)Yox = local mass fraction of oxidant in the gasρg = gas density (kg/m3)Sb = stoichiometry of Equation 19.3-64

Equation 19.3-65 is derived from the model of Baum and Street [13]with the kinetic contribution to the surface reaction rate ignored. Thediffusion-limited rate model assumes that the diameter of the particlesdoes not change. Since the mass of the particles is decreasing, the effec-tive density decreases, and the char particles become more porous.



Kinetic/Diffusion Surface Reaction Rate Model

The kinetic/diffusion-limited rate model assumes that the surface reac-tion rate is determined either by kinetics or by a diffusion rate. FLUENTuses the model of Baum and Street [13] and Field [65], in which a diffu-sion rate coefficient

D0 = C1[(Tp + T∞)/2]0.75

dp(19.3-66)

and a kinetic rate

R = C2e−(E/RTp) (19.3-67)

are weighted to yield a char combustion rate of

dmp

dt= −πd2

ppoxD0RD0 + R (19.3-68)

where pox is the partial pressure of oxidant species in the gas surroundingthe combusting particle, and the kinetic rate, R, incorporates the effectsof chemical reaction on the internal surface of the char particle (intrinsicreaction) and pore diffusion. In FLUENT, Equation 19.3-68 is recast interms of the oxidant mass fraction, Yox, as

dmp

dt= −πd2

p

ρRTYox

Mw,ox

D0RD0 + R (19.3-69)

The particle size is assumed to remain constant in this model while thedensity is allowed to decrease.

When this model is enabled, the rate constants used in Equations 19.3-66and 19.3-67 are entered in the Materials panel, as described in Sec-tion 19.11.



Intrinsic Model

The intrinsic model in FLUENT is based on Smith’s model [218], assum-ing the order of reaction is equal to unity. Like the kinetic/diffusionmodel, the intrinsic model assumes that the surface reaction rate in-cludes the effects of both bulk diffusion and chemical reaction (see Equa-tion 19.3-69). The intrinsic model uses Equation 19.3-66 to compute thediffusion rate coefficient, D0, but the chemical rate, R, is explicitly ex-pressed in terms of the intrinsic chemical and pore diffusion rates:

R = ηdp

6ρpAgki (19.3-70)

η is the effectiveness factor, or the ratio of the actual combustion rate tothe rate attainable if no pore diffusion resistance existed [130]:

η =3φ2

(φ coth φ− 1) (19.3-71)

where φ is the Thiele modulus:

φ =dp

2

[SbρpAgkipox

DeCox

]1/2

(19.3-72)

Cox is the concentration of oxidant in the bulk gas (kg/m3) and De isthe effective diffusion coefficient in the particle pores. Assuming that thepore size distribution is unimodal and the bulk and Knudsen diffusionproceed in parallel, De is given by

De =θ

τ2

[1

DKn+

1D0

]−1

(19.3-73)

where D0 is the bulk molecular diffusion coefficient and θ is the porosityof the char particle:

θ = 1 − ρp

ρt(19.3-74)



ρp and ρt are, respectively, the apparent and true densities of the pyrol-ysis char.

τ (in Equation 19.3-73) is the tortuosity of the pores. The default valuefor τ in FLUENT is

√2, which corresponds to an average intersecting

angle between the pores and the external surface of 45 [130].

DKn is the Knudsen diffusion coefficient:

DKn = 97.0rp

√Tp

Mw,ox(19.3-75)

where Tp is the particle temperature and rp is the mean pore radiusof the char particle, which can be measured by mercury porosimetry.Note that macropores (rp > 150 A) dominate in low-rank chars whilemicropores (rp < 10 A) dominate in high-rank chars [130].

Ag (in Equations 19.3-70 and 19.3-72) is the specific internal surfacearea of the char particle, which is assumed in this model to remainconstant during char combustion. Internal surface area data for variouspyrolysis chars can be found in [217]. The mean value of the internalsurface area during char combustion is higher than that of the pyrolysischar [130]. For example, an estimated mean value for bituminous charsis 300 m2/g [33].

ki (in Equations 19.3-70 and 19.3-72) is the intrinsic reactivity, which isof Arrhenius form:

ki = Aie−(Ei/RTp) (19.3-76)

where the pre-exponential factor Ai and the activation energy Ei canbe measured for each char. In the absence of such measurements, thedefault values provided by FLUENT (which are taken from a least squaresfit of data of a wide range of porous carbons, including chars [217]) canbe used.

To allow a more adequate description of the char particle size (and hencedensity) variation during combustion, you can specify the burning mode



α, relating the char particle diameter to the fractional degree of burnoutU (where U = 1 −mp/mp,0) by [216]

dp

dp,0= (1 − U)α (19.3-77)

where mp is the char particle mass and the subscript zero refers to initialconditions (i.e., at the start of char combustion). Note that 0 ≤ α ≤1/3 where the limiting values 0 and 1/3 correspond, respectively, to aconstant size with decreasing density (zone 1) and a decreasing size withconstant 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 found to work well for a variety of chars [216].

When this model is enabled, the rate constants used in Equations 19.3-66,19.3-70, 19.3-72, 19.3-73, 19.3-75, 19.3-76, and 19.3-77 are entered in theMaterials panel, as described in Section 19.11.

The Multiple Surface Reactions Model

Modeling multiple char reactions follows the same pattern as the wallsurface reaction models, where the surface species is now a “particle sur-face species”. The particle surface species can be depleted or producedby the stoichiometry of the particle surface reaction (defined in the Reac-tions panel) for the mixture material defined in the Species Model panel.If a particle surface species is depleted, the reactive “char” content ofthe particle is consumed. In turn, if a surface species is produced bythe particle surface reaction, the species is added to the particle residual“ash” mass. Any number of particle surface species and any number ofparticle surface reactions can be defined for any given combusting par-ticle; however, you must have only one particle surface species in thereactants list of a particle reaction.

Multiple injections can be accommodated, and combusting particles re-acting according to the multiple surface reactions model can coexist inthe calculation with combusting particles following other char combus-tion laws. The model is based on oxidation studies of char particles,



but it is applicable to gas-solid reactions in general, not only to charoxidation reactions.

See Section 13.3 for information about particle surface reactions.

Limitations

Note the following limitations of the multiple surface reactions model:

• The model is not available together with the unsteady trackingoption.

• The model is available only with the species transport model forvolumetric reactions, and not with the non-premixed, premixed, orpartially premixed combustion models.

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 thetransport equation for this species. Similarly, the surface reaction is asource of species in the gas phase: the product of the heterogeneoussurface reaction appears in the gas phase as a user-selected chemicalspecies. The surface reaction also consumes or produces energy, in anamount determined by the heat of reaction defined by you.

The particle heat balance during surface reaction is

mpcpdTp

dt= hAp(T∞ − Tp)− fh

dmp

dtHreac +Apεpσ(θR

4 − Tp4) (19.3-78)

where Hreac is the heat released by the surface reaction. Note that onlya portion (1−fh) of the energy produced by the surface reaction appearsas a heat source in the gas-phase energy equation: the particle absorbs afraction fh of this heat directly. For coal combustion, it is recommendedthat fh be set at 1.0 if the char burnout product is CO and 0.3 if thechar burnout product is CO2 [24].




By default, Equation 19.3-78 is solved analytically, by assuming that thetemperature and mass of the particle do not change significantly betweentime steps. FLUENT can also solve Equation 19.3-78 in conjunction withthe equivalent mass transfer equation using a stiff coupled solver. SeeSection 19.7.3 for details.

19.3.7 Using Combusting Particles for General HeterogeneousSurface Reactions

The combusting particle type in FLUENT is presented with a focus onmodeling of coal particle combustion. You can, however, use this particletype to model general heterogeneous reactions on particles in which asolid particle reacts with a gas-phase component to form a single gas-phase product. For example,

4Al(s) + 3Cl2(g) → 2Al2Cl3(g)

This can be accomplished by simply omitting the devolatilization process(Law 4) by setting the fraction of volatiles to zero. In this case the surfacereaction law, Law 5, provides a general heterogeneous surface reactionthat consumes a gas-phase “oxidant” and produces a gas-phase productspecies defined by you.


19.4 Spray Models

19.4 Spray Models

In addition to the simple injection types described in Section 19.9.2,FLUENT also provides more complex injection types for sprays. Formost types of injections, you will need to provide the initial diameter,position, and velocity of the particles. For sprays, however, there aremodels available for droplet breakup and collision, as well as a dragcoefficient that accounts for variation in droplet shape. These modelsfor realistic spray simulations are described in this section.

Information is organized into the following subsections:

• Section 19.4.1: Atomizer Models

• Section 19.4.2: Droplet Collision Model

• Section 19.4.3: Spray Breakup Models

• Section 19.4.4: Dynamic Drag Model

19.4.1 Atomizer Models

Five atomizer models are available in FLUENT:

• 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 param-eters in the Set Injection Properties panel, as described in Section 19.9.2.Details about the atomizer models are provided below.



General Information

All of the models use physical and numerical atomizer parameters, suchas orifice diameter and mass flow rate, to calculate initial droplet size,velocity, and position.

For realistic atomizer simulations, the droplets must be randomly dis-tributed, both through a dispersion angle and in their time of release.For the other types of injections in FLUENT (non-atomizer), all of thedroplets are released along fixed trajectories and at the beginning of thetime step. The atomizer models use stochastic trajectory selection andstaggering to attain random distribution.

Stochastic trajectory selection is the random dispersion of initial dropletdirections. All of the atomizer models provide an initial dispersion angle,and the stochastic trajectory selection picks an initial direction withinthis angle. This approach improves the accuracy of the results for spray-dominated flows. The droplets will be more evenly spread among thecomputational cells near the atomizer, which improves the coupling tothe gas phase by spreading drag more smoothly over the cells near theinjection.

The Plain-Orifice Atomizer Model

The plain-orifice is the most common type of atomizer and the mostsimply made. However there is nothing simple about the physics of theinternal nozzle flow and the external atomization. In the plain-orificeatomizer, the liquid is accelerated through a nozzle, forms a liquid jet,and then forms droplets. This apparently simple process is dauntinglycomplex. The plain orifice may operate in three different regimes: single-phase, cavitating, and flipped [225]. The transition between regimes isabrupt, producing dramatically different sprays. The internal regimedetermines the velocity at the orifice exit, as well as the initial dropletsize and the angle of droplet dispersion. Diagrams of each case are shownin Figures 19.4.1, 19.4.2, and 19.4.3.


19.4 Spray Models

downstreamgas

liquid jet

orifice walls

d

L

p1

p2

r

Figure 19.4.1: Single-Phase Nozzle Flow (Liquid completely fills the ori-fice.)

vapor

vapor

downstreamgas

liquid jet

orifice walls

Figure 19.4.2: Cavitating Nozzle Flow (Vapor pockets form just afterthe inlet corners.)



downstreamgas

liquid jet

orifice walls

Figure 19.4.3: Flipped Nozzle Flow (Downstream gas surrounds the liq-uid jet inside the nozzle.)

Internal Nozzle State

The plain-orifice model must identify the correct state for the nozzle flow,because the internal nozzle state has a tremendous effect on the externalspray. Unfortunately, there is no established theory for determining thenozzle state. One must rely on empirical models that fix experimentaldata. A suggested list of the governing parameters for the internal nozzleflow is given in Table 19.4.1.

Table 19.4.1: List of Governing Parameters for Internal Nozzle Flow

nozzle diameter d

nozzle length L

radius of curvature of the inlet corner r

upstream pressure p1

downstream pressure p2

viscosity µ

liquid density ρl

vapor pressure pv

These may be combined to form geometric non-dimensional groups suchas r/d and L/d, as well as the Reynolds number based on “head” (Reh)


19.4 Spray Models

and a cavitation parameter (K).

Reh =dρl

µ

√2(p1 − p2)

ρl(19.4-1)

K =p1 − pv

p1 − p2(19.4-2)

The liquid flow often contracts in the nozzle, as can be seen in Fig-ures 19.4.2 and 19.4.3. Nurick [166] found it helpful to use a coefficientof contraction (Cc) that represents the area of the stream of contractingliquid over the total cross-sectional area of the nozzle. FLUENT usesNurick’s fit for the coefficient of contraction:

Cc =1√

1Cct

− 11.4rd

(19.4-3)

Cct is a theoretical constant equal to 0.611, which comes from potentialflow analysis of flipped nozzles.

Another important parameter used to describe the performance of noz-zles is the coefficient of discharge (Cd). The coefficient of discharge is aratio of the mass flow rate through the nozzle, divided by the theoreticalmaximum mass flow rate:

Cd =m

A√

2ρl(p1 − p2)(19.4-4)

The cavitation number (K in Equation 19.4-2) is an essential parameterfor predicting the inception of cavitation. The inception of cavitation isknown to occur at a value of Kincep ≈ 1.9 for short, sharp-edged nozzles.However, to include some of the effects of inlet rounding and viscosity,an empirical relationship is used:

Kincep = 1.9(

1 − r

d

)2

− 1000Reh

(19.4-5)



Similarly, a critical value of K where flip occurs is defined as Kcrit:

Kcrit = 1 +1(

1 + L4d

) (1 + 2000

Reh

)e70r/d

(19.4-6)

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

These variables are then used in a decision tree to identify the nozzlestate. The decision tree is shown in Figure 19.4.4. Depending on thestate of the nozzle, a unique closure is chosen for the above equations.

For a single-phase nozzle [137],

Cdu = 0.827 − 0.0085L

d(19.4-7)

Cd =1

1Cdu

+ 20 (1+2.25L/d)Reh

(19.4-8)

Equation 19.4-7 is for the ultimate coefficient of discharge, Cdu. Equa-tion 19.4-8 corrects this ultimate coefficient of discharge for the effectsof viscosity.

For a cavitating nozzle [166],

Cd = Cc

√K (19.4-9)

For a flipped nozzle [166],

Cd = Cct (19.4-10)

All of the nozzle flow equations are solved iteratively, along with theappropriate relationship for coefficient of discharge as given by the nozzlestate. The nozzle state may change as the upstream or downstreampressures change. Once the nozzle state is determined, the exit velocityis found, and appropriate correlations for spray angle and initial dropletsize distribution are determined.


19.4 Spray Models

flipped cavitating flipped single phase

K < K K ≥ K K <K K ≥ K

K ≤ K K > Kincep

crit

incep

crit crit crit

Figure 19.4.4: Decision Tree for the State of the Cavitating Nozzle

Exit Velocity

The estimate of exit velocity (u) for the single-phase nozzle comes fromconservation of mass and the assumption of a uniform exit velocity:

u =m

ρlA(19.4-11)

For the cavitating nozzle, Schmidt and Corradini [204] have shown thatthe uniform exit velocity is not accurate. Instead, they derived an ex-pression for a higher velocity over a reduced area:

u =2Ccp1 − p2 + (1 − 2Cc)pv

Cc

√2ρl(p1 − pv)

(19.4-12)

This analytical relation is used for cavitating nozzles in FLUENT.

For the case of flip, the exit velocity is found from conservation of massand the value of the reduced flow area:

u =m

ρlCctA(19.4-13)



Spray Angle

The correlation for spray angle (θ) comes from the work of Ranz [184]:

θ

2= tan−1

[4πCA

√ρg

ρl

√3

6

](19.4-14)

θ

2= 0.01 (19.4-15)

Equation 19.4-14 describes the spray angle for both single-phase andcavitating nozzles. For flipped nozzles, the spray angle has a constantvalue (Equation 19.4-15).

CA is thought to be a constant for a given nozzle geometry. You mustchoose the value for CA. The larger the value, the narrower the spray.Reitz [189] suggests the following correlation for CA:

CA = 3 +L

3.6d(19.4-16)

The spray angle is sensitive to the internal flow regime of the nozzle.Hence, you may wish to choose smaller values of CA for cavitating nozzlesthan for single-phase nozzles. Typical values are from 4.0 to 6.0. Thespray angle for flipped nozzles is a small, arbitrary value that representsthe lack of any turbulence or initial disturbance from the nozzle.

Droplet Diameter Distribution

Finally, there must be a droplet diameter distribution for the injection.The droplet diameter distribution is closely related to the nozzle state.FLUENT’s spray models use the most probable droplet size and a spreadparameter to define the Rosin-Rammler distribution. For more informa-tion about the Rosin-Rammler size distribution, see Section 19.9.7.

For single-phase nozzle flows, the correlation of Wu et al. [270] is used.This correlation relates the initial drop size to the estimated turbulencequantities of the liquid jet:


19.4 Spray Models

d32 = 133.0λWe−0.74 (19.4-17)

where d32 is the Sauter mean diameter, λ is the length scale, and We isthe Weber number, which, in this case, is defined to be

We ≡ ρlu2λ

σ(19.4-18)

where λ = d/8 and σ is the droplet surface tension. For a more de-tailed discussion of droplet surface tension and the Weber number, seeSection 19.4.3.

For cavitating nozzles, FLUENT uses a slight modification to Equa-tion 19.4-17. The initial jet diameter used in Wu’s correlation is cal-culated from the effective area of the cavitating orifice exit. For anexplanation of effective area of cavitating nozzles, see Schmidt and Cor-radini [204].

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

deff =

√4mπρlu

(19.4-19)

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

d0 = d√Cct (19.4-20)

where d0 is defined as the most probable diameter.

The values for the spread parameter, s, are chosen from past modelingexperience and from a review of experimental observations. Table 19.4.2lists the values of s for the three kinds of nozzles:

The larger the value of the spread parameter, the narrower the dropletsize distribution. The function that samples the Rosin-Rammler dis-tribution uses the most probable diameter and the spread parameter.



Table 19.4.2: 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 must be converted to the most probable diameter, d0. Lefeb-vre [132] gives the most general relationship between the Sauter meandiameter 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

(19.4-21)

At this point, the initialization of the droplets is complete.

The Pressure-Swirl Atomizer Model

Another important type of atomizer is the pressure-swirl atomizer, some-times referred to by the gas-turbine community as a simplex atomizer.This type of atomizer accelerates the liquid through nozzles known asswirl ports into a central swirl chamber. The swirling liquid pushesagainst the walls of the swirl chamber and develops a hollow air core.It then emerges from the orifice as a thinning sheet, which is unstable,breaking up into ligaments and droplets. The pressure-swirl atomizer isvery widely used for liquid-fuel combustion in gas turbines, oil furnaces,and direct-injection spark-ignited automobile engines. The transitionfrom internal injector flow to fully-developed spray can be divided intothree steps: film formation, sheet breakup, and atomization. A sketchof how this process is thought to occur is shown in Figure 19.4.5.

The interaction between the air and the sheet is not well understood. Itis generally accepted that an aerodynamic instability causes the sheet


19.4 Spray Models

film formation

atomization

sheet breakup

Figure 19.4.5: Theoretical Progression from the Internal Atomizer Flowto the External Spray

to break up. The mathematical analysis below assumes that Kelvin-Helmholtz waves grow on the sheet and eventually break the liquid intoligaments. It is then assumed that the ligaments break up into dropletsdue to varicose instability. Once the liquid forms droplets, the spraybehavior is determined by drag, collision, coalescence, and secondarybreakup.

The model used in this study is called the Linearized Instability SheetAtomization (LISA) model of Schmidt et al. [206]. The LISA model isdivided into two stages:

1. film formation

2. sheet breakup and atomization

Both parts of the model are described below. The implementation isslightly improved from that of Schmidt et al. [206].



Film Formation

The centrifugal motion of the liquid within the injector creates an aircore surrounded by a liquid film. The thickness of this film, t, is relatedto the mass flow rate by

m = πρut(dinj − t) (19.4-22)

where dinj is the injector exit diameter, and m is the mass flow rate,which must be measured experimentally. The other unknown in Equa-tion 19.4-22 is u, the axial component of velocity at the injector exit.This quantity depends on internal details of the injector and is diffi-cult to calculate from first principles. Instead, the approach of Han etal. [86] is used. The total velocity is assumed to be related to the injectorpressure by

U = kv

√2∆pρl

(19.4-23)

Lefebvre [132] has noted that kv is a function of the injector designand injection pressure. If the swirl ports are treated as nozzles, Equa-tion 19.4-23 is then an expression for the coefficient of discharge for theswirl ports, assuming that the majority of the pressure drop through theinjector occurs at the ports. The coefficient of discharge (Cd) for single-phase nozzles with sharp inlet corners and an L/d of 4 is typically 0.78or less [137]. If the nozzles are cavitating, the value of Cd may be as lowas 0.61. Hence, 0.78 should be a practical upper bound for kv. Reducingkv by 10% to allow for other momentum losses in the injector gives anestimate of 0.7.

Physical limits on kv are such that it must be less than unity by con-servation of energy, and it must be large enough to permit sufficientmass flow. To guarantee that the size of the air core is non-negative, thefollowing expression is used for kv:

kv = max[0.7,

4mπd2

0ρl cos θ

√ρl

2∆p

](19.4-24)


19.4 Spray Models

Assuming that ∆p is known, Equation 19.4-23 can be used to find U .Once U is determined, u is found from

u = U cos θ (19.4-25)

where θ is the spray angle, which is assumed to be known. The tan-gential component of velocity is assumed equal to the radial componentfurther downstream. The axial component of velocity is assumed to bea constant value.

Sheet Breakup and Atomization

The pressure-swirl atomizer includes the effects of the surrounding gas,liquid viscosity, and surface tension on the breakup of the liquid sheet.Details of the theoretical development of the model are given in Senecalet al. [207] and are only briefly presented here. For a more accurate androbust implementation, the gas-phase velocity is neglected in calculatingthe relative liquid-gas velocity. This decision avoids depending on theusually under-resolved gas-phase velocity field around the injector.

The model assumes that a two-dimensional, viscous, incompressible liq-uid sheet of thickness 2h moves with velocity U through a quiescent,inviscid, incompressible gas medium. The liquid and gas have densitiesof ρl and ρg, respectively, and the viscosity of the liquid is µl. A co-ordinate system is used that moves with the sheet, and a spectrum ofinfinitesimal disturbances of the form

η = η0eikx+ωt (19.4-26)

is imposed on the initially steady, motion-producing fluctuating velocitiesand pressures for both the liquid and the gas. In Equation 19.4-26 η0

is the initial wave amplitude, k = 2π/λ is the wave number, and ω =ωr + iωi is the complex growth rate. The most unstable disturbancehas the largest value of ωr, denoted here by Ω, and is assumed to beresponsible for sheet breakup. Thus, it is desired to obtain a dispersionrelation ω = ω(k) from which the most unstable disturbance can bededuced.



Squire [229] and Hagerty and Shea [84] have shown that two solutions,or modes, exist that satisfy the liquid governing equations subject tothe boundary conditions at the upper and lower interfaces. For thefirst solution, called the sinuous mode, the waves at the upper and lowerinterfaces are exactly in phase. On the other hand, for the varicose mode,the waves are π radians out of phase. It has been shown by numerousauthors (e.g., Senecal et al. [207]) that the sinuous mode dominates thegrowth of varicose waves for low velocities and low gas-to-liquid densityratios. In addition, it can be shown that the sinuous and varicose modesbecome indistinguishable for high-velocity flows. As a result, the presentdiscussion focuses on the growth of sinuous waves on the liquid sheet.

As derived in Senecal et al. [207], the dispersion relation for the sinuousmode is given by

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

4νlk4 tanh(kh) − 4ν2

l k3` tanh(`h) −QU2k2 +

σk3

ρl= 0 (19.4-27)

where Q = ρg/ρl and `2 = k2 + ω/νl.

It can be shown that above a critical Weber number of Weg = 27/16(based on the relative velocity, the gas density, and the sheet half-thick-ness), the fastest-growing waves are short. Below 27/16, the wavelengthsare long compared to the sheet thickness. The speed of modern fuelinjection is high enough that the film Weber number is often well abovethis critical limit.

Li and Tankin [136] derived a dispersion relation similar to Equation19.4-27 for a viscous sheet from a linear analysis with a stationary coor-dinate system. While Li and Tankin’s dispersion relation is quite general,a simplified relation has been presented in Senecal et al. [207] for use inmulti-dimensional simulations of pressure-swirl atomizers. The resultingexpression for the growth rate is given by

ωr =1

tanh(kh) +Q

−2νlk

2 tanh(kh) +


19.4 Spray Models

√4ν2

l k4 tanh2(kh) −Q2U2k2 − [tanh(kh) +Q]

[−QU2k2 +

σk3

ρl

]

(19.4-28)

Two main assumptions have been made to reduce Equation 19.4-27 toEquation 19.4-28. First, an order-of-magnitude analysis using typicalvalues from the inviscid solutions shows that the terms of second orderin viscosity can be neglected in comparison to the other terms in Equa-tion 19.4-28. In addition, the density ratio Q is on the order of 10−3 intypical applications and hence it is assumed that Q 1.

The physical mechanism of sheet disintegration proposed by Dombrowskiand Johns [52] is adopted only for long waves. For long waves, ligamentsare assumed to form from the sheet breakup process once the unstablewaves reach a critical amplitude. If the surface disturbance has reacheda value of ηb at breakup, a breakup time, τ , can be evaluated:

ηb = η0eΩτ ⇒ 1

Ωln(ηb

η0

)(19.4-29)

where Ω, the maximum growth rate, is found by numerically maximizingEquation 19.4-28 as a function of k. The maximum is found using abinary search that checks the sign of the derivative. The sheet breaksup and ligaments will be formed at a length given by

Lb = Uτ =U

Ωln(ηb

η0

)(19.4-30)

where the quantity ln( ηbη0

) is an empirical constant from 3 to 12. Youmust specify the value for this constant, which has a default value of 12.

The diameter of the ligaments formed at the point of breakup can beobtained from a mass balance. If it is assumed that the ligaments areformed from tears in the sheet once per wavelength, the resulting diam-eter is given by

dL =

√8hKs

(19.4-31)



where Ks is the wave number corresponding to the maximum growthrate, Ω. The ligament diameter depends on the sheet thickness, which isa function of the breakup length. The film thickness is calculated fromthe breakup length and the radial distance from the center line to themid-line of the sheet at the atomizer exit, r0:

hend =r0h0

r0 + Lb sin(

θ2

) (19.4-32)

This mechanism is not logical for short waves. For short waves, thedetermination of the ligament diameter is simpler. The value of dL isassumed to be linearly proportional to the wavelength that breaks upthe sheet. FLUENT allows you to control the constant of proportional-ity. The wavelength is calculated from the wave number, Ks. In eitherthe long wave or the short wave case, the breakup from ligaments todroplets is assumed to behave according to Weber’s [257] analysis forcapillary instability. The variable Oh is the Ohnesorge number and is acombination of Reynolds number and Weber number (see Section 19.4.3for more details about Oh):

d0 = 1.88dL(1 + 3Oh)1/6 (19.4-33)

This procedure determines the most probable droplet size. The spreadparameter is assumed to be 3.5, based on past modeling experience [205].You will specify the spray cone angle. The dispersion angle of the sprayis assumed to be a fixed value of 6.

The Air-Blast/Air-Assist Atomizer Model

In order to accelerate the breakup of liquid sheets from an atomizer, anadditional air stream is often directed through the atomizer. The liquidis formed into a sheet by a nozzle, and the air is then directed againstthe sheet to promote atomization. This technique is called air-assistedatomization or air-blast atomization, depending on the quantity of airand its velocity. The addition of the external air stream past the sheetproduces smaller droplets than without the air. The exact mechanism for


19.4 Spray Models

this enhanced performance is not completely understood. It is thoughtthat the assisting air may accelerate the sheet instability. The air mayalso help disperse the droplets, preventing collisions between them. Air-assisted atomization is used in many of the same fields as pressure-swirlatomization, where especially fine atomization is required.

FLUENT’s air-blast atomization model is a variation of the pressure-swirlmodel. One difference is that you set the sheet thickness directly in theair-blast atomizer model. This input is necessary because of the varietyof sheet formation mechanisms used in air-blast atomizers. Hence the air-blast atomizer model does not contain the sheet formation equations thatwere included in the pressure-swirl atomizer model (Equations 19.4-22–19.4-25). You will also specify the maximum relative velocity that isproduced by the sheet and air. Though this quantity could be calculated,specifying a value relieves you from the necessity of finely resolving theatomizer internal flow. This feature is convenient for simulations in largedomains, where the atomizer is very small by comparison.

Another difference is that the air-blast atomizer model assumes thatthe sheet breakup is always due to short waves. This assumption isa consequence of the greater sheet thickness commonly found in air-blast atomizers. Hence the ligament diameter is assumed to be linearlyproportional to the wavelength of the fastest-growing wave on the sheet.

Other inputs are similar to the pressure-swirl model. You must providethe mass flow rate and spray angle. The angle in the case of the air-blastatomizer is the initial trajectory of the film as it leaves the end of theorifice. The value of the angle is negative if the initial film trajectory isinward, towards the centerline. You will also provide the inner and outerdiameter of the film at the atomizer exit.

The air-blast atomizer model does not include the internal gas flows. Youmust create the atomizing air streams as a boundary condition withinthe FLUENT case. These streams are ordinary continuous-phase flowsand require no special treatment.



The Flat-Fan Atomizer Model

The flat-fan atomizer is very similar to the pressure-swirl atomizer, butit makes a flat sheet and does not use swirl. The liquid emerges from awide, thin orifice as a flat liquid sheet that breaks up into droplets. Theprimary atomization process is thought to be similar to the pressure-swirlatomizer. Some researchers believe that flat-fan atomization, because ofjet impingement, is very similar to the atomization of a flat sheet. Theflat-fan model could serve doubly for this application.

The flat-fan atomizer is available only for 3D models. An image of thethree-dimensional flat fan is shown in Figure 19.4.6. The model assumesthat the fan originates from a virtual origin. You will provide the locationof this origin, which is the intersection of the lines that mark the sides ofthe fan. You will also provide the location of the center point of the arcfrom which the fan originates. FLUENT will find the vector that pointsfrom the origin to the center point in order to determine the directionof the injection. You will also provide the half-angle of the fan arc, thewidth of the orifice (in the normal direction), and the mass flow rate ofthe liquid.

normal vector

virtual origin

center point

Figure 19.4.6: Flat Fan Viewed From Above and From the Side


19.4 Spray Models

The breakup of the flat fan is calculated very much like the breakupof the sheet in the pressure-swirl atomizer. The sheet breaks up intoligaments that then form droplets. The only difference is that for shortwaves, the flat fan sheet is assumed to form ligaments at half-wavelengthintervals. Hence the ligament diameter for short waves is given by

dL =

√16hKs

(19.4-34)

The Rosin-Rammler spread parameter is assumed to be 3.5 and thedispersion angle is set to 6. In all other respects, the flat-fan atomizermodel is like the sheet breakup portion of the pressure-swirl atomizer.

Effervescent Atomizer Model

Effervescent atomization is the injection of liquid infused with a super-heated (with respect to downstream conditions) liquid or propellant.As the volatile liquid exits the nozzle, it rapidly changes phase. Thisphase change quickly breaks up the stream into small droplets with awide dispersion angle. The model also applies to cases where a very hotliquid is discharged.

Since the physics of effervescence is not well understood, the model mustrely on rough empirical fits. The photographs of Reitz [189] provide somebasic insights. These photographs show a dense liquid core to the spray,surrounded by a wide shroud of smaller droplets.

The initial velocity of the droplets is computed from conservation ofmass, assuming the exiting jet has a cross-sectional area that is Cct

times the nozzle area, where Cct is a constant that you specify duringthe problem setup:

u =m

ρlCctA(19.4-35)

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



dmax = d√Cct (19.4-36)

The droplet size is then sampled from a Rosin-Rammler distributionwith a spread parameter of 4.0 (see Section 19.9.7). The most probabledroplet size depends on the angle, θ, between the droplet’s stochastictrajectory and the injection direction:

d0 = dmaxe−(θ/Θs)

2

(19.4-37)

The dispersion angle multiplier, Θs, is computed from the quality, x, andthe specified value for the dispersion constant, Ceff :

Θs =x

Ceff(19.4-38)

This technique creates a spray with large droplets in the central coreand a shroud of smaller surrounding droplets. The droplet temperatureis initialized to the initial temperature fraction, f , times the saturationtemperature of the droplets. f should be slightly less than 1.0, becausethe droplet temperatures should be close to boiling. To complete themodel, the flashing vapor must also be included in the calculation. Thisvapor is part of the continuous phase and not part of the discrete phasemodel. You must create an inlet at the point of injection when youspecify boundary conditions for the continuous phase.

When the effervescent atomizer model is selected, you will need to specifythe nozzle diameter, mass flow rate, mixture quality, saturation temper-ature of the volatile substance, temperature fraction, spray half-angle,and dispersion constant.

19.4.2 Droplet Collision Model

Introduction

When your simulation includes unsteady tracking of droplets, FLUENTprovides an option for estimating the number of droplet collisions andtheir outcomes in a computationally efficient manner. The difficulty in


19.4 Spray Models

any collision calculation is that for N droplets, each droplet has N − 1possible collision partners. Thus, the number of possible collision pairs isapproximately 1

2N2. (The factor of 1

2 appears because droplet A collidingwith droplet B is identical to droplet B colliding with droplet A. Thissymmetry reduces the number of possible collision events by half.)

An important consideration is that the collision algorithm must calculate12N

2 possible collision events at every time step. Since a spray can consistof several million droplets, the computational cost of a collision calcu-lation from first principles is prohibitive. This motivates the concept ofparcels. Parcels are statistical representations of a number of individualdroplets. For example, if FLUENT tracks a set of parcels, each of whichrepresents 1000 droplets, the cost of the collision calculation is reducedby a factor of 106. Because the cost of the collision calculation still scaleswith the square of N , the reduction of cost is significant; however, theeffort to calculate the possible intersection of so many parcel trajectorieswould still be prohibitively expensive.

The algorithm of O’Rourke [168] efficiently reduces the computationalcost of the spray calculation. Rather than using geometry to see if parcelpaths intersect, O’Rourke’s method is a stochastic estimate of collisions.O’Rourke also makes the assumption that two parcels may collide onlyif they are located in the same continuous-phase cell. These two as-sumptions are valid only when the continuous-phase cell size is smallcompared to the size of the spray. For these conditions, the method ofO’Rourke is second-order accurate at estimating the chance of collisions.The concept of parcels together with the algorithm of O’Rourke makesthe calculation of collision possible for practical spray problems.

Once it is decided that two parcels of droplets collide, the algorithmfurther determines the type of collision. Only coalescence and bouncingoutcomes are considered. The probability of each outcome is calculatedfrom the collisional Weber number and a fit to experimental observations.The properties of the two colliding parcels are modified based on theoutcome of the collision.



Use and Limitations

The collision model assumes that the frequency of collisions is much lessthan the particle time step. If the particle time step is too large, thenthe results may be time-step-dependent. You should adjust the particlelength scale accordingly. Additionally, the model is most applicable forlow-Weber-number collisions where collisions result in bouncing and co-alescence. Above a collisional Weber number of about 100, the outcomeof collision could be shattering.

Sometimes the collision model can cause grid-dependent artifacts to ap-pear in the spray. This is a result of the assumption that droplets cancollide only within the same cell. These tend to be visible when thesource of injection is at a mesh vertex. The coalescence of droplets tendsto cause the spray to pull away from cell boundaries. In two dimensions,a finer mesh and more computational droplets can be used to reducethese effects. In three dimensions, best results are achieved when thespray is modeled using a polar mesh with the spray at the center.

Theory

As noted above, O’Rourke’s algorithm assumes that two droplets maycollide only if they are in the same continuous-phase cell. This assump-tion can prevent droplets that are quite close to each other, but not inthe same cell, from colliding, although the effect of this error is lessenedby allowing some droplets that are farther apart to collide. The overallaccuracy of the scheme is second-order in space.

Probability of Collision

The probability of collision of two droplets is derived from the point ofview of the larger droplet, called the collector droplet and identified belowwith the number 1. The smaller droplet is identified in the followingderivation with the number 2. The calculation is in the frame of referenceof the larger droplet so that the velocity of the collector droplet is zero.Only the relative distance between the collector and the smaller droplet isimportant in this derivation. If the smaller droplet is on a collision coursewith the collector, the centers will pass within a distance of r1+r2. More


19.4 Spray Models

precisely, if the smaller droplet center passes within a flat circle centeredaround the collector of area π(r1 + r2)2 perpendicular to the trajectoryof the smaller droplet, a collision will take place. This disk can be usedto define the collision volume, which is the area of the aforementioneddisk multiplied by the distance traveled by the smaller droplet in onetime step, namely π(r1 + r2)2vrel∆t.

The algorithm of O’Rourke uses the concept of a collision volume tocalculate the probability of collision. Rather than calculate if the positionof the smaller droplet center is within the collision volume, the algorithmcalculates the probability of the smaller droplet being within the collisionvolume. It is known that the smaller droplet is somewhere within thecontinuous-phase cell of volume V . If there is a uniform probability ofthe 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 probability of the collector colliding with the smaller droplet is

P1 =π(r1 + r2)2vrel∆t

V(19.4-39)

Equation 19.4-39 can be generalized for parcels, where there are n1 andn2 droplets in the collector and smaller droplet parcels, respectively. Thecollector undergoes a mean expected number of collisions given by

n =n2π(r1 + r2)2vrel∆t

V(19.4-40)

The actual number of collisions that the collector experiences is not gen-erally the mean expected number of collisions. The probability distribu-tion of the number of collisions follows a Poisson distribution, accordingto O’Rourke, which is given by

P (n) = e−n nn

n!(19.4-41)

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 col-lision must be determined. In general, the outcome tends to be coales-cence if the droplets collide head-on, and bouncing if the collision is moreoblique. The critical offset is a function of the collisional Weber numberand 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,

2.4fWe

)(19.4-42)

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

f

(r1r2

)=(r1r2

)3

− 2.4(r1r2

)2

+ 2.7(r1r2

)(19.4-43)

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

a uniform deviate. The calculated value of b is compared to bcrit, and ifb < bcrit, the result of the collision is coalescence. Equation 19.4-41 givesthe number of smaller droplets that coalesce with the collector. Theproperties of the coalesced droplets are found from the basic conservationlaws.

In the case of a grazing collision, the new velocities are calculated basedon conservation of momentum and kinetic energy. It is assumed thatsome fraction of the kinetic energy of the droplets is lost to viscousdissipation and angular momentum generation. This fraction is relatedto 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

)(19.4-44)

This relation is used for each of the components of velocity. No otherdroplet properties are altered in grazing collisions.


19.4 Spray Models

19.4.3 Spray Breakup Models

FLUENT offers two spray breakup models: the Taylor Analogy Breakup(TAB) model and the wave model. The TAB model is recommended forlow-Weber-number injections and is well suited for low-speed sprays intoa standard atmosphere. For Weber numbers greater than 100, the wavemodel is more applicable. The wave model is popular for use in high-speed fuel-injection applications. Details for each model are providedbelow.

Taylor Analogy Breakup (TAB) Model

Introduction

The Taylor Analogy Breakup (TAB) model is a classic method for calcu-lating droplet breakup, which is applicable to many engineering sprays.This method is based upon Taylor’s analogy [239] between an oscillatingand distorting droplet and a spring mass system. Table 19.4.3 illustratesthe analogous components.

Table 19.4.3: Comparison of a Spring-Mass System to a DistortingDroplet

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 oscillatingand distorting droplet, can be solved to determine the droplet oscillationand distortion at any given time. As described in detail below, whenthe droplet oscillations grow to a critical value the “parent” droplet willbreak up into a number of smaller “child” droplets. As a droplet is dis-torted from a spherical shape, the drag coefficient changes. A drag modelthat incorporates the distorting droplet effects is available in FLUENT.See Section 19.4.4 for details.



Use and Limitations

The TAB model is best for low-Weber-number sprays. Extremely high-Weber-number sprays result in shattering of droplets, which is not de-scribed well by the spring-mass analogy.

Droplet Distortion

The equation governing a damped, forced oscillator is [169]

F − kx− ddx

dt= m

d2x

dt2(19.4-45)

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

F

m= CF

ρgu2

ρlr(19.4-46)

k

m= Ck

σ

ρlr3(19.4-47)

d

m= Cd

µl

ρlr2(19.4-48)

where ρl and ρg are the discrete phase and continuous phase densities,u is the relative velocity of the droplet, r is the undisturbed dropletradius, σ is the droplet surface tension, and µl is the droplet viscosity.The dimensionless constants CF , Ck, and Cd will be defined later.

The droplet is assumed to break up if the distortion grows to a criticalratio of the droplet radius. This breakup requirement is given as

x > Cbr (19.4-49)

where Cb is a constant equal to 0.5 if breakup is assumed to occur whenthe distortion is equal to the droplet radius, i.e., the north and south


19.4 Spray Models

poles of the droplet meet at the droplet center. This implicitly assumesthat the droplet is undergoing only one (fundamental) oscillation mode.Equation 19.4-45 is non-dimensionalized by setting y = x/(Cbr) andsubstituting the relationships in Equations 19.4-46–19.4-48:

d2y

dt2=CF

Cb

ρg

ρl

u2

r2− Ckσ

ρlr3y − Cdµl

ρlr2dy

dt(19.4-50)

where breakup now occurs for y > 1. For under-damped droplets, theequation governing y can easily be determined from Equation 19.4-50 ifthe relative velocity is assumed to be constant:

y(t) = Wec+e−(t/td)[(y0 − Wec) cos(ωt) +

1ω

(dy0

dt+y0 − Wec

td

)sin(ωt)

](19.4-51)

where

We =ρgu

2r

σ(19.4-52)

Wec =CF

CkCbWe (19.4-53)

y0 = y(0) (19.4-54)dy0

dt=

dy

dt(0) (19.4-55)

1td

=Cd

2µl

ρlr2(19.4-56)

ω2 = Ckσ

ρlr3− 1t2d

(19.4-57)

In Equation 19.4-51, u is the relative velocity between the droplet andthe gas phase and We is the droplet Weber number, a dimensionlessparameter defined as the ratio of aerodynamic forces to surface tensionforces. The droplet oscillation frequency is represented by ω. The con-stants have been chosen to match experiments and theory [122]:



Ck = 8Cd = 5

CF =13

If Equation 19.4-51 is solved for all droplets, those with y > 1 are as-sumed to break up. The size and velocity of the new child droplets mustbe determined.

Size of Child Droplets

The size of the child droplets is determined by equating the energy ofthe parent droplet to the combined energy of the child droplets. Theenergy of the parent droplet is [169]

Eparent = 4πr2σ +Kπ

5ρlr

5

[(dy

dt

)2

+ ω2y2

](19.4-58)

where K is the ratio of the total energy in distortion and oscillationto the energy in the fundamental mode, of the order (10

3 ). The childdroplets are assumed to be non-distorted and non-oscillating. Thus, theenergy of the child droplets can be shown to be

Echild = 4πr2σr

r32+π

6ρlr

5(dy

dt

)2

(19.4-59)

where r32 is the Sauter mean radius of the droplet size distribution. r32can be found by equating the energy of the parent and child droplets(i.e., Equations 19.4-58 and 19.4-59), setting y = 1, and ω2 = 8σ/ρlr

3:

r32 =r

1 + 8Ky2

20 + ρlr3(dy/dt)2

σ

(6K−5120

) (19.4-60)

Once the size of the child droplets is determined, the number of childdroplets can easily be determined by mass conservation.


19.4 Spray Models

Velocity of Child Droplets

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

vnormal = CvCbrdy

dt(19.4-61)

where Cv is a constant of order (1). Although this imposed velocity isassumed to be in a plane normal to the path of the parent droplet, theexact direction in this plane cannot be specified. Therefore, the directionof this imposed velocity is selected randomly, yet is confined in a planenormal to the parent relative velocity vector.

Droplet Breakup

To model droplet breakup, the TAB model first determines the amplitudefor an undamped oscillation (td ≈ ∞) for each droplet at time step nusing the following:

A =

√(yn − Wec)2 +

((dy/dt)n

ω

)2

(19.4-62)

According to Equation 19.4-62, breakup is possible only if the followingcondition is satisfied:

Wec +A > 1 (19.4-63)

This is the limiting case, as damping will only reduce the chance ofbreakup. If a droplet fails the above criterion, breakup does not occur.The only additional calculations required, then, are to update y using adiscretized form of Equation 19.4-51 and its derivative, which are both



based on work done by O’Rourke and Amsden [169]:

yn+1 = Wec+

e−(∆t/td)

(yn − Wec) cos(ωt) +1ω

[(dy

dt

)n

+yn − Wec

td

]sin(ωt)

(19.4-64)(

dy

dt

)n+1

=Wec − yn+1

td+

ωe−(∆t/td)

1ω

[(dy

dt

)n

+yn − Wec

td

]cos(ω∆t) − (yn − Wec) sin(ω∆t)

(19.4-65)

All of the constants in these expressions are assumed to be constantthroughout the time step.

If the criterion of Equation 19.4-63 is met, then breakup is possible.The breakup time, tbu, must be determined to see if breakup occurswithin the time step ∆t. The value of tbu is set to the time required foroscillations to grow sufficiently large that the magnitude of the dropletdistortion, y, is equal to unity. The breakup time is determined under theassumption that the droplet oscillation is undamped for its first period.The breakup time is therefore the smallest root greater than tn of anundamped version of Equation 19.4-51:

Wec +A cos[ω(t− tn) + φ] = 1 (19.4-66)

where

cos φ =yn − Wec

A(19.4-67)

and

sinφ = −(dy/dt)n

Aω(19.4-68)


19.4 Spray Models

If tbu > tn+1 , then breakup will not occur during the current time step,and y and (dy/dt) are updated by Equations 19.4-64 and 19.4-65. Thebreakup calculation then continues with the next droplet. Conversely,if tn < tbu < tn+1, then breakup will occur and the child droplet radiiare determined by Equation 19.4-60. The number of child droplets, N ,is determined by mass conservation:

Nn+1 = Nn(rn

rn+1

)3

(19.4-69)

A velocity component normal to the relative velocity vector, with magni-tude computed by Equation 19.4-61, is imposed upon the child droplets.It is assumed that the child droplets are neither distorted nor oscillating;i.e., y = (dy/dt) = 0.

The breakup process is applied to all of the droplets in the parcel (seeSection 19.4.2 for a description of parcels). Hence, there is no need tocreate another computational droplet after breakup. The TAB model inFLUENT changes the mass, size, and velocity of the current droplet only.

Wave Breakup Model

Introduction

An alternative to the TAB model is the wave breakup model of Re-itz [188], which considers the breakup of the injected liquid to be inducedby the relative velocity between the gas and liquid phases. The modelassumes that the time of breakup and the resulting droplet size are re-lated to the fastest-growing Kelvin-Helmholtz instability, derived fromthe jet stability analysis described below. The wavelength and growthrate of this instability are used to predict details of the newly-formeddroplets.

Use and Limitations

The wave model is appropriate for very-high-speed injection, where theKelvin-Helmholtz instability is believed to dominate spray breakup (We >100). Because breakup can increase the number of computational droplets,



you may wish to inject a modest number of droplets. You must also spec-ify the model constants, which are thought to depend on the internal flowof the spray nozzle.

Jet Stability Analysis

The jet stability analysis described in detail by Reitz and Bracco [187]is presented briefly here. The analysis considers the stability of a cylin-drical, viscous, liquid jet of radius a issuing from a circular orifice at avelocity v into a stagnant, incompressible, inviscid gas of density ρ2. Theliquid has a density, ρ1, and viscosity, µ1, and a cylindrical polar coordi-nate system is used which moves with the jet. An arbitrary infinitesimalaxisymmetric surface displacement of the form

η = η0eikz+ωt (19.4-70)

is imposed on the initially steady motion and it is thus desired to findthe dispersion relation ω = ω(k) which relates the real part of the growthrate, ω, to its wave number, k = 2π/λ.

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

φ1 = C1I0(kr)eikz+ωt (19.4-71)ψ1 = C2I1(Lr)eikz+ωt (19.4-72)

where φ1 and ψ1 are the velocity potential and stream function, respec-tively, C1 and C2 are integration constants, I0 and I1 are modified Besselfunctions of the first kind, L2 = k2 +ω/ν1, and ν1 is the liquid kinematicviscosity [188]. The liquid pressure is obtained from the inviscid partof the liquid equations. In addition, the inviscid gas equations can besolved to obtain the fluctuating gas pressure at r = a:

−p21 = −ρ2(U − iωk)2kηK0(ka)K1(ka)

(19.4-73)


19.4 Spray Models

where K0 and K1 are modified Bessel functions of the second kind andu is the relative velocity between the liquid and the gas. The linearizedboundary conditions are

v1 =∂η

∂t(19.4-74)

∂u1

∂r= −∂v1

∂z(19.4-75)

and

−p1 + 2µ1 − σ

a2

(η + a2∂

2η

∂z2

)+ p2 = 0 (19.4-76)

which are mathematical statements of the liquid kinematic free surfacecondition, continuity of shear stress, and continuity of normal stress,respectively. Note that u1 is the axial perturbation liquid velocity, v1is the radial perturbation liquid velocity, and σ is the surface tension.Also note that Equation 19.4-75 was obtained under the assumption thatv2 = 0.

As described by Reitz [188], Equations 19.4-74 and 19.4-75 can be usedto eliminate the integration constants C1 and C2 in Equation 19.4-72.Thus, when the pressure and velocity solutions are substituted into Equa-tion 19.4-76, the desired dispersion relation is obtained:

ω2 + 2ν1k2ω

[I ′1(ka)I0(ka)

− 2kLk2 + 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 − i

ω

k

)2(L2 − a2

L2 + a2

)I1(ka)I0(ka)

K0(ka)K1(ka)

(19.4-77)

As shown by Reitz [188], Equation 19.4-77 predicts that a maximumgrowth rate (or most unstable wave) exists for a given set of flow condi-tions. Curve fits of numerical solutions to Equation 19.4-77 were gener-



ated for the maximum growth rate, Ω, and the corresponding wavelength,Λ, and are given by Reitz [188]:

Λa

= 9.02(1 + 0.45Oh0.5)(1 + 0.4Ta0.7)

(1 + 0.87We1.672 )0.6

(19.4-78)

Ω

(ρ1a

3

σ

)=

(0.34 + 0.38We1.52 )

(1 + Oh)(1 + 1.4Ta0.6)(19.4-79)

where Oh =√

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

We2 isthe Taylor number. Furthermore, We1 = ρ1U

2a/σ and We2 = ρ2U2a/σ

are the liquid and gas Weber numbers, respectively, and Re1 = Ua/ν1 isthe Reynolds number.

Droplet Breakup

In the wave model, the initial parcel diameters of the relatively largeinjected droplets are modeled using the stability analysis for liquid jetsas described above. The breakup of the parcels and resulting dropletsof radius a is calculated by assuming that the breakup droplet radius, r,is proportional to the wavelength of the fastest-growing unstable surfacewave given by Equation 19.4-78. In other words,

r = B0Λ (19.4-80)

where B0 is a model constant set equal to 0.61 based on the work ofReitz [188]. Furthermore, the rate of change of droplet radius in a parentparcel is given by

da

dt= −(a− r)

τ, r ≤ a (19.4-81)

where the breakup time, τ , is given by

τ =3.726B1a

ΛΩ(19.4-82)


19.4 Spray Models

and Λ and Ω are obtained from Equations 19.4-78 and 19.4-79, respec-tively. The breakup time constant, B1, is related to the initial distur-bance level on the liquid jet and has been found to vary from one nozzleto another [118].

19.4.4 Dynamic Drag Model

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

Use and Limitations

The dynamic drag model is applicable in almost any circumstance. Itis compatible with both the TAB and wave models for spray breakup.When the collision model is turned on, collisions reset the distortion anddistortion velocities of the colliding droplets.

Theory

Many droplet drag models assume the droplet remains spherical through-out the domain. With this assumption, the drag of a spherical object isdetermined by the following [142]:

Cd,sphere =

0.424 Re > 1000

24Re

(1 + 1

6Re2/3)

Re ≤ 1000(19.4-83)

However, as an initially spherical droplet moves through a gas, its shapeis distorted significantly when the Weber number is large. In the extremecase, the droplet shape will approach that of a disk. The drag of a disk,however, is significantly higher than that of a sphere. Since the dropletdrag coefficient is highly dependent upon the droplet shape, a drag modelthat assumes the droplet is spherical is unsatisfactory. The dynamic dragmodel accounts for the effects of droplet distortion, linearly varying thedrag between that of a sphere (Equation 19.4-83) and a value of 1.52corresponding to a disk [142]. The drag coefficient is given by



Cd = Cd,sphere(1 + 2.632y) (19.4-84)

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

d2y

dt2=CF

Cb

ρg

ρl

u2

r2− Ckσ

ρlr3y − Cdµl

ρlr2dy

dt(19.4-85)

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

Note that Equation 19.4-85 is obtained from the TAB model for spraybreakup, described in Section 19.4.3, but the dynamic drag model canbe used with either of the breakup models.

19.5 Coupling Between the Discrete and ContinuousPhases

As the trajectory of a particle is computed, FLUENT keeps track of theheat, mass, and momentum gained or lost by the particle stream that fol-lows that trajectory and these quantities can be incorporated in the sub-sequent continuous phase calculations. Thus, while the continuous phasealways impacts the discrete phase, you can also incorporate the effect ofthe discrete phase trajectories on the continuum. This two-way couplingis accomplished by alternately solving the discrete and continuous phaseequations until the solutions in both phases have stopped changing. Thisinterphase exchange of heat, mass, and momentum from the particle tothe continuous phase is depicted qualitatively in Figure 19.5.1.

Momentum Exchange

The momentum transfer from the continuous phase to the discrete phaseis computed in FLUENT by examining the change in momentum of aparticle as it passes through each control volume in the FLUENT model.This momentum change is computed as


19.5 Coupling Between the Discrete and Continuous Phases

mass-exchangeheat-exchangemomentum-exchange

typical

particle

trajectory

typical continuous

phase control volume

Figure 19.5.1: Heat, Mass, and Momentum Transfer Between the Dis-crete and Continuous Phases

F =∑(

18µCDReρpd2

p24(up − u) + Fother

)mp∆t (19.5-1)

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 continu-ous phase momentum balance in any subsequent calculations of the con-tinuous phase flow field and can be reported by FLUENT as described in



Section 19.13.

Heat Exchange

The heat transfer from the continuous phase to the discrete phase iscomputed in FLUENT by examining the change in thermal energy of aparticle as it passes through each control volume in the FLUENT model.In the absence of chemical reaction (i.e., for all particle laws except Law5) this heat exchange is computed as

Q =

[mp

mp,0cp∆Tp +

∆mp

mp,0

(−hfg + hpyrol +

∫ Tp

Tref

cp,idT

)]mp,0

(19.5-2)

wheremp = average mass of the particle in the control volume

(kg)mp,0 = initial mass of the particle (kg)cp = heat capacity of the particle (J/kg-K)∆Tp = temperature change of the particle in the control

volume (K)∆mp = change in the mass of the particle in the control

volume (kg)hfg = latent heat of volatiles evolved (J/kg)hpyrol = heat of pyrolysis as volatiles are evolved (J/kg)cp,i = heat capacity of the volatiles evolved (J/kg-K)Tp = temperature of the particle upon exit of the con-

trol volume (K)Tref = reference temperature for enthalpy (K)mp,0 = initial mass flow rate of the particle injection

tracked (kg/s)

This heat exchange appears as a source or sink of energy in the con-tinuous phase energy balance during any subsequent calculations of thecontinuous phase flow field and is reported by FLUENT as described inSection 19.13. A similar equation governs heat exchange under Law 5,in which the heat of surface combustion is incorporated.


19.5 Coupling Between the Discrete and Continuous Phases

Mass Exchange

The mass transfer from the discrete phase to the continuous phase iscomputed in FLUENT by examining the change in mass of a particle asit passes through each control volume in the FLUENT model. The masschange is computed simply as

M =∆mp

mp,0mp,0 (19.5-3)

This mass exchange appears as a source of mass in the continuous phasecontinuity equation and as a source of a chemical species defined byyou. The mass sources are included in any subsequent calculations ofthe continuous phase flow field and are reported by FLUENT as describedin Section 19.13.

Under-Relaxation of the Interphase Exchange Terms

Note that the interphase exchange of momentum, heat, and mass isunder-relaxed during the calculation, so that

Fnew = Fold + α(Fcalculated − Fold) (19.5-4)

Qnew = Qold + α(Qcalculated −Qold) (19.5-5)

Mnew = Mold + α(Mcalculated −Mold) (19.5-6)

where α is the under-relaxation factor for particles/droplets that youcan set in the Solution Controls panel. The default value for α is 0.5.This value may be reduced in order to improve the stability of coupledcalculations. Note that the value of α does not influence the predictionsobtained in the final converged solution.



Interphase Exchange During Stochastic Tracking

When stochastic tracking is performed, the interphase exchange terms,computed via Equations 19.5-1 to 19.5-6, are computed for each stochas-tic trajectory with the particle mass flow rate, mp0, divided by the num-ber of stochastic tracks computed. This implies that an equal mass flowof particles follows each stochastic trajectory.

Interphase Exchange During Cloud Tracking

When the particle cloud model is used, the interphase exchange termsare computed via Equations 19.5-1 to 19.5-6 based on ensemble-averagedflow properties in the particle cloud. The exchange terms are then dis-tributed to all the cells in the cloud based on the weighting factor definedin Equation 19.2-61.

19.6 Overview of Using the Discrete Phase Models

The procedure for setting up and solving a problem involving a discretephase is outlined below, and described in detail in Sections 19.7–19.13.Only the steps related specifically to discrete phase modeling are shownhere. For information about inputs related to other models that you areusing in conjunction with the discrete phase models, see the appropriatesections for those models.

1. Enable any of the discrete phase modeling options, if relevant, asdescribed in Section 19.7.

2. If you are using unsteady particle tracking, define the unsteadyparameters as described in Section 19.8.

3. Specify the initial conditions, as described in Section 19.9.

4. Define the boundary conditions, as described in Section 19.10.

5. Define the material properties, as described in Section 19.11.

6. Set the solution parameters and solve the problem, as described inSection 19.12.

7. Examine the results, as described in Section 19.13.


19.7 Discrete Phase Model Options


This section provides instructions for using the optional discrete phasemodels available in FLUENT. All of them can be turned on in the DiscretePhase Model panel (Figure 19.7.1).

Define −→ Models −→Discrete Phase...

19.7.1 Including Radiation Heat Transfer to the Particles

If you want to include the effect of radiation heat transfer to the particles(Equation 11.3-20), you must turn on the Particle Radiation Interactionoption in the Discrete Phase Model panel. You will also need to defineadditional properties for the particle materials (emissivity and scatteringfactor), as described in Section 19.11.2. This option is available onlywhen the P-1 or discrete ordinates radiation model is used.

19.7.2 Including the Thermophoretic Force on the Particles

If you want to include the effect of the thermophoretic force on the par-ticle trajectories (Equation 19.2-14), turn on the Thermophoretic Forceoption in the Discrete Phase Model panel. You will also need to definethe thermophoretic coefficient for the particle material, as described inSection 19.11.2.

19.7.3 Including a Coupled Heat-Mass Solution on the Particles

By default, the solution of the particle heat and mass equations are solvedin a segregated manner. If you enable the Coupled Heat-Mass Solutionoption, FLUENT will solve this pair of equations pair using a stiff, cou-pled ODE solver with error tolerance control. The increased accuracy,however, comes at the expense of increased computational expense.

19.7.4 Including Brownian Motion Effects on the Particles

For sub-micron particles in laminar flow, you may want to include theeffects of Brownian motion (described in Section 19.2.1) on the particletrajectories. To do so, turn on the Brownian Motion option in the DiscretePhase Model panel. When Brownian motion effects are included, it is



Figure 19.7.1: The Discrete Phase Model Panel



recommended that you also select the Stokes-Cunningham drag law inthe Drag Law drop-down list under Drag Parameters, and specify theCunningham Correction (Cc in Equation 19.2-9).

19.7.5 Including Saffman Lift Force Effects on the Particles

For sub-micron particles, you can also model the lift due to shear (theSaffman lift force, described in Section 19.2.1) in the particle trajectory.To do this, turn on the Saffman Lift Force option in the Discrete PhaseModel panel.

19.7.6 Monitoring Erosion/Accretion of Particles at Walls

Particle erosion and accretion rates can be monitored at wall bound-aries. These rate calculations can be enabled in the Discrete PhaseModel panel when the discrete phase is coupled with the continuousphase (i.e., when Interaction with Continuous Phase is selected). Turn-ing on the Erosion/Accretion option will cause the erosion and accretionrates to be calculated at wall boundary faces when particle tracks areupdated. You will also need to set the Impact Angle Function (f(α) inEquation 19.2-62), Diameter Function (C(dp) in Equation 19.2-62), andVelocity Exponent Function (b(v) in Equation 19.2-62) in the Wall bound-ary conditions panel for each wall zone (as described in Section 19.10.2).

19.7.7 Alternate Drag Laws

There are five drag laws for the particles that can be selected in the DragLaw drop-down list under Drag Parameters.

The spherical, non-spherical, Stokes-Cunningham, and high-Mach-numberlaws described in Section 19.2.1 are always available, and the dynamic-drag law described in Section 19.4.4 is available only when one of thedroplet breakup models is used in conjunction with unsteady tracking.See Section 19.8.2 for information about enabling the droplet breakupmodels.

If the spherical law, the high-Mach-number law, or the dynamic-drag lawis selected, no further inputs are required. If the nonspherical law is se-lected, the particle Shape Factor (φ in Equation 19.2-7) must be specified.



For the Stokes-Cunningham law, the Cunningham Correction factor (Cc inEquation 19.2-9) must be specified.

19.7.8 User-Defined Functions

User-defined functions can be used to customize the discrete phase modelto include additional body forces, modify interphase exchange terms(sources), calculate or integrate scalar values along the particle trajec-tory, and incorporate non-standard erosion rate definitions. See the sep-arate UDF Manual for information about user-defined functions.

In the Discrete Phase Model panel, under User-Defined Functions, thereare drop-down lists labeled Body Force, Source, and Scalar Update. IfErosion/Accretion is enabled under Options, there will be an additionaldrop-down list labeled Erosion/Accretion. These lists will show availableuser-defined functions that can be selected to customize the discretephase model.

19.8 Unsteady Particle Tracking

This section contains information about unsteady particle tracking withthe discrete phase model. Note that you cannot use adaptive time step-ping for an unsteady discrete phase calculation.

19.8.1 Inputs for Unsteady Particle Tracking

For transient flow simulations, particle trajectories can also be advancedin time with the flow simulation. If you select the Unsteady Trackingoption under Unsteady Parameters in the Discrete Phase Model panel,particles will be advanced by the flow time step each time the flow so-lution is advanced in time. Coupled calculations are also allowed fortransient flow simulations. Particle sub-iterations are done during eachtime step based on the value of the Number Of Continuous Phase IterationsPer DPM Iteration.

When the coupled explicit solver is used with the explicit unsteady for-!mulation, the particles are advanced once per time step, and are calcu-lated at the start of the time step (before the flow is updated).


19.8 Unsteady Particle Tracking

Additional inputs are required for each injection in the Set Injection Prop-erties panel. The injection Start Time and Stop Time must be specifiedunder Point Properties. Injections with start and stop times set to zerowill be injected only at the start of the calculation (t = 0). Changinginjection settings during the transient simulation will not affect parti-cles currently released in the domain. At any point during the transientsimulation, you can clear particles that are currently in the domain byclicking on the Clear Particles button in the Discrete Phase Model panel.

If you want to save the particle history during the unsteady calculation,you can use the File/Write/Start Particle History... menu item to specifya particle history filename.

File −→ Write −→Start Particle History...

During the calculation, FLUENT will write the position, velocity, andother data for each particle at each time step. To turn the particlehistory off, select the File/Write/Stop Particle History menu item.

File −→ Write −→Stop Particle History

19.8.2 Options for Spray Modeling

When you enable unsteady tracking, the Discrete Phase Model panel willexpand to show options related to spray modeling.

Modeling Spray Breakup

To enable the modeling of spray breakup, select the Droplet Breakupoption under Spray Models and then select the desired model (TAB orWave). A detailed description of these models can be found in Sec-tion 19.4.3.

For the TAB model, you will need to specify a value for y0 (the initialdistortion at time equal to zero in Equation 19.4-51) in the y0 field.

For the wave model, you will need to specify values for C0 and C1,which are the integration constants of the velocity potential and streamfunction models represented in Equation 19.4-72, in the C0 and C1 fields.You will generally not need to modify the value of B0. This is the model



constant B0 in Equation 19.4-80, and the default value 0.61 is acceptablefor nearly all cases.

Note that you may want to use the dynamic drag law when you use oneof the spray breakup models. See Section 19.7.7 for information aboutchoosing the drag law.

Modeling Droplet Collisions

To include the effect of droplet collisions, as described in Section 19.4.2,select the Droplet Collision option under Spray Models. There are nofurther inputs for this model.

19.9 Setting Initial Conditions for the Discrete Phase

19.9.1 Overview of Initial Conditions

The primary inputs that you must provide for the discrete phase calcu-lations in FLUENT are the initial conditions that define the starting po-sitions, velocities, and other parameters for each particle stream. Theseinitial conditions provide the starting values for all of the dependent dis-crete phase variables that describe the instantaneous conditions of anindividual particle:

• Position (x, y, z coordinates) of the particle.

• Velocities (u, v, w) of the particle. Velocity magnitudes and spraycone angle can also be used (in 3D) to define the initial velocities(see Section 19.9.8). For moving reference frames, relative veloci-ties should be specified.

• Diameter of the particle, dp.

• Temperature of the particle, Tp.

• Mass flow rate of the particle stream that will follow the trajectoryof the individual particle/droplet, mp (required only for coupledcalculations).



• Additional parameters if one of the atomizer models described inSection 19.4.1 is used for the injection.

When an atomizer model is selected, you will not input initial!diameter, velocity, and position quantities for the particles due tothe complexities of sheet and ligament breakup. Instead of initialconditions, the quantities you will input for the atomizer modelsare global parameters.

These dependent variables are updated according to the equations ofmotion (Section 19.2) and according to the heat/mass transfer relationsapplied (Section 19.3) as the particle/droplet moves along its trajectory.You can define any number of different sets of initial conditions for dis-crete phase particles/droplets provided that your computer has sufficientmemory.

19.9.2 Injection Types

You will define the initial conditions for a particle/droplet stream bycreating an “injection” and assigning properties to it. FLUENT provides10 types of injections:

• single

• group

• cone (only in 3D)

• surface

• plain-orifice atomizer

• pressure-swirl atomizer

• flat-fan atomizer

• air-blast atomizer

• effervescent atomizer

• read from a file



For each non-atomizer injection type, you will specify each of the initialconditions listed in Section 19.9.1, the type of particle that possessesthese initial conditions, and any other relevant parameters for the parti-cle type chosen.

You should create a single injection when you want to specify a singlevalue for each of the initial conditions (Figure 19.9.1). Create a groupinjection (Figure 19.9.2) when you want to define a range for one ormore of the initial conditions (e.g., a range of diameters or a range ofinitial positions). To define hollow spray cone injections in 3D problems,create a cone injection (Figure 19.9.3). To release particles from a surface(either a zone surface or a surface you have defined using the items inthe Surface menu), you will create a surface injection. (If you create asurface injection, a particle stream will be released from each facet of thesurface. You can use the Bounded and Sample Points options in the PlaneSurface panel to create injections from a rectangular grid of particles in3D (see Section 24.6 for details).

Particle initial conditions (position, velocity, diameter, temperature, andmass flow rate) can also be read from an external file if none of the injec-tion types listed above can be used 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 is optional.

The inputs for setting injections are described in detail in Section 19.9.5.

19.9.3 Particle Types

When you define a set of initial conditions (as described in Section 19.9.5),you will need to specify the type of particle. The particle types availableto you depend on the range of physical models that you have defined inthe Models family of panels.

• An “inert” particle is a discrete phase element (particle, droplet,or bubble) that obeys the force balance (Equation 19.2-1) and is



Figure 19.9.1: Particle Injection Defining a Single Particle Stream

Figure 19.9.2: Particle Injection Defining an Initial Spatial Distributionof the Particle Streams

Figure 19.9.3: Particle Injection Defining an Initial Spray Distributionof the Particle Velocity



subject to heating or cooling via Law 1 (Section 19.3.2). The inerttype is available for all FLUENT models.

• A “droplet” particle is a liquid droplet in a continuous-phase gasflow that obeys the force balance (Equation 19.2-1) and that ex-periences heating/cooling via Law 1 followed by vaporization andboiling via Laws 2 and 3 (Sections 19.3.3 and 19.3.4). The droplettype is available when heat transfer is being modeled and at leasttwo chemical species are active or the non-premixed or partiallypremixed combustion model is active. You should use the idealgas law to define the gas-phase density (in the Materials panel, asdiscussed in Section 7.2.5) when you select the droplet type.

• A “combusting” particle is a solid particle that obeys the force bal-ance (Equation 19.2-1) and experiences heating/cooling via Law 1followed by devolatilization via Law 4 (Section 19.3.5), and a het-erogeneous surface reaction via Law 5 (Section 19.3.6). Finally,the non-volatile portion of a combusting particle is subject to inertheating via Law 6. You can also include an evaporating mate-rial with the combusting particle by selecting the Wet Combustionoption in the Set Injection Properties panel. This allows you toinclude a material that evaporates and boils via Laws 2 and 3(Sections 19.3.3 and 19.3.4) before devolatilization of the particlematerial begins. The combusting type is available when heat trans-fer is being modeled and at least three chemical species are activeor the non-premixed combustion model is active. You should usethe ideal gas law to define the gas-phase density (in the Materialspanel) when you select the combusting particle type.

19.9.4 Creating, Copying, Deleting, and Listing Injections

You will use the Injections panel (Figure 19.9.4) to create, copy, delete,and list injections.

Define −→Injections...

(You can also click on the Injections... button in the Discrete Phase Modelpanel to open the Injections panel.)



Figure 19.9.4: The Injections Panel

Creating Injections

To create an injection, click on the Create button. A new injection willappear in the Injections list and the Set Injection Properties panel will openautomatically to allow you to set the injection properties (as describedin Section 19.9.5).

Modifying Injections

To modify an existing injection, select its name in the Injections list andclick on the Set... button. The Set Injection Properties panel will open,and you can modify the properties as needed.

If you have two or more injections for which you want to set some ofthe same properties, select their names in the Injections list and click onthe Set... button. The Set Multiple Injection Properties panel will open,which will allow you to set the common properties. For instructions



about using this panel, see Section 19.9.17.

Copying Injections

To copy an existing injection to a new injection, select the existing injec-tion in the Injections list and click on the Copy button. The Set InjectionProperties panel will open with a new injection that has the same prop-erties as the injection you selected. This is useful if you want to setanother injection with similar properties.

Deleting Injections

You can delete an injection by selecting its name in the Injections listand clicking on the Delete button.

Listing Injections

To list the initial conditions for the particle streams in the selected in-jection, click on the List button. The list reported by FLUENT in theconsole window contains, for each particle stream that you have defined,the following (in SI units):

• Particle stream number in the column headed NO

• Particle type (IN for inert, DR for droplet, or CP for combustingparticle) in the column headed TYP

• x, y, and z position in the columns headed (X), (Y), and (Z)

• x, y, and z velocity in the columns headed (U), (V), and (W)

• Temperature in the column headed (T)

• Diameter in the column headed (DIAM)

• Mass flow rate in the column headed (MFLOW)



Shortcuts for Selecting Injections

FLUENT provides a shortcut for selecting injections with names thatmatch a specified pattern. To use this shortcut, enter the pattern un-der Injection Name Pattern and then click Match to select the injectionswith names that match the specified pattern. For example, if you specifydrop*, all injections that have names beginning with drop (e.g., drop-1,droplet) will be selected automatically. If they are all selected already,they will be deselected. If you specify drop?, all surfaces with namesconsisting of drop followed by a single character will be selected (or des-elected, if they are all selected already).

19.9.5 Defining Injection Properties

Once you have created an injection (using the Injections panel, as de-scribed in Section 19.9.4), you will use the Set Injection Properties panel(Figure 19.9.5) to define the injection properties. (Remember that thispanel will open when you create a new injection, or when you select anexisting injection and click on the Set... button in the Injections panel.)

The procedure for defining an injection is as follows:

1. If you want to change the name of the injection from its defaultname, enter a new one in the Injection Name field. This is recom-mended if you are defining a large number of injections so you caneasily distinguish them. When assigning names to your injections,keep in mind the selection shortcut described in Section 19.9.4.

2. Choose the type of injection in the Injection Type drop-down list.The ten choices (single, group, cone, surface, plain-orifice-atomizer,pressure-swirl-atomizer, air-blast-atomizer, flat-fan-atomizer,effervescent-atomizer, and file) are described in Section 19.9.2. Notethat if you select any of the atomizer models, you will also need toset the Viscosity and Droplet Surface Tension in the Materials panel.

If you are using sliding or moving/deforming meshes in your sim-!ulation, you should not use surface injections because they are notcompatible with moving meshes.



Figure 19.9.5: The Set Injection Properties Panel



3. If you are defining a single injection, go to the next step. For agroup, cone, or any of the atomizer injections, set the Number ofParticle Streams in the group, spray cone, or atomizer. If you aredefining a surface injection, choose the surface(s) from which theparticles will be released in the Release From Surfaces list. If youare reading the injection from a file, click on the File... button atthe bottom of the Set Injection Properties panel and specify the fileto be read in the resulting Select File dialog box. The parametersin the injection file must be in SI units.

4. Select Inert, Droplet, or Combusting as the Particle Type. The avail-able types are described in Section 19.9.3.

5. Choose the material for the particle(s) in the Material drop-downlist. If this is the first time you have created a particle of this type,you can choose from all of the materials of this type defined inthe database. If you have already created a particle of this type,the only available material will be the material you selected forthat particle. You can define additional materials by copying themfrom the database or creating them from scratch, as discussed inSection 19.11.2 and described in detail in Section 7.1.2.

6. If you are defining a group or surface injection and you want tochange from the default linear (for group injections) or uniform (forsurface injections) interpolation method used to determine the sizeof the particles, select rosin-rammler or rosin-rammler-logarithmicin the Diameter Distribution drop-down list. The Rosin-Rammlermethod for determining the range of diameters for a group injectionis described in Section 19.9.7.

7. If you have created a customized particle law using user-definedfunctions, turn on the Custom option under Laws and specify theappropriate laws as described in Section 19.9.16.

8. If your particle type is Inert, go to the next step. If you are defin-ing Droplet particles, select the gas phase species created by thevaporization and boiling laws (Laws 2 and 3) in the EvaporatingSpecies drop-down list.



If you are defining Combusting particles, select the gas phase speciescreated by the devolatilization law (Law 4) in the DevolatilizingSpecies drop-down list, the gas phase species that participatesin the surface char combustion reaction (Law 5) in the OxidizingSpecies list, and the gas phase species created by the surface charcombustion reaction (Law 5) in the Product Species list. Note thatif the Combustion Model for the selected combusting particle mate-rial (in the Materials panel) is the multiple-surface-reaction model,then the Oxidizing Species and Product Species lists will be disabledbecause the reaction stoichiometry has been defined in the mixturematerial.

9. Click the Point Properties tab (the default), and specify the pointproperties (position, velocity, diameter, temperature, and—ifappropriate—mass flow rate and any atomizer-related parameters)as described for each injection type in Sections 19.9.6–19.9.14.

10. If the flow is turbulent and you wish to include the effects of turbu-lence on the particle dispersion, click the Turbulent Dispersion tab,turn on the Stochastic Model and/or the Cloud Model, and set therelated parameters as described in Section 19.9.15.

11. If your combusting particle includes an evaporating material, clickthe Wet Combustion tab, select the Wet Combustion option, andthen select the material that is evaporating/boiling from the parti-cle before devolatilization begins in the Liquid Material drop-downlist. You should also set the volume fraction of the liquid presentin the particle by entering the value of the Liquid Fraction. Finally,select the gas phase species created by the evaporating and boilinglaws in the Evaporating Species drop-down list in the top part ofthe panel.

12. If you want to use a user-defined function to initialize the injec-tion properties, click the UDF tab to access the UDF inputs. Youcan select an Initialization function under User-Defined Functions tomodify injection properties at the time the particles are injectedinto the domain. This allows the position and/or properties of theinjection to be set as a function of flow conditions. See the separateUDF Manual for information about user-defined functions.



19.9.6 Point Properties for Single Injections

For a single injection, you will define the following initial conditionsfor the particle stream under the Point Properties heading (in the SetInjection Properties panel):

• Position: Set the x, y, and z positions of the injected stream alongthe Cartesian axes of the problem geometry in the X-, Y-, andZ-Position fields. (Z-Position will appear only for 3D problems.)

• Velocity: Set the x, y, and z components of the stream’s initialvelocity in the X-, Y-, and Z-Velocity fields. (Z-Velocity will appearonly for 3D problems.)

• Diameter: Set the initial diameter of the injected particle streamin the Diameter field.

• Temperature: Set the initial (absolute) temperature of the injectedparticle stream in the Temperature field.

• Mass flow rate: For coupled phase calculations (see Section 19.12),set the mass of particles per unit time that follows the trajectorydefined by the injection in the Flow Rate field. Note that in axisym-metric problems the mass flow rate is defined per 2π radians andin 2D problems per unit meter depth (regardless of the referencevalue for length).

• Duration of injection: For unsteady particle tracking (see Sec-tion 19.8), set the starting and ending time for the injection inthe Start Time and Stop Time fields.

19.9.7 Point Properties for Group Injections

For group injections, you will define the properties described in Sec-tion 19.9.6 for single injections for the First Point and Last Point in thegroup. That is, you will define a range of values, φ1 through φN , foreach initial condition φ by setting values for φ1 and φN . FLUENT as-signs a value of φ to the ith injection in the group using a linear variationbetween the first and last values for φ:



φi = φ1 +φN − φ1

N − 1(i− 1) (19.9-1)

Thus, for example, if your group consists of 5 particle streams and youdefine a range for the initial x location from 0.2 to 0.6 meters, the initialx location of each stream is as follows:

• Stream 1: x = 0.2 meters





In general, you should supply a range for only one of the initial condi-!tions in a given group—leaving all other conditions fixed while a singlecondition varies among the stream numbers of the group. Otherwiseyou may find, for example, that your simultaneous inputs of a spatialdistribution and a size distribution have placed the small droplets at thebeginning of the spatial range and the large droplets at the end of thespatial range.

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

Using the Rosin-Rammler Diameter Distribution Method

By default, you will define the size distribution of particles by inputtinga diameter for the first and last points and using the linear equation(19.9-1) to vary the diameter of each particle stream in the group. Whenyou want a different mass flow rate for each particle/droplet size, how-ever, the linear variation may not yield the distribution you need. Yourparticle size distribution may be defined most easily by fitting the sizedistribution data to the Rosin-Rammler equation. In this approach, thecomplete range of particle sizes is divided into a set of discrete size ranges,



each to be defined by a single stream that is part of the group. Assume,for example, that the particle size data obeys the following distribution:

Diameter Mass FractionRange (µm ) in Range

0–70 0.0570–100 0.10100–120 0.35120–150 0.30150–180 0.15180–200 0.05

The Rosin-Rammler distribution function is based on the assumptionthat an exponential relationship exists between the droplet diameter, d,and the mass fraction of droplets with diameter greater than d, Yd:

Yd = e−(d/d)n(19.9-2)

FLUENT refers to the quantity d in Equation 19.9-2 as the Mean Diameterand to n as the Spread Parameter. These parameters are input by you (inthe Set Injection Properties panel under the First Point heading) to definethe Rosin-Rammler size distribution. To solve for these parameters, youmust fit your particle size data to the Rosin-Rammler exponential equa-tion. To determine these inputs, first recast the given droplet size datain terms of the Rosin-Rammler format. For the example data providedabove, this yields the following pairs of d and Yd:

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

70 0.95100 0.85120 0.50150 0.20180 0.05200 (0.00)



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

s Fr

acti

on >

d, Y

d

Diameter, d ( µm)

Figure 19.9.6: Example of Cumulative Size Distribution of Particles



A plot of Yd vs. d is shown in Figure 19.9.6.

Next, derive values of d and n such that the data in Figure 19.9.6 fitEquation 19.9-2. The value for d is obtained by noting that this is thevalue of d at which Yd = e−1 ≈ 0.368. From Figure 19.9.6, you canestimate that this occurs for d ≈ 131 µm. The numerical value for n isgiven by

n =ln(− lnYd)

ln(d/d

)

By substituting the given data pairs for Yd and d/d into this equation,you can obtain values for n and find an average. Doing so yields an aver-age value of n = 4.52 for the example data above. The resulting Rosin-Rammler curve fit is compared to the example data in Figure 19.9.7.You can input values for Yd and n, as well as the diameter range of thedata and the total mass flow rate for the combined individual size ranges,using the Set Injection Properties panel.

A second Rosin-Rammler distribution is also available based on the nat-ural logarithm of the particle diameter. If in your case, the smaller-diameter particles in a Rosin-Rammler distribution have higher massflows in comparison with the larger-diameter particles, you may wantbetter resolution of the smaller-diameter particle streams, or “bins”.You can therefore choose to have the diameter increments in the Rosin-Rammler distribution done uniformly by ln d.

In the standard Rosin-Rammler distribution, a particle injection mayhave a diameter range of 1 to 200 µm. In the logarithmic Rosin-Rammlerdistribution, the same diameter range would be converted to a range ofln 1 to ln 200, or about 0 to 5.3. In this way, the mass flow in one binwould be less-heavily skewed as compared to the other bins.

When a Rosin-Rammler size distribution is being defined for the groupof streams, you should define (in addition to the initial velocity, posi-tion, and temperature) the following parameters, which appear underthe heading for the First Point:



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

s Fr

acti

on >

d, Y

d

Diameter, d ( µm)

Figure 19.9.7: Rosin-Rammler Curve Fit for the Example Particle SizeData



• Total Flow Rate: the total mass flow rate of the N streams in thegroup. Note that in axisymmetric problems this mass flow rate isdefined per 2π radians and in 2D problems per unit meter depth.

• Min. Diameter: the smallest diameter to be considered in the sizedistribution.

• Max. Diameter: the largest diameter to be considered in the sizedistribution.

• Mean Diameter: the size parameter, d, in the Rosin-Rammler equa-tion (19.9-2).

• Spread Parameter: the exponential parameter, n, in Equation 19.9-2.

19.9.8 Point Properties for Cone Injections

In 3D problems, you can conveniently define a hollow spray cone ofparticle streams using the cone injection type. For this injection type,the inputs are as follows:

• Position: Set the coordinates of the origin of the spray cone in theX-, Y-, and Z-Position fields.

• Diameter: Set the diameter of the particles in the stream in theDiameter field.

• Temperature: Set the temperature of the streams in the Tempera-ture field.

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

• Velocity: Set the velocity magnitude of the particle streams thatwill be oriented along the specified spray cone angle in the VelocityMag. field.

• Cone angle: Set the included half-angle, θ, of the hollow spray conein the Cone Angle field, as shown in Figure 19.9.8.



• Radius: A non-zero inner radius can be specified to model injectorsthat do not emanate from a single point. Set the radius r (definedas shown in Figure 19.9.8) in the Radius field. The particles willbe distributed about the axis with the specified radius.

θ

r origin axis

Figure 19.9.8: Cone Half Angle and Radius

• Swirl fraction: Set the fraction of the velocity magnitude to go intothe swirling component of the flow in the Swirl Fraction field. Thedirection of the swirl component is defined using the right-handrule about the axis (a negative value for the swirl fraction can beused to reverse the swirl direction).

• Mass flow rate: For coupled calculations, set the total mass flowrate for the streams in the spray cone in the Total Flow Rate field.

Note that you may want to define multiple spray cones emanating fromthe same initial location in order to include a size distribution of thespray or to include a range of cone angles.

19.9.9 Point Properties for Surface Injections

For surface injections, you will define all the properties described inSection 19.9.6 for single injections except for the initial position of theparticle streams. The initial positions of the particles will be the location



of the data points on the specified surface(s). Note that you will set theTotal Flow Rate of all particles released from the surface (required forcoupled calculations only). If you want, you can scale the individualmass flow rates of the particles by the ratio of the area of the face theyare released from to the total area of the surface. To scale the massflow rates, select the Scale Flow Rate By Face Area option under PointProperties.

Note that many surfaces have non-uniform distributions of points. Ifyou want to generate a uniform spatial distribution of particle streamsreleased from a surface in 3D, you can create a bounded plane surfacewith a uniform distribution using the Plane Surface panel, as describedin Section 24.6. In 2D, you can create a rake using the Line/Rake Surfacepanel, as described in Section 24.5.

A non-uniform size distribution can be used for surface injections, asdescribed below.

Using the Rosin-Rammler Diameter Distribution Method

The Rosin-Rammler size distributions described in Section 19.9.7 forgroup injections is also available for surface injections. If you selectone of the Rosin-Rammler distributions, you will need to specify thefollowing parameters under Point Properties, in addition to the initialvelocity, temperature, and total flow rate:

• Min. Diameter: the smallest diameter to be considered in the sizedistribution.

• Max. Diameter: the largest diameter to be considered in the sizedistribution.

• Mean Diameter: the size parameter, d, in the Rosin-Rammler equa-tion (Equation 19.9-2).

• Spread Parameter: the exponential parameter, n, in Equation 19.9-2.

• Number of Diameters: the number of diameters in each distribution(i.e., the number of different diameters in the stream injected fromeach face of the surface).



FLUENT will inject streams of particles from each face on the surface,with diameters defined by the Rosin-Rammler distribution function. Thetotal number of injection streams tracked for the surface injection willbe equal to the number of diameters in each distribution (Number ofDiameters) multiplied by the number of faces on the surface.

19.9.10 Point Properties for Plain-Orifice Atomizer Injections

For a plain-orifice atomizer injection, you will define the following initialconditions under Point Properties:

• Position: Set the x, y, and z positions of the injected stream alongthe Cartesian axes of the problem geometry in the X-Position, Y-Position, and Z-Position fields. (Z-Position will appear only for 3Dproblems.

• Axis (3D only): Set the x, y, and z components of the vectordefining the axis of the orifice in the X-Axis, Y-Axis, and Z-Axisfields.


• Mass flow rate: Set the mass flow rate for the streams in the at-omizer in the Flow Rate field.


• Vapor pressure: Set the vapor pressure governing the flow throughthe internal orifice (pv in Table 19.4.1) in the Vapor Pressure field.

• Diameter: Set the diameter of the orifice in the Injector Inner Diam.field (d in Table 19.4.1).

• Orifice length: Set the length of the orifice in the Orifice Lengthfield (L in Table 19.4.1).



• Radius of curvature: Set the radius of curvature of the inlet cornerin the Corner Radius of Curv. field (r in Table 19.4.1).

• Nozzle parameter: Set the constant for the spray angle correlationin the Constant A field (CA in Equation 19.4-16).

• Azimuthal angles: For 3D sectors, set the Azimuthal Start Angleand Azimuthal Stop Angle.

See Section 19.4.1 for details about how these inputs are used.

19.9.11 Point Properties for Pressure-Swirl Atomizer Injections

For a pressure-swirl atomizer injection, you will specify some of the sameproperties as for a plain-orifice atomizer. In addition to the position, axis(if 3D), temperature, mass flow rate, duration of injection (if unsteady),injector inner diameter, and azimuthal angles (if relevant) described inSection 19.9.10, you will need to specify the following parameters underPoint Properties:

• Spray angle: Set the value of the spray angle of the injected streamin the Spray Half Angle field (θ in Equation 19.4-25).

• Pressure: Set the pressure upstream of the injection in the Up-stream Pressure field (p1 in Table 19.4.1).

• Sheet breakup: Set the value of the empirical constant that de-termines the length of the ligaments that are formed after sheetbreakup in the Sheet Constant field (ln( ηb

η0) in Equation 19.4-30).

• Ligament diameter: For short waves, set the proportionality con-stant that linearly relates the ligament diameter, dL, to the wave-length that breaks up the sheet in the Ligament Constant field (seeEquations 19.4-31–19.4-33).




19.9.12 Point Properties for Air-Blast/Air-Assist AtomizerInjections

For an air-blast/air-assist atomizer, you will specify some of the sameproperties as for a plain-orifice atomizer. In addition to the position, axis(if 3D), temperature, mass flow rate, duration of injection (if unsteady),injector inner diameter, and azimuthal angles (if relevant) described inSection 19.9.10, you will need to specify the following parameters underPoint Properties:

• Outer diameter: Set the outer diameter of the injector in the In-jector Outer Diam. field. This value is used in conjunction with theInjector Inner Diam. to set the thickness of the liquid sheet (t inEquation 19.4-22).

• Spray angle: Set the initial trajectory of the film as it leaves the endof the orifice in the Spray Half Angle field (θ in Equation 19.4-25).

• Relative velocity: Set the maximum relative velocity that is pro-duced by the sheet and air in the Relative Velocity field.

• Sheet breakup: Set the value of the empirical constant that de-termines the length of the ligaments that are formed after sheetbreakup in the Sheet Constant field (ln( ηb

η0) in Equation 19.4-30).

• Ligament diameter: For short waves, set the proportionality con-stant that linearly relates the ligament diameter, dL, to the wave-length that breaks up the sheet in the Ligament Constant field (seeEquations 19.4-31–19.4-33).


19.9.13 Point Properties for Flat-Fan Atomizer Injections

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



• Arc position: Set the coordinates of the center point of the arc fromwhich the fan originates in the X-Center, Y-Center, and Z-Centerfields (see Figure 19.4.6).

• Virtual position: Set the coordinates of the virtual origin of the fanin the X-Virtual Origin, Y-Virtual Origin, and Z-Virtual Origin fields.This point is the intersection of the lines that mark the sides of thefan (see Figure 19.4.6).

• Normal vector: Set the direction that is normal to the fan in the X-Fan Normal Vector, Y-Fan Normal Vector, and Z-Fan Normal Vectorfields.


• Mass flow rate: Set the mass flow rate for the streams in the at-omizer in the Flow Rate field.


• Spray half angle: Set the initial half angle of the drops as theyleave the end of the orifice in the Spray 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 de-termines the length of the ligaments that are formed after sheetbreakup in the Flat Fan Sheet Constant field (see Equation 19.4-30).


19.9.14 Point Properties for Effervescent Atomizer Injections

For an effervescent atomizer injection, you will specify some of the sameproperties as for a plain-orifice atomizer. In addition to the position, axis



(if 3D), temperature, mass flow rate (including both flashing and non-flashing components), duration of injection (if unsteady), vapor pressure,injector inner diameter, and azimuthal angles (if relevant) described inSection 19.9.10, you will need to specify the following parameters underPoint Properties:

• Mixture quality: Set the mass fraction of the injected mixture thatvaporizes in the Mixture Quality field (x in Equation 19.4-38).

• Saturation temperature: Set the saturation temperature of thevolatile substance in the Saturation Temp. field.

• Droplet dispersion: Set the parameter that controls the spatialdispersion of the droplet sizes in the Dispersion Constant field (Ceff

in Equation 19.4-38).

• Spray angle: Set the initial trajectory of the film as it leaves theend of the orifice in the Maximum Half Angle field.


19.9.15 Modeling Turbulent Dispersion of Particles

As mentioned in Section 19.9.5, you can choose stochastic tracking and/orcloud tracking as the method for modeling turbulent dispersion of par-ticles.

Stochastic Tracking

For turbulent flows, if you choose to use the stochastic tracking tech-nique, you must enable it and specify the “number of tries”. Stochastictracking includes the effect of turbulent velocity fluctuations on the par-ticle trajectories using the DRW model described in Section 19.2.2.

1. Click the Turbulent Dispersion tab in the Set Injection Propertiespanel.

2. Enable stochastic tracking by turning on the Stochastic Model un-der Stochastic Tracking.



3. Specify the Number Of Tries:

• An input of zero tells FLUENT to compute the particle trajec-tory based on the mean continuous phase velocity field (Equa-tion 19.2-1), ignoring the effects of turbulence on the particletrajectories.

• An input of 1 or greater tells FLUENT to include turbulentvelocity fluctuations in the particle force balance as in Equa-tion 19.2-20. The trajectory is computed more than once ifyour input exceeds 1: two trajectory calculations are per-formed if you input 2, three trajectory calculations are per-formed if you input 3, etc. Each trajectory calculation in-cludes a new stochastic representation of the turbulent con-tributions to the trajectory equation.

When a sufficient number of tries is requested, the trajecto-ries computed will include a statistical representation of thespread of the particle stream due to turbulence. Note that forunsteady particle tracking, the Number of Tries is set to 1 ifStochastic Tracking is enabled.

If you want the characteristic lifetime of the eddy to be random (Equa-tion 19.2-37), enable the Random Eddy Lifetime option. You will generallynot need to change the Time Scale Constant (CL in Equation 19.2-28)from its default value of 0.15, unless you are using the Reynolds Stressturbulence model (RSM), in which case a value of 0.3 is recommended.

Figure 19.9.9 illustrates a discrete phase trajectory calculation computedvia the “mean” tracking (number of tries = 0) and Figure 19.9.10 illus-trates the “stochastic” tracking (number of tries > 1) option.

When multiple stochastic trajectory calculations are performed, the mo-mentum and mass defined for the injection are divided evenly among themultiple particle/droplet tracks, and are thus spread out in terms of theinterphase momentum, heat, and mass transfer calculations. Includingturbulent dispersion in your model can thus have a significant impact onthe effect of the particles on the continuous phase when coupled calcu-lations are 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 19.9.9: 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 19.9.10: Stochastic Trajectories in a Turbulent Flow



Cloud Tracking

For turbulent flows, you can also include the effects of turbulent disper-sion on the injection. When cloud tracking is used, the trajectory willbe tracked as a cloud of particles about a mean trajectory, as describedin Section 19.2.2.

1. Click the Turbulent Dispersion tab in the Set Injection Propertiespanel.

2. Enable cloud tracking by turning on the Cloud Model under CloudTracking.

3. Specify the minimum and maximum cloud diameters. Particlesenter the domain with an initial cloud diameter equal to the Min.Cloud Diameter. The particle cloud’s maximum allowed diameteris specified by the Max. Cloud Diameter.

You may want to restrict the Max. Cloud Diameter to a relevantlength scale for the problem to improve computational efficiencyin complex domains where the mean trajectory may become stuckin recirculation regions.

19.9.16 Custom Particle Laws

If the standard FLUENT laws, Laws 1 through 6, do not adequatelydescribe the physics of your discrete phase model, you can modify themby creating custom laws with user-defined functions. See the separateUDF Manual for information about user-defined functions. You can alsocreate custom laws by using a subset of the existing FLUENT laws (e.g.,Laws 1, 2, and 4), or a combination of existing laws and user-definedfunctions.

Once you have defined and loaded your user-defined function(s), you cancreate a custom law by enabling the Custom option under Laws in theSet Injection Properties panel. This will open the Custom Laws panel.In the drop-down list to the left of each of the six particle laws, youcan select the appropriate particle law for your custom law. Each list



Figure 19.9.11: The Custom Laws Panel

contains the available options that can be chosen (the standard laws plusany user-defined functions you have loaded).

There is a seventh drop-down list in the Custom Laws panel labeledSwitching. You may wish to have FLUENT vary the laws used dependingon conditions in the model. You can customize the way FLUENT switchesbetween laws by selecting a user-defined function from this drop-downlist.

An example of when you might want to use a custom law might be toreplace the standard devolatilization law with a specialized devolatiliza-tion law that more accurately describes some unique aspects of yourmodel. After creating and loading a user-defined function that detailsthe physics of your devolatilization law, you would visit the Custom Lawspanel and replace the standard devolatilization law (Law 2) with youruser-defined function.



19.9.17 Defining Properties Common to More Than OneInjection

If you have a number of injections for which you want to set the sameproperties, FLUENT provides a shortcut so that you do not need to visitthe Set Injection Properties panel for each injection to make the samechanges.

As described in Section 19.9.5, if you select more than one injection inthe Injections panel, clicking the Set... button will open the Set MultipleInjection Properties panel (Figure 19.9.12) instead of the Set InjectionProperties panel.

Depending on the type of injections you have selected (single, group,atomizers, etc.), there will be different categories of properties listedunder Injections Setup. The names of these categories correspond to theheadings within the Set Injection Properties panel (e.g., Particle Type andStochastic Tracking). Only those categories that are appropriate for allof your selected injections (which are shown in the Injections list) willbe listed. If all of these injections are of the same type, more categoriesof properties will be available for you to modify. If the injections are ofdifferent types, you will have fewer categories to select from.

Modifying Properties

To modify a property, follow these steps:

1. Select the appropriate category in the Injections Setup list. Forexample, if you want to set the same flow rate for all of the selectedinjections, select Point Properties. The panel will expand to showthe properties that appear under that heading in the Set InjectionProperties 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.

You must click Apply to save the property settings within each!category. If, for example, you want to modify the flow rate andthe stochastic tracking parameters, you will need to select Point



Figure 19.9.12: The Set Multiple Injection Properties Panel



Properties in the Injections Setup list, specify the flow rate, andclick Apply. You would then repeat the process for the stochastictracking parameters, clicking Apply again when you are done.

There are two types of properties that can be modified using the SetMultiple Injection Properties 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 SetInjection Properties panel. For example, if you select Particle Type in theInjections Setup list, the panel will expand to show the portion of the SetInjection Properties panel where you choose the particle type. You cansimply choose the desired type and click Apply.

Setting the second type of property requires an additional step. If youselect a category in the Injections Setup list that contains this type ofproperty, the expanded portion of the panel will look like the corre-sponding part of the Set Injection Properties panel, with the addition ofModify check buttons (see Figure 19.9.12). To change one of the prop-erties, first turn on the Modify check button to its left, and then specifythe desired status or value.

For example, if you would like to enable stochastic tracking, first turn onthe Modify check button to the left of Stochastic Model. This will makethe property active so you can modify its status. Then, under Property,turn on the Stochastic Model check button. (Be sure to click Apply whenyou are done setting stochastic tracking parameters.)



If you would like to change the value of Number of Tries, select the Modifycheck button to its left to make it active, and then enter the new valuein the field. Make sure you click Apply when you have finished modifyingthe stochastic tracking properties.

The setting for a property that has not been activated with the Modify!check 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 youwant them before you do this. If you make a mistake, you will have toreturn to the Set Injection Properties panel for each injection to fix theincorrect setting, if it is not possible to do so in the Set Multiple InjectionProperties panel.

Modifying Properties Common to a Subset of Selected Injections

Note that it is possible to change a property that is relevant for only asubset of the selected injections. For example, if some of the selectedinjections are using stochastic tracking and some are not, enabling theRandom Eddy Lifetime option and clicking Apply will turn this option ononly for those injections that are using stochastic tracking. The otherinjections will be unaffected.


19.10 Setting Boundary Conditions for the Discrete Phase


When a particle reaches a physical boundary (e.g., a wall or inlet bound-ary) in your model, FLUENT applies a discrete phase boundary conditionto determine the fate of the trajectory at that boundary. The boundarycondition, or trajectory fate, can be defined separately for each zone inyour FLUENT model.

19.10.1 Discrete Phase Boundary Condition Types

The available boundary conditions, as noted in Section 19.2, include thefollowing:

• “reflect” rebounds the particle off the boundary in question with achange in its momentum as defined by the coefficient of restitution.(See Figure 19.10.1.)

coefficientofrestitution

=V

2,nV1,n

θ1 θ2

Figure 19.10.1: “Reflect” Boundary Condition for the Discrete Phase

The normal coefficient of restitution defines the amount of mo-mentum in the direction normal to the wall that is retained by theparticle after the collision with the boundary [236]:

en =v2,n

v1,n(19.10-1)

where vn is the particle velocity normal to the wall and the sub-scripts 1 and 2 refer to before and after collision, respectively. Simi-



larly, the tangential coefficient of restitution, et, defines the amountof momentum in the direction tangential to the wall that is retainedby the particle.

A normal or tangential coefficient of restitution equal to 1.0 impliesthat the particle retains all of its normal or tangential momentumafter the rebound (an elastic collision). A normal or tangential co-efficient of restitution equal to 0.0 implies that the particle retainsnone of its normal or tangential momentum after the rebound.

Non-constant coefficients of restitution can be specified for wallzones with the “reflect” type boundary condition. The coefficientsare set as a function of the impact angle, θ1, in Figure 19.10.1.

Note that the default setting for both coefficients of restitutionis a constant value of 1.0 (all normal and tangential momentumretained).

• “trap” terminates the trajectory calculations and records the fateof the particle as “trapped”. In the case of evaporating droplets,their entire mass instantaneously passes into the vapor phase andenters the cell adjacent to the boundary. See Figure 19.10.2. In thecase of combusting particles, the remaining volatile mass is passedinto the vapor phase.

θ1

volatile fractionflashes to vapor

Figure 19.10.2: “Trap” Boundary Condition for the Discrete Phase

• “escape” reports the particle as having “escaped” when it encoun-ters the boundary in question. Trajectory calculations are termi-nated. See Figure 19.10.3.



particle vanishes

Figure 19.10.3: “Escape” Boundary Condition for the Discrete Phase

• “interior” means that the particles will pass through the inter-nal boundary. This option is available only for internal boundaryzones, such as a radiator or a porous jump.

Because you can stipulate any of these conditions at flow boundaries, itis possible to incorporate mixed discrete phase boundary conditions inyour FLUENT model.

Default Discrete Phase Boundary Conditions

FLUENT assumes the following boundary conditions:

• “reflect” at wall, symmetry, and axis boundaries, with both coeffi-cients of restitution equal to 1.0

• “escape” at all flow boundaries (pressure and velocity inlets, pres-sure outlets, etc.)

• “interior” at all internal boundaries (radiator, porous jump, etc.)

The coefficient of restitution can be modified only for wall boundaries.



19.10.2 Inputs for Discrete Phase Boundary Conditions

Discrete phase boundary conditions can be set for boundaries in thepanels opened from the Boundary Conditions panel. When one or moreinjections have been defined, inputs for the discrete phase will appear inthe panels (e.g., Figure 19.10.4).

Figure 19.10.4: Discrete Phase Boundary Conditions in the Wall Panel

Select reflect, trap, or escape in the Boundary Cond. Type drop-downlist under Discrete Phase Model Conditions. (In the Walls panel, youwill need to click on the DPM tab to access the Discrete Phase ModelConditions.) These conditions are described in Section 19.10.1. Youcan also select a user-defined function in this list. For internal boundary


19.11 Setting Material Properties for the Discrete Phase

zones, such as a radiator or a porous jump, you can also choose an interiorboundary condition. The interior condition means that the particles willpass through 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 Nor-mal and Tangent coefficients of restitution under Discrete Phase ReflectionCoefficients. See Section 19.10.1 for details about the boundary condi-tion types and the coefficients of restitution. The panels for defining thepolynomial, piecewise-linear, and piecewise-polynomial functions are thesame as those used for defining temperature-dependent properties. SeeSection 7.1.3 for details.

If the Erosion/Accretion option is selected in the Discrete Phase Modelpanel, the erosion rate expression must be specified at the walls. Theerosion rate is defined in Equation 19.2-62 as a product of the massflux and specified functions for the particle diameter, impact angle, andvelocity exponent. Under Erosion Model in the Wall panel, you can definea constant, polynomial, piecewise-linear, or piecewise-polynomial functionfor the Impact Angle Function, Diameter Function, and Velocity ExponentFunction (f(α), C(dp), and b(v) in Equation 19.2-62). See Section 19.7.6for a detailed description of these functions and Section 7.1.3 for detailsabout using the panels for defining polynomial, piecewise-linear, andpiecewise-polynomial functions.


In order to apply the physical models described in earlier sections tothe prediction of the discrete phase trajectories and heat/mass transfer,FLUENT requires many physical property inputs.

19.11.1 Summary of Property Inputs

Tables 19.11.1–19.11.4 summarize which of these property inputs areused for each particle type and in which of the equations for heat andmass transfer each property input is used. Detailed descriptions of eachinput are provided in Section 19.11.2.



Table 19.11.1: Property Inputs for Inert ParticlesProperty Symboldensity ρp in Eq. 19.2-1specific heat cp in Eq. 19.3-3particle emissivity εp in Eq. 19.3-3particle scattering factor f in Eq. 11.3-20thermophoretic coefficient DT,p in Eq. 19.2-14

Table 19.11.2: Property Inputs for Droplet ParticlesProperties Symboldensity ρp in Eq. 19.2-1specific heat cp in Eq. 19.3-17thermal conductivity kp in Eq. 19.2-15viscosity µ in Eq. 19.4-48latent heat hfg in Eq. 19.3-17vaporization temperature Tvap in Eq. 19.3-10boiling point Tbp in Eq. 19.3-10, 19.3-18volatile component fraction fv0 in Eq. 19.3-11, 19.3-19binary diffusivity Di,m in Eq. 19.3-15saturation vapor pressure psat(T ) in Eq. 19.3-13heat of pyrolysis hpyrol in Eq. 19.5-2droplet surface tension σ in Eq. 19.4-18, 19.4-47particle emissivity εp in Eq. 19.3-17, 19.3-23particle scattering factor f in Eq. 11.3-20thermophoretic coefficient DT,p in Eq. 19.2-14



Table 19.11.3: Property Inputs for Combusting Particles (Laws 1–4)

Properties Symboldensity ρp in Eq. 19.2-1specific heat cp in Eq. 19.3-3latent heat hfg in Eq. 19.5-2vaporization temperature Tvap = Tbp in Eq. 19.3-24volatile component fraction fv0 in Eq. 19.3-25swelling coefficient Csw in Eq. 19.3-57burnout stoichiometric ratio Sb in Eq. 19.3-64combustible fraction fcomb in Eq. 19.3-63heat of reaction for burnout Hreac in Eq. 19.3-64 19.3-78fraction of reaction heat given to solid fh in Eq. 19.3-78particle emissivity εp in Eq. 19.3-58, 19.3-78particle scattering factor f in Eq. 11.3-20thermophoretic coefficient DT,p in Eq. 19.2-14devolatilization model–law 4, constant rateconstant A0 in Eq. 19.3-26

–law 4, single ratepre-exponential factor A1 in Eq. 19.3-27activation energy E in Eq. 19.3-27

–law 4, two ratespre-exponential factors A1, A2 in Eq. 19.3-30, 19.3-31activation energies E1, E2 in Eq. 19.3-30, 19.3-31weighting factors α1, α2 in Eq. 19.3-32

–law 4, CPDinitial fraction of bridges in coal lattice p0 in Eq. 19.3-43initial fraction of char bridges c0 in Eq. 19.3-42lattice coordination number σ + 1 in Eq. 19.3-54cluster molecular weight Mw,1 in Eq. 19.3-54side chain molecular weight Mw,δ in Eq. 19.3-53



Table 19.11.4: Property Inputs for Combusting Particles (Law 5)Properties Symbolcombustion model–law 5, diffusion rate

binary diffusivity Di,m in Eq. 19.3-65–law 5, diffusion/kinetic rate

mass diffusion limited rate constant C1 in Eq. 19.3-66kinetics limited rate pre-exp. factor C2 in Eq. 19.3-67kinetics limited rate activ. energy E in Eq. 19.3-67

–law 5, intrinsic ratemass diffusion limited rate constant C1 in Eq. 19.3-66kinetics limited rate pre-exp. factor Ai in Eq. 19.3-76kinetics limited rate activ. energy Ei in Eq. 19.3-76char porosity θ in Eq. 19.3-73mean pore radius rp in Eq. 19.3-75specific internal surface area Ag in Eq. 19.3-70, 19.3-72tortuosity τ in Eq. 19.3-73burning mode α in Eq. 19.3-77

–law 5, multiple surface reactionbinary diffusivity Di,m in Eq. 19.3-65



19.11.2 Setting Discrete-Phase Physical Properties

The Concept of Discrete-Phase Materials

When you create a particle injection and define the initial conditions forthe discrete phase (as described in Section 19.9), you choose a particularmaterial as the particle’s material. All particle streams of that materialwill have the same physical properties.

Discrete-phase materials are divided into three categories, correspondingto the three types of particles available. These material types are inert-particle, droplet-particle, and combusting-particle. Each material type willbe added to the Material Type list in the Materials panel when an injectionof that type of particle is defined (in the Set Injection Properties or SetMultiple Injection Properties panel, as described in Section 19.9). Thefirst time you create an injection of each particle type, you will be ableto choose a material from the database, and this will become the defaultmaterial for that type of particle. That is, if you create another injectionof the same type of particle, your selected material will be used for thatinjection as well. You may choose to modify the predefined properties foryour selected particle material, if you want (as described in Section 7.1.2).If you need only one set of properties for each type of particle, you neednot define any new materials; you can simply use the same material forall particles.

If you do not find the material you want in the database, you can select!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 7.1.2.

Note that a discrete-phase material type will not appear in the Material!Type list in the Materials panel until you have defined an injection ofthat type of particles. This means, for example, that you cannot defineor modify any combusting-particle materials until you have defined acombusting particle injection (as described in Section 19.9).



Defining Additional Discrete-Phase Materials

In many cases, a single set of physical properties (density, heat capacity,etc.) is appropriate for each type of discrete phase particle consideredin a given model. Sometimes, however, a single model may contain twodifferent types of inert, droplet, or combusting particles (e.g., heavy par-ticles and gaseous bubbles or two different types of evaporating liquiddroplets). In such cases, it is necessary to assign a different set of prop-erties to the two (or more) different types of particles. This is easilyaccomplished by defining two or more inert, droplet, or combusting par-ticle materials and using the appropriate one for each particle injection.

You can define additional discrete-phase materials either by copyingthem from the database or by creating them from scratch. See Sec-tion 7.1.2 for instructions on using the Materials panel to perform theseactions.

Recall that you must define at least one injection (as described in Sec-!tion 19.9) containing particles of a certain type before you will be ableto define additional materials for that particle type.

Description of Properties

The properties that appear in the Materials panel vary depending onthe particle type (selected in the Set Injection Properties or Set MultipleInjection Properties panel, as described in Sections 19.9.5 and 19.9.17) andthe physical models you are using in conjunction with the discrete-phasemodel.

Below, all properties you may need to define for a discrete-phase materialare listed. See Tables 19.11.1–19.11.4 to see which properties are definedfor each type of particle.

Density is the density of the particulate phase in units of mass per unitvolume of the discrete phase. This density is the mass density andnot the volumetric density. Since certain particles may swell dur-ing the trajectory calculations, your input is actually an “initial”density.



Cp is the specific heat, cp, of the particle. The specific heat may bedefined as a function of temperature by selecting one of the functiontypes from the drop-down list to the right of Cp. See Section 7.1.3for details about temperature-dependent properties.

Thermal Conductivity is the thermal conductivity of the particle. Thisinput is specified in units of W/m-K in SI units or Btu/ft-h-F inBritish units and is treated as a constant by FLUENT.

Latent Heat is the latent heat of vaporization, hfg, required for phasechange from an evaporating liquid droplet (Equation 19.3-17) orfor the evolution of volatiles from a combusting particle (Equa-tion 19.3-58). This input is supplied in units of J/kg in SI units orof Btu/lbm in British units and is treated as a constant by FLU-ENT.

Thermophoretic Coefficient is the coefficient DT,p in Equation 19.2-14,and appears when the thermophoretic force (which is described inSection 19.2.1) is included in the trajectory calculation (i.e., whenthe Thermophoretic Force option is enabled in the Discrete PhaseModel panel). The default is the expression developed by Tal-bot [237] (talbot-diffusion-coeff) and requires no input from you.You can also define the thermophoretic coefficient as a functionof temperature by selecting one of the function types from thedrop-down list to the right of Thermophoretic Coefficient. See Sec-tion 7.1.3 for details about temperature-dependent properties.

Vaporization Temperature is the temperature, Tvap, at which the calcula-tion of vaporization from a liquid droplet or devolatilization froma combusting particle is initiated by FLUENT. Until the particletemperature reaches Tvap, the particle is heated via Law 1, Equa-tion 19.3-3. This temperature input represents a modeling decisionrather than any physical characteristic of the discrete phase.

Boiling Point is the temperature, Tbp, at which the calculation of theboiling rate equation (19.3-20) is initiated by FLUENT. When adroplet particle reaches the boiling point, FLUENT applies Law 3and assumes that the droplet temperature is constant at Tbp. Theboiling point should be defined as the saturated vapor temperature



at the system pressure that you defined in the Operating Conditionspanel.

Volatile Component Fraction (fv0) is the fraction of a droplet particlethat may vaporize via Laws 2 and/or 3 (Sections 19.3.3 and 19.3.4).For combusting particles, it is the fraction of volatiles that may beevolved via Law 4 (Section 19.3.5).

Binary Diffusivity is the mass diffusion coefficient, Di,m, used in the va-porization law, Law 2 (Equation 19.3-15). This input is also usedto define the mass diffusion of the oxidizing species to the sur-face of a combusting particle, Di,m, as given in Equation 19.3-65.(Note that the diffusion coefficient inputs that you supply for thecontinuous phase are not used for the discrete phase.)

Saturation Vapor Pressure is the saturated vapor pressure, psat, definedas a function of temperature, which is used in the vaporizationlaw, Law 2 (Equation 19.3-13). The saturated vapor pressure maybe defined as a function of temperature by selecting one of thefunction types from the drop-down list to the right of its name. (SeeSection 7.1.3 for details about temperature-dependent properties.)In the case of unrealistic inputs, FLUENT restricts the range ofPsat to between 0.0 and the operating pressure. Correct input of arealistic vapor pressure curve is essential for accurate results fromthe vaporization model.

Heat of Pyrolysis is the heat of the instantaneous pyrolysis reaction, hpyrol,that the evaporating/boiling species may undergo when released tothe continuous phase. This input represents the conversion of theevaporating species to lighter components during the evaporationprocess. The heat of pyrolysis should be input as a positive numberfor exothermic reaction and as a negative number for endothermicreaction. The default value of zero implies that the heat of pyrol-ysis is not considered. This input is used in Equation 19.5-2.

Swelling Coefficient is the coefficient Csw in Equation 19.3-57, which gov-erns the swelling of the coal particle during the devolatilization law,Law 4 (Section 19.3.5). A swelling coefficient of unity (the default)



implies that the coal particle stays at constant diameter during thedevolatilization process.

Burnout Stoichiometric Ratio is the stoichiometric requirement, Sb, forthe burnout reaction, Equation 19.3-64, in terms of mass of oxidantper mass of char in the particle.

Combustible Fraction is the mass fraction of char, fcomb, in the coal par-ticle, i.e., the fraction of the initial combusting particle that willreact in the surface reaction, Law 5 (Equation 19.3-63).

Heat of Reaction for Burnout is the heat released by the surface charcombustion reaction, Law 5 (Equation 19.3-64). This parameteris input in terms of heat release (e.g., Joules) per unit mass of charconsumed in the surface reaction.

React. Heat Fraction Absorbed by Solid is the parameter fh (Equation19.3-78), which controls the distribution of the heat of reactionbetween the particle and the continuous phase. The default valueof zero implies that the entire heat of reaction is released to thecontinuous phase.

Devolatilization Model defines which version of the devolatilization model,Law 4, is being used. If you want to use the default constant ratedevolatilization model, Equation 19.3-26, retain the selection ofconstant in the drop-down list to the right of Devolatilization Modeland input the rate constant A0 in the field below the list.

You can activate one of the optional devolatilization models (thesingle kinetic rate, two kinetic rates, or CPD model, as describedin Section 19.3.5) by choosing single rate, two-competing-rates, orcpd-model in the drop-down list.

When the single kinetic rate model (single-rate) is selected, theSingle Rate Devolatilization Model panel will appear and you willenter the Pre-exponential Factor, A1, and the Activation Energy, E,to be used in Equation 19.3-28 for the computation of the kineticrate.

When the two competing rates model (two-competing-rates) is se-lected, the Two Competing Rates Model panel will appear and



you will enter, for the First Rate and the Second Rate, the Pre-exponential Factor (A1 in Equation 19.3-30 and A2 in Equation19.3-31), Activation Energy (E1 in Equation 19.3-30 and E2 inEquation 19.3-31), and Weighting Factor (α1 and α2 in Equation19.3-32). The constants you input are used in Equations 19.3-30through 19.3-32.

When the CPD model (cpd-model) is selected, the CPD Model panelwill appear and you will enter the Initial Fraction of Bridges in CoalLattice (p0 in Equation 19.3-43), Initial Fraction of Char Bridges (c0in Equation 19.3-42), Lattice Coordination Number (σ + 1 in Equa-tion 19.3-54), Cluster Molecular Weight (Mw,1 in Equation 19.3-54),and Side Chain Molecular Weight (Mw,δ in Equation 19.3-53).

Note that the Single Rate Devolatilization Model, Two CompetingRates Model, and CPD Model panels are modal panels, which meansthat you must tend to them immediately before continuing theproperty definitions.

Combustion Model defines which version of the surface char combustionlaw (Law 5) is being used. If you want to use the default diffusion-limited rate model, retain the selection of diffusion-limited in thedrop-down list to the right of Combustion Model. No additionalinputs are necessary, because the binary diffusivity defined abovewill be used in Equation 19.3-65.

To use the kinetics/diffusion-limited rate model for the surfacecombustion model, select kinetics/diffusion-limited in the drop-downlist. The Kinetics/Diffusion Limited Combustion Model panel willappear and you will enter the Mass Diffusion Limited Rate Constant(C1 in Equation 19.3-66), Kinetics Limited Rate Pre-exponential Fac-tor (C2 in Equation 19.3-67), and Kinetics Limited Rate ActivationEnergy (E in Equation 19.3-67).

Note that the Kinetics/Diffusion Limited Combustion Model panel isa modal panel, which means that you must tend to it immediatelybefore continuing the property definitions.

To use the intrinsic model for the surface combustion model, se-lect intrinsic-model in the drop-down list. The Intrinsic CombustionModel panel will appear and you will enter the Mass Diffusion Lim-



ited Rate Constant (C1 in Equation 19.3-66), Kinetics Limited RatePre-exponential Factor (Ai in Equation 19.3-76), Kinetics LimitedRate Activation Energy (Ei in Equation 19.3-76), Char Porosity (θin Equation 19.3-73), Mean Pore Radius (rp in Equation 19.3-75),Specific Internal Surface Area (Ag in Equations 19.3-70 and 19.3-72),Tortuosity (τ in Equation 19.3-73), and Burning Mode, alpha (α inEquation 19.3-77).

Note that the Intrinsic Combustion Model panel is a modal panel,which means that you must tend to it immediately before contin-uing the property definitions.

To use the multiple surface reactions model, select multiple-surface-reactions in the drop-down list. FLUENT will display a dialog boxinforming you that you will need to open the Reactions panel, whereyou can review or modify the particle surface reactions that youspecified as described in Section 13.1.2.

If you have not yet defined any particle surface reactions, you must!be sure to define them now. See Section 13.3.3 for more informationabout using the multiple surface reactions model.

You will notice that the Burnout Stoichiometric Ratio and Heat ofReaction for Burnout are no longer available in the Materials panel,as these parameters are now computed from the particle surfacereactions you defined in the Reactions panel.

Note that the multiple surface reactions model is available only ifthe Particle Surface option for Reactions is enabled in the SpeciesModel panel. See Section 13.3.2 for details.

When the effect of particles on radiation is enabled (for the P-1 or dis-crete ordinates radiation model only) in the Discrete Phase Model panel,you will need to define the following additional parameters:

Particle Emissivity is the emissivity of particles in your model, εp, used tocompute radiation heat transfer to the particles (Equations 19.3-3,19.3-17, 19.3-23, 19.3-58, and 19.3-78) when the P-1 or discreteordinates radiation model is active. Note that you must enableradiation to particles, using the Particle Radiation Interaction option



in the Discrete Phase Model panel. Recommended values of particleemissivity are 1.0 for coal particles and 0.5 for ash [143].

Particle Scattering Factor is the scattering factor, fp, due to particles inthe P-1 or discrete ordinates radiation model (Equation 11.3-20).Note that you must enable particle effects in the radiation model,using the Particle Radiation Interaction option in the Discrete PhaseModel panel. The recommended value of fp for coal combustionmodeling is 0.9 [143]. Note that if the effect of particles on radiationis enabled, scattering in the continuous phase will be ignored in theradiation model.

When an atomizer injection model and/or the spray breakup or collisionmodel is enabled in the Set Injection Properties panel (atomizers) and/orDiscrete Phase Model panel (spray breakup/collision), you will need todefine the following additional parameters:

Viscosity is the droplet viscosity, µl. The viscosity may be defined asa function of temperature by selecting one of the function typesfrom the drop-down list to the right of Viscosity. See Section 7.1.3for details about temperature-dependent properties. You also havethe option of implementing a user-defined function to model thedroplet viscosity. See the separate UDF Manual for informationabout user-defined functions.

Droplet Surface Tension is the droplet surface tension, σ. The surfacetension may be defined as a function of temperature by selectingone of the function types from the drop-down list to the rightof Droplet Surface Tension. See Section 7.1.3 for details abouttemperature-dependent properties. You also have the option ofimplementing a user-defined function to model the droplet surfacetension. See the separate UDF Manual for information about user-defined functions.


19.12 Calculation Procedures for the Discrete Phase


Solution of the discrete phase implies integration in time of the forcebalance on the particle (Equation 19.2-1) to yield the particle trajectory.As the particle is moved along its trajectory, heat and mass transferbetween the particle and the continuous phase are also computed viathe heat/mass transfer laws (Section 19.3). The accuracy of the dis-crete phase calculation thus depends on the time accuracy of the in-tegration and upon the appropriate coupling between the discrete andcontinuous phases when required. Numerical controls are described inSection 19.12.1. Coupling and performing trajectory calculations aredescribed in Section 19.12.2. Sections 19.12.3 and 19.12.4 provide infor-mation about resetting interphase exchange terms and using the parallelsolver for a discrete phase calculation.

19.12.1 Parameters Controlling the Numerical Integration

You will use two parameters to control the time integration of the particletrajectory equations:

• the length scale or step length factor, used to set the time step forintegration within each control volume

• the maximum number of time steps, used to abort trajectory cal-culations when the particle never exits the flow domain

Each of these parameters is set in the Discrete Phase Model panel (Fig-ure 19.12.1) under Tracking Parameters.


Max. Number Of Steps is the maximum number of time steps used tocompute a single particle trajectory via integration of Equations19.2-1 and 19.2-21. When the maximum number of steps is ex-ceeded, FLUENT abandons the trajectory calculation for the cur-rent particle injection and reports the trajectory fate as “incom-plete”. The limit on the number of integration time steps elimi-nates the possibility of a particle being caught in a recirculating



Figure 19.12.1: The Discrete Phase Model Panel



region of the continuous phase flow field and being tracked in-finitely. Note that you may easily create problems in which thedefault value of 500 time steps is insufficient for completion of thetrajectory calculation. In this case, when trajectories are reportedas incomplete within the domain and the particles are not recircu-lating indefinitely, you can increase the maximum number of steps(up to a limit of 109).

Length Scale controls the integration time step size used to integrate theequations of motion for the particle. The integration time step iscomputed by FLUENT based on a specified length scale L, and thevelocity of the particle (up) and of the continuous phase (uc):

∆t =L

up + uc(19.12-1)

where L is the Length Scale that you define. As defined by Equa-tion 19.12-1, L is proportional to the integration time step and isequivalent to the distance that the particle will travel before itsmotion equations are solved again and its trajectory is updated. Asmaller value for the Length Scale increases the accuracy of the tra-jectory and heat/mass transfer calculations for the discrete phase.

(Note that particle positions are always computed when particlesenter/leave a cell; even if you specify a very large length scale, thetime step used for integration will be such that the cell is traversedin one step.)

Length Scale will appear in the Discrete Phase Model panel whenthe Specify Length Scale option is on (the default setting).

Step Length Factor also controls the time step size used to integrate theequations of motion for the particle. It differs from the LengthScale in that it allows FLUENT to compute the time step in termsof the number of time steps required for a particle to traverse acomputational cell. To set this parameter instead of the LengthScale, turn off the Specify Length Scale option.

The integration time step is computed by FLUENT based on a char-acteristic time that is related to an estimate of the time required



for the particle to traverse the current continuous phase controlvolume. If this estimated transit time is defined as ∆t∗, FLUENTchooses a time step ∆t as

∆t =∆t∗

λ(19.12-2)

where λ is the Step Length Factor. As defined by Equation 19.12-2,λ is inversely proportional to the integration time step and isroughly equivalent to the number of time steps required to tra-verse the current continuous phase control volume. A larger valuefor the Step Length Factor decreases the discrete phase integrationtime step. The default value for the Step Length Factor is 20.

One simple rule of thumb to follow when setting the parameters aboveis that if you want the particles to advance through a domain of lengthD, the Length Scale times the Max. Number Of Steps should be approx-imately equal to D.

19.12.2 Performing Trajectory Calculations

The trajectories of your discrete phase injections are computed whenyou display the trajectories using graphics or when you perform solu-tion iterations. That is, you can display trajectories without impactingthe continuous phase, or you can include their effect on the continuum(termed a coupled calculation). In turbulent flows, trajectories can bebased on mean (time-averaged) continuous phase velocities or they canbe impacted by instantaneous velocity fluctuations in the fluid. This sec-tion describes the procedures and commands you use to perform coupledor uncoupled trajectory calculations, with or without stochastic trackingor 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 injec-tions of interest.

In the uncoupled approach, this two-step procedure completes the mod-eling effort, as illustrated in Figure 19.12.2. The particle trajectories arecomputed as they are displayed, based on a fixed continuous-phase flowfield. Graphical and reporting options are detailed in Section 19.13.

continuous phase flow field calculation

particle trajectory calculation

Figure 19.12.2: Uncoupled Discrete Phase Calculations

This procedure is adequate when the discrete phase is present at a lowmass and momentum loading, in which case the continuous phase is notimpacted by the presence of the discrete phase.

Coupled Calculations

In a coupled two-phase simulation, FLUENT modifies the two-step pro-cedure above as follows:

1. Solve the continuous phase flow field (prior to introduction of thediscrete phase).

2. Introduce the discrete phase by calculating the particle trajectoriesfor each discrete phase injection.

3. Recalculate the continuous phase flow, using the interphase ex-change of momentum, heat, and mass determined during the pre-vious particle calculation.

4. Recalculate the discrete phase trajectories in the modified contin-uous phase flow field.



5. Repeat the previous two steps until a converged solution is achievedin which both the continuous phase flow field and the discrete phaseparticle trajectories are unchanged with each additional calcula-tion.

This coupled calculation procedure is illustrated in Figure 19.12.3. Whenyour FLUENT model includes a high mass and/or momentum loading inthe discrete phase, the coupled procedure must be followed in order toinclude the important impact of the discrete phase on the continuousphase flow field.

particle trajectory calculation

update continuous phase source terms

continuous phase flow field calculation

Figure 19.12.3: Coupled Discrete Phase Calculations

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 begin with a partially (or fully) converged continuous phase



flow field. You will then create your injection(s) and set up the coupledcalculation.

For each discrete-phase iteration, FLUENT computes the particle/droplettrajectories and updates the interphase exchange of momentum, heat,and mass in each control volume. These interphase exchange terms thenimpact the continuous phase when the continuous phase iteration is per-formed. During the coupled calculation, FLUENT will perform the dis-crete phase iteration at specified intervals during the continuous-phasecalculation. The coupled calculation continues until the continuous phaseflow field no longer changes with further calculations (i.e., all convergencecriteria are satisfied). When convergence is reached, the discrete phasetrajectories no longer change either, since changes in the discrete phasetrajectories 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 19.12.1), enable the In-teraction with Continuous Phase option.

3. Set the frequency with which the particle trajectory calculationsare introduced in the Number Of Continuous Phase Iterations PerDPM Iteration field. If you set this parameter to 5, for example,a discrete phase iteration will be performed every fifth continu-ous phase iteration. The optimum number of iterations betweentrajectory calculations depends upon the physics of your FLUENTmodel.

Note that if you set this parameter to 0, FLUENT will not perform!any discrete phase iterations.

During the coupled calculation (which you initiate using the Iterate panelin the usual manner) you will see the following information in the FLU-ENT console as the continuous and discrete phase iterations are per-formed:



iter continuity x-velocity y-velocity k epsilon energy time/ite

314 2.5249e-01 2.8657e-01 1.0533e+00 7.6227e-02 2.9771e-02 9.8181e-03 0:00:05

315 2.7955e-01 2.5867e-01 9.2736e-01 6.4516e-02 2.6545e-02 4.2314e-03 0:00:03

DPM Iteration ....

number tracked= 9, number escaped= 1, aborted= 0, trapped= 0, evaporated = 8, i

Done.

316 1.9206e-01 1.1860e-01 6.9573e-01 5.2692e-02 2.3997e-02 2.4532e-03 0:00:02

317 2.0729e-01 3.2982e-02 8.3036e-01 4.1649e-02 2.2111e-02 2.5369e-01 0:00:01

318 3.2820e-01 5.5508e-02 6.0900e-01 5.9018e-02 2.6619e-02 4.0394e-02 0:00:00

Note that you can perform a discrete phase calculation at any time byusing the solve/dpm-update text command.

Stochastic Tracking in Coupled Calculations

If you include the stochastic prediction of turbulent dispersion in thecoupled two-phase flow calculations, the number of stochastic tries ap-plied each time the discrete phase trajectories are introduced duringcoupled calculations will be equal to the Number of Tries specified in theSet Injection Properties panel. Input of this parameter is described inSection 19.9.15.

Note that the number of tries should be set to 0 if you want to per-form the coupled calculation based on the mean continuous phase flowfield. An input of n ≥ 1 requests n stochastic trajectory calculations foreach particle in the injection. Note that when the number of stochastictracks included is small, you may find that the ensemble average of thetrajectories is quite different each time the trajectories are computed.These differences may, in turn, impact the convergence of your coupledsolution. For this reason, you should include an adequate number ofstochastic tracks in order to avoid convergence troubles in coupled cal-culations.

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, FLU-ENT applies under-relaxation to the momentum, heat, and mass trans-fer terms. This under-relaxation serves to increase the stability of the



coupled calculation procedure by letting the impact of the discrete phasechange only gradually:

Enew = Eold + α(Ecalculated − Eold) (19.12-3)

where Enew is the exchange term, Eold is the previous value, Ecalculated isthe newly computed value, and α is the particle/droplet under-relaxationfactor. FLUENT uses a default value of 0.5 for α. You can modify αby changing the value in the Discrete Phase Sources field under Under-Relaxation Factors in the Solution Controls panel. You may need to de-crease α in order to improve the stability of coupled discrete phase cal-culations.

19.12.3 Resetting the Interphase Exchange Terms

If you have performed coupled calculations, resulting in non-zero inter-phase sources/sinks of momentum, heat, and/or mass that you do notwant to include in subsequent calculations, you can reset these sourcesto zero.

Solve −→ Initialize −→Reset DPM Sources

When you select the Reset DPM Sources menu item, the sources willimmediately be reset to zero without any further confirmation from you.

19.12.4 Parallel Processing for the Discrete Phase Model

If you are running FLUENT on a shared-memory multiprocessor machine(see the Release Notes for platform limitations), you will need to specifyexplicitly that you want to perform the discrete phase calculation in par-allel. In the Discrete Phase Model panel, turn on the Workpile Algorithmoption under Parallel and specify the Number of Threads. By default,the Number of Threads is equal to the number of compute nodes youspecified for the parallel solver. You can modify this value based on thecomputational requirements of the particle calculations. If, for example,the particle calculations require more computation than the flow calcu-lation, you can increase the Number of Threads (up to the number ofavailable processors) to improve performance.



Note that the discrete phase model is also available when solving inparallel on a distributed memory machine or compute cluster. However,as when running on a shared-memory machine, the particle calculationswill take place entirely within the Host process. Therefore, you will needto make sure that there is enough memory to store the entire grid on themachine executing the Host process. In such a situation, the number ofthreads should not exceed the number of CPUs on the host machine.

19.13 Postprocessing for the Discrete Phase

After you have completed your discrete phase inputs and any coupledtwo-phase calculations of interest, you can display and store the particletrajectory predictions. FLUENT provides both graphical and alphanu-meric reporting facilities for the discrete phase, including 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 ex-change of momentum, heat, and mass

• Sampling of trajectories at boundaries and lines/planes

• Histograms of trajectory data at sample planes

• Display of erosion/accretion rates

This section provides detailed descriptions of each of these postprocessingoptions.

(Note that plotting or reporting trajectories does not change the sourceterms.)



19.13.1 Graphical Display of Trajectories

When you have defined discrete phase particle injections, as described inSection 19.9, you can display the trajectories of these discrete particlesusing the Particle Tracks panel (Figure 19.13.1).

Display −→Particle Tracks...

Figure 19.13.1: The Particle Tracks Panel

The procedure for drawing trajectories for particle injections is as follows:

1. Select the particle injection(s) you wish to track in the Release FromInjections 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 theDiscrete Phase Model panel, as described in Section 19.12.1.


If stochastic and/or cloud tracking is desired, set the related pa-rameters in the Set Injection Properties panel, as described in Sec-tion 19.9.15.

3. Set any of the display options described below.

4. Click on the Display button to draw the trajectories or click on thePulse button to animate the particle positions. The Pulse buttonwill become the Stop ! button during the animation, and you mustclick on Stop ! to stop the pulsing.

For unsteady particle tracking simulations, clicking on Display will!show only the current location of the particles. Typically, youshould select point in the Style drop-down list when displaying tran-sient particle locations since individual positions will be displayed.The Pulse button option is not available for unsteady tracking.

Specifying Individual Particles for Display

It is also possible to display the trajectory for an individual particlestream instead of for all the streams in a given injection. To do so, youwill first need to determine which particle is of interest. Use the Injectionspanel to list the particle streams in the desired injection, as described inSection 19.9.4.

Define −→Injections...

Note the ID numbers listed in the first column of the listing printedin the FLUENT console. Then perform the following steps after step 1above:

1. Enable the Track Single Particle Stream option in the Particle Trackspanel.

2. In the Stream ID field, specify the ID number of the particle streamfor which you want to plot the trajectory.



Options for Particle Trajectory Plots

The options mentioned above include the following: You can includethe grid in the trajectory display, control the style of the trajectories(including the twisting of ribbon-style trajectories), and color them bydifferent scalar fields and control the color scale. You can also choosenode or cell values for display. If you are “pulsing” the trajectories, youcan control the pulse mode. Finally, you can generate an XY plot of theparticle trajectory data (e.g., residence time) as a function of time orpath length and save this XY plot data to a file.

These options are controlled in exactly the same way that pathline-plotting options are controlled. See Section 25.1.4 for details about set-ting the trajectory plotting options mentioned above.

Note that in addition to coloring the trajectories by continuous phasevariables, you can also color them according to the following discretephase variables: particle time, particle velocity, particle diameter, par-ticle density, particle mass, particle temperature, particle law number,particle time step, and particle Reynolds number. These variables areincluded in the Particle Variables... category of the Color By list. To dis-play the minimum and maximum values in the domain, click the UpdateMin/Max button.

Graphical Display in Axisymmetric Geometries

For axisymmetric problems in which the particle has a non-zero circum-ferential velocity component, the trajectory of an individual particle isoften a spiral about the centerline of rotation. FLUENT displays the rand x components of the trajectory (but not the θ component) projectedin the axisymmetric plane.

19.13.2 Reporting of Trajectory Fates

When you perform trajectory calculations by displaying the trajecto-ries (as described in Section 19.13.1), FLUENT will provide informationabout the trajectories as they are completed. By default, the number oftrajectories with each possible fate (escaped, aborted, evaporated, etc.)is reported:



DPM Iteration ....

number tracked = 7, escaped = 4, aborted = 0, trapped = 0, evaporated = 3, inco

Done.

You can also track particles through the domain without displaying thetrajectories by clicking on the Track button at the bottom of the panel.This allows the listing of reports without 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 boundaryfor which the “escape” condition is set.

• “Incomplete” trajectories are those that were terminated when themaximum allowed number of time steps—as defined by the Max.Number Of Steps input in the Discrete Phase Model panel (see Sec-tion 19.12.1)—was exceeded.

• “Trapped” trajectories are those that terminate at a flow boundarywhere the “trap” condition has been set.

• “Evaporated” trajectories include those trajectories along whichthe particles were evaporated within the domain.

• “Aborted” trajectories are those that fail to complete due to round-off reasons. You may want to retry the calculation with a modifiedlength scale and/or different initial conditions.

Summary Reports

You can request additional detail about the trajectory fates as the parti-cles exit the domain, including the mass flow rates through each bound-ary zone, mass flow rate of evaporated droplets, and composition of theparticles.

1. Follow steps 1 and 2 in Section 19.13.1 for displaying trajectories.



2. Select Summary as the Report Type and click Display or Track.

A detailed report similar to the following example will appear in theconsole window. (You may also choose to write this report to a file byselecting File as the Report to option, clicking on the Write... button(which was originally the Display button), and specifying a file name forthe summary report file in the resulting Select File dialog box.)

DPM Iteration ....

number 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+00

Escaped - 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-03

Escaped - Zone 7 1.502e-03 2.481e-04 -1.254e-03

(*)- Energy Transfer Summary -(*)

Fate Heat Content (W)


---- ---------- ---------- ----------

Incomplete 4.051e+02 3.088e+02 -9.630e+01

Escaped - 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 1

Escaped - 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 re-ports the number of particles, the time elapsed during trajectories, and



the mass and energy transfer. This information can be very useful forobtaining information such as where particles are escaping from the do-main, where particles are colliding with surfaces, and the extent of heatand mass transfer to/from the particles within the domain. Additionalinformation is reported for combusting particles.

Elapsed Time

The number of particles with each fate is listed under the Number head-ing. (Particles that escape through different zones or are trapped atdifferent zones are considered to have different fates, and are thereforelisted separately.) The minimum, maximum, and average time elapsedduring the trajectories of these particles, as well as the standard devi-ation about the average time, are listed in the Min, Max, Avg, and StdDev columns. This information indicates how much time the particle(s)spent in the domain before they escaped, aborted, evaporated, or weretrapped.

Fate Number Elapsed Time (s)

Min Max Avg Std Dev

---- ------ ---------- ---------- ---------- ---------- ---

Incomplete 2 1.485e+01 2.410e+01 1.947e+01 4.623e+00

Escaped - 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 andindex of the trajectories with the minimum and maximum elapsed times.(You may need to use the scroll bar to view this information.)

Elapsed Time (s) Injection, Index

Min 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 initialand final mass flow rates and the change in mass flow rate are reported



in the Initial, Final, and Change columns. With this information, youcan determine how much mass was transferred to the continuous phasefrom the particles.

(*)- Mass Transfer Summary -(*)

Fate Mass Flow (kg/s)


---- ---------- ---------- ----------

Incomplete 1.388e-03 1.943e-04 -1.194e-03

Escaped - 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 contentand the change in heat content are reported in the Initial, Final, andChange columns. This report tells you how much heat was transferredfrom the continuous phase to the particles.

(*)- Energy Transfer Summary -(*)

Fate Heat Content (W)


---- ---------- ---------- ----------

Incomplete 4.051e+02 3.088e+02 -9.630e+01

Escaped - Zone 7 4.383e+02 3.914e+02 -4.696e+01

Combusting Particles

If combusting particles are present, FLUENT will include additional re-porting on the volatiles and char converted. These reports are intendedto help you identify the composition of the combusting particles as theyexit 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+00 100.00

Escaped - 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 re-ported in the Initial and Final columns under Volatile Content.The percentage of volatiles that has been devolatilized is reported in the%Conv column.

The total reactive portion (char) at the start and end of the trajectory isreported in the Initial and Final columns under Char Content. Thepercentage of char that reacted is reported in the %Conv column.

19.13.3 Step-by-Step Reporting of Trajectories

At times, you may want to obtain a detailed, step-by-step report ofthe particle trajectory/trajectories. Such reports can be obtained inalphanumeric format. This capability allows you to monitor the particleposition, velocity, temperature, or diameter as the trajectory proceeds.

The procedure for generating files containing step-by-step reports islisted below:

1. Follow steps 1 and 2 in Section 19.13.1 for displaying trajectories.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.

3. Select File as the Report to option. (The Display button will becomethe Write... button.)

4. In the Significant Figures field, enter the number of significant fig-ures to be used in the step-by-step report.

5. Click on the Write... button and specify a file name for the step-by-step report file in the resulting Select File dialog box.

A detailed report similar to the following example will be saved to thespecified file before the trajectories are plotted. (You may also chooseto print the report in the console by choosing Console as the Report tooption and clicking on Display or Track, but the report is so long that itis unlikely to be of use to you in that form.)



The step-by-step report lists the particle position and velocity of theparticle at selected time steps along the trajectory:

Time X-Position Y-Position Z-Velocity X-Velocity Y-Velocity Z-Veloc

0.000e+00 1.411e-03 3.200e-03 0.000e+00 2.650e+01 0.000e+00 0.000e

3.773e-05 2.411e-03 3.200e-03 0.000e+00 2.648e+01 0.000e+00 0.000e

5.403e-05 2.822e-03 3.192e-03 0.000e+00 2.647e+01 0.000e+00 0.000e

9.181e-05 3.822e-03 3.192e-03 0.000e+00 2.644e+01 0.000e+00 0.000e

1.296e-04 4.821e-03 3.192e-03 0.000e+00 2.642e+01 0.000e+00 0.000e

1.608e-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 par-ticle. (You may need to use the scroll bar to view this information.)

elocity Y-Velocity Z-Velocity Diameter Temperature Density Mass

650e+01 0.000e+00 0.000e+00 2.000e-04 3.000e+02 1.300e+03 5.445e-09

648e+01 0.000e+00 0.000e+00 2.000e-04 3.006e+02 1.300e+03 5.445e-09

647e+01 0.000e+00 0.000e+00 2.000e-04 3.009e+02 1.300e+03 5.445e-09

644e+01 0.000e+00 0.000e+00 2.000e-04 3.015e+02 1.300e+03 5.445e-09

642e+01 0.000e+00 0.000e+00 2.000e-04 3.022e+02 1.300e+03 5.445e-09

639e+01 0.000e+00 0.000e+00 2.000e-04 3.027e+02 1.300e+03 5.445e-09

. . . . . . .

. . . . . . .

. . . . . . .

19.13.4 Reporting Current Positions for Unsteady Tracking

When using unsteady tracking, you may want to obtain a report of theparticle trajectory/trajectories showing the current positions of the par-ticles. Selecting Current Positions under Report Type in the ParticleTrackspanel enables the display of the current positions of the particles.

The procedure for generating files containing current position reports islisted below:

1. Follow steps 1 and 2 in Section 19.13.1 for displaying trajectories.You may want to track only one particle stream at a time, usingthe Track Single Particle Stream option.



2. Select Current Position as the Report Type.

3. Select File as the Report to option. (The Display button will becomethe Write... button.)

4. In the Significant Figures field, enter the number of significant fig-ures to be used in the step-by-step report.

5. Click on the Write... button and specify a file name for the currentposition report file in the resulting Select File dialog box.

The current position report lists the positions and velocities of all parti-cles that are currently in the domain:

Time X-Position Y-Position Z-Position X-Velocity Y-Velocity Z-Veloc

0.000e+00 1.000e-03 3.120e-02 0.000e+00 1.000e+01 5.000e+00 0.000e

1.672e-05 1.168e-03 3.128e-02 0.000e+00 1.010e+01 4.988e+00 0.000e

3.342e-05 1.337e-03 3.137e-02 0.000e+00 1.019e+01 4.977e+00 0.000e

5.010e-05 1.508e-03 3.145e-02 0.000e+00 1.028e+01 4.965e+00 0.000e

6.675e-05 1.680e-03 3.153e-02 0.000e+00 1.038e+01 4.954e+00 0.000e

8.338e-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, and mass of the par-ticles. (You may need to use the scroll bar to view this information.)

elocity Y-Velocity Z-Velocity Diameter Temperature Density Mass

000e+01 5.000e+00 0.000e+00 7.000e-05 3.000e+02 1.300e+03 2.335e-10

010e+01 4.988e+00 0.000e+00 7.000e-05 3.009e+02 1.300e+03 2.335e-10

019e+01 4.977e+00 0.000e+00 7.000e-05 3.019e+02 1.300e+03 2.335e-10

028e+01 4.965e+00 0.000e+00 7.000e-05 3.028e+02 1.300e+03 2.335e-10

038e+01 4.954e+00 0.000e+00 7.000e-05 3.037e+02 1.300e+03 2.335e-10

047e+01 4.942e+00 0.000e+00 7.000e-05 3.046e+02 1.300e+03 2.335e-10

. . . . . . .

. . . . . . .

. . . . . . .



19.13.5 Reporting of Interphase Exchange Terms and DiscretePhase Concentration

FLUENT reports the magnitudes of the interphase exchange of momen-tum, heat, and mass in each control volume in your FLUENT model. Itcan also report the total concentration of the discrete phase. You candisplay these variables graphically, by drawing contours, profiles, etc.They are all contained in the Discrete Phase Model... category of thevariable 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

See Chapter 27 for definitions of these variables.

Note that these exchange terms are updated and displayed only whencoupled calculations are performed. Displaying and reporting particletrajectories (as described in Sections 19.13.1 and 19.13.2) will not affectthe values of these exchange terms.



19.13.6 Trajectory Sampling

Particle states (position, velocity, diameter, temperature, and mass flowrate) can be written to files at various boundaries and planes (lines in2D) using the Sample Trajectories panel (Figure 19.13.2).

Report −→ Discrete Phase −→Sample...

Figure 19.13.2: The Sample Trajectories Panel

The procedure for generating files containing the particle samples is listedbelow:

1. Select the injections to be tracked in the Release From Injectionslist.

2. Select the surfaces at which samples will be written. These can be



boundaries from the 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 particletracking, the Compute button will become the Start button (toinitiate sampling) or a Stop button (to stop sampling).

Clicking on the Compute button will cause the particles to be tracked andtheir status to be written to files when they encounter selected surfaces.The file names will be formed by appending .dpm to the surface name.

For unsteady particle tracking, clicking on the Start button will openthe files and write the file header sections. If the solution is advancedin time by computing some time steps, the particle trajectories will beupdated and the particle states will be written to the files as they crossthe selected planes or boundaries. Clicking on the Stop button will closethe files and end the sampling.

For stochastic tracking, it may be useful to repeat this process multipletimes and append the results to the same file, while monitoring thesample statistics at each update. To do this, enable the Append Filesoption before repeating the calculation (clicking on Compute). Similarly,you can cause erosion and accretion rates to be accumulated for repeatedtrajectory calculations by turning on the Accumulate Erosion/AccretionRates option. (See also Section 19.13.8.) The format and the informationwritten for the sample output can also be controlled through a user-defined function, which can be selected in the Output drop-down list. Seethe separate UDF Manual for information about user-defined functions.

19.13.7 Histogram Reporting of Samples

Histograms can be plotted from sample files created in the Sample Trajec-tories panel (as described in Section 19.13.6) using the Trajectory SampleHistograms panel (Figure 19.13.3).

Report −→ Discrete Phase −→Histogram...

The procedure for plotting histograms from data in a sample file is listedbelow:



Figure 19.13.3: The Trajectory Sample Histograms Panel

1. Select a file to be read by clicking on the Read... button. Afteryou read in the sample file, the boundary name will appear in theSample list.

2. Select the data sample in the Sample list, and then select the datato be plotted from the Fields list.

3. Click on the Plot button at the bottom of the panel to display thehistogram.

By default, the percent of particles will be plotted on the y axis. You canplot the actual number of particles by deselecting Percent under Options.The number of “bins” or intervals in the plot can be set in the Divisionsfield. You can delete samples from the list with the Delete button andupdate the Min/Max values with the Compute button.



19.13.8 Postprocessing of Erosion/Accretion Rates

You can calculate the erosion and accretion rates in a cumulative man-ner (over a series of injections) by using the Sample Trajectories panel.First select an injection in the Release From Injections list and computeits trajectory. Then turn on the Accumulate Erosion/Accretion Rates op-tion, select the next injection (after deselecting the first one), and clickCompute again. The rates will accumulate at the surfaces each time youclick Compute.

Both the erosion rate and the accretion rate are defined at wall face!surfaces only, so they cannot be displayed at node values.


fluent dpm chp19

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