Chapter 19.
Discrete Phase Models
This chapter describes the Lagrangian discrete phase capabilities available in FLUENT and how to use them.
Information is organized into the following sections:
• Section 19.1: Overview and Limitations of the Discrete Phase Models
• 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
19.1.1
Overview and Limitations of the Discrete Phase
Models
Introduction
In addition to solving transport equations for the continuous phase, FLUENT allows you to simulate a discrete second phase in a Lagrangian frame
of reference. This second phase consists of spherical particles (which may
be taken to represent droplets or bubbles) dispersed in the continuous
phase. FLUENT computes the trajectories of these discrete phase entities, as well as heat and mass transfer to/from them. The coupling
between the phases and its impact on both the discrete phase trajectories and the continuous phase flow can be included.
FLUENT provides the following discrete phase modeling options:
• Calculation of the discrete phase trajectory using a Lagrangian
formulation that includes the discrete phase inertia, hydrodynamic
drag, and the force of gravity, for both steady and unsteady flows
• Prediction of the effects of turbulence on the dispersion of particles
due to turbulent eddies present in the continuous phase
• Heating/cooling of the discrete phase
• Vaporization and boiling of liquid droplets
• Combusting particles, including volatile evolution and char combustion to simulate coal combustion
• Optional coupling of the continuous phase flow field prediction to
the discrete phase calculations
• Droplet breakup and coalescence
These modeling capabilities allow FLUENT to simulate a wide range
of discrete phase problems including particle separation and classification, spray drying, aerosol dispersion, bubble stirring of liquids, liquid
fuel combustion, and coal combustion. The physical equations used for
these 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 Sections 19.6–19.13.
19.1.2
Particles in Turbulent Flows
The dispersion of particles due to turbulence in the fluid phase can be
predicted using the stochastic tracking model or the particle cloud model
(see Section 19.2.2). The stochastic tracking (random walk) model includes the effect of instantaneous turbulent velocity fluctuations on the
particle trajectories through the use of stochastic methods (see Section 19.2.2). The particle cloud model tracks the statistical evolution
of a cloud of particles about a mean trajectory (see Section 19.2.2). The
concentration of particles within the cloud is represented by a Gaussian probability density function (PDF) about the mean trajectory. In
both models, the particles have no direct impact on the generation or
dissipation 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 assumption
that the second phase is sufficiently dilute that particle-particle interactions and the effects of the particle volume fraction on the gas phase are
negligible. In practice, these issues imply that the discrete phase must be
present at a fairly low volume fraction, usually less than 10–12%. Note
that the mass loading of the discrete phase may greatly exceed 10–12%:
you may solve problems in which the mass flow of the discrete phase
equals or exceeds that of the continuous phase. See Chapters 18 and 20
for information about when you might want to use one of the general
multiphase models instead of the discrete phase model.
Limitation on Modeling Continuous Suspensions of Particles
The steady-particle Lagrangian discrete phase model described in this
chapter is suited for flows in which particle streams are injected into a
continuous phase flow with a well-defined entrance and exit condition.
c Fluent Inc. December 3, 2001
19-3
Discrete Phase Models
The Lagrangian model does not effectively model flows in which particles are suspended indefinitely in the continuum, as occurs in solid
suspensions 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 Chapters 18 and 20 for information about when you might want to use one of
the general multiphase models instead of the discrete phase models.
Limitations on Using the Discrete Phase Model with Other
FLUENT Models
The following restrictions exist on the use of other models with the discrete phase model:
• Streamwise periodic flow (either specified mass flow rate or specified pressure drop) cannot be modeled when the discrete phase
model is used.
• Adaptive time stepping cannot be used with the discrete phase
model.
• Only non-reacting particles can be included when the premixed
combustion model is used.
• When multiple reference frames are used in conjunction with the
discrete phase model, the display of particle tracks will not, by default, be meaningful. Similarly, coupled discrete-phase calculations
are not meaningful.
An alternative approach for particle tracking and coupled discretephase calculations with multiple reference frames is to track particles based on absolute velocity instead of relative velocity. To make
this change, use the define/models/dpm/tracking/track-inabsolute-frame text command. Note, however, that tracking particles based on absolute velocity may result in incorrect particlewall interaction.
The particle injection velocities (specified in the Set Injection Properties panel) are defined relative to the frame of reference in which
the particles are tracked. By default, the injection velocities are
19-4
c Fluent Inc. December 3, 2001
19.1 Overview and Limitations of the Discrete Phase Models
specified relative to the local reference frame. If you enable the
track-in-absolute-frame option, the injection velocities are specified 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 defining
the initial position, velocity, size, and temperature of individual particles. These initial conditions, along with your inputs defining the physical properties of the discrete phase, are used to initiate trajectory and
heat/mass transfer calculations. The trajectory and heat/mass transfer
calculations are based on the force balance on the particle and on the
convective/radiative heat and mass transfer from the particle, using the
local continuous phase conditions as the particle moves through the flow.
The predicted trajectories and the associated heat and mass transfer can
be viewed graphically and/or alphanumerically.
You can use FLUENT to predict the discrete phase patterns based on
a fixed continuous phase flow field (an uncoupled approach), or you can
include the effect of the discrete phase on the continuum (a coupled
approach). In the coupled approach, the continuous phase flow pattern
is impacted by the discrete phase (and vice versa), and you can alternate
calculations of the continuous phase and discrete phase equations until
a 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 discretephase 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.
c Fluent Inc. December 3, 2001
19-5
Discrete Phase Models
Outline of Unsteady Problem Setup and Solution Procedure
The general procedure for setting up and solving an unsteady discretephase 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 time
steps. Particle positions will be updated as the solution advances in
time. If you are solving an uncoupled flow, the particle position will
be updated at the end of each time step. For a coupled calculation,
the positions are iterated on within each time step.
19.2
19.2.1
Trajectory Calculations
Equations of Motion for Particles
Particle Force Balance
FLUENT predicts the trajectory of a discrete phase particle (or droplet or
bubble) by integrating the force balance on the particle, which is written
in a Lagrangian reference frame. This force balance equates the particle
inertia with the forces acting on the particle, and can be written (for the
x direction in Cartesian coordinates) as
dup
gx (ρp − ρ)
+ Fx
= FD (u − up ) +
dt
ρp
(19.2-1)
where FD (u − up ) is the drag force per unit particle mass and
FD =
18µ CD Re
ρp d2p
24
(19.2-2)
Here, u is the fluid phase velocity, up is the particle velocity, µ is the
molecular viscosity of the fluid, ρ is the fluid density, ρp is the density of
19-6
c Fluent Inc. December 3, 2001
19.2 Trajectory Calculations
the particle, and dp is the particle diameter. Re is the relative Reynolds
number, which is defined as
Re ≡
ρdp |up − u|
µ
(19.2-3)
The drag coefficient, CD , can be taken from either
CD = a1 +
a2
a3
+
Re Re2
(19.2-4)
where a1 , a2 , and a3 are constants that apply for smooth spherical particles over several ranges of Re given by Morsi and Alexander [163], or
CD =
24 b3 Re
1 + b1 Reb2 +
Re
b4 + Re
(19.2-5)
where
b1 = exp(2.3288 − 6.4581φ + 2.4486φ2 )
b2 = 0.0964 + 0.5565φ
b3 = exp(4.905 − 13.8944φ + 18.4222φ2 − 10.2599φ3 )
b4 = exp(1.4681 + 12.2584φ − 20.7322φ2 + 15.8855φ3 )(19.2-6)
which is taken from Haider and Levenspiel [85]. The shape factor, φ, is
defined as
s
φ=
(19.2-7)
S
where s is the surface area of a sphere having the same volume as the
particle, and S is the actual surface area of the particle.
For sub-micron particles, a form of Stokes’ drag law is available [170]. In
this case, FD is defined as
FD =
c Fluent Inc. December 3, 2001
18µ
dp 2 ρp Cc
(19.2-8)
19-7
Discrete Phase Models
The factor Cc is the Cunningham correction to Stokes’ drag law, which
you can compute from
Cc = 1 +
2λ
(1.257 + 0.4e−(1.1dp /2λ) )
dp
(19.2-9)
where λ is the molecular mean free path.
A high-Mach-number drag law is also available. This drag law is similar
to the spherical law (Equation 19.2-4) with corrections [38] to account for
a particle Mach number greater than 0.4 or a particle Reynolds number
greater than 20.
For unsteady models involving discrete phase droplet breakup, a dynamic
drag law option is also available. See Section 19.4.4 for a description of
this 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 is
important to note that in FLUENT the default gravitational acceleration
is zero. If you want to include the gravity force, you must remember to
define the magnitude and direction of the gravity vector in the Operating
Conditions panel.
Other Forces
Equation 19.2-1 incorporates additional forces (Fx ) in the particle force
balance that can be important under special circumstances. The first
of these is the “virtual mass” force, the force required to accelerate the
fluid surrounding the particle. This force can be written as
Fx =
1 ρ d
(u − up )
2 ρp dt
(19.2-10)
and is important when ρ > ρp . An additional force arises due to the
pressure gradient in the fluid:
19-8
c Fluent Inc. December 3, 2001
19.2 Trajectory Calculations
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 on
particles that arise due to rotation of the reference frame. These forces
arise when you are modeling flows in rotating frames of reference (see
Section 9.2). For rotation defined about the z axis, for example, the
forces on the particles in the Cartesian x and y directions can be written
as
ρ
1−
ρp
!
ρ
Ω x + 2Ω uy,p − uy
ρp
!
2
(19.2-12)
where uy,p and uy are the particle and fluid velocities in the Cartesian y
direction, and
ρ
1−
ρp
!
ρ
Ω y − 2Ω ux,p − ux
ρp
2
!
(19.2-13)
where ux,p and ux are the particle and fluid velocities in the Cartesian x
direction.
Thermophoretic Force
Small particles suspended in a gas that has a temperature gradient experience a force in the direction opposite to that of the gradient. This
phenomenon is known as thermophoresis. FLUENT can optionally include a thermophoretic force on particles in the additional force term,
Fx , in Equation 19.2-1:
Fx = −DT,p
c Fluent Inc. December 3, 2001
1 ∂T
mp T ∂x
(19.2-14)
19-9
Discrete Phase Models
where DT,p is the thermophoretic coefficient. You can define the coefficient to be constant, polynomial, or a user-defined function, or you can
use the form suggested by Talbot [237]:
Fx = −
where:
6πdp µ2 Cs (K + Ct Kn)
1 ∂T
ρ(1 + 3Cm Kn)(1 + 2K + 2Ct Kn) mp T ∂x
Kn
λ
K
k
=
=
=
=
kp
CS
Ct
Cm
mp
T
µ
=
=
=
=
=
=
=
(19.2-15)
Knudsen number = 2 λ/dp
mean free path of the fluid
k/kp
fluid thermal conductivity based on translational
energy only = (15/4) µR
particle thermal conductivity
1.17
2.18
1.14
particle mass
local fluid temperature
fluid viscosity
This expression assumes that the particle is a sphere and that the fluid
is an ideal gas.
Brownian Force
For sub-micron particles, the effects of Brownian motion can optionally
be included in the additional force term. The components of the Brownian force are modeled as a Gaussian white noise process with spectral
intensity Sn,ij given by [135]
Sn,ij = S0 δij
(19.2-16)
where δij is the Kronecker delta function, and
S0 =
216νσT
π 2 ρd5p
19-10
ρp 2
Cc
ρ
(19.2-17)
c Fluent Inc. December 3, 2001
19.2 Trajectory Calculations
T is the absolute temperature of the fluid, ν is the kinematic viscosity,
and σ is the Stefan-Boltzmann constant. Amplitudes of the Brownian
force components are of the form
s
Fbi = ζi
πSo
∆t
(19.2-18)
where ζi are zero-mean, unit-variance-independent Gaussian random numbers. The amplitudes of the Brownian force components are evaluated
at each time step. The energy equation must be enabled in order for
the Brownian force to take effect. Brownian force is intended only for
non-turbulent models.
Saffman’s Lift Force
The Saffman’s lift force, or lift due to shear, can also be included in
the additional force term as an option. The lift force used is from Li
and Ahmadi [135] and is a generalization of the expression provided by
Saffman [196]:
F~ =
2Kν 1/2 ρdij
(~v − ~vp )
ρp dp (dlk dkl )1/4
(19.2-19)
where K = 2.594 and dij is the deformation tensor. This form of the lift
force is intended for small particle Reynolds numbers. Also, the particle
Reynolds number based on the particle-fluid velocity difference must be
smaller than the square root of the particle Reynolds number based on
the shear field. Since this restriction is valid for submicron particles, it
is 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 particles using the mean fluid phase velocity, u, in the trajectory equations
(Equation 19.2-1). Optionally, you can include the instantaneous value
of the fluctuating gas flow velocity,
c Fluent Inc. December 3, 2001
19-11
Discrete Phase Models
u = u + u0
(19.2-20)
to predict the dispersion of the particles due to turbulence. FLUENT
uses a stochastic method (random walk model) to determine the instantaneous gas velocity, as detailed in Section 19.2.2.
Particle Cloud Tracking in Turbulent Flow
Particle dispersion due to turbulent fluctuations can also be modeled
with the particle cloud model [14, 15, 99, 141]. The turbulent dispersion of particles about a mean trajectory is calculated using statistical
methods. The concentration of particles about the mean trajectory is
represented by a Gaussian probability density function (PDF) whose
variance is based on the degree of particle dispersion due to turbulent
fluctuations. The mean trajectory is obtained by solving the ensembleaveraged 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 heat
or mass transfer to/from the particle, are solved by stepwise integration
over discrete time steps. Integration in time of Equation 19.2-1 yields
the velocity of the particle at each point along the trajectory, with the
trajectory itself predicted by
dx
= up
dt
(19.2-21)
Equations similar to 19.2-1 and 19.2-21 are solved in each coordinate
direction to predict the trajectories of the discrete phase.
Assuming that the term containing the body force remains constant over
each small time interval, and linearizing any other forces acting on the
particle, the trajectory equation can be rewritten in simplified form as
dup
1
= (u − up )
dt
τp
19-12
(19.2-22)
c Fluent Inc. December 3, 2001
19.2 Trajectory Calculations
where τp is the particle relaxation time. FLUENT uses a trapezoidal
scheme for integrating Equation 19.2-22:
un+1
− unp
1
p
) + ...
= (u∗ − un+1
p
∆t
τ
(19.2-23)
where n represents the iteration number and
1 n
(u + un+1 )
2
= un + ∆tunp · ∇un
u∗ =
un+1
(19.2-24)
(19.2-25)
Equations 19.2-21 and 19.2-22 are solved simultaneously to determine
the velocity and position of the particle at any given time. For rotating
reference frames, the integration is carried out in the rotating frame
with 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 that
the time step used for integration is sufficiently small that the trajectory
integration is accurate in time. (See Section 19.12.)
Droplet Size Distributions
For liquid sprays, a convenient representation of the droplet size distribution is the Rosin-Rammler expression. The complete range of sizes is
divided into an adequate number of discrete intervals; each represented
by a mean diameter for which trajectory calculations are performed. If
the size distribution is of the Rosin-Rammler type, the mass fraction of
droplets of diameter greater than d is given by
¯n
Yd = e−(d/d)
(19.2-26)
where d¯ is the size constant and n is the size distribution parameter. Use
of the Rosin-Rammler size distribution is detailed in Section 19.9.7.
c Fluent Inc. December 3, 2001
19-13
Discrete Phase Models
Discrete Phase Boundary Conditions
When a particle strikes a boundary face, one of several contingencies
may arise:
• Reflection via an elastic or inelastic collision.
• Escape through the boundary. (The particle is lost from the calculation at the point where it impacts the boundary.)
• Trap at the wall. Non-volatile material is lost from the calculation
at the point of impact with the boundary; volatile material present
in the particle or droplet is released to the vapor phase at this point.
• Passing through an internal boundary zone, such as radiator or
porous jump.
You also have the option of implementing a user-defined function to
model the particle path. See the separate UDF Manual for information
about 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 stochastic discrete-particle approach or a “cloud” representation of a group of
particles about a mean trajectory. In addition, these approaches can
be combined to model a set of “clouds” about a mean trajectory that
includes 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 dispersion of particles by integrating the trajectory equations for individual
0
particles, using the instantaneous fluid velocity, u + u (t), along the particle path during the integration. By computing the trajectory in this
19-14
c Fluent Inc. December 3, 2001
19.2 Trajectory Calculations
manner for a sufficient number of representative particles (termed the
“number of tries”), the random effects of turbulence on the particle dispersion may be accounted for. In FLUENT, the Discrete Random Walk
(DRW) model is used. In this model, the fluctuating velocity components are discrete piecewise constant functions of time. Their random
value is kept constant over an interval of time given by the characteristic
lifetime of the eddies.
The DRW model may give non-physical results in strongly inhomogeneous diffusion-dominated flows, where small particles should become
uniformly distributed. Instead, the DRW will show a tendency for such
particles to concentrate in low-turbulence regions of the flow.
The Integral Time
Prediction of particle dispersion makes use of the concept of the integral
time scale, T , which describes the time spent in turbulent motion along
the particle path, ds:
Z
T =
0
∞
up 0 (t)up 0 (t + s)
up 0 2
ds
(19.2-27)
The integral time is proportional to the particle dispersion rate, as larger
values indicate more turbulent motion in the flow. It can be shown that
the particle diffusivity is given by ui 0 uj 0 T .
For small “tracer” particles that move with the fluid (zero drift velocity),
the integral time becomes the fluid Lagrangian integral time, TL . This
time scale can be approximated as
TL = CL
k
(19.2-28)
where CL is to be determined and is not well known. By matching
the diffusivity of tracer particles, ui 0 uj 0 TL , to the scalar diffusion rate
predicted by the turbulence model, νt /σ, one can obtain
TL ≈ 0.15
c Fluent Inc. December 3, 2001
k
(19.2-29)
19-15
Discrete Phase Models
for the k- model and its variants, and
TL ≈ 0.30
k
(19.2-30)
when the Reynolds stress model (RSM) is used [48]. For the k-ω models,
substitute ω = /k into Equation 19.2-28. The LES model uses the
equivalent LES time scales.
The Discrete Random Walk Model
In the Discrete Random Walk (DRW) model, or “eddy lifetime” model,
the interaction of a particle with a succession of discrete stylized fluid
phase turbulent eddies is simulated. Each eddy is characterized by
• a Gaussian distributed random velocity fluctuation, u0 , v 0 , and w0
• a time scale, τe
The values of u0 , v 0 , and w0 that prevail during the lifetime of the turbulent eddy are sampled by assuming that they obey a Gaussian probability
distribution, so that
0
q
u = ζ u0 2
(19.2-31)
where ζ is a normally distributed random number, and the remainder of
the right-hand side is the local RMS value of the velocity fluctuations.
Since the kinetic energy of turbulence is known at each point in the
flow, these values of the RMS fluctuating components can be obtained
(assuming isotropy) as
q
u0 2 =
q
v0 2 =
q
w0 2 =
q
2k/3
(19.2-32)
for the k- model, the k-ω model, and their variants. When the RSM
is used, non-isotropy of the stresses is included in the derivation of the
velocity fluctuations:
19-16
c Fluent Inc. December 3, 2001
19.2 Trajectory Calculations
q
u0 = ζ u0 2
(19.2-33)
v0 = ζ v0 2
(19.2-34)
w0 = ζ w0 2
(19.2-35)
q
q
when viewed in a reference frame in which the second moment of the
turbulence is diagonal [274]. For the LES model, the velocity fluctuations
are 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 by
default), 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 by
Equation 19.2-29. The option of random calculation of τe yields a more
realistic 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 the
smaller of the eddy lifetime and the eddy crossing time. When this
time is reached, a new value of the instantaneous velocity is obtained by
applying a new value of ζ in Equation 19.2-31.
c Fluent Inc. December 3, 2001
19-17
Discrete Phase Models
Using the DRW Model
The only inputs required for the DRW model are the value for the integral
time-scale constant, CL (see Equations 19.2-28 and 19.2-36) and the
choice of the method used for the prediction of the eddy lifetime. You can
choose to use either a constant value or a random value by selecting the
appropriate 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 particles
model developed by Litchford and Jeng [141], Baxter and Smith [15], and
Jain [99]. The approach uses statistical methods to trace the turbulent
dispersion of particles about a mean trajectory. The mean trajectory is
calculated from the ensemble average of the equations of motion for the
particles represented by the cloud. The cloud enters the domain either
as a point source or with an initial diameter. The cloud expands due
to turbulent dispersion as it is transported through the domain until it
exits. The distribution of particles in the cloud is defined by a probability
density function (PDF) based on the position in the cloud relative to the
cloud center. The value of the PDF represents the probability of finding
particles represented by that cloud with residence time t at location xi
in the flow field. The average particle number density can be obtained
by weighting the total flow rate of particles represented by that cloud,
ṁ, as
hn(xi )i = ṁP (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 , and
are of the form
19-18
c Fluent Inc. December 3, 2001
19.2 Trajectory Calculations
1
P (xi , t) =
(8π)3/2
3
Y
e−(s
2 /2)
(19.2-40)
σi
i=1
where
s=
3
X
xi − µ i
i=1
σi
(19.2-41)
The mean of the PDF, or the center of the cloud, at a given time represents the most likely location of the particles in the cloud. The mean
location is obtained by integrating a particle velocity as defined by an
equation of motion for the cloud of particles:
µi (t) ≡ hxi (t)i =
Z
t
0
hVi (t1 )idt1 + hxi (0)i
(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. The
variance, σi2 (t), of the PDF can be expressed in terms of two particle
turbulence statistical quantities:
Z
σi2 (t) = 2
t
0
hu02
p,i (t2 )i
Z
t2
0
Rp,ii (t2 , t1 )dt1 dt2
(19.2-43)
0
2 i are the mean square velocity fluctuations, and R
where hup,i
p,ij (t2 , t1 )
is the particle velocity correlation function:
hu0p,i (t2 )u0p,j (t1 )i
Rp,ij (t2 , t1 ) = h
i1/2
02 (t )i
hu02
(t
)u
2
2
p,i
p,j
(19.2-44)
By using the substitution τ = |t2 − t1 |, and the fact that
Rp,ij (t2 , t1 ) = Rp,ij (t4 , t3 )
c Fluent Inc. December 3, 2001
(19.2-45)
19-19
Discrete Phase Models
whenever |t2 − t1 | = |t4 − t3 |, we can write
Z
σi2 (t) = 2
t
0
hu02
p,i (t2 )i
Z
t2
0
Rp,ii (τ )dτ dt2
(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 determines
the particle dispersion in the cloud model. FLUENT uses a correlation
function first proposed by Wang [254], and used by Jain [99]. When the
gravity vector is aligned with the z-coordinate direction, Rij takes the
form:
Rp,11 =
+
u02
p −(τ /τa )
StT
e
θ
St2 B 2 + 1
B − 0.5mT γ T
θ
!
u02 −(τ B/T )
mT St2T γB
τ
−1 +
e
+ 0.5mT γ
θ
θ
T
!
(19.2-47)
Rp,22 = Rp,11
u02 StT B −(τ /τa ) u02 −(τ B/T )
Rp,33 =
−
e
e
θ
θ
(19.2-48)
(19.2-49)
q
1 + m2T γ 2 and τa is the aerodynamic response time of the
where B =
particle:
τa =
ρp d2p
18µ
(19.2-50)
and
19-20
T
=
Tf E
=
mT TmE
m
3/4 3/2
Cµ k
( 23 k)1/2
(19.2-51)
(19.2-52)
c Fluent Inc. December 3, 2001
19.2 Trajectory Calculations
γ =
St =
StT
=
θ =
m =
TmE
=
mT
=
G(m) =
τa g
u0
τa
TmE
τa
T
St2T (1 + m2T γ 2 ) − 1
ū
u0
ū
Tf E 0
u
G(m)
m 1−
(1 + St)0.4(1+0.01St)
2
√
π
Z
e−y dy
(19.2-54)
(19.2-55)
(19.2-56)
(19.2-57)
(19.2-58)
(19.2-59)
2
∞
0
(19.2-53)
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 of
the cloud. At any given time, the cloud radius is set to three standard
deviations in the coordinate directions. The cloud radius is limited to
three standard deviations since at least 99.2% of the area under a Gaussian PDF is accounted for at this distance. Once the cells within the
cloud are established, the fluid properties are ensemble-averaged for the
mean trajectory, and the mean path is integrated in time. This is done
with a weighting factor defined as
Z
W (xi , t) ≡ Z
Vcell
P (xi , t)dV
(19.2-61)
P (xi , t)dV
Vcloud
If coupled calculations are performed, sources are distributed to the cells
in 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 minimum and maximum cloud diameters. The cloud model is enabled in
c Fluent Inc. December 3, 2001
19-21
Discrete Phase Models
the Set Injection Properties panel for each injection, as described in Section 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
Nparticles
Rerosion =
X
p=1
ṁp C(dp )f (α)v b(v)
Aface
(19.2-62)
where C(dp ) is a function of particle diameter, α is the impact angle of
the particle path with the wall face, f (α) is a function of impact angle, v
is the relative particle velocity, and b(v) is a function of relative particle
velocity. Default values are C = 1, f = 1, and b = 0.
Since C, f , and b are defined as boundary conditions at a wall, rather
than properties of a material, the default values are not updated to reflect
the material being used. You will need to specify appropriate values at
all walls. Values of these functions for sand eroding both carbon steel
and aluminum are given by Edwards et al. [60].
Note that the erosion rate as calculated above is displayed as dimensionless (that is, no units are listed) to provide some flexibility. The functions
C 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 are
mass of material removed/(area-time).
Note that the particle erosion and accretion rates can be displayed only
when coupled calculations are enabled.
The accretion rate is defined as
Nparticles
Raccretion =
X
p=1
19-22
ṁp
Aface
(19.2-63)
c Fluent Inc. December 3, 2001
19.3 Heat and Mass Transfer Calculations
19.3
Heat and Mass Transfer Calculations
Using FLUENT’s discrete phase modeling capability, reacting particles
or droplets can be modeled and their impact on the continuous phase
can be examined. Several heat and mass transfer relationships, termed
“laws”, are available in FLUENT and the physical models employed in
these 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 you
select. In the Set Injection Properties panel you will specify the Particle
Type, and FLUENT will use a given set of heat and mass transfer laws
for the chosen type. All particle types have pre-defined sequences of
physical laws as shown in the table below:
Particle Type
Inert
Droplet
Combusting
Description
inert/heating or cooling
heating/evaporation/boiling
heating;
evolution of volatiles/swelling;
heterogeneous surface reaction
Laws Activated
1, 6
1, 2, 3, 6
1, 4, 5, 6
In addition to the above laws, you can define your own laws using a userdefined function. See the separate UDF Manual for information about
user-defined functions.
You can also extend combusting particles to include an evaporating/boiling
material by selecting Wet Combustion in the Set Injection Properties panel.
FLUENT’s physical laws (Laws 1 through 6), which describe the heat
and mass transfer conditions listed in this table, are explained in detail
in 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 the
particle temperature is less than the vaporization temperature that you
c Fluent Inc. December 3, 2001
19-23
Discrete Phase Models
define, Tvap , and after the volatile fraction, fv,0 , of a particle has been
consumed. These conditions may be written as
Law 1:
Tp < Tvap
(19.3-1)
mp ≤ (1 − fv,0 )mp,0
(19.3-2)
Law 6:
where Tp is the particle temperature, mp,0 is the initial mass of the
particle, and mp is its current mass.
Law 1 is applied until the temperature of the particle/droplet reaches
the vaporization temperature. At this point a non-inert particle/droplet
may 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/droplet
has been consumed. (Note that the vaporization temperature, Tvap ,
is thus an arbitrary modeling constant used to define the onset of the
vaporization/boiling/volatilization laws.)
When using Law 1 or Law 6, FLUENT uses a simple heat balance to
relate the particle temperature, Tp (t), to the convective heat transfer
and the absorption/emission of radiation at the particle surface:
mp cp
dTp
4
− Tp4 )
= hAp (T∞ − Tp ) + p Ap σ(θR
dt
mp
cp
Ap
T∞
h
p
σ
θR
=
=
=
=
=
=
=
=
(19.3-3)
where
mass of the particle (kg)
heat capacity of the particle (J/kg-K)
surface area of the particle (m2 )
local temperature of the continuous phase (K)
convective heat transfer coefficient (W/m2 -K)
particle emissivity (dimensionless)
Stefan-Boltzmann constant (5.67 x 10−8 W/m2 -K4 )
G 1/4
radiation temperature, ( 4σ
)
Equation 19.3-3 assumes that there is negligible internal resistance to
heat transfer, i.e., the particle is at uniform temperature throughout.
19-24
c Fluent Inc. December 3, 2001
19.3 Heat and Mass Transfer Calculations
G is the incident radiation in W/m2 :
Z
G=
IdΩ
(19.3-4)
Ω=4π
where I is the radiation intensity and Ω is the solid angle.
Radiation heat transfer to the particle is included only if you have enabled the P-1 or discrete ordinates radiation model and you have activated radiation heat transfer to particles using the Particle Radiation
Interaction option in the Discrete Phase Model panel.
Equation 19.3-3 is integrated in time using an approximate, linearized
form that assumes that the particle temperature changes slowly from
one time value to the next:
mp cp
n h
i
h
io
dTp
4
= Ap − h + p σTp3 Tp + hT∞ + p σθR
dt
(19.3-5)
As the particle trajectory is computed, FLUENT integrates Equation 19.3-5
to 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
4
hT∞ + p σθR
h + p σTp3 (t)
(19.3-7)
Ap (h + p σTp3 (t))
mp cp
(19.3-8)
αp =
and
βp =
FLUENT can also solve Equation 19.3-5 in conjunction with the equivalent mass transfer equation using a stiff coupled solver. See Section 19.7.3
for details.
c Fluent Inc. December 3, 2001
19-25
Discrete Phase Models
The heat transfer coefficient, h, is evaluated using the correlation of Ranz
and Marshall [185, 186]:
Nu =
hdp
1/2
= 2.0 + 0.6Red Pr1/3
k∞
(19.3-9)
where
dp
k∞
Red
=
=
=
Pr
=
particle diameter (m)
thermal conductivity of the continuous phase (W/m-K)
Reynolds number based on the particle diameter and
the relative velocity (Equation 19.2-3)
Prandtl number of the continuous phase (cp µ/k∞ )
Finally, the heat lost or gained by the particle as it traverses each computational cell appears as a source or sink of heat in subsequent calculations
of the continuous phase energy equation. During Laws 1 and 6, particles/droplets do not exchange mass with the continuous phase and do
not 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 vaporization temperature, Tvap , and continues until the droplet reaches the
boiling point, Tbp , or until the droplet’s volatile fraction is completely
consumed:
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 once
vaporization is initiated (by the droplet reaching this threshold temperature), it will continue even if the droplet temperature falls below Tvap .
19-26
c Fluent Inc. December 3, 2001
19.3 Heat and Mass Transfer Calculations
Vaporization will be halted only if the droplet temperature falls below
the dew point. In such cases, the droplet will remain in Law 2 but no
evaporation will be predicted. When the boiling point is reached, the
droplet vaporization is predicted by a boiling rate, Law 3, as described
in 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 gradient
of the vapor concentration between the droplet surface and the bulk gas:
Ni = kc (Ci,s − Ci,∞ )
(19.3-12)
where
Ni
kc
Ci,s
Ci,∞
=
=
=
=
molar flux of vapor (kgmol/m2 -s)
mass transfer coefficient (m/s)
vapor concentration at the droplet surface (kgmol/m3 )
vapor concentration in the bulk gas (kgmol/m3 )
Note that FLUENT’s vaporization law assumes that Ni is positive (evaporation). If conditions exist in which Ni is negative (i.e., the droplet
temperature falls below the dew point and condensation conditions exist), FLUENT treats the droplet as inert (Ni = 0.0).
The concentration of vapor at the droplet surface is evaluated by assuming that the partial pressure of vapor at the interface is equal to the
saturated vapor pressure, psat , at the particle droplet temperature, Tp :
Ci,s =
psat (Tp )
RTp
(19.3-13)
where R is the universal gas constant.
The concentration of vapor in the bulk gas is known from solution of
the transport equation for species i or from the PDF look-up table for
non-premixed or partially premixed combustion calculations:
c Fluent Inc. December 3, 2001
19-27
Discrete Phase Models
Ci,∞ = Xi
pop
RT∞
(19.3-14)
where Xi is the local bulk mole fraction of species i, pop is the operating
pressure, and T∞ is the local bulk temperature in the gas.
The mass transfer coefficient in Equation 19.3-12 is calculated from a
Nusselt correlation [185, 186]:
NuAB =
where
Di,m
Sc
dp
=
=
=
kc dp
1/2
= 2.0 + 0.6Red Sc1/3
Di,m
(19.3-15)
diffusion coefficient of vapor in the bulk (m2 /s)
the Schmidt number, ρDµi,m
particle (droplet) diameter (m)
The vapor flux given by Equation 19.3-12 becomes a source of species i
in the gas phase species transport equation, as specified by you (see Section 19.11) or from the PDF look-up table for non-premixed combustion
calculations.
The mass of the droplet is reduced according to
mp (t + ∆t) = mp (t) − Ni Ap Mw,i ∆t
where
Mw,i
mp
Ap
=
=
=
(19.3-16)
molecular weight of species i (kg/kgmol)
mass of the droplet (kg)
surface area of the droplet (m2 )
FLUENT can also solve Equation 19.3-16 in conjunction with the equivalent heat transfer equation using a stiff coupled solver. See Section 19.7.3
for details.
Defining the Vapor Pressure and Diffusion Coefficient
You define the vapor pressure as a polynomial or piecewise linear function
of temperature (psat (T )) during the problem definition. Note that the
vapor pressure definition is critical, as psat is used to obtain the driving
19-28
c Fluent Inc. December 3, 2001
19.3 Heat and Mass Transfer Calculations
force for the evaporation process (Equations 19.3-12 and 19.3-13). You
should provide accurate vapor pressure values for temperatures over the
entire range of possible droplet temperatures in your problem. Vapor
pressure 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 the
discrete phase material properties. Note that the diffusion coefficient
inputs that you supply for the continuous phase are not used in the
discrete phase model.
Heat Transfer to the Droplet
Finally, the droplet temperature is updated according to a heat balance
that relates the sensible heat change in the droplet to the convective and
latent heat transfer between the droplet and the continuous phase:
mp cp
where
dTp
dmp
= hAp (T∞ − Tp ) +
hfg + Ap p σ(θR 4 − Tp 4 )
dt
dt
cp
Tp
h
T∞
dmp
dt
hfg
p
σ
θR
=
=
=
=
=
=
=
=
=
(19.3-17)
droplet heat capacity (J/kg-K)
droplet temperature (K)
convective heat transfer coefficient (W/m2 -K)
temperature of continuous phase (K)
rate of evaporation (kg/s)
latent heat (J/kg)
particle emissivity (dimensionless)
Stefan-Boltzmann constant (5.67 x 10−8 W/m2 -K4 )
I 1/4
radiation temperature, ( 4σ
) , where I is the
radiation intensity
Radiation heat transfer to the particle is included only if you have enabled the P-1 or discrete ordinates radiation model and you have activated radiation heat transfer to particles using the Particle Radiation
Interaction option in the Discrete Phase Model panel.
The heat transferred to or from the gas phase becomes a source/sink
of energy during subsequent calculations of the continuous phase energy
c Fluent Inc. December 3, 2001
19-29
Discrete Phase Models
equation.
19.3.4
Law 3: Droplet Boiling
Law 3 is applied to predict the convective boiling of a discrete phase
droplet when the temperature of the droplet has reached the boiling
temperature, Tbp , and while the mass of the droplet exceeds the nonvolatile fraction, (1 − fv,0 ):
Tp ≥ Tbp
(19.3-18)
mp > (1 − fv,0 )mp,0
(19.3-19)
and
When the droplet temperature reaches the boiling point, a boiling rate
equation is applied [120]:
"
p
d(dp )
cp,∞ (T∞ − Tp )
4k∞
(1 + 0.23 Red ) ln 1 +
=
dt
ρp cp,∞ dp
hfg
where
cp,∞
ρp
k∞
=
=
=
#
(19.3-20)
heat capacity of the gas (J/kg-K)
droplet density (kg/m3 )
thermal conductivity of the gas (W/m-K)
Equation 19.3-20 has been derived assuming steady flow at constant pressure. Note that the model requires T∞ > Tbp in order for boiling to occur
and that the droplet remains at fixed temperature (Tbp ) throughout the
boiling law.
When radiation heat transfer is active, FLUENT uses a slight modification of Equation 19.3-20, derived by starting from Equation 19.3-17 and
assuming that the droplet temperature is constant. This yields
−
19-30
dmp
hfg = hAp (T∞ − Tp ) + Ap p σ(θR 4 − Tp 4 )
dt
(19.3-21)
c Fluent Inc. December 3, 2001
19.3 Heat and Mass Transfer Calculations
or
"
#
d(dp )
k∞ Nu
2
4
−
(T∞ − Tp ) + p σ(θR
− Tp4 )
=
dt
ρp hfg
dp
(19.3-22)
Using Equation 19.3-9 for the Nusselt number correlation and replacing
the Prandtl number term with an empirical constant, Equation 19.3-22
becomes
"
#
√
d(dp )
2k∞ [1 + 0.23 Red ]
2
4
−
(T∞ − Tp ) + p σ(θR
− Tp4 )
=
dt
ρp hfg
dp
(19.3-23)
In the absence of radiation, this result matches that of Equation 19.3-20
in the limit that the argument of the logarithm is close to unity. FLUENT uses Equation 19.3-23 when radiation is active in your model and
Equation 19.3-20 when radiation is not active. Radiation heat transfer
to the particle is included only if you have enabled the P-1 or discrete
ordinates radiation model and you have activated radiation heat transfer
to particles using the Particle Radiation Interaction option in the Discrete
Phase Model panel.
The droplet is assumed to stay at constant temperature while the boiling
rate is applied. Once the boiling law is entered it is applied for the
duration of the particle trajectory. The energy required for vaporization
appears as a (negative) source term in the energy equation for the gas
phase. The evaporated liquid enters the gas phase as species i, as defined
by 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 the
temperature of the particle reaches the vaporization temperature, Tvap ,
and remains in effect while the mass of the particle, mp , exceeds the
mass of the non-volatiles in the particle:
Tp ≥ Tvap and Tp ≥ Tbp
c Fluent Inc. December 3, 2001
(19.3-24)
19-31
Discrete Phase Models
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 if
Wet Combustion is selected (otherwise, fw,0 = 0). As implied by Equation 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 wet
combustion is active, Tbp and Tvap refer to the boiling and evaporation
temperatures for the combusting material only.
FLUENT provides a choice of four devolatilization models:
• the constant rate model (the default model)
• the single kinetic rate model
• the two competing rates model (the Kobayashi model)
• the chemical percolation devolatilization (CPD) model
Each of these models is described, in turn, below.
Choosing the Devolatilization Model
You will choose the devolatilization model when you are setting physical
properties for 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 released
at a constant rate [13]:
−
19-32
1
dmp
= A0
fv,0 (1 − fw,0 )mp,0 dt
(19.3-26)
c Fluent Inc. December 3, 2001
19.3 Heat and Mass Transfer Calculations
where
mp
fv,0
mp,0
A0
=
=
=
=
particle mass (kg)
fraction of volatiles initially present in the particle
initial particle mass (kg)
rate constant (s−1 )
The rate constant A0 is defined as part of your modeling inputs, with a
default value of 12 s−1 derived from the work of Pillai [180] on coal combustion. Proper use of the constant devolatilization rate requires that
the vaporization temperature, which controls the onset of devolatilization, be set appropriately. Values in the literature show this temperature
to be about 600 K [13].
The volatile fraction of the particle enters the gas phase as the devolatilizing 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 phase
chemistry.
The Single Kinetic Rate Model
The single kinetic rate devolatilization model assumes that the rate of
devolatilization is first-order dependent on the amount of volatiles remaining in the particle [5]:
−
where
dmp
= k[mp − (1 − fv,0 )(1 − fw,0 )mp,0 ]
dt
mp
fv,0
=
=
fw,0
=
mp,0
k
=
=
(19.3-27)
particle mass (kg)
mass fraction of volatiles initially present in the
particle
mass fraction of evaporating/boiling material (if
wet combustion is modeled)
initial particle mass (kg)
kinetic rate (s−1 )
Note that fv,0 , the fraction of volatiles
fined using a value slightly in excess of
analysis. The kinetic rate, k, is defined
pre-exponential factor and an activation
c Fluent Inc. December 3, 2001
in the particle, should be dethat determined by proximate
by input of an Arrhenius type
energy:
19-33
Discrete Phase Models
k = A1 e−(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 only
slightly between discrete time integration steps.
FLUENT can also solve Equation 19.3-29 in conjunction with the equivalent heat transfer equation using a stiff coupled solver. See Section 19.7.3
for details.
The Two Competing Rates Kobayashi Model
FLUENT also provides the kinetic devolatilization rate expressions of the
form proposed by Kobayashi [117]:
R1 = A1 e−(E1 /RTp )
(19.3-30)
R2 = A2 e−(E2 /RTp )
(19.3-31)
where R1 and R2 are competing rates that may control the devolatilization over different temperature ranges. The two kinetic rates are weighted
to yield an expression for the devolatilization as
mv (t)
=
(1 − fw,0)mp,0 − ma
Z
0
t
(α1 R1 + α2 R2 ) exp −
Z
0
t
(R1 + R2 ) dt dt
(19.3-32)
19-34
c Fluent Inc. December 3, 2001
19.3 Heat and Mass Transfer Calculations
where
mv (t)
mp,0
α1 , α2
ma
=
=
=
=
volatile yield up to time t
initial particle mass at injection
yield factors
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 and
α2 . 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 recommended
in the literature [117] that α1 be set to the fraction of volatiles determined
by proximate analysis, since this rate represents devolatilization at low
temperature. The second yield parameter, α2 , should be set close to
unity, which is the yield of volatiles at very high temperature.
By default, Equation 19.3-32 is integrated in time analytically, assuming
the particle temperature to be constant over the discrete time integration step. FLUENT can also solve Equation 19.3-32 in conjunction with
the equivalent heat transfer equation using a stiff coupled solver. See
Section 19.7.3 for details.
The CPD Model
In contrast to the coal devolatilization models presented above, which
are based on empirical rate relationships, the chemical percolation devolatilization (CPD) model characterizes the devolatilization behavior of
rapidly heated coal based on the physical and chemical transformations
of the coal structure [68, 69, 81].
General Description
During coal pyrolysis, the labile bonds between the aromatic clusters in
the coal structure lattice are cleaved, resulting in two general classes of
fragments. One set of fragments has a low molecular weight (and correspondingly high vapor pressure) and escapes from the coal particle as a
light gas. The other set of fragments consists of tar gas precursors that
have a relatively high molecular weight (and correspondingly low vapor
pressure) and tend to remain in the coal for a long period of time during
typical devolatilization conditions. During this time, reattachment with
the coal lattice (which is referred to as crosslinking) can occur. The high
c Fluent Inc. December 3, 2001
19-35
Discrete Phase Models
molecular weight compounds plus the residual lattice are referred to as
metaplast. The softening behavior of a coal particle is determined by the
quantity and nature of the metaplast generated during devolatilization.
The portion of the lattice structure that remains after devolatilization is
comprised of char and mineral-compound-based ash.
The CPD model characterizes the chemical and physical processes by
considering the coal structure as a simplified lattice or network of chemical bridges that link the aromatic clusters. Modeling the cleavage of
the bridges and the generation of light gas, char, and tar precursors is
then considered to be analogous to the chemical reaction scheme shown
in Figure 19.3.1.
kδ
£
kb
2δ
kg
2g 1
£*
kc
c+ 2g 2
Figure 19.3.1: Coal Bridge
The variable £ represents the original population of labile bridges in
the coal lattice. Upon heating, these bridges become the set of reactive
bridges, £∗ . For the reactive bridges, two competing paths are available.
In one path, the bridges react to form side chains, δ. The side chains
may detach from the aromatic clusters to form light gas, g1 . As bridges
between neighboring aromatic clusters are cleaved, a certain fraction of
the coal becomes detached from the coal lattice. These detached aromatic clusters are the heavy-molecular-weight tar precursors that form
the metaplast. The metaplast vaporizes to form coal tar. While waiting for vaporization, the metaplast can also reattach to the coal lattice
matrix (crosslinking). In the other path, the bridges react and become
a 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
19-36
c Fluent Inc. December 3, 2001
19.3 Heat and Mass Transfer Calculations
represented by the variable p, where p = £ + c.
Reaction Rates
Given this set of variables that characterizes the coal lattice structure
during devolatilization, the following set of reaction rate expressions
can be defined for each, starting with the assumption that the reactive bridges are destroyed at the same rate at which they are created
∗
( ∂£
∂t = 0):
d£
dt
dc
dt
dδ
dt
dg1
dt
dg2
dt
= −kb £
£
ρ+1
£
− kg δ
= 2ρkb
ρ+1
= kb
(19.3-33)
(19.3-34)
(19.3-35)
= kg δ
(19.3-36)
dc
dt
(19.3-37)
= 2
where the rate constants for bridge breaking and gas release steps, kb
and kg , are expressed in Arrhenius form with a distributed activation
energy:
k = Ae−(E±Eσ )/RT
(19.3-38)
where A, E, and Eσ are, respectively, the pre-exponential factor, the activation energy, and the distributed variation in the activation energy, R
is the universal gas constant, and T is the temperature. The ratio of rate
constants, ρ = kδ /kc , is set to 0.9 in this model based on experimental
data.
Mass Conservation
The following mass conservation relationships are imposed:
c Fluent Inc. December 3, 2001
19-37
Discrete Phase Models
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 conditions 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 of
bridges in the coal lattice, and £0 is the initial fraction of labile bridges
in the coal lattice.
Fractional Change in the Coal Mass
Given the set of reaction equations for the coal structure parameters, it
is necessary to relate these quantities to changes in coal mass and the
related release of volatile products. To accomplish this, the fractional
change in the coal mass as a function of time is divided into three parts:
light gas (fgas ), tar precursor fragments (ffrag ), and char (fchar ). This
is accomplished by using the following relationships, which are obtained
using percolation lattice statistics:
r(g1 + g2 )(σ + 1)
(19.3-46)
4 + 2r(1 − c0 )(σ + 1)
2
ffrag (t) =
[ΦF (p) + rΩK(p)] (19.3-47)
2 + r(1 − c0 )(σ + 1)
fchar (t) = 1 − fgas (t) − ffrag (t)
(19.3-48)
fgas (t) =
19-38
c Fluent Inc. December 3, 2001
19.3 Heat and Mass Transfer Calculations
The variables Φ, Ω, F (p), and K(p) are the statistical relationships related to the cleaving of bridges based on the percolation lattice statistics,
and are given by the following equations:
£ (σ − 1)δ
+
p
4(1 − p)
δ
£
−
2(1 − p)
p
Φ = 1+r
(19.3-49)
Ω =
(19.3-50)
F (p) =
p0
p
σ+1
σ−1
K(p) =
1−
(19.3-51)
σ+1 0
p
2
p0
p
σ+1
σ−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)
where Mw,δ and Mw,1 are the side chain and cluster molecular weights respectively. σ + 1 is the lattice coordination number, which is determined
from solid-state Nuclear Magnetic Resonance (NMR) measurements related to coal structure parameters, and p0 is the root of the following
equation in p (the total number of bridges in the coal lattice matrix):
p0 (1 − p0 )σ−1 = p(1 − p)σ−1
(19.3-55)
In accounting for mass in the metaplast (tar precursor fragments), the
part that vaporizes is treated in a manner similar to flash vaporization,
where it is assumed that the finite fragments undergo vapor/liquid phase
equilibration on a time scale that is rapid with respect to the bridge
reactions. 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 is
used. The vapor pressure treatment is largely responsible for predicting
c Fluent Inc. December 3, 2001
19-39
Discrete Phase Models
pressure-dependent devolatilization yields. For the part of the metaplast
that reattaches to the coal lattice, a cross-linking rate expression given
by the following equation is used:
dmcross
= mfrag Across e−(Ecross /RT )
dt
(19.3-56)
where mcross is the amount of mass reattaching to the matrix, mfrag is the
amount 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 introduced
for the CPD model, the number of constants that must be defined to use
the 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 not
input or modified during problem setup.
There are an additional five parameters that are coal-specific and must
be 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
19-40
c Fluent Inc. December 3, 2001
19.3 Heat and Mass Transfer Calculations
The first four of these are coal structure quantities that are obtained
from NMR experimental data. The last quantity, representing the char
bridges that either exist in the parent coal or are formed very early in the
devolatilization process, is estimated based on the coal rank. These quantities 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 listed
in Table 19.3.1.
Table 19.3.1: Chemical Structure Parameters for
Coal Type
Zap (AR)
Wyodak (AR)
Utah (AR)
Ill6 (AR)
Pitt8 (AR)
Stockton (AR)
Freeport (AR)
Pocahontas (AR)
Blue (Sandia)
Rose (AFR)
1443 (lignite, ACERC)
1488 (subbituminous, ACERC)
1468 (anthracite, ACERC)
σ+1
3.9
5.6
5.1
5.0
4.5
4.8
5.3
4.4
5.0
5.8
4.8
4.7
4.7
p0
.63
.55
.49
.63
.62
.69
.67
.74
.42
.57
.59
.54
.89
13 C
NMR for 13 Coals
Mw,1
277
410
359
316
294
275
302
299
410
459
297
310
656
Mw,δ
40
42
36
27
24
20
17
14
47
48
36
37
12
c0
.20
.14
0
0
0
0
0
.20
.15
.10
.20
.15
.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 to
the swelling coefficient, Csw , which is defined by you and applied in the
following relationship:
c Fluent Inc. December 3, 2001
19-41
Discrete Phase Models
dp
(1 − fw,0 )mp,0 − mp
= 1 + (Csw − 1)
dp,0
fv,0 (1 − fw,0 )mp,0
where
dp,0
dp
(1−f
=
=
(19.3-57)
particle diameter at the start of devolatilization
current particle diameter
)m
−m
p
w,0
p,0
The term fv,0 (1−f
is the ratio of the mass that has been dew,0 )mp,0
volatilized to the total volatile mass of the particle. This quantity approaches a value of 1.0 as the devolatilization law is applied. When the
swelling coefficient is equal to 1.0, the particle diameter stays constant.
When the swelling coefficient is equal to 2.0, the final particle diameter
doubles when all of the volatile component has vaporized, and when the
swelling coefficient is equal to 0.5 the final particle diameter is half of its
initial diameter.
Heat Transfer to the Particle During Devolatilization
Heat transfer to the particle during the devolatilization process includes
contributions from convection, radiation (if active), and the heat consumed during devolatilization:
mp cp
dTp
dmp
= hAp (T∞ − Tp ) +
hfg + Ap p σ(θR 4 − Tp 4 )
dt
dt
(19.3-58)
Radiation heat transfer to the particle is included only if you have enabled the P-1 or discrete ordinates radiation model and you have activated radiation heat transfer to particles using the Particle Radiation
Interaction option in the Discrete Phase Model panel.
By default, Equation 19.3-58 is solved analytically, by assuming that the
temperature and mass of the particle do not change significantly between
time steps:
Tp (t + ∆t) = αp + [Tp (t) − αp ]e−βp t
(19.3-59)
where
19-42
c Fluent Inc. December 3, 2001
19.3 Heat and Mass Transfer Calculations
dm
hAp T∞ + dtp hfg + Ap p σθR 4
αp =
hAp + p Ap σTp 3
(19.3-60)
and
βp =
Ap (h + p σTp 3 )
mp cp
(19.3-61)
FLUENT can also solve Equation 19.3-58 in conjunction with the equivalent mass transfer equation using a stiff coupled solver. See Section 19.7.3
for details.
19.3.6
Law 5: Surface Combustion
After the volatile component of the particle is completely evolved, a
surface reaction begins, which consumes the combustible fraction, fcomb ,
of the particle. Law 5 is thus active (for a combusting particle) after the
volatiles 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, the
combusting particle may contain residual “ash” that reverts to the inert
heating law, Law 6 (see Section 19.3.2).
With the exception of the multiple surface reactions model, the surface
combustion law consumes the reactive content of the particle as governed
by the stoichiometric requirement, Sb , of the surface “burnout” reaction:
char(s) + Sb ox(g) −→ products(g)
c Fluent Inc. December 3, 2001
(19.3-64)
19-43
Discrete Phase Models
where Sb is defined in terms of mass of oxidant per mass of char, and the
oxidant and product species are defined in the Set Injection Properties
panel.
FLUENT provides a choice of four heterogeneous surface reaction rate
models 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 the
surface combustion model when you are setting physical properties for
the combusting-particle material in the Materials panel, as described in
Section 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 in
FLUENT, assumes that the surface reaction proceeds at a rate determined
by the diffusion of the gaseous oxidant to the surface of the particle:
dmp
Yox T∞ ρg
= −4πdp Di,m
dt
Sb (Tp + T∞ )
where
Di,m
Yox
ρg
Sb
=
=
=
=
(19.3-65)
diffusion coefficient for oxidant in the bulk (m2 /s)
local mass fraction of oxidant in the gas
gas density (kg/m3 )
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. The
diffusion-limited rate model assumes that the diameter of the particles
does not change. Since the mass of the particles is decreasing, the effective density decreases, and the char particles become more porous.
19-44
c Fluent Inc. December 3, 2001
19.3 Heat and Mass Transfer Calculations
Kinetic/Diffusion Surface Reaction Rate Model
The kinetic/diffusion-limited rate model assumes that the surface reaction rate is determined either by kinetics or by a diffusion rate. FLUENT
uses the model of Baum and Street [13] and Field [65], in which a diffusion rate coefficient
D0 = C1
[(Tp + T∞ )/2]0.75
dp
(19.3-66)
and a kinetic rate
R = C2 e−(E/RTp )
(19.3-67)
are weighted to yield a char combustion rate of
dmp
D0 R
= −πd2p pox
dt
D0 + R
(19.3-68)
where pox is the partial pressure of oxidant species in the gas surrounding
the combusting particle, and the kinetic rate, R, incorporates the effects
of chemical reaction on the internal surface of the char particle (intrinsic
reaction) and pore diffusion. In FLUENT, Equation 19.3-68 is recast in
terms of the oxidant mass fraction, Yox , as
dmp
ρRT Yox D0 R
= −πd2p
dt
Mw,ox D0 + R
(19.3-69)
The particle size is assumed to remain constant in this model while the
density is allowed to decrease.
When this model is enabled, the rate constants used in Equations 19.3-66
and 19.3-67 are entered in the Materials panel, as described in Section 19.11.
c Fluent Inc. December 3, 2001
19-45
Discrete Phase Models
Intrinsic Model
The intrinsic model in FLUENT is based on Smith’s model [218], assuming the order of reaction is equal to unity. Like the kinetic/diffusion
model, the intrinsic model assumes that the surface reaction rate includes the effects of both bulk diffusion and chemical reaction (see Equation 19.3-69). The intrinsic model uses Equation 19.3-66 to compute the
diffusion rate coefficient, D0 , but the chemical rate, R, is explicitly expressed in terms of the intrinsic chemical and pore diffusion rates:
R=η
dp
ρp Ag ki
6
(19.3-70)
η is the effectiveness factor, or the ratio of the actual combustion rate to
the rate attainable if no pore diffusion resistance existed [130]:
η=
3
(φ coth φ − 1)
φ2
(19.3-71)
where φ is the Thiele modulus:
dp Sb ρp Ag ki pox
φ=
2
De Cox
1/2
(19.3-72)
Cox is the concentration of oxidant in the bulk gas (kg/m3 ) and De is
the effective diffusion coefficient in the particle pores. Assuming that the
pore size distribution is unimodal and the bulk and Knudsen diffusion
proceed in parallel, De is given by
1
θ
1
De = 2
+
τ DKn D0
−1
(19.3-73)
where D0 is the bulk molecular diffusion coefficient and θ is the porosity
of the char particle:
θ =1−
19-46
ρp
ρt
(19.3-74)
c Fluent Inc. December 3, 2001
19.3 Heat and Mass Transfer Calculations
ρp and ρt are, respectively, the apparent and true densities of the pyrolysis char.
τ (in Equation 19.3-73)
√ is the tortuosity of the pores. The default value
for 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:
s
DKn = 97.0r p
Tp
Mw,ox
(19.3-75)
where Tp is the particle temperature and rp is the mean pore radius
of the char particle, which can be measured by mercury porosimetry.
Note that macropores (r p > 150 Å) dominate in low-rank chars while
micropores (rp < 10 Å) dominate in high-rank chars [130].
Ag (in Equations 19.3-70 and 19.3-72) is the specific internal surface
area of the char particle, which is assumed in this model to remain
constant during char combustion. Internal surface area data for various
pyrolysis chars can be found in [217]. The mean value of the internal
surface area during char combustion is higher than that of the pyrolysis
char [130]. For example, an estimated mean value for bituminous chars
is 300 m2 /g [33].
ki (in Equations 19.3-70 and 19.3-72) is the intrinsic reactivity, which is
of Arrhenius form:
ki = Ai e−(Ei /RTp )
(19.3-76)
where the pre-exponential factor Ai and the activation energy Ei can
be measured for each char. In the absence of such measurements, the
default values provided by FLUENT (which are taken from a least squares
fit of data of a wide range of porous carbons, including chars [217]) can
be used.
To allow a more adequate description of the char particle size (and hence
density) variation during combustion, you can specify the burning mode
c Fluent Inc. December 3, 2001
19-47
Discrete Phase Models
α, relating the char particle diameter to the fractional degree of burnout
U (where U = 1 − mp /mp,0 ) by [216]
dp
= (1 − U )α
dp,0
(19.3-77)
where mp is the char particle mass and the subscript zero refers to initial
conditions (i.e., at the start of char combustion). Note that 0 ≤ α ≤
1/3 where the limiting values 0 and 1/3 correspond, respectively, to a
constant size with decreasing density (zone 1) and a decreasing size with
constant density (zone 3) during burnout. In zone 2, an intermediate
value 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 the
Materials panel, as described in Section 19.11.
The Multiple Surface Reactions Model
Modeling multiple char reactions follows the same pattern as the wall
surface reaction models, where the surface species is now a “particle surface species”. The particle surface species can be depleted or produced
by the stoichiometry of the particle surface reaction (defined in the Reactions panel) for the mixture material defined in the Species Model panel.
If a particle surface species is depleted, the reactive “char” content of
the particle is consumed. In turn, if a surface species is produced by
the particle surface reaction, the species is added to the particle residual
“ash” mass. Any number of particle surface species and any number of
particle surface reactions can be defined for any given combusting particle; however, you must have only one particle surface species in the
reactants list of a particle reaction.
Multiple injections can be accommodated, and combusting particles reacting according to the multiple surface reactions model can coexist in
the calculation with combusting particles following other char combustion laws. The model is based on oxidation studies of char particles,
19-48
c Fluent Inc. December 3, 2001
19.3 Heat and Mass Transfer Calculations
but it is applicable to gas-solid reactions in general, not only to char
oxidation 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 tracking
option.
• The model is available only with the species transport model for
volumetric reactions, and not with the non-premixed, premixed, or
partially 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 the
transport equation for this species. Similarly, the surface reaction is a
source of species in the gas phase: the product of the heterogeneous
surface reaction appears in the gas phase as a user-selected chemical
species. The surface reaction also consumes or produces energy, in an
amount determined by the heat of reaction defined by you.
The particle heat balance during surface reaction is
mp cp
dTp
dmp
= hAp (T∞ − Tp ) − fh
Hreac + Ap p σ(θR 4 − Tp 4 ) (19.3-78)
dt
dt
where Hreac is the heat released by the surface reaction. Note that only
a portion (1− fh ) of the energy produced by the surface reaction appears
as a heat source in the gas-phase energy equation: the particle absorbs a
fraction fh of this heat directly. For coal combustion, it is recommended
that fh be set at 1.0 if the char burnout product is CO and 0.3 if the
char burnout product is CO2 [24].
c Fluent Inc. December 3, 2001
19-49
Discrete Phase Models
Radiation heat transfer to the particle is included only if you have enabled the P-1 or discrete ordinates radiation model and you have activated radiation heat transfer to particles using the Particle Radiation
Interaction option in the Discrete Phase Model panel.
By default, Equation 19.3-78 is solved analytically, by assuming that the
temperature and mass of the particle do not change significantly between
time steps. FLUENT can also solve Equation 19.3-78 in conjunction with
the equivalent mass transfer equation using a stiff coupled solver. See
Section 19.7.3 for details.
19.3.7
Using Combusting Particles for General Heterogeneous
Surface Reactions
The combusting particle type in FLUENT is presented with a focus on
modeling of coal particle combustion. You can, however, use this particle
type to model general heterogeneous reactions on particles in which a
solid particle reacts with a gas-phase component to form a single gasphase product. For example,
4Al(s) + 3Cl2 (g) → 2Al2 Cl3 (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 surface
reaction law, Law 5, provides a general heterogeneous surface reaction
that consumes a gas-phase “oxidant” and produces a gas-phase product
species defined by you.
19-50
c Fluent Inc. December 3, 2001
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. For
most types of injections, you will need to provide the initial diameter,
position, and velocity of the particles. For sprays, however, there are
models available for droplet breakup and collision, as well as a drag
coefficient that accounts for variation in droplet shape. These models
for 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 parameters in the Set Injection Properties panel, as described in Section 19.9.2.
Details about the atomizer models are provided below.
c Fluent Inc. December 3, 2001
19-51
Discrete Phase Models
General Information
All of the models use physical and numerical atomizer parameters, such
as orifice diameter and mass flow rate, to calculate initial droplet size,
velocity, and position.
For realistic atomizer simulations, the droplets must be randomly distributed, both through a dispersion angle and in their time of release.
For the other types of injections in FLUENT (non-atomizer), all of the
droplets are released along fixed trajectories and at the beginning of the
time step. The atomizer models use stochastic trajectory selection and
staggering to attain random distribution.
Stochastic trajectory selection is the random dispersion of initial droplet
directions. All of the atomizer models provide an initial dispersion angle,
and the stochastic trajectory selection picks an initial direction within
this angle. This approach improves the accuracy of the results for spraydominated flows. The droplets will be more evenly spread among the
computational cells near the atomizer, which improves the coupling to
the gas phase by spreading drag more smoothly over the cells near the
injection.
The Plain-Orifice Atomizer Model
The plain-orifice is the most common type of atomizer and the most
simply made. However there is nothing simple about the physics of the
internal nozzle flow and the external atomization. In the plain-orifice
atomizer, the liquid is accelerated through a nozzle, forms a liquid jet,
and then forms droplets. This apparently simple process is dauntingly
complex. The plain orifice may operate in three different regimes: singlephase, cavitating, and flipped [225]. The transition between regimes is
abrupt, producing dramatically different sprays. The internal regime
determines the velocity at the orifice exit, as well as the initial droplet
size and the angle of droplet dispersion. Diagrams of each case are shown
in Figures 19.4.1, 19.4.2, and 19.4.3.
19-52
c Fluent Inc. December 3, 2001
19.4 Spray Models
r
p
p
1
d
2
liquid jet
orifice walls
downstream
gas
L
Figure 19.4.1: Single-Phase Nozzle Flow (Liquid completely fills the orifice.)
vapor
liquid jet
vapor
orifice walls
downstream
gas
Figure 19.4.2: Cavitating Nozzle Flow (Vapor pockets form just after
the inlet corners.)
c Fluent Inc. December 3, 2001
19-53
Discrete Phase Models
liquid jet
orifice walls
downstream
gas
Figure 19.4.3: Flipped Nozzle Flow (Downstream gas surrounds the liquid 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 external
spray. Unfortunately, there is no established theory for determining the
nozzle state. One must rely on empirical models that fix experimental
data. A suggested list of the governing parameters for the internal nozzle
flow is given in Table 19.4.1.
Table 19.4.1: List of Governing Parameters for Internal Nozzle Flow
nozzle diameter
nozzle length
radius of curvature of the inlet corner
upstream pressure
downstream pressure
viscosity
liquid density
vapor pressure
d
L
r
p1
p2
µ
ρl
pv
These may be combined to form geometric non-dimensional groups such
as r/d and L/d, as well as the Reynolds number based on “head” (Reh )
19-54
c Fluent Inc. December 3, 2001
19.4 Spray Models
and a cavitation parameter (K).
s
dρl
Reh =
µ
2(p1 − p2 )
ρl
p1 − pv
p1 − p2
K=
(19.4-1)
(19.4-2)
The liquid flow often contracts in the nozzle, as can be seen in Figures 19.4.2 and 19.4.3. Nurick [166] found it helpful to use a coefficient
of contraction (Cc ) that represents the area of the stream of contracting
liquid over the total cross-sectional area of the nozzle. FLUENT uses
Nurick’s fit for the coefficient of contraction:
Cc = q
1
1
Cct
−
(19.4-3)
11.4r
d
Cct is a theoretical constant equal to 0.611, which comes from potential
flow analysis of flipped nozzles.
Another important parameter used to describe the performance of nozzles is the coefficient of discharge (Cd ). The coefficient of discharge is a
ratio of the mass flow rate through the nozzle, divided by the theoretical
maximum mass flow rate:
Cd =
ṁ
A 2ρl (p1 − p2 )
p
(19.4-4)
The cavitation number (K in Equation 19.4-2) is an essential parameter
for predicting the inception of cavitation. The inception of cavitation is
known 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 −
c Fluent Inc. December 3, 2001
r
d
2
−
1000
Reh
(19.4-5)
19-55
Discrete Phase Models
Similarly, a critical value of K where flip occurs is defined as Kcrit :
Kcrit = 1 + 1+
L
4d
1
1+
2000
Reh
e70r/d
(19.4-6)
If r/d is greater than 0.05, then flip is deemed impossible and Kcrit is
set to 1.0.
These variables are then used in a decision tree to identify the nozzle
state. The decision tree is shown in Figure 19.4.4. Depending on the
state of the nozzle, a unique closure is chosen for the above equations.
For a single-phase nozzle [137],
Cdu = 0.827 − 0.0085
Cd =
L
d
(19.4-7)
1
1
Cdu
+ 20 (1+2.25L/d)
Reh
(19.4-8)
Equation 19.4-7 is for the ultimate coefficient of discharge, Cdu . Equation 19.4-8 corrects this ultimate coefficient of discharge for the effects
of viscosity.
For a cavitating nozzle [166],
√
Cd = Cc K
(19.4-9)
Cd = Cct
(19.4-10)
For a flipped nozzle [166],
All of the nozzle flow equations are solved iteratively, along with the
appropriate relationship for coefficient of discharge as given by the nozzle
state. The nozzle state may change as the upstream or downstream
pressures change. Once the nozzle state is determined, the exit velocity
is found, and appropriate correlations for spray angle and initial droplet
size distribution are determined.
19-56
c Fluent Inc. December 3, 2001
19.4 Spray Models
K ≤ Kincep
K > K incep
K ≥ K crit
K < K crit
flipped
K ≥ Kcrit
K <K crit
cavitating
flipped
single phase
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 from
conservation of mass and the assumption of a uniform exit velocity:
u=
ṁ
ρl A
(19.4-11)
For the cavitating nozzle, Schmidt and Corradini [204] have shown that
the uniform exit velocity is not accurate. Instead, they derived an expression for a higher velocity over a reduced area:
u=
2Cc p1 − p2 + (1 − 2Cc )pv
p
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 mass
and the value of the reduced flow area:
u=
c Fluent Inc. December 3, 2001
ṁ
ρl Cct A
(19.4-13)
19-57
Discrete Phase Models
Spray Angle
The correlation for spray angle (θ) comes from the work of Ranz [184]:
θ
2
θ
2
"
= tan−1
4π
CA
r
√ #
ρg 3
ρl 6
= 0.01
(19.4-14)
(19.4-15)
Equation 19.4-14 describes the spray angle for both single-phase and
cavitating nozzles. For flipped nozzles, the spray angle has a constant
value (Equation 19.4-15).
CA is thought to be a constant for a given nozzle geometry. You must
choose 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 nozzles
than for single-phase nozzles. Typical values are from 4.0 to 6.0. The
spray angle for flipped nozzles is a small, arbitrary value that represents
the lack of any turbulence or initial disturbance from 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 spread
parameter to define the Rosin-Rammler distribution. For more information 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 turbulence
quantities of the liquid jet:
19-58
c Fluent Inc. December 3, 2001
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 is
the Weber number, which, in this case, is defined to be
We ≡
ρ l u2 λ
σ
(19.4-18)
where λ = d/8 and σ is the droplet surface tension. For a more detailed discussion of droplet surface tension and the Weber number, see
Section 19.4.3.
For cavitating nozzles, FLUENT uses a slight modification to Equation 19.4-17. The initial jet diameter used in Wu’s correlation is calculated from the effective area of the cavitating orifice exit. For an
explanation of effective area of cavitating nozzles, see Schmidt and Corradini [204].
The length scale for a cavitating nozzle is λ = deff /8, where
s
deff =
4ṁ
πρl u
(19.4-19)
For the case of the flipped nozzle, the initial droplet diameter is set to
the diameter of the liquid jet:
p
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 modeling
experience and from a review of experimental observations. Table 19.4.2
lists the values of s for the three kinds of nozzles:
The larger the value of the spread parameter, the narrower the droplet
size distribution. The function that samples the Rosin-Rammler distribution uses the most probable diameter and the spread parameter.
c Fluent Inc. December 3, 2001
19-59
Discrete Phase Models
Table 19.4.2: Values of Spread Parameter for Different Nozzle States
State
single phase
cavitating
flipped
Spread Parameter
3.5
1.5
∞
Since the correlations of Wu et al. provide the Sauter mean diameter,
d32 , these must be converted to the most probable diameter, d0 . Lefebvre [132] gives the most general relationship between the Sauter mean
diameter and most probable diameter for a Rosin-Rammler distribution.
The simplified version for s=3.5 is as follows:
d0 = 1.2726d32
1
1−
s
1/s
(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, sometimes referred to by the gas-turbine community as a simplex atomizer.
This type of atomizer accelerates the liquid through nozzles known as
swirl ports into a central swirl chamber. The swirling liquid pushes
against 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 is
very widely used for liquid-fuel combustion in gas turbines, oil furnaces,
and direct-injection spark-ignited automobile engines. The transition
from internal injector flow to fully-developed spray can be divided into
three steps: film formation, sheet breakup, and atomization. A sketch
of 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. It
is generally accepted that an aerodynamic instability causes the sheet
19-60
c Fluent Inc. December 3, 2001
19.4 Spray Models
film formation
sheet breakup
atomization
Figure 19.4.5: Theoretical Progression from the Internal Atomizer Flow
to the External Spray
to break up. The mathematical analysis below assumes that KelvinHelmholtz waves grow on the sheet and eventually break the liquid into
ligaments. It is then assumed that the ligaments break up into droplets
due to varicose instability. Once the liquid forms droplets, the spray
behavior is determined by drag, collision, coalescence, and secondary
breakup.
The model used in this study is called the Linearized Instability Sheet
Atomization (LISA) model of Schmidt et al. [206]. The LISA model is
divided into two stages:
1. film formation
2. sheet breakup and atomization
Both parts of the model are described below. The implementation is
slightly improved from that of Schmidt et al. [206].
c Fluent Inc. December 3, 2001
19-61
Discrete Phase Models
Film Formation
The centrifugal motion of the liquid within the injector creates an air
core surrounded by a liquid film. The thickness of this film, t, is related
to the mass flow rate by
ṁ = πρut(dinj − t)
(19.4-22)
where dinj is the injector exit diameter, and ṁ is the mass flow rate,
which must be measured experimentally. The other unknown in Equation 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 difficult to calculate from first principles. Instead, the approach of Han et
al. [86] is used. The total velocity is assumed to be related to the injector
pressure by
s
U = kv
2∆p
ρl
(19.4-23)
Lefebvre [132] has noted that kv is a function of the injector design
and injection pressure. If the swirl ports are treated as nozzles, Equation 19.4-23 is then an expression for the coefficient of discharge for the
swirl ports, assuming that the majority of the pressure drop through the
injector occurs at the ports. The coefficient of discharge (Cd ) for singlephase nozzles with sharp inlet corners and an L/d of 4 is typically 0.78
or less [137]. If the nozzles are cavitating, the value of Cd may be as low
as 0.61. Hence, 0.78 should be a practical upper bound for kv . Reducing
kv by 10% to allow for other momentum losses in the injector gives an
estimate of 0.7.
Physical limits on kv are such that it must be less than unity by conservation of energy, and it must be large enough to permit sufficient
mass flow. To guarantee that the size of the air core is non-negative, the
following expression is used for kv :
4ṁ
kv = max 0.7, 2
πd0 ρl cos θ
19-62
r
ρl
2∆p
(19.4-24)
c Fluent Inc. December 3, 2001
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 tangential component of velocity is assumed equal to the radial component
further downstream. The axial component of velocity is assumed to be
a 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 Senecal
et al. [207] and are only briefly presented here. For a more accurate and
robust implementation, the gas-phase velocity is neglected in calculating
the relative liquid-gas velocity. This decision avoids depending on the
usually under-resolved gas-phase velocity field around the injector.
The model assumes that a two-dimensional, viscous, incompressible liquid sheet of thickness 2h moves with velocity U through a quiescent,
inviscid, incompressible gas medium. The liquid and gas have densities
of ρl and ρg , respectively, and the viscosity of the liquid is µl . A coordinate system is used that moves with the sheet, and a spectrum of
infinitesimal disturbances of the form
η = η0 eikx+ωt
(19.4-26)
is imposed on the initially steady, motion-producing fluctuating velocities
and 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 disturbance
has the largest value of ωr , denoted here by Ω, and is assumed to be
responsible for sheet breakup. Thus, it is desired to obtain a dispersion
relation ω = ω(k) from which the most unstable disturbance can be
deduced.
c Fluent Inc. December 3, 2001
19-63
Discrete Phase Models
Squire [229] and Hagerty and Shea [84] have shown that two solutions,
or modes, exist that satisfy the liquid governing equations subject to
the boundary conditions at the upper and lower interfaces. For the
first solution, called the sinuous mode, the waves at the upper and lower
interfaces 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 numerous
authors (e.g., Senecal et al. [207]) that the sinuous mode dominates the
growth of varicose waves for low velocities and low gas-to-liquid density
ratios. In addition, it can be shown that the sinuous and varicose modes
become indistinguishable for high-velocity flows. As a result, the present
discussion focuses on the growth of sinuous waves on the liquid sheet.
As derived in Senecal et al. [207], the dispersion relation for the sinuous
mode is given by
ω 2 [tanh(kh) + Q] + [4νl k2 tanh(kh) + 2iQkU ]+
4νl k4 tanh(kh) − 4νl2 k3 ` tanh(`h) − QU 2 k2 +
σk3
=0
ρl
(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-thickness), the fastest-growing waves are short. Below 27/16, the wavelengths
are long compared to the sheet thickness. The speed of modern fuel
injection is high enough that the film Weber number is often well above
this critical limit.
Li and Tankin [136] derived a dispersion relation similar to Equation
19.4-27 for a viscous sheet from a linear analysis with a stationary coordinate system. While Li and Tankin’s dispersion relation is quite general,
a simplified relation has been presented in Senecal et al. [207] for use in
multi-dimensional simulations of pressure-swirl atomizers. The resulting
expression for the growth rate is given by
1
ωr =
tanh(kh) + Q
19-64
(
−2νl k2 tanh(kh) +
c Fluent Inc. December 3, 2001
19.4 Spray Models
s

σk3 
2
2
4
2
2
2
2
2
4νl k tanh (kh) − Q U k − [tanh(kh) + Q] −QU k +
ρl 
(19.4-28)
Two main assumptions have been made to reduce Equation 19.4-27 to
Equation 19.4-28. First, an order-of-magnitude analysis using typical
values from the inviscid solutions shows that the terms of second order
in viscosity can be neglected in comparison to the other terms in Equation 19.4-28. In addition, the density ratio Q is on the order of 10−3 in
typical applications and hence it is assumed that Q 1.
The physical mechanism of sheet disintegration proposed by Dombrowski
and Johns [52] is adopted only for long waves. For long waves, ligaments
are assumed to form from the sheet breakup process once the unstable
waves reach a critical amplitude. If the surface disturbance has reached
a value of ηb at breakup, a breakup time, τ , can be evaluated:
Ωτ
ηb = η0 e
ηb
1
⇒ ln
Ω
η0
(19.4-29)
where Ω, the maximum growth rate, is found by numerically maximizing
Equation 19.4-28 as a function of k. The maximum is found using a
binary search that checks the sign of the derivative. The sheet breaks
up and ligaments will be formed at a length given by
ηb
U
Lb = U τ = ln
Ω
η0
(19.4-30)
where the quantity ln( ηη0b ) is an empirical constant from 3 to 12. You
must 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 be
obtained from a mass balance. If it is assumed that the ligaments are
formed from tears in the sheet once per wavelength, the resulting diameter is given by
s
dL =
c Fluent Inc. December 3, 2001
8h
Ks
(19.4-31)
19-65
Discrete Phase Models
where Ks is the wave number corresponding to the maximum growth
rate, Ω. The ligament diameter depends on the sheet thickness, which is
a function of the breakup length. The film thickness is calculated from
the breakup length and the radial distance from the center line to the
mid-line of the sheet at the atomizer exit, r0 :
hend =
r0 h0
r0 + Lb sin
θ
2
(19.4-32)
This mechanism is not logical for short waves. For short waves, the
determination of the ligament diameter is simpler. The value of dL is
assumed to be linearly proportional to the wavelength that breaks up
the sheet. FLUENT allows you to control the constant of proportionality. The wavelength is calculated from the wave number, Ks . In either
the long wave or the short wave case, the breakup from ligaments to
droplets is assumed to behave according to Weber’s [257] analysis for
capillary instability. The variable Oh is the Ohnesorge number and is a
combination of Reynolds number and Weber number (see Section 19.4.3
for more details about Oh):
d0 = 1.88dL (1 + 3Oh)1/6
(19.4-33)
This procedure determines the most probable droplet size. The spread
parameter is assumed to be 3.5, based on past modeling experience [205].
You will specify the spray cone angle. The dispersion angle of the spray
is 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, an
additional air stream is often directed through the atomizer. The liquid
is formed into a sheet by a nozzle, and the air is then directed against
the sheet to promote atomization. This technique is called air-assisted
atomization or air-blast atomization, depending on the quantity of air
and its velocity. The addition of the external air stream past the sheet
produces smaller droplets than without the air. The exact mechanism for
19-66
c Fluent Inc. December 3, 2001
19.4 Spray Models
this enhanced performance is not completely understood. It is thought
that the assisting air may accelerate the sheet instability. The air may
also help disperse the droplets, preventing collisions between them. Airassisted atomization is used in many of the same fields as pressure-swirl
atomization, where especially fine atomization is required.
FLUENT’s air-blast atomization model is a variation of the pressure-swirl
model. One difference is that you set the sheet thickness directly in the
air-blast atomizer model. This input is necessary because of the variety
of sheet formation mechanisms used in air-blast atomizers. Hence the airblast atomizer model does not contain the sheet formation equations that
were included in the pressure-swirl atomizer model (Equations 19.4-22–
19.4-25). You will also specify the maximum relative velocity that is
produced by the sheet and air. Though this quantity could be calculated,
specifying a value relieves you from the necessity of finely resolving the
atomizer internal flow. This feature is convenient for simulations in large
domains, where the atomizer is very small by comparison.
Another difference is that the air-blast atomizer model assumes that
the sheet breakup is always due to short waves. This assumption is
a consequence of the greater sheet thickness commonly found in airblast atomizers. Hence the ligament diameter is assumed to be linearly
proportional to the wavelength of the fastest-growing wave on the sheet.
Other inputs are similar to the pressure-swirl model. You must provide
the mass flow rate and spray angle. The angle in the case of the air-blast
atomizer is the initial trajectory of the film as it leaves the end of the
orifice. The value of the angle is negative if the initial film trajectory is
inward, towards the centerline. You will also provide the inner and outer
diameter of the film at the atomizer exit.
The air-blast atomizer model does not include the internal gas flows. You
must create the atomizing air streams as a boundary condition within
the FLUENT case. These streams are ordinary continuous-phase flows
and require no special treatment.
c Fluent Inc. December 3, 2001
19-67
Discrete Phase Models
The Flat-Fan Atomizer Model
The flat-fan atomizer is very similar to the pressure-swirl atomizer, but
it makes a flat sheet and does not use swirl. The liquid emerges from a
wide, thin orifice as a flat liquid sheet that breaks up into droplets. The
primary atomization process is thought to be similar to the pressure-swirl
atomizer. Some researchers believe that flat-fan atomization, because of
jet impingement, is very similar to the atomization of a flat sheet. The
flat-fan model could serve doubly for this application.
The flat-fan atomizer is available only for 3D models. An image of the
three-dimensional flat fan is shown in Figure 19.4.6. The model assumes
that the fan originates from a virtual origin. You will provide the location
of this origin, which is the intersection of the lines that mark the sides of
the fan. You will also provide the location of the center point of the arc
from which the fan originates. FLUENT will find the vector that points
from the origin to the center point in order to determine the direction
of the injection. You will also provide the half-angle of the fan arc, the
width of the orifice (in the normal direction), and the mass flow rate of
the liquid.
center point
virtual origin
normal vector
Figure 19.4.6: Flat Fan Viewed From Above and From the Side
19-68
c Fluent Inc. December 3, 2001
19.4 Spray Models
The breakup of the flat fan is calculated very much like the breakup
of the sheet in the pressure-swirl atomizer. The sheet breaks up into
ligaments that then form droplets. The only difference is that for short
waves, the flat fan sheet is assumed to form ligaments at half-wavelength
intervals. Hence the ligament diameter for short waves is given by
s
dL =
16h
Ks
(19.4-34)
The Rosin-Rammler spread parameter is assumed to be 3.5 and the
dispersion angle is set to 6◦ . In all other respects, the flat-fan atomizer
model is like the sheet breakup portion of the pressure-swirl atomizer.
Effervescent Atomizer Model
Effervescent atomization is the injection of liquid infused with a superheated (with respect to downstream conditions) liquid or propellant.
As the volatile liquid exits the nozzle, it rapidly changes phase. This
phase change quickly breaks up the stream into small droplets with a
wide dispersion angle. The model also applies to cases where a very hot
liquid is discharged.
Since the physics of effervescence is not well understood, the model must
rely on rough empirical fits. The photographs of Reitz [189] provide some
basic 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 of
mass, assuming the exiting jet has a cross-sectional area that is Cct
times the nozzle area, where Cct is a constant that you specify during
the problem setup:
u=
ṁ
ρl Cct A
(19.4-35)
The maximum droplet diameter is set to the effective diameter of the
exiting jet:
c Fluent Inc. December 3, 2001
19-69
Discrete Phase Models
p
dmax = d Cct
(19.4-36)
The droplet size is then sampled from a Rosin-Rammler distribution
with a spread parameter of 4.0 (see Section 19.9.7). The most probable
droplet size depends on the angle, θ, between the droplet’s stochastic
trajectory and the injection direction:
d0 = dmax e−(θ/Θs )
2
(19.4-37)
The dispersion angle multiplier, Θs , is computed from the quality, x, and
the specified value for the dispersion constant, Ceff :
Θs =
x
Ceff
(19.4-38)
This technique creates a spray with large droplets in the central core
and a shroud of smaller surrounding droplets. The droplet temperature
is initialized to the initial temperature fraction, f , times the saturation
temperature of the droplets. f should be slightly less than 1.0, because
the droplet temperatures should be close to boiling. To complete the
model, the flashing vapor must also be included in the calculation. This
vapor is part of the continuous phase and not part of the discrete phase
model. You must create an inlet at the point of injection when you
specify boundary conditions for the continuous phase.
When the effervescent atomizer model is selected, you will need to specify
the nozzle diameter, mass flow rate, mixture quality, saturation temperature 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, FLUENT
provides an option for estimating the number of droplet collisions and
their outcomes in a computationally efficient manner. The difficulty in
19-70
c Fluent Inc. December 3, 2001
19.4 Spray Models
any collision calculation is that for N droplets, each droplet has N − 1
possible collision partners. Thus, the number of possible collision pairs is
approximately 12 N 2 . (The factor of 12 appears because droplet A colliding
with droplet B is identical to droplet B colliding with droplet A. This
symmetry reduces the number of possible collision events by half.)
An important consideration is that the collision algorithm must calculate
1 2
2 N possible collision events at every time step. Since a spray can consist
of several million droplets, the computational cost of a collision calculation from first principles is prohibitive. This motivates the concept of
parcels. Parcels are statistical representations of a number of individual
droplets. For example, if FLUENT tracks a set of parcels, each of which
represents 1000 droplets, the cost of the collision calculation is reduced
by a factor of 106 . Because the cost of the collision calculation still scales
with the square of N , the reduction of cost is significant; however, the
effort to calculate the possible intersection of so many parcel trajectories
would still be prohibitively expensive.
The algorithm of O’Rourke [168] efficiently reduces the computational
cost of the spray calculation. Rather than using geometry to see if parcel
paths intersect, O’Rourke’s method is a stochastic estimate of collisions.
O’Rourke also makes the assumption that two parcels may collide only
if they are located in the same continuous-phase cell. These two assumptions are valid only when the continuous-phase cell size is small
compared to the size of the spray. For these conditions, the method of
O’Rourke is second-order accurate at estimating the chance of collisions.
The concept of parcels together with the algorithm of O’Rourke makes
the calculation of collision possible for practical spray problems.
Once it is decided that two parcels of droplets collide, the algorithm
further determines the type of collision. Only coalescence and bouncing
outcomes are considered. The probability of each outcome is calculated
from the collisional Weber number and a fit to experimental observations.
The properties of the two colliding parcels are modified based on the
outcome of the collision.
c Fluent Inc. December 3, 2001
19-71
Discrete Phase Models
Use and Limitations
The collision model assumes that the frequency of collisions is much less
than the particle time step. If the particle time step is too large, then
the results may be time-step-dependent. You should adjust the particle
length scale accordingly. Additionally, the model is most applicable for
low-Weber-number collisions where collisions result in bouncing and coalescence. Above a collisional Weber number of about 100, the outcome
of collision could be shattering.
Sometimes the collision model can cause grid-dependent artifacts to appear in the spray. This is a result of the assumption that droplets can
collide only within the same cell. These tend to be visible when the
source of injection is at a mesh vertex. The coalescence of droplets tends
to cause the spray to pull away from cell boundaries. In two dimensions,
a finer mesh and more computational droplets can be used to reduce
these effects. In three dimensions, best results are achieved when the
spray is modeled using a polar mesh with the spray at the center.
Theory
As noted above, O’Rourke’s algorithm assumes that two droplets may
collide only if they are in the same continuous-phase cell. This assumption can prevent droplets that are quite close to each other, but not in
the same cell, from colliding, although the effect of this error is lessened
by allowing some droplets that are farther apart to collide. The overall
accuracy of the scheme is second-order in space.
Probability of Collision
The probability of collision of two droplets is derived from the point of
view of the larger droplet, called the collector droplet and identified below
with the number 1. The smaller droplet is identified in the following
derivation with the number 2. The calculation is in the frame of reference
of the larger droplet so that the velocity of the collector droplet is zero.
Only the relative distance between the collector and the smaller droplet is
important in this derivation. If the smaller droplet is on a collision course
with the collector, the centers will pass within a distance of r1 +r2 . More
19-72
c Fluent Inc. December 3, 2001
19.4 Spray Models
precisely, if the smaller droplet center passes within a flat circle centered
around the collector of area π(r1 + r2 )2 perpendicular to the trajectory
of the smaller droplet, a collision will take place. This disk can be used
to define the collision volume, which is the area of the aforementioned
disk multiplied by the distance traveled by the smaller droplet in one
time step, namely π(r1 + r2 )2 vrel ∆t.
The algorithm of O’Rourke uses the concept of a collision volume to
calculate the probability of collision. Rather than calculate if the position
of the smaller droplet center is within the collision volume, the algorithm
calculates the probability of the smaller droplet being within the collision
volume. It is known that the smaller droplet is somewhere within the
continuous-phase cell of volume V . If there is a uniform probability of
the droplet being anywhere within the cell, then the chance of the droplet
being 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 )2 vrel ∆t
V
(19.4-39)
Equation 19.4-39 can be generalized for parcels, where there are n1 and
n2 droplets in the collector and smaller droplet parcels, respectively. The
collector undergoes a mean expected number of collisions given by
n̄ =
n2 π(r1 + r2 )2 vrel ∆t
V
(19.4-40)
The actual number of collisions that the collector experiences is not generally the mean expected number of collisions. The probability distribution of the number of collisions follows a Poisson distribution, according
to O’Rourke, which is given by
P (n) = e−n̄
n̄n
n!
(19.4-41)
where n is the number of collisions between a collector and other droplets.
c Fluent Inc. December 3, 2001
19-73
Discrete Phase Models
Collision Outcomes
Once it is determined that two parcels collide, the outcome of the collision must be determined. In general, the outcome tends to be coalescence if the droplets collide head-on, and bouncing if the collision is more
oblique. The critical offset is a function of the collisional Weber number
and the relative radii of the collector and the smaller droplet.
The critical offset is calculated by O’Rourke using the expression
s
bcrit
2.4f
= (r1 + r2 ) min 1.0,
We
(19.4-42)
where f is a function of r1 /r2 , defined as
f
r1
r2
=
r1
r2
3
r1
− 2.4
r2
2
r1
+ 2.7
r2
(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 if
b < bcrit , the result of the collision is coalescence. Equation 19.4-41 gives
the number of smaller droplets that coalesce with the collector. The
properties of the coalesced droplets are found from the basic conservation
laws.
In the case of a grazing collision, the new velocities are calculated based
on conservation of momentum and kinetic energy. It is assumed that
some fraction of the kinetic energy of the droplets is lost to viscous
dissipation and angular momentum generation. This fraction is related
to b, the collision offset parameter. Using assumed forms for the energy
loss, O’Rourke derived the following expression for the new velocity:
v10 =
m1 v1 + m2 v2 + 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 other
droplet properties are altered in grazing collisions.
19-74
c Fluent Inc. December 3, 2001
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 for
low-Weber-number injections and is well suited for low-speed sprays into
a standard atmosphere. For Weber numbers greater than 100, the wave
model is more applicable. The wave model is popular for use in highspeed fuel-injection applications. Details for each model are provided
below.
Taylor Analogy Breakup (TAB) Model
Introduction
The Taylor Analogy Breakup (TAB) model is a classic method for calculating droplet breakup, which is applicable to many engineering sprays.
This method is based upon Taylor’s analogy [239] between an oscillating
and distorting droplet and a spring mass system. Table 19.4.3 illustrates
the analogous components.
Table 19.4.3: Comparison of a Spring-Mass System to a Distorting
Droplet
Spring-Mass System
restoring force of spring
external force
damping force
Distorting and Oscillating Droplet
surface tension forces
droplet drag force
droplet viscosity forces
The resulting TAB model equation set, which governs the oscillating
and distorting droplet, can be solved to determine the droplet oscillation
and distortion at any given time. As described in detail below, when
the droplet oscillations grow to a critical value the “parent” droplet will
break up into a number of smaller “child” droplets. As a droplet is distorted from a spherical shape, the drag coefficient changes. A drag model
that incorporates the distorting droplet effects is available in FLUENT.
See Section 19.4.4 for details.
c Fluent Inc. December 3, 2001
19-75
Discrete Phase Models
Use and Limitations
The TAB model is best for low-Weber-number sprays. Extremely highWeber-number sprays result in shattering of droplets, which is not described well by the spring-mass analogy.
Droplet Distortion
The equation governing a damped, forced oscillator is [169]
F − kx − d
dx
d2 x
=m 2
dt
dt
(19.4-45)
where x is the displacement of the droplet equator from its spherical
(undisturbed) position. The coefficients of this equation are taken from
Taylor’s analogy:
F
m
k
m
d
m
ρg u2
ρl r
σ
= Ck 3
ρl r
µl
= Cd 2
ρl r
= CF
(19.4-46)
(19.4-47)
(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 droplet
radius, 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 critical
ratio of the droplet radius. This breakup requirement is given as
x > Cb r
(19.4-49)
where Cb is a constant equal to 0.5 if breakup is assumed to occur when
the distortion is equal to the droplet radius, i.e., the north and south
19-76
c Fluent Inc. December 3, 2001
19.4 Spray Models
poles of the droplet meet at the droplet center. This implicitly assumes
that the droplet is undergoing only one (fundamental) oscillation mode.
Equation 19.4-45 is non-dimensionalized by setting y = x/(Cb r) and
substituting the relationships in Equations 19.4-46–19.4-48:
d2 y
C F ρg u2 C k σ
Cd µl dy
=
−
y−
2
2
3
dt
C b ρl r
ρl r
ρl r 2 dt
(19.4-50)
where breakup now occurs for y > 1. For under-damped droplets, the
equation governing y can easily be determined from Equation 19.4-50 if
the relative velocity is assumed to be constant:
−(t/td )
y(t) = Wec +e
1
(y0 − Wec ) cos(ωt) +
ω
dy0 y0 − Wec
sin(ωt)
+
dt
td
(19.4-51)
where
We =
Wec =
y0 =
dy0
=
dt
1
=
td
ω2 =
ρg u2 r
σ
CF
We
Ck Cb
y(0)
dy
(0)
dt
Cd µ l
2 ρl r 2
σ
1
Ck 3 − 2
ρl r
td
(19.4-52)
(19.4-53)
(19.4-54)
(19.4-55)
(19.4-56)
(19.4-57)
In Equation 19.4-51, u is the relative velocity between the droplet and
the gas phase and We is the droplet Weber number, a dimensionless
parameter defined as the ratio of aerodynamic forces to surface tension
forces. The droplet oscillation frequency is represented by ω. The constants have been chosen to match experiments and theory [122]:
c Fluent Inc. December 3, 2001
19-77
Discrete Phase Models
Ck = 8
Cd = 5
1
CF =
3
If Equation 19.4-51 is solved for all droplets, those with y > 1 are assumed to break up. The size and velocity of the new child droplets must
be determined.
Size of Child Droplets
The size of the child droplets is determined by equating the energy of
the parent droplet to the combined energy of the child droplets. The
energy of the parent droplet is [169]
Eparent
π
= 4πr σ + K ρl r 5
5
2
"
dy
dt
#
2
2 2
+ω y
(19.4-58)
where K is the ratio of the total energy in distortion and oscillation
to the energy in the fundamental mode, of the order ( 10
3 ). The child
droplets are assumed to be non-distorted and non-oscillating. Thus, the
energy of the child droplets can be shown to be
Echild
dy
r
π
= 4πr σ
+ ρl r 5
r32
6
dt
2
2
(19.4-59)
where r32 is the Sauter mean radius of the droplet size distribution. r32
can 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 ω 2 = 8σ/ρl r 3 :
r32 =
r
1+
8Ky 2
20
+
ρl r 3 (dy/dt)2
σ
6K−5
120
(19.4-60)
Once the size of the child droplets is determined, the number of child
droplets can easily be determined by mass conservation.
19-78
c Fluent Inc. December 3, 2001
19.4 Spray Models
Velocity of Child Droplets
The TAB model allows for a velocity component normal to the parent
droplet velocity to be imposed upon the child droplets. When breakup
occurs, the equator of the parent droplet is traveling at a velocity of
dx/dt = Cb r(dy/dt). Therefore, the child droplets will have a velocity
normal to the parent droplet velocity given by
vnormal = Cv Cb r
dy
dt
(19.4-61)
where Cv is a constant of order (1). Although this imposed velocity is
assumed to be in a plane normal to the path of the parent droplet, the
exact direction in this plane cannot be specified. Therefore, the direction
of this imposed velocity is selected randomly, yet is confined in a plane
normal to the parent relative velocity vector.
Droplet Breakup
To model droplet breakup, the TAB model first determines the amplitude
for an undamped oscillation (td ≈ ∞) for each droplet at time step n
using the following:
s
A=
(y n
− Wec
)2
+
(dy/dt)n
ω
2
(19.4-62)
According to Equation 19.4-62, breakup is possible only if the following
condition is satisfied:
Wec + A > 1
(19.4-63)
This is the limiting case, as damping will only reduce the chance of
breakup. If a droplet fails the above criterion, breakup does not occur.
The only additional calculations required, then, are to update y using a
discretized form of Equation 19.4-51 and its derivative, which are both
c Fluent Inc. December 3, 2001
19-79
Discrete Phase Models
based on work done by O’Rourke and Amsden [169]:
y n+1 = Wec +
e−(∆t/td ) (y n − Wec ) cos(ωt) +
dy
dt
n+1
=
−(∆t/td )
ωe
Wec − y n+1
+
td
1
ω
dy
dt
n
1
ω
dy
dt
n
+
y n − Wec
sin(ωt)
td
(19.4-64)
y n − Wec
+
cos(ω∆t) − (y n − Wec ) sin(ω∆t)
td
(19.4-65)
All of the constants in these expressions are assumed to be constant
throughout 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 occurs
within the time step ∆t. The value of tbu is set to the time required for
oscillations to grow sufficiently large that the magnitude of the droplet
distortion, y, is equal to unity. The breakup time is determined under the
assumption that the droplet oscillation is undamped for its first period.
The breakup time is therefore the smallest root greater than tn of an
undamped version of Equation 19.4-51:
Wec + A cos[ω(t − tn ) + φ] = 1
(19.4-66)
where
cos φ =
y n − Wec
A
(19.4-67)
(dy/dt)n
Aω
(19.4-68)
and
sin φ = −
19-80
c Fluent Inc. December 3, 2001
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. The
breakup calculation then continues with the next droplet. Conversely,
if tn < tbu < tn+1 , then breakup will occur and the child droplet radii
are determined by Equation 19.4-60. The number of child droplets, N ,
is determined by mass conservation:
N
n+1
=N
n
rn
r n+1
3
(19.4-69)
A velocity component normal to the relative velocity vector, with magnitude 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 (see
Section 19.4.2 for a description of parcels). Hence, there is no need to
create another computational droplet after breakup. The TAB model in
FLUENT 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 Reitz [188], which considers the breakup of the injected liquid to be induced
by the relative velocity between the gas and liquid phases. The model
assumes that the time of breakup and the resulting droplet size are related to the fastest-growing Kelvin-Helmholtz instability, derived from
the jet stability analysis described below. The wavelength and growth
rate of this instability are used to predict details of the newly-formed
droplets.
Use and Limitations
The wave model is appropriate for very-high-speed injection, where the
Kelvin-Helmholtz instability is believed to dominate spray breakup (We >
100). Because breakup can increase the number of computational droplets,
c Fluent Inc. December 3, 2001
19-81
Discrete Phase Models
you may wish to inject a modest number of droplets. You must also specify the model constants, which are thought to depend on the internal flow
of 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 cylindrical, viscous, liquid jet of radius a issuing from a circular orifice at a
velocity v into a stagnant, incompressible, inviscid gas of density ρ2 . The
liquid has a density, ρ1 , and viscosity, µ1 , and a cylindrical polar coordinate system is used which moves with the jet. An arbitrary infinitesimal
axisymmetric surface displacement of the form
η = η0 eikz+ωt
(19.4-70)
is imposed on the initially steady motion and it is thus desired to find
the dispersion relation ω = ω(k) which relates the real part of the growth
rate, ω, to its wave number, k = 2π/λ.
In order to determine the dispersion relation, the linearized hydrodynamic equations for the liquid are solved with wave solutions of the form
φ1 = C1 I0 (kr)eikz+ωt
(19.4-71)
ikz+ωt
(19.4-72)
ψ1 = C2 I1 (Lr)e
where φ1 and ψ1 are the velocity potential and stream function, respectively, C1 and C2 are integration constants, I0 and I1 are modified Bessel
functions of the first kind, L2 = k2 + ω/ν1 , and ν1 is the liquid kinematic
viscosity [188]. The liquid pressure is obtained from the inviscid part
of the liquid equations. In addition, the inviscid gas equations can be
solved to obtain the fluctuating gas pressure at r = a:
−p21 = −ρ2 (U − iωk)2 kη
19-82
K0 (ka)
K1 (ka)
(19.4-73)
c Fluent Inc. December 3, 2001
19.4 Spray Models
where K0 and K1 are modified Bessel functions of the second kind and
u is the relative velocity between the liquid and the gas. The linearized
boundary conditions are
∂η
∂t
∂v1
= −
∂z
v1 =
∂u1
∂r
(19.4-74)
(19.4-75)
and
σ
−p1 + 2µ1 − 2
a
2∂
η+a
2η
∂z 2
!
+ p2 = 0
(19.4-76)
which are mathematical statements of the liquid kinematic free surface
condition, continuity of shear stress, and continuity of normal stress,
respectively. Note that u1 is the axial perturbation liquid velocity, v1
is the radial perturbation liquid velocity, and σ is the surface tension.
Also note that Equation 19.4-75 was obtained under the assumption that
v2 = 0.
As described by Reitz [188], Equations 19.4-74 and 19.4-75 can be used
to eliminate the integration constants C1 and C2 in Equation 19.4-72.
Thus, when the pressure and velocity solutions are substituted into Equation 19.4-76, the desired dispersion relation is obtained:
I 0 (ka)
2kL I1 (ka) I10 (La)
ω + 2ν1 k ω 1
=
− 2
I0 (ka) k + L2 I0 (ka) I1 (La)
2
2
σk
L2 − a2
2 2
(1−k
a
)
ρ1 a2
L2 + a2
!
I1 (ka) ρ2
ω
U −i
+
I0 (ka) ρ1
k
2
L2 − a2
L2 + a2
!
I1 (ka) K0 (ka)
I0 (ka) K1 (ka)
(19.4-77)
As shown by Reitz [188], Equation 19.4-77 predicts that a maximum
growth rate (or most unstable wave) exists for a given set of flow conditions. Curve fits of numerical solutions to Equation 19.4-77 were gener-
c Fluent Inc. December 3, 2001
19-83
Discrete Phase Models
ated for the maximum growth rate, Ω, and the corresponding wavelength,
Λ, and are given by Reitz [188]:
Λ
a
Ω
ρ1 a3
σ
= 9.02
!
=
(1 + 0.45Oh0.5 )(1 + 0.4Ta0.7 )
0.6
(1 + 0.87We1.67
2 )
(0.34 + 0.38We1.5
2 )
(1 + Oh)(1 + 1.4Ta0.6 )
(19.4-78)
(19.4-79)
p
√
where Oh = We1 /Re1 is the Ohnesorge number and Ta = Oh We2 is
the Taylor number. Furthermore, We1 = ρ1 U 2 a/σ and We2 = ρ2 U 2 a/σ
are the liquid and gas Weber numbers, respectively, and Re1 = U a/ν1 is
the Reynolds number.
Droplet Breakup
In the wave model, the initial parcel diameters of the relatively large
injected droplets are modeled using the stability analysis for liquid jets
as described above. The breakup of the parcels and resulting droplets
of radius a is calculated by assuming that the breakup droplet radius, r,
is proportional to the wavelength of the fastest-growing unstable surface
wave given by Equation 19.4-78. In other words,
r = B0 Λ
(19.4-80)
where B0 is a model constant set equal to 0.61 based on the work of
Reitz [188]. Furthermore, the rate of change of droplet radius in a parent
parcel is given by
da
(a − r)
=−
, r≤a
dt
τ
(19.4-81)
where the breakup time, τ , is given by
τ=
19-84
3.726B1 a
ΛΩ
(19.4-82)
c Fluent Inc. December 3, 2001
19.4 Spray Models
and Λ and Ω are obtained from Equations 19.4-78 and 19.4-79, respectively. The breakup time constant, B1 , is related to the initial disturbance level on the liquid jet and has been found to vary from one nozzle
to another [118].
19.4.4
Dynamic Drag Model
Accurate determination of droplet drag coefficients is crucial for accurate spray modeling. FLUENT provides a method that determines the
droplet drag coefficient dynamically, accounting for variations in the
droplet shape.
Use and Limitations
The dynamic drag model is applicable in almost any circumstance. It
is compatible with both the TAB and wave models for spray breakup.
When the collision model is turned on, collisions reset the distortion and
distortion velocities of the colliding droplets.
Theory
Many droplet drag models assume the droplet remains spherical throughout the domain. With this assumption, the drag of a spherical object is
determined by the following [142]:



Cd,sphere =


0.424
24
Re
1 + 16 Re2/3
Re > 1000
(19.4-83)
Re ≤ 1000
However, as an initially spherical droplet moves through a gas, its shape
is distorted significantly when the Weber number is large. In the extreme
case, 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 droplet
drag coefficient is highly dependent upon the droplet shape, a drag model
that assumes the droplet is spherical is unsatisfactory. The dynamic drag
model accounts for the effects of droplet distortion, linearly varying the
drag between that of a sphere (Equation 19.4-83) and a value of 1.52
corresponding to a disk [142]. The drag coefficient is given by
c Fluent Inc. December 3, 2001
19-85
Discrete Phase Models
Cd = Cd,sphere (1 + 2.632y)
(19.4-84)
where y is the droplet distortion, as determined by the solution of
d2 y
C F ρg u2 C k σ
Cd µl dy
=
−
y−
2
2
3
dt
C b ρl r
ρl r
ρl r 2 dt
(19.4-85)
In the limit of no distortion (y = 0), the drag coefficient of a sphere will
be obtained, while at maximum distortion (y = 1) the drag coefficient
corresponding to a disk will be obtained.
Note that Equation 19.4-85 is obtained from the TAB model for spray
breakup, described in Section 19.4.3, but the dynamic drag model can
be used with either of the breakup models.
19.5
Coupling Between the Discrete and Continuous
Phases
As the trajectory of a particle is computed, FLUENT keeps track of the
heat, mass, and momentum gained or lost by the particle stream that follows that trajectory and these quantities can be incorporated in the subsequent continuous phase calculations. Thus, while the continuous phase
always impacts the discrete phase, you can also incorporate the effect of
the discrete phase trajectories on the continuum. This two-way coupling
is accomplished by alternately solving the discrete and continuous phase
equations until the solutions in both phases have stopped changing. This
interphase exchange of heat, mass, and momentum from the particle to
the continuous phase is depicted qualitatively in Figure 19.5.1.
Momentum Exchange
The momentum transfer from the continuous phase to the discrete phase
is computed in FLUENT by examining the change in momentum of a
particle as it passes through each control volume in the FLUENT model.
This momentum change is computed as
19-86
c Fluent Inc. December 3, 2001
19.5 Coupling Between the Discrete and Continuous Phases
typical
particle
trajectory
mass-exchange
heat-exchange
momentum-exchange
typical continuous
phase control volume
Figure 19.5.1: Heat, Mass, and Momentum Transfer Between the Discrete and Continuous Phases
F =
!
X
18µCD Re
(up − u) + Fother ṁp ∆t
ρp d2p 24
(19.5-1)
where
µ
ρp
dp
Re
up
u
CD
ṁp
∆t
Fother
=
=
=
=
=
=
=
=
=
=
viscosity of the fluid
density of the particle
diameter of the particle
relative Reynolds number
velocity of the particle
velocity of the fluid
drag coefficient
mass flow rate of the particles
time step
other interaction forces
This momentum exchange appears as a momentum sink in the continuous phase momentum balance in any subsequent calculations of the continuous phase flow field and can be reported by FLUENT as described in
c Fluent Inc. December 3, 2001
19-87
Discrete Phase Models
Section 19.13.
Heat Exchange
The heat transfer from the continuous phase to the discrete phase is
computed in FLUENT by examining the change in thermal energy of a
particle as it passes through each control volume in the FLUENT model.
In the absence of chemical reaction (i.e., for all particle laws except Law
5) this heat exchange is computed as
"
m̄p
∆mp
Q=
cp ∆Tp +
mp,0
mp,0
−hfg + hpyrol +
Z
Tp
!#
cp,i dT
ṁp,0
Tref
(19.5-2)
where
m̄p
=
mp,0
cp
∆Tp
=
=
=
∆mp
=
hfg
hpyrol
cp,i
Tp
=
=
=
=
Tref
ṁp,0
=
=
average mass of the particle in the control volume
(kg)
initial mass of the particle (kg)
heat capacity of the particle (J/kg-K)
temperature change of the particle in the control
volume (K)
change in the mass of the particle in the control
volume (kg)
latent heat of volatiles evolved (J/kg)
heat of pyrolysis as volatiles are evolved (J/kg)
heat capacity of the volatiles evolved (J/kg-K)
temperature of the particle upon exit of the control volume (K)
reference temperature for enthalpy (K)
initial mass flow rate of the particle injection
tracked (kg/s)
This heat exchange appears as a source or sink of energy in the continuous phase energy balance during any subsequent calculations of the
continuous phase flow field and is reported by FLUENT as described in
Section 19.13. A similar equation governs heat exchange under Law 5,
in which the heat of surface combustion is incorporated.
19-88
c Fluent Inc. December 3, 2001
19.5 Coupling Between the Discrete and Continuous Phases
Mass Exchange
The mass transfer from the discrete phase to the continuous phase is
computed in FLUENT by examining the change in mass of a particle as
it passes through each control volume in the FLUENT model. The mass
change is computed simply as
M=
∆mp
ṁp,0
mp,0
(19.5-3)
This mass exchange appears as a source of mass in the continuous phase
continuity equation and as a source of a chemical species defined by
you. The mass sources are included in any subsequent calculations of
the continuous phase flow field and are reported by FLUENT as described
in Section 19.13.
Under-Relaxation of the Interphase Exchange Terms
Note that the interphase exchange of momentum, heat, and mass is
under-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 you
can 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 coupled
calculations. Note that the value of α does not influence the predictions
obtained in the final converged solution.
c Fluent Inc. December 3, 2001
19-89
Discrete Phase Models
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 stochastic trajectory with the particle mass flow rate, ṁp0 , divided by the number of stochastic tracks computed. This implies that an equal mass flow
of particles follows each stochastic trajectory.
Interphase Exchange During Cloud Tracking
When the particle cloud model is used, the interphase exchange terms
are computed via Equations 19.5-1 to 19.5-6 based on ensemble-averaged
flow properties in the particle cloud. The exchange terms are then distributed to all the cells in the cloud based on the weighting factor defined
in Equation 19.2-61.
19.6
Overview of Using the Discrete Phase Models
The procedure for setting up and solving a problem involving a discrete
phase is outlined below, and described in detail in Sections 19.7–19.13.
Only the steps related specifically to discrete phase modeling are shown
here. For information about inputs related to other models that you are
using in conjunction with the discrete phase models, see the appropriate
sections for those models.
1. Enable any of the discrete phase modeling options, if relevant, as
described in Section 19.7.
2. If you are using unsteady particle tracking, define the unsteady
parameters 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 in
Section 19.12.
7. Examine the results, as described in Section 19.13.
19-90
c Fluent Inc. December 3, 2001
19.7 Discrete Phase Model Options
19.7
Discrete Phase Model Options
This section provides instructions for using the optional discrete phase
models available in FLUENT. All of them can be turned on in the Discrete
Phase 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 Interaction
option in the Discrete Phase Model panel. You will also need to define
additional properties for the particle materials (emissivity and scattering
factor), as described in Section 19.11.2. This option is available only
when 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 particle trajectories (Equation 19.2-14), turn on the Thermophoretic Force
option in the Discrete Phase Model panel. You will also need to define
the thermophoretic coefficient for the particle material, as described in
Section 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 solved
in a segregated manner. If you enable the Coupled Heat-Mass Solution
option, FLUENT will solve this pair of equations pair using a stiff, coupled 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 the
effects of Brownian motion (described in Section 19.2.1) on the particle
trajectories. To do so, turn on the Brownian Motion option in the Discrete
Phase Model panel. When Brownian motion effects are included, it is
c Fluent Inc. December 3, 2001
19-91
Discrete Phase Models
Figure 19.7.1: The Discrete Phase Model Panel
19-92
c Fluent Inc. December 3, 2001
19.7 Discrete Phase Model Options
recommended that you also select the Stokes-Cunningham drag law in
the Drag Law drop-down list under Drag Parameters, and specify the
Cunningham 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 (the
Saffman 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 Phase
Model panel.
19.7.6
Monitoring Erosion/Accretion of Particles at Walls
Particle erosion and accretion rates can be monitored at wall boundaries. These rate calculations can be enabled in the Discrete Phase
Model panel when the discrete phase is coupled with the continuous
phase (i.e., when Interaction with Continuous Phase is selected). Turning on the Erosion/Accretion option will cause the erosion and accretion
rates to be calculated at wall boundary faces when particle tracks are
updated. You will also need to set the Impact Angle Function (f (α) in
Equation 19.2-62), Diameter Function (C(dp ) in Equation 19.2-62), and
Velocity Exponent Function (b(v) in Equation 19.2-62) in the Wall boundary 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 Drag
Law drop-down list under Drag Parameters.
The spherical, non-spherical, Stokes-Cunningham, and high-Mach-number
laws described in Section 19.2.1 are always available, and the dynamicdrag law described in Section 19.4.4 is available only when one of the
droplet breakup models is used in conjunction with unsteady tracking.
See Section 19.8.2 for information about enabling the droplet breakup
models.
If the spherical law, the high-Mach-number law, or the dynamic-drag law
is selected, no further inputs are required. If the nonspherical law is selected, the particle Shape Factor (φ in Equation 19.2-7) must be specified.
c Fluent Inc. December 3, 2001
19-93
Discrete Phase Models
For the Stokes-Cunningham law, the Cunningham Correction factor (Cc in
Equation 19.2-9) must be specified.
19.7.8
User-Defined Functions
User-defined functions can be used to customize the discrete phase model
to include additional body forces, modify interphase exchange terms
(sources), calculate or integrate scalar values along the particle trajectory, and incorporate non-standard erosion rate definitions. See the separate UDF Manual for information about user-defined functions.
In the Discrete Phase Model panel, under User-Defined Functions, there
are drop-down lists labeled Body Force, Source, and Scalar Update. If
Erosion/Accretion is enabled under Options, there will be an additional
drop-down list labeled Erosion/Accretion. These lists will show available
user-defined functions that can be selected to customize the discrete
phase model.
19.8
Unsteady Particle Tracking
This section contains information about unsteady particle tracking with
the discrete phase model. Note that you cannot use adaptive time stepping for an unsteady discrete phase calculation.
19.8.1
Inputs for Unsteady Particle Tracking
For transient flow simulations, particle trajectories can also be advanced
in time with the flow simulation. If you select the Unsteady Tracking
option under Unsteady Parameters in the Discrete Phase Model panel,
particles will be advanced by the flow time step each time the flow solution is advanced in time. Coupled calculations are also allowed for
transient flow simulations. Particle sub-iterations are done during each
time step based on the value of the Number Of Continuous Phase Iterations
Per DPM Iteration.
! When the coupled explicit solver is used with the explicit unsteady formulation, the particles are advanced once per time step, and are calculated at the start of the time step (before the flow is updated).
19-94
c Fluent Inc. December 3, 2001
19.8 Unsteady Particle Tracking
Additional inputs are required for each injection in the Set Injection Properties panel. The injection Start Time and Stop Time must be specified
under Point Properties. Injections with start and stop times set to zero
will be injected only at the start of the calculation (t = 0). Changing
injection settings during the transient simulation will not affect particles currently released in the domain. At any point during the transient
simulation, you can clear particles that are currently in the domain by
clicking 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 specify
a particle history filename.
File −→ Write −→Start Particle History...
During the calculation, FLUENT will write the position, velocity, and
other data for each particle at each time step. To turn the particle
history 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 will
expand to show options related to spray modeling.
Modeling Spray Breakup
To enable the modeling of spray breakup, select the Droplet Breakup
option under Spray Models and then select the desired model (TAB or
Wave). A detailed description of these models can be found in Section 19.4.3.
For the TAB model, you will need to specify a value for y0 (the initial
distortion 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 stream
function 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
c Fluent Inc. December 3, 2001
19-95
Discrete Phase Models
constant B0 in Equation 19.4-80, and the default value 0.61 is acceptable
for nearly all cases.
Note that you may want to use the dynamic drag law when you use one
of the spray breakup models. See Section 19.7.7 for information about
choosing 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 no
further inputs for this model.
19.9
19.9.1
Setting Initial Conditions for the Discrete Phase
Overview of Initial Conditions
The primary inputs that you must provide for the discrete phase calculations in FLUENT are the initial conditions that define the starting positions, velocities, and other parameters for each particle stream. These
initial conditions provide the starting values for all of the dependent discrete phase variables that describe the instantaneous conditions of an
individual particle:
• Position (x, y, z coordinates) of the particle.
• Velocities (u, v, w) of the particle. Velocity magnitudes and spray
cone angle can also be used (in 3D) to define the initial velocities
(see Section 19.9.8). For moving reference frames, relative velocities should be specified.
• Diameter of the particle, dp .
• Temperature of the particle, Tp .
• Mass flow rate of the particle stream that will follow the trajectory
of the individual particle/droplet, ṁp (required only for coupled
calculations).
19-96
c Fluent Inc. December 3, 2001
19.9 Setting Initial Conditions for the Discrete Phase
• Additional parameters if one of the atomizer models described in
Section 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 to
the complexities of sheet and ligament breakup. Instead of initial
conditions, the quantities you will input for the atomizer models
are global parameters.
These dependent variables are updated according to the equations of
motion (Section 19.2) and according to the heat/mass transfer relations
applied (Section 19.3) as the particle/droplet moves along its trajectory.
You can define any number of different sets of initial conditions for discrete phase particles/droplets provided that your computer has sufficient
memory.
19.9.2
Injection Types
You will define the initial conditions for a particle/droplet stream by
creating an “injection” and assigning properties to it. FLUENT provides
10 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
c Fluent Inc. December 3, 2001
19-97
Discrete Phase Models
For each non-atomizer injection type, you will specify each of the initial
conditions listed in Section 19.9.1, the type of particle that possesses
these initial conditions, and any other relevant parameters for the particle type chosen.
You should create a single injection when you want to specify a single
value for each of the initial conditions (Figure 19.9.1). Create a group
injection (Figure 19.9.2) when you want to define a range for one or
more of the initial conditions (e.g., a range of diameters or a range of
initial 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 in
the Surface menu), you will create a surface injection. (If you create a
surface injection, a particle stream will be released from each facet of the
surface. You can use the Bounded and Sample Points options in the Plane
Surface panel to create injections from a rectangular grid of particles in
3D (see Section 24.6 for details).
Particle initial conditions (position, velocity, diameter, temperature, and
mass flow rate) can also be read from an external file if none of the injection types listed above can 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 available
to you depend on the range of physical models that you have defined in
the 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
19-98
c Fluent Inc. December 3, 2001
19.9 Setting Initial Conditions for the Discrete Phase
➞
●
Figure 19.9.1: Particle Injection Defining a Single Particle Stream
➞
➞
➞
➞
●
●
●
●
Figure 19.9.2: Particle Injection Defining an Initial Spatial Distribution
of the Particle Streams
▼
▼
▼
●
▼
▼
Figure 19.9.3: Particle Injection Defining an Initial Spray Distribution
of the Particle Velocity
c Fluent Inc. December 3, 2001
19-99
Discrete Phase Models
subject to heating or cooling via Law 1 (Section 19.3.2). The inert
type is available for all FLUENT models.
• A “droplet” particle is a liquid droplet in a continuous-phase gas
flow that obeys the force balance (Equation 19.2-1) and that experiences heating/cooling via Law 1 followed by vaporization and
boiling via Laws 2 and 3 (Sections 19.3.3 and 19.3.4). The droplet
type is available when heat transfer is being modeled and at least
two chemical species are active or the non-premixed or partially
premixed combustion model is active. You should use the ideal
gas law to define the gas-phase density (in the Materials panel, as
discussed in Section 7.2.5) when you select the droplet type.
• A “combusting” particle is a solid particle that obeys the force balance (Equation 19.2-1) and experiences heating/cooling via Law 1
followed by devolatilization via Law 4 (Section 19.3.5), and a heterogeneous surface reaction via Law 5 (Section 19.3.6). Finally,
the non-volatile portion of a combusting particle is subject to inert
heating via Law 6. You can also include an evaporating material with the combusting particle by selecting the Wet Combustion
option in the Set Injection Properties panel. This allows you to
include a material that evaporates and boils via Laws 2 and 3
(Sections 19.3.3 and 19.3.4) before devolatilization of the particle
material begins. The combusting type is available when heat transfer is being modeled and at least three chemical species are active
or the non-premixed combustion model is active. You should use
the ideal gas law to define the gas-phase density (in the Materials
panel) 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 Model
panel to open the Injections panel.)
19-100
c Fluent Inc. December 3, 2001
19.9 Setting Initial Conditions for the Discrete Phase
Figure 19.9.4: The Injections Panel
Creating Injections
To create an injection, click on the Create button. A new injection will
appear in the Injections list and the Set Injection Properties panel will open
automatically to allow you to set the injection properties (as described
in Section 19.9.5).
Modifying Injections
To modify an existing injection, select its name in the Injections list and
click on the Set... button. The Set Injection Properties panel will open,
and you can modify the properties as needed.
If you have two or more injections for which you want to set some of
the same properties, select their names in the Injections list and click on
the Set... button. The Set Multiple Injection Properties panel will open,
which will allow you to set the common properties. For instructions
c Fluent Inc. December 3, 2001
19-101
Discrete Phase Models
about using this panel, see Section 19.9.17.
Copying Injections
To copy an existing injection to a new injection, select the existing injection in the Injections list and click on the Copy button. The Set Injection
Properties panel will open with a new injection that has the same properties as the injection you selected. This is useful if you want to set
another injection with similar properties.
Deleting Injections
You can delete an injection by selecting its name in the Injections list
and clicking on the Delete button.
Listing Injections
To list the initial conditions for the particle streams in the selected injection, click on the List button. The list reported by FLUENT in the
console 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 combusting
particle) 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)
19-102
c Fluent Inc. December 3, 2001
19.9 Setting Initial Conditions for the Discrete Phase
Shortcuts for Selecting Injections
FLUENT provides a shortcut for selecting injections with names that
match a specified pattern. To use this shortcut, enter the pattern under Injection Name Pattern and then click Match to select the injections
with names that match the specified pattern. For example, if you specify
drop*, all injections that have names beginning with drop (e.g., drop-1,
droplet) will be selected automatically. If they are all selected already,
they will be deselected. If you specify drop?, all surfaces with names
consisting of drop followed by a single character will be selected (or deselected, if they are all selected already).
19.9.5
Defining Injection Properties
Once you have created an injection (using the Injections panel, as described in Section 19.9.4), you will use the Set Injection Properties panel
(Figure 19.9.5) to define the injection properties. (Remember that this
panel will open when you create a new injection, or when you select an
existing 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 default
name, enter a new one in the Injection Name field. This is recommended if you are defining a large number of injections so you can
easily 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. Note
that if you select any of the atomizer models, you will also need to
set the Viscosity and Droplet Surface Tension in the Materials panel.
!
If you are using sliding or moving/deforming meshes in your simulation, you should not use surface injections because they are not
compatible with moving meshes.
c Fluent Inc. December 3, 2001
19-103
Discrete Phase Models
Figure 19.9.5: The Set Injection Properties Panel
19-104
c Fluent Inc. December 3, 2001
19.9 Setting Initial Conditions for the Discrete Phase
3. If you are defining a single injection, go to the next step. For a
group, cone, or any of the atomizer injections, set the Number of
Particle Streams in the group, spray cone, or atomizer. If you are
defining a surface injection, choose the surface(s) from which the
particles will be released in the Release From Surfaces list. If you
are reading the injection from a file, click on the File... button at
the bottom of the Set Injection Properties panel and specify the file
to be read in the resulting Select File dialog box. The parameters
in the injection file must be in SI units.
4. Select Inert, Droplet, or Combusting as the Particle Type. The available types are described in Section 19.9.3.
5. Choose the material for the particle(s) in the Material drop-down
list. 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 in
the database. If you have already created a particle of this type,
the only available material will be the material you selected for
that particle. You can define additional materials by copying them
from the database or creating them from scratch, as discussed in
Section 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 to
change from the default linear (for group injections) or uniform (for
surface injections) interpolation method used to determine the size
of the particles, select rosin-rammler or rosin-rammler-logarithmic
in the Diameter Distribution drop-down list. The Rosin-Rammler
method for determining the range of diameters for a group injection
is described in Section 19.9.7.
7. If you have created a customized particle law using user-defined
functions, turn on the Custom option under Laws and specify the
appropriate laws as described in Section 19.9.16.
8. If your particle type is Inert, go to the next step. If you are defining Droplet particles, select the gas phase species created by the
vaporization and boiling laws (Laws 2 and 3) in the Evaporating
Species drop-down list.
c Fluent Inc. December 3, 2001
19-105
Discrete Phase Models
If you are defining Combusting particles, select the gas phase species
created by the devolatilization law (Law 4) in the Devolatilizing
Species drop-down list, the gas phase species that participates
in the surface char combustion reaction (Law 5) in the Oxidizing
Species list, and the gas phase species created by the surface char
combustion reaction (Law 5) in the Product Species list. Note that
if the Combustion Model for the selected combusting particle material (in the Materials panel) is the multiple-surface-reaction model,
then the Oxidizing Species and Product Species lists will be disabled
because the reaction stoichiometry has been defined in the mixture
material.
9. Click the Point Properties tab (the default), and specify the point
properties (position, velocity, diameter, temperature, and—if
appropriate—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 turbulence on the particle dispersion, click the Turbulent Dispersion tab,
turn on the Stochastic Model and/or the Cloud Model, and set the
related parameters as described in Section 19.9.15.
11. If your combusting particle includes an evaporating material, click
the Wet Combustion tab, select the Wet Combustion option, and
then select the material that is evaporating/boiling from the particle before devolatilization begins in the Liquid Material drop-down
list. You should also set the volume fraction of the liquid present
in the particle by entering the value of the Liquid Fraction. Finally,
select the gas phase species created by the evaporating and boiling
laws in the Evaporating Species drop-down list in the top part of
the panel.
12. If you want to use a user-defined function to initialize the injection properties, click the UDF tab to access the UDF inputs. You
can select an Initialization function under User-Defined Functions to
modify injection properties at the time the particles are injected
into the domain. This allows the position and/or properties of the
injection to be set as a function of flow conditions. See the separate
UDF Manual for information about user-defined functions.
19-106
c Fluent Inc. December 3, 2001
19.9 Setting Initial Conditions for the Discrete Phase
19.9.6
Point Properties for Single Injections
For a single injection, you will define the following initial conditions
for the particle stream under the Point Properties heading (in the Set
Injection Properties panel):
• Position: Set the x, y, and z positions of the injected stream along
the Cartesian axes of the problem geometry in the X-, Y-, and
Z-Position fields. (Z-Position will appear only for 3D problems.)
• Velocity: Set the x, y, and z components of the stream’s initial
velocity in the X-, Y-, and Z-Velocity fields. (Z-Velocity will appear
only for 3D problems.)
• Diameter: Set the initial diameter of the injected particle stream
in the Diameter field.
• Temperature: Set the initial (absolute) temperature of the injected
particle stream in the Temperature field.
• Mass flow rate: For coupled phase calculations (see Section 19.12),
set the mass of particles per unit time that follows the trajectory
defined by the injection in the Flow Rate field. Note that in axisymmetric problems the mass flow rate is defined per 2π radians and
in 2D problems per unit meter depth (regardless of the reference
value for length).
• Duration of injection: For unsteady particle tracking (see Section 19.8), set the starting and ending time for the injection in
the Start Time and Stop Time fields.
19.9.7
Point Properties for Group Injections
For group injections, you will define the properties described in Section 19.9.6 for single injections for the First Point and Last Point in the
group. That is, you will define a range of values, φ1 through φN , for
each initial condition φ by setting values for φ1 and φN . FLUENT assigns a value of φ to the ith injection in the group using a linear variation
between the first and last values for φ:
c Fluent Inc. December 3, 2001
19-107
Discrete Phase Models
φi = φ1 +
φN − φ1
(i − 1)
N −1
(19.9-1)
Thus, for example, if your group consists of 5 particle streams and you
define a range for the initial x location from 0.2 to 0.6 meters, the initial
x location of each stream is as follows:
• Stream 1: x = 0.2 meters
• Stream 2: x = 0.3 meters
• Stream 3: x = 0.4 meters
• Stream 4: x = 0.5 meters
• Stream 5: x = 0.6 meters
! In general, you should supply a range for only one of the initial conditions in a given group—leaving all other conditions fixed while a single
condition varies among the stream numbers of the group. Otherwise
you may find, for example, that your simultaneous inputs of a spatial
distribution and a size distribution have placed the small droplets at the
beginning of the spatial range and the large droplets at the end of the
spatial range.
Note that you can use a different method for defining the size distribution
of the particles, as discussed below.
Using the Rosin-Rammler Diameter Distribution Method
By default, you will define the size distribution of particles by inputting
a diameter 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. When
you want a different mass flow rate for each particle/droplet size, however, the linear variation may not yield the distribution you need. Your
particle size distribution may be defined most easily by fitting the size
distribution data to the Rosin-Rammler equation. In this approach, the
complete range of particle sizes is divided into a set of discrete size ranges,
19-108
c Fluent Inc. December 3, 2001
19.9 Setting Initial Conditions for the Discrete Phase
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
Range (µm )
0–70
70–100
100–120
120–150
150–180
180–200
Mass Fraction
in Range
0.05
0.10
0.35
0.30
0.15
0.05
The Rosin-Rammler distribution function is based on the assumption
that 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 Diameter
and to n as the Spread Parameter. These parameters are input by you (in
the Set Injection Properties panel under the First Point heading) to define
the Rosin-Rammler size distribution. To solve for these parameters, you
must fit your particle size data to the Rosin-Rammler exponential equation. To determine these inputs, first recast the given droplet size data
in terms of the Rosin-Rammler format. For the example data provided
above, this yields the following pairs of d and Yd :
Diameter, d (µm)
70
100
120
150
180
200
c Fluent Inc. December 3, 2001
Mass Fraction with
Diameter Greater than d, Yd
0.95
0.85
0.50
0.20
0.05
(0.00)
19-109
Discrete Phase Models
1.0
0.9
0.8
Mass Fraction > d, Yd
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
Diameter, d ( µm)
Figure 19.9.6: Example of Cumulative Size Distribution of Particles
19-110
c Fluent Inc. December 3, 2001
19.9 Setting Initial Conditions for the Discrete Phase
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 fit
Equation 19.9-2. The value for d is obtained by noting that this is the
value of d at which Yd = e−1 ≈ 0.368. From Figure 19.9.6, you can
estimate that this occurs for d ≈ 131 µm. The numerical value for n is
given by
n=
ln(− ln Yd )
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 average value of n = 4.52 for the example data above. The resulting RosinRammler 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 the
data 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 natural logarithm of the particle diameter. If in your case, the smallerdiameter particles in a Rosin-Rammler distribution have higher mass
flows in comparison with the larger-diameter particles, you may want
better resolution of the smaller-diameter particle streams, or “bins”.
You can therefore choose to have the diameter increments in the RosinRammler distribution done uniformly by ln d.
In the standard Rosin-Rammler distribution, a particle injection may
have a diameter range of 1 to 200 µm. In the logarithmic Rosin-Rammler
distribution, the same diameter range would be converted to a range of
ln 1 to ln 200, or about 0 to 5.3. In this way, the mass flow in one bin
would be less-heavily skewed as compared to the other bins.
When a Rosin-Rammler size distribution is being defined for the group
of streams, you should define (in addition to the initial velocity, position, and temperature) the following parameters, which appear under
the heading for the First Point:
c Fluent Inc. December 3, 2001
19-111
Discrete Phase Models
1.0
0.9
0.8
Mass Fraction > d, Yd
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
Diameter, d ( µm)
Figure 19.9.7: Rosin-Rammler Curve Fit for the Example Particle Size
Data
19-112
c Fluent Inc. December 3, 2001
19.9 Setting Initial Conditions for the Discrete Phase
• Total Flow Rate: the total mass flow rate of the N streams in the
group. Note that in axisymmetric problems this mass flow rate is
defined per 2π radians and in 2D problems per unit meter depth.
• Min. Diameter: the smallest diameter to be considered in the size
distribution.
• Max. Diameter: the largest diameter to be considered in the size
distribution.
• Mean Diameter: the size parameter, d, in the Rosin-Rammler equation (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 of
particle 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 the
X-, Y-, and Z-Position fields.
• Diameter: Set the diameter of the particles in the stream in the
Diameter field.
• Temperature: Set the temperature of the streams in the Temperature field.
• Axis: Set the x, y, and z components of the vector defining the
cone’s axis in the X-Axis, Y-Axis, and Z-Axis fields.
• Velocity: Set the velocity magnitude of the particle streams that
will be oriented along the specified spray cone angle in the Velocity
Mag. field.
• Cone angle: Set the included half-angle, θ, of the hollow spray cone
in the Cone Angle field, as shown in Figure 19.9.8.
c Fluent Inc. December 3, 2001
19-113
Discrete Phase Models
• Radius: A non-zero inner radius can be specified to model injectors
that do not emanate from a single point. Set the radius r (defined
as shown in Figure 19.9.8) in the Radius field. The particles will
be 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 into
the swirling component of the flow in the Swirl Fraction field. The
direction of the swirl component is defined using the right-hand
rule about the axis (a negative value for the swirl fraction can be
used to reverse the swirl direction).
• Mass flow rate: For coupled calculations, set the total mass flow
rate for the streams in the spray cone in the Total Flow Rate field.
Note that you may want to define multiple spray cones emanating from
the same initial location in order to include a size distribution of the
spray 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 in
Section 19.9.6 for single injections except for the initial position of the
particle streams. The initial positions of the particles will be the location
19-114
c Fluent Inc. December 3, 2001
19.9 Setting Initial Conditions for the Discrete Phase
of the data points on the specified surface(s). Note that you will set the
Total Flow Rate of all particles released from the surface (required for
coupled calculations only). If you want, you can scale the individual
mass flow rates of the particles by the ratio of the area of the face they
are released from to the total area of the surface. To scale the mass
flow rates, select the Scale Flow Rate By Face Area option under Point
Properties.
Note that many surfaces have non-uniform distributions of points. If
you want to generate a uniform spatial distribution of particle streams
released from a surface in 3D, you can create a bounded plane surface
with a uniform distribution using the Plane Surface panel, as described
in Section 24.6. In 2D, you can create a rake using the Line/Rake Surface
panel, as described in Section 24.5.
A non-uniform size distribution can be used for surface injections, as
described below.
Using the Rosin-Rammler Diameter Distribution Method
The Rosin-Rammler size distributions described in Section 19.9.7 for
group injections is also available for surface injections. If you select
one of the Rosin-Rammler distributions, you will need to specify the
following parameters under Point Properties, in addition to the initial
velocity, temperature, and total flow rate:
• Min. Diameter: the smallest diameter to be considered in the size
distribution.
• Max. Diameter: the largest diameter to be considered in the size
distribution.
• Mean Diameter: the size parameter, d, in the Rosin-Rammler equation (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 from
each face of the surface).
c Fluent Inc. December 3, 2001
19-115
Discrete Phase Models
FLUENT will inject streams of particles from each face on the surface,
with diameters defined by the Rosin-Rammler distribution function. The
total number of injection streams tracked for the surface injection will
be equal to the number of diameters in each distribution (Number of
Diameters) 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 initial
conditions under Point Properties:
• Position: Set the x, y, and z positions of the injected stream along
the Cartesian axes of the problem geometry in the X-Position, YPosition, and Z-Position fields. (Z-Position will appear only for 3D
problems.
• Axis (3D only): Set the x, y, and z components of the vector
defining the axis of the orifice in the X-Axis, Y-Axis, and Z-Axis
fields.
• Temperature: Set the temperature of the streams in the Temperature field.
• Mass flow rate: Set the mass flow rate for the streams in the atomizer in the Flow Rate field.
• Duration of injection: For unsteady particle tracking (see Section 19.8), set the starting and ending time for the injection in
the Start Time and Stop Time fields.
• Vapor pressure: Set the vapor pressure governing the flow through
the 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 Length
field (L in Table 19.4.1).
19-116
c Fluent Inc. December 3, 2001
19.9 Setting Initial Conditions for the Discrete Phase
• Radius of curvature: Set the radius of curvature of the inlet corner
in the Corner Radius of Curv. field (r in Table 19.4.1).
• Nozzle parameter: Set the constant for the spray angle correlation
in the Constant A field (CA in Equation 19.4-16).
• Azimuthal angles: For 3D sectors, set the Azimuthal Start Angle
and 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 same
properties 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 in
Section 19.9.10, you will need to specify the following parameters under
Point Properties:
• Spray angle: Set the value of the spray angle of the injected stream
in the Spray Half Angle field (θ in Equation 19.4-25).
• Pressure: Set the pressure upstream of the injection in the Upstream Pressure field (p1 in Table 19.4.1).
• Sheet breakup: Set the value of the empirical constant that determines the length of the ligaments that are formed after sheet
breakup in the Sheet Constant field (ln( ηη0b ) in Equation 19.4-30).
• Ligament diameter: For short waves, set the proportionality constant that linearly relates the ligament diameter, dL , to the wavelength that breaks up the sheet in the Ligament Constant field (see
Equations 19.4-31–19.4-33).
See Section 19.4.1 for details about how these inputs are used.
c Fluent Inc. December 3, 2001
19-117
Discrete Phase Models
19.9.12
Point Properties for Air-Blast/Air-Assist Atomizer
Injections
For an air-blast/air-assist atomizer, you will specify some of the same
properties as for 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 in
Section 19.9.10, you will need to specify the following parameters under
Point Properties:
• Outer diameter: Set the outer diameter of the injector in the Injector Outer Diam. field. This value is used in conjunction with the
Injector Inner Diam. to set the thickness of the liquid sheet (t in
Equation 19.4-22).
• Spray angle: Set the initial trajectory of the film as it leaves the end
of the orifice in the Spray Half Angle field (θ in Equation 19.4-25).
• Relative velocity: Set the maximum relative velocity that is produced by the sheet and air in the Relative Velocity field.
• Sheet breakup: Set the value of the empirical constant that determines the length of the ligaments that are formed after sheet
breakup in the Sheet Constant field (ln( ηη0b ) in Equation 19.4-30).
• Ligament diameter: For short waves, set the proportionality constant that linearly relates the ligament diameter, dL , to the wavelength that breaks up the sheet in the Ligament Constant field (see
Equations 19.4-31–19.4-33).
See Section 19.4.1 for details about how these inputs are used.
19.9.13
Point Properties for Flat-Fan Atomizer Injections
The flat-fan atomizer model is available only for 3D models. For this
type of injection, you will define the following initial conditions under
Point Properties:
19-118
c Fluent Inc. December 3, 2001
19.9 Setting Initial Conditions for the Discrete Phase
• Arc position: Set the coordinates of the center point of the arc from
which the fan originates in the X-Center, Y-Center, and Z-Center
fields (see Figure 19.4.6).
• Virtual position: Set the coordinates of the virtual origin of the fan
in 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 the
fan (see Figure 19.4.6).
• Normal vector: Set the direction that is normal to the fan in the XFan Normal Vector, Y-Fan Normal Vector, and Z-Fan Normal Vector
fields.
• Temperature: Set the temperature of the streams in the Temperature field.
• Mass flow rate: Set the mass flow rate for the streams in the atomizer in the Flow Rate field.
• Duration of injection: For unsteady particle tracking (see Section 19.8), set the starting and ending time for the injection in
the Start Time and Stop Time fields.
• Spray half angle: Set the initial half angle of the drops as they
leave 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 determines the length of the ligaments that are formed after sheet
breakup in the Flat Fan Sheet Constant field (see Equation 19.4-30).
See Section 19.4.1 for details about how these inputs are used.
19.9.14
Point Properties for Effervescent Atomizer Injections
For an effervescent atomizer injection, you will specify some of the same
properties as for a plain-orifice atomizer. In addition to the position, axis
c Fluent Inc. December 3, 2001
19-119
Discrete Phase Models
(if 3D), temperature, mass flow rate (including both flashing and nonflashing components), duration of injection (if unsteady), vapor pressure,
injector inner diameter, and azimuthal angles (if relevant) described in
Section 19.9.10, you will need to specify the following parameters under
Point Properties:
• Mixture quality: Set the mass fraction of the injected mixture that
vaporizes in the Mixture Quality field (x in Equation 19.4-38).
• Saturation temperature: Set the saturation temperature of the
volatile substance in the Saturation Temp. field.
• Droplet dispersion: Set the parameter that controls the spatial
dispersion of the droplet sizes in the Dispersion Constant field (Ceff
in Equation 19.4-38).
• Spray angle: Set the initial trajectory of the film as it leaves the
end of the orifice in the Maximum Half Angle field.
See Section 19.4.1 for details about how these inputs are used.
19.9.15
Modeling Turbulent Dispersion of Particles
As mentioned in Section 19.9.5, you can choose stochastic tracking and/or
cloud tracking as the method for modeling turbulent dispersion of particles.
Stochastic Tracking
For turbulent flows, if you choose to use the stochastic tracking technique, you must enable it and specify the “number of tries”. Stochastic
tracking includes the effect of turbulent velocity fluctuations on the particle trajectories using the DRW model described in Section 19.2.2.
1. Click the Turbulent Dispersion tab in the Set Injection Properties
panel.
2. Enable stochastic tracking by turning on the Stochastic Model under Stochastic Tracking.
19-120
c Fluent Inc. December 3, 2001
19.9 Setting Initial Conditions for the Discrete Phase
3. Specify the Number Of Tries:
• An input of zero tells FLUENT to compute the particle trajectory based on the mean continuous phase velocity field (Equation 19.2-1), ignoring the effects of turbulence on the particle
trajectories.
• An input of 1 or greater tells FLUENT to include turbulent
velocity fluctuations in the particle force balance as in Equation 19.2-20. The trajectory is computed more than once if
your input exceeds 1: two trajectory calculations are performed if you input 2, three trajectory calculations are performed if you input 3, etc. Each trajectory calculation includes a new stochastic representation of the turbulent contributions to the trajectory equation.
When a sufficient number of tries is requested, the trajectories computed will include a statistical representation of the
spread of the particle stream due to turbulence. Note that for
unsteady particle tracking, the Number of Tries is set to 1 if
Stochastic Tracking is enabled.
If you want the characteristic lifetime of the eddy to be random (Equation 19.2-37), enable the Random Eddy Lifetime option. You will generally
not 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 Stress
turbulence model (RSM), in which case a value of 0.3 is recommended.
Figure 19.9.9 illustrates a discrete phase trajectory calculation computed
via the “mean” tracking (number of tries = 0) and Figure 19.9.10 illustrates the “stochastic” tracking (number of tries > 1) option.
When multiple stochastic trajectory calculations are performed, the momentum and mass defined for the injection are divided evenly among the
multiple particle/droplet tracks, and are thus spread out in terms of the
interphase momentum, heat, and mass transfer calculations. Including
turbulent dispersion in your model can thus have a significant impact on
the effect of the particles on the continuous phase when coupled calculations are performed.
c Fluent Inc. December 3, 2001
19-121
Discrete Phase Models
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
Particle Traces Colored by Particle Time (s)
Figure 19.9.9: Mean Trajectory in a Turbulent Flow
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
Particle Traces Colored by Particle Time (s)
Figure 19.9.10: Stochastic Trajectories in a Turbulent Flow
19-122
c Fluent Inc. December 3, 2001
19.9 Setting Initial Conditions for the Discrete Phase
Cloud Tracking
For turbulent flows, you can also include the effects of turbulent dispersion on the injection. When cloud tracking is used, the trajectory will
be tracked as a cloud of particles about a mean trajectory, as described
in Section 19.2.2.
1. Click the Turbulent Dispersion tab in the Set Injection Properties
panel.
2. Enable cloud tracking by turning on the Cloud Model under Cloud
Tracking.
3. Specify the minimum and maximum cloud diameters. Particles
enter the domain with an initial cloud diameter equal to the Min.
Cloud Diameter. The particle cloud’s maximum allowed diameter
is specified by the Max. Cloud Diameter.
You may want to restrict the Max. Cloud Diameter to a relevant
length scale for the problem to improve computational efficiency
in complex domains where the mean trajectory may become stuck
in recirculation regions.
19.9.16
Custom Particle Laws
If the standard FLUENT laws, Laws 1 through 6, do not adequately
describe the physics of your discrete phase model, you can modify them
by creating custom laws with user-defined functions. See the separate
UDF Manual for information about user-defined functions. You can also
create 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-defined
functions.
Once you have defined and loaded your user-defined function(s), you can
create a custom law by enabling the Custom option under Laws in the
Set 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, you
can select the appropriate particle law for your custom law. Each list
c Fluent Inc. December 3, 2001
19-123
Discrete Phase Models
Figure 19.9.11: The Custom Laws Panel
contains the available options that can be chosen (the standard laws plus
any user-defined functions you have loaded).
There is a seventh drop-down list in the Custom Laws panel labeled
Switching. You may wish to have FLUENT vary the laws used depending
on conditions in the model. You can customize the way FLUENT switches
between laws by selecting a user-defined function from this drop-down
list.
An example of when you might want to use a custom law might be to
replace the standard devolatilization law with a specialized devolatilization law that more accurately describes some unique aspects of your
model. After creating and loading a user-defined function that details
the physics of your devolatilization law, you would visit the Custom Laws
panel and replace the standard devolatilization law (Law 2) with your
user-defined function.
19-124
c Fluent Inc. December 3, 2001
19.9 Setting Initial Conditions for the Discrete Phase
19.9.17
Defining Properties Common to More Than One
Injection
If you have a number of injections for which you want to set the same
properties, FLUENT provides a shortcut so that you do not need to visit
the Set Injection Properties panel for each injection to make the same
changes.
As described in Section 19.9.5, if you select more than one injection in
the Injections panel, clicking the Set... button will open the Set Multiple
Injection Properties panel (Figure 19.9.12) instead of the Set Injection
Properties panel.
Depending on the type of injections you have selected (single, group,
atomizers, etc.), there will be different categories of properties listed
under Injections Setup. The names of these categories correspond to the
headings within the Set Injection Properties panel (e.g., Particle Type and
Stochastic Tracking). Only those categories that are appropriate for all
of your selected injections (which are shown in the Injections list) will
be listed. If all of these injections are of the same type, more categories
of properties will be available for you to modify. If the injections are of
different 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. For
example, if you want to set the same flow rate for all of the selected
injections, select Point Properties. The panel will expand to show
the properties that appear under that heading in the Set Injection
Properties panel.
2. Set the property (or properties) to be modified, as described below.
3. Click Apply. FLUENT will report the change in the console window.
!
You must click Apply to save the property settings within each
category. If, for example, you want to modify the flow rate and
the stochastic tracking parameters, you will need to select Point
c Fluent Inc. December 3, 2001
19-125
Discrete Phase Models
Figure 19.9.12: The Set Multiple Injection Properties Panel
19-126
c Fluent Inc. December 3, 2001
19.9 Setting Initial Conditions for the Discrete Phase
Properties in the Injections Setup list, specify the flow rate, and
click Apply. You would then repeat the process for the stochastic
tracking parameters, clicking Apply again when you are done.
There are two types of properties that can be modified using the Set
Multiple 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 Set
Injection Properties panel. For example, if you select Particle Type in the
Injections Setup list, the panel will expand to show the portion of the Set
Injection Properties panel where you choose the particle type. You can
simply choose the desired type and click Apply.
Setting the second type of property requires an additional step. If you
select a category in the Injections Setup list that contains this type of
property, the expanded portion of the panel will look like the corresponding part of the Set Injection Properties panel, with the addition of
Modify check buttons (see Figure 19.9.12). To change one of the properties, first turn on the Modify check button to its left, and then specify
the desired status or value.
For example, if you would like to enable stochastic tracking, first turn on
the Modify check button to the left of Stochastic Model. This will make
the property active so you can modify its status. Then, under Property,
turn on the Stochastic Model check button. (Be sure to click Apply when
you are done setting stochastic tracking parameters.)
c Fluent Inc. December 3, 2001
19-127
Discrete Phase Models
If you would like to change the value of Number of Tries, select the Modify
check button to its left to make it active, and then enter the new value
in the field. Make sure you click Apply when you have finished modifying
the 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 selected
injections when you click Apply. After you turn on Modify for a particular
property, clicking Apply will modify that property for all of the selected
injections, so make sure that you have the settings the way that you
want them before you do this. If you make a mistake, you will have to
return to the Set Injection Properties panel for each injection to fix the
incorrect setting, if it is not possible to do so in the Set Multiple Injection
Properties panel.
Modifying Properties Common to a Subset of Selected Injections
Note that it is possible to change a property that is relevant for only a
subset of the selected injections. For example, if some of the selected
injections are using stochastic tracking and some are not, enabling the
Random Eddy Lifetime option and clicking Apply will turn this option on
only for those injections that are using stochastic tracking. The other
injections will be unaffected.
19-128
c Fluent Inc. December 3, 2001
19.10 Setting Boundary Conditions for the Discrete Phase
19.10
Setting Boundary Conditions for the Discrete Phase
When a particle reaches a physical boundary (e.g., a wall or inlet boundary) in your model, FLUENT applies a discrete phase boundary condition
to determine the fate of the trajectory at that boundary. The boundary
condition, or trajectory fate, can be defined separately for each zone in
your FLUENT model.
19.10.1
Discrete Phase Boundary Condition Types
The available boundary conditions, as noted in Section 19.2, include the
following:
• “reflect” rebounds the particle off the boundary in question with a
change in its momentum as defined by the coefficient of restitution.
(See Figure 19.10.1.)
V2,n
coefficient
of
=
V
restitution
1,n
θ1
θ
2
Figure 19.10.1: “Reflect” Boundary Condition for the Discrete Phase
The normal coefficient of restitution defines the amount of momentum in the direction normal to the wall that is retained by the
particle 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 subscripts 1 and 2 refer to before and after collision, respectively. Simi-
c Fluent Inc. December 3, 2001
19-129
Discrete Phase Models
larly, the tangential coefficient of restitution, et , defines the amount
of momentum in the direction tangential to the wall that is retained
by the particle.
A normal or tangential coefficient of restitution equal to 1.0 implies
that the particle retains all of its normal or tangential momentum
after the rebound (an elastic collision). A normal or tangential coefficient of restitution equal to 0.0 implies that the particle retains
none of its normal or tangential momentum after the rebound.
Non-constant coefficients of restitution can be specified for wall
zones with the “reflect” type boundary condition. The coefficients
are set as a function of the impact angle, θ1 , in Figure 19.10.1.
Note that the default setting for both coefficients of restitution
is a constant value of 1.0 (all normal and tangential momentum
retained).
• “trap” terminates the trajectory calculations and records the fate
of the particle as “trapped”. In the case of evaporating droplets,
their entire mass instantaneously passes into the vapor phase and
enters the cell adjacent to the boundary. See Figure 19.10.2. In the
case of combusting particles, the remaining volatile mass is passed
into the vapor phase.
volatile fraction
flashes to vapor
θ1
Figure 19.10.2: “Trap” Boundary Condition for the Discrete Phase
• “escape” reports the particle as having “escaped” when it encounters the boundary in question. Trajectory calculations are terminated. See Figure 19.10.3.
19-130
c Fluent Inc. December 3, 2001
19.10 Setting Boundary Conditions for the Discrete Phase
particle vanishes
Figure 19.10.3: “Escape” Boundary Condition for the Discrete Phase
• “interior” means that the particles will pass through the internal boundary. This option is available only for internal boundary
zones, such as a radiator or a porous jump.
Because you can stipulate any of these conditions at flow boundaries, it
is possible to incorporate mixed discrete phase boundary conditions in
your FLUENT model.
Default Discrete Phase Boundary Conditions
FLUENT assumes the following boundary conditions:
• “reflect” at wall, symmetry, and axis boundaries, with both coefficients of restitution equal to 1.0
• “escape” at all flow boundaries (pressure and velocity inlets, pressure outlets, etc.)
• “interior” at all internal boundaries (radiator, porous jump, etc.)
The coefficient of restitution can be modified only for wall boundaries.
c Fluent Inc. December 3, 2001
19-131
Discrete Phase Models
19.10.2
Inputs for Discrete Phase Boundary Conditions
Discrete phase boundary conditions can be set for boundaries in the
panels opened from the Boundary Conditions panel. When one or more
injections have been defined, inputs for the discrete phase will appear in
the 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-down
list under Discrete Phase Model Conditions. (In the Walls panel, you
will need to click on the DPM tab to access the Discrete Phase Model
Conditions.) These conditions are described in Section 19.10.1. You
can also select a user-defined function in this list. For internal boundary
19-132
c Fluent Inc. December 3, 2001
19.11 Setting Material Properties for the Discrete Phase
zones, such as a radiator or a porous jump, you can also choose an interior
boundary condition. The interior condition means that the particles will
pass 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 Normal and Tangent coefficients of restitution under Discrete Phase Reflection
Coefficients. See Section 19.10.1 for details about the boundary condition types and the coefficients of restitution. The panels for defining the
polynomial, piecewise-linear, and piecewise-polynomial functions are the
same as those used for defining temperature-dependent properties. See
Section 7.1.3 for details.
If the Erosion/Accretion option is selected in the Discrete Phase Model
panel, the erosion rate expression must be specified at the walls. The
erosion rate is defined in Equation 19.2-62 as a product of the mass
flux and specified functions for the particle diameter, impact angle, and
velocity exponent. Under Erosion Model in the Wall panel, you can define
a constant, polynomial, piecewise-linear, or piecewise-polynomial function
for the Impact Angle Function, Diameter Function, and Velocity Exponent
Function (f (α), C(dp ), and b(v) in Equation 19.2-62). See Section 19.7.6
for a detailed description of these functions and Section 7.1.3 for details
about using the panels for defining polynomial, piecewise-linear, and
piecewise-polynomial functions.
19.11
Setting Material Properties for the Discrete Phase
In order to apply the physical models described in earlier sections to
the 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 are
used for each particle type and in which of the equations for heat and
mass transfer each property input is used. Detailed descriptions of each
input are provided in Section 19.11.2.
c Fluent Inc. December 3, 2001
19-133
Discrete Phase Models
Table 19.11.1: Property Inputs for Inert Particles
Property
Symbol
density
ρp in Eq. 19.2-1
specific heat
cp in Eq. 19.3-3
particle emissivity
p in Eq. 19.3-3
particle scattering factor
f in Eq. 11.3-20
thermophoretic coefficient DT,p in Eq. 19.2-14
Table 19.11.2: Property Inputs for Droplet Particles
Properties
Symbol
density
ρp in Eq. 19.2-1
specific heat
cp in Eq. 19.3-17
thermal conductivity
kp in Eq. 19.2-15
viscosity
µ in Eq. 19.4-48
latent heat
hfg in Eq. 19.3-17
vaporization temperature
Tvap in Eq. 19.3-10
boiling point
Tbp in Eq. 19.3-10, 19.3-18
volatile component fraction fv0 in Eq. 19.3-11, 19.3-19
binary diffusivity
Di,m in Eq. 19.3-15
saturation vapor pressure
psat (T ) in Eq. 19.3-13
heat of pyrolysis
hpyrol in Eq. 19.5-2
droplet surface tension
σ in Eq. 19.4-18, 19.4-47
particle emissivity
p in Eq. 19.3-17, 19.3-23
particle scattering factor
f in Eq. 11.3-20
thermophoretic coefficient
DT,p in Eq. 19.2-14
19-134
c Fluent Inc. December 3, 2001
19.11 Setting Material Properties for the Discrete Phase
Table 19.11.3: Property Inputs for Combusting Particles (Laws 1–4)
Properties
density
specific heat
latent heat
vaporization temperature
volatile component fraction
swelling coefficient
burnout stoichiometric ratio
combustible fraction
heat of reaction for burnout
fraction of reaction heat given to solid
particle emissivity
particle scattering factor
thermophoretic coefficient
devolatilization model
–law 4, constant rate
constant
–law 4, single rate
pre-exponential factor
activation energy
–law 4, two rates
pre-exponential factors
activation energies
weighting factors
–law 4, CPD
initial fraction of bridges in coal lattice
initial fraction of char bridges
lattice coordination number
cluster molecular weight
side chain molecular weight
c Fluent Inc. December 3, 2001
Symbol
ρp in Eq. 19.2-1
cp in Eq. 19.3-3
hfg in Eq. 19.5-2
Tvap = Tbp in Eq. 19.3-24
fv0 in Eq. 19.3-25
Csw in Eq. 19.3-57
Sb in Eq. 19.3-64
fcomb in Eq. 19.3-63
Hreac in Eq. 19.3-64 19.3-78
fh in Eq. 19.3-78
p in Eq. 19.3-58, 19.3-78
f in Eq. 11.3-20
DT,p in Eq. 19.2-14
A0 in Eq. 19.3-26
A1 in Eq. 19.3-27
E in Eq. 19.3-27
A1 , A2 in Eq. 19.3-30, 19.3-31
E1 , E2 in Eq. 19.3-30, 19.3-31
α1 , α2 in Eq. 19.3-32
p0 in Eq. 19.3-43
c0 in Eq. 19.3-42
σ + 1 in Eq. 19.3-54
Mw,1 in Eq. 19.3-54
Mw,δ in Eq. 19.3-53
19-135
Discrete Phase Models
Table 19.11.4: Property Inputs for Combusting Particles (Law 5)
Properties
Symbol
combustion 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. 19.3-66
kinetics limited rate pre-exp. factor
C2 in Eq. 19.3-67
kinetics limited rate activ. energy
E in Eq. 19.3-67
–law 5, intrinsic rate
mass diffusion limited rate constant
C1 in Eq. 19.3-66
kinetics limited rate pre-exp. factor
Ai in Eq. 19.3-76
kinetics limited rate activ. energy
Ei in Eq. 19.3-76
char porosity
θ in Eq. 19.3-73
mean pore radius
r p in Eq. 19.3-75
specific internal surface area
Ag in Eq. 19.3-70, 19.3-72
tortuosity
τ in Eq. 19.3-73
burning mode
α in Eq. 19.3-77
–law 5, multiple surface reaction
binary diffusivity
Di,m in Eq. 19.3-65
19-136
c Fluent Inc. December 3, 2001
19.11 Setting Material Properties for the Discrete Phase
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 for
the discrete phase (as described in Section 19.9), you choose a particular
material as the particle’s material. All particle streams of that material
will have the same physical properties.
Discrete-phase materials are divided into three categories, corresponding
to the three types of particles available. These material types are inertparticle, droplet-particle, and combusting-particle. Each material type will
be added to the Material Type list in the Materials panel when an injection
of that type of particle is defined (in the Set Injection Properties or Set
Multiple Injection Properties panel, as described in Section 19.9). The
first time you create an injection of each particle type, you will be able
to choose a material from the database, and this will become the default
material for that type of particle. That is, if you create another injection
of the same type of particle, your selected material will be used for that
injection as well. You may choose to modify the predefined properties for
your 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 need
not define any new materials; you can simply use the same material for
all 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 modify
the properties and give the material a new name, as described in Section 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 of
that type of particles. This means, for example, that you cannot define
or modify any combusting-particle materials until you have defined a
combusting particle injection (as described in Section 19.9).
c Fluent Inc. December 3, 2001
19-137
Discrete Phase Models
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 considered
in a given model. Sometimes, however, a single model may contain two
different types of inert, droplet, or combusting particles (e.g., heavy particles and gaseous bubbles or two different types of evaporating liquid
droplets). In such cases, it is necessary to assign a different set of properties to the two (or more) different types of particles. This is easily
accomplished by defining two or more inert, droplet, or combusting particle materials and using the appropriate one for each particle injection.
You can define additional discrete-phase materials either by copying
them from the database or by creating them from scratch. See Section 7.1.2 for instructions on using the Materials panel to perform these
actions.
! Recall that you must define at least one injection (as described in Section 19.9) containing particles of a certain type before you will be able
to define additional materials for that particle type.
Description of Properties
The properties that appear in the Materials panel vary depending on
the particle type (selected in the Set Injection Properties or Set Multiple
Injection Properties panel, as described in Sections 19.9.5 and 19.9.17) and
the physical models you are using in conjunction with the discrete-phase
model.
Below, all properties you may need to define for a discrete-phase material
are listed. See Tables 19.11.1–19.11.4 to see which properties are defined
for each type of particle.
Density is the density of the particulate phase in units of mass per unit
volume of the discrete phase. This density is the mass density and
not the volumetric density. Since certain particles may swell during the trajectory calculations, your input is actually an “initial”
density.
19-138
c Fluent Inc. December 3, 2001
19.11 Setting Material Properties for the Discrete Phase
Cp is the specific heat, cp , of the particle. The specific heat may be
defined as a function of temperature by selecting one of the function
types from the drop-down list to the right of Cp. See Section 7.1.3
for details about temperature-dependent properties.
Thermal Conductivity is the thermal conductivity of the particle. This
input is specified in units of W/m-K in SI units or Btu/ft-h-◦ F in
British units and is treated as a constant by FLUENT.
Latent Heat is the latent heat of vaporization, hfg , required for phase
change from an evaporating liquid droplet (Equation 19.3-17) or
for the evolution of volatiles from a combusting particle (Equation 19.3-58). This input is supplied in units of J/kg in SI units or
of Btu/lbm in British units and is treated as a constant by FLUENT.
Thermophoretic Coefficient is the coefficient DT,p in Equation 19.2-14,
and appears when the thermophoretic force (which is described in
Section 19.2.1) is included in the trajectory calculation (i.e., when
the Thermophoretic Force option is enabled in the Discrete Phase
Model panel). The default is the expression developed by Talbot [237] (talbot-diffusion-coeff) and requires no input from you.
You can also define the thermophoretic coefficient as a function
of temperature by selecting one of the function types from the
drop-down list to the right of Thermophoretic Coefficient. See Section 7.1.3 for details about temperature-dependent properties.
Vaporization Temperature is the temperature, Tvap , at which the calculation of vaporization from a liquid droplet or devolatilization from
a combusting particle is initiated by FLUENT. Until the particle
temperature reaches Tvap , the particle is heated via Law 1, Equation 19.3-3. This temperature input represents a modeling decision
rather than any physical characteristic of the discrete phase.
Boiling Point is the temperature, Tbp , at which the calculation of the
boiling rate equation (19.3-20) is initiated by FLUENT. When a
droplet particle reaches the boiling point, FLUENT applies Law 3
and assumes that the droplet temperature is constant at Tbp . The
boiling point should be defined as the saturated vapor temperature
c Fluent Inc. December 3, 2001
19-139
Discrete Phase Models
at the system pressure that you defined in the Operating Conditions
panel.
Volatile Component Fraction (fv0 ) is the fraction of a droplet particle
that 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 be
evolved via Law 4 (Section 19.3.5).
Binary Diffusivity is the mass diffusion coefficient, Di,m , used in the vaporization law, Law 2 (Equation 19.3-15). This input is also used
to define the mass diffusion of the oxidizing species to the surface of a combusting particle, Di,m , as given in Equation 19.3-65.
(Note that the diffusion coefficient inputs that you supply for the
continuous phase are not used for the discrete phase.)
Saturation Vapor Pressure is the saturated vapor pressure, psat , defined
as a function of temperature, which is used in the vaporization
law, Law 2 (Equation 19.3-13). The saturated vapor pressure may
be defined as a function of temperature by selecting one of the
function types from the drop-down list to the right of its name. (See
Section 7.1.3 for details about temperature-dependent properties.)
In the case of unrealistic inputs, FLUENT restricts the range of
Psat to between 0.0 and the operating pressure. Correct input of a
realistic vapor pressure curve is essential for accurate results from
the vaporization model.
Heat of Pyrolysis is the heat of the instantaneous pyrolysis reaction, hpyrol ,
that the evaporating/boiling species may undergo when released to
the continuous phase. This input represents the conversion of the
evaporating species to lighter components during the evaporation
process. The heat of pyrolysis should be input as a positive number
for exothermic reaction and as a negative number for endothermic
reaction. The default value of zero implies that the heat of pyrolysis is not considered. This input is used in Equation 19.5-2.
Swelling Coefficient is the coefficient Csw in Equation 19.3-57, which governs the swelling of the coal particle during the devolatilization law,
Law 4 (Section 19.3.5). A swelling coefficient of unity (the default)
19-140
c Fluent Inc. December 3, 2001
19.11 Setting Material Properties for the Discrete Phase
implies that the coal particle stays at constant diameter during the
devolatilization process.
Burnout Stoichiometric Ratio is the stoichiometric requirement, Sb , for
the burnout reaction, Equation 19.3-64, in terms of mass of oxidant
per mass of char in the particle.
Combustible Fraction is the mass fraction of char, fcomb , in the coal particle, i.e., the fraction of the initial combusting particle that will
react in the surface reaction, Law 5 (Equation 19.3-63).
Heat of Reaction for Burnout is the heat released by the surface char
combustion reaction, Law 5 (Equation 19.3-64). This parameter
is input in terms of heat release (e.g., Joules) per unit mass of char
consumed in the surface reaction.
React. Heat Fraction Absorbed by Solid is the parameter fh (Equation
19.3-78), which controls the distribution of the heat of reaction
between the particle and the continuous phase. The default value
of zero implies that the entire heat of reaction is released to the
continuous phase.
Devolatilization Model defines which version of the devolatilization model,
Law 4, is being used. If you want to use the default constant rate
devolatilization model, Equation 19.3-26, retain the selection of
constant in the drop-down list to the right of Devolatilization Model
and input the rate constant A0 in the field below the list.
You can activate one of the optional devolatilization models (the
single kinetic rate, two kinetic rates, or CPD model, as described
in Section 19.3.5) by choosing single rate, two-competing-rates, or
cpd-model in the drop-down list.
When the single kinetic rate model (single-rate) is selected, the
Single Rate Devolatilization Model panel will appear and you will
enter the Pre-exponential Factor, A1 , and the Activation Energy, E,
to be used in Equation 19.3-28 for the computation of the kinetic
rate.
When the two competing rates model (two-competing-rates) is selected, the Two Competing Rates Model panel will appear and
c Fluent Inc. December 3, 2001
19-141
Discrete Phase Models
you will enter, for the First Rate and the Second Rate, the Preexponential Factor (A1 in Equation 19.3-30 and A2 in Equation
19.3-31), Activation Energy (E1 in Equation 19.3-30 and E2 in
Equation 19.3-31), and Weighting Factor (α1 and α2 in Equation
19.3-32). The constants you input are used in Equations 19.3-30
through 19.3-32.
When the CPD model (cpd-model) is selected, the CPD Model panel
will appear and you will enter the Initial Fraction of Bridges in Coal
Lattice (p0 in Equation 19.3-43), Initial Fraction of Char Bridges (c0
in Equation 19.3-42), Lattice Coordination Number (σ + 1 in Equation 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 Competing
Rates Model, and CPD Model panels are modal panels, which means
that you must tend to them immediately before continuing the
property definitions.
Combustion Model defines which version of the surface char combustion
law (Law 5) is being used. If you want to use the default diffusionlimited rate model, retain the selection of diffusion-limited in the
drop-down list to the right of Combustion Model. No additional
inputs are necessary, because the binary diffusivity defined above
will be used in Equation 19.3-65.
To use the kinetics/diffusion-limited rate model for the surface
combustion model, select kinetics/diffusion-limited in the drop-down
list. The Kinetics/Diffusion Limited Combustion Model panel will
appear and you will enter the Mass Diffusion Limited Rate Constant
(C1 in Equation 19.3-66), Kinetics Limited Rate Pre-exponential Factor (C2 in Equation 19.3-67), and Kinetics Limited Rate Activation
Energy (E in Equation 19.3-67).
Note that the Kinetics/Diffusion Limited Combustion Model panel is
a modal panel, which means that you must tend to it immediately
before continuing the property definitions.
To use the intrinsic model for the surface combustion model, select intrinsic-model in the drop-down list. The Intrinsic Combustion
Model panel will appear and you will enter the Mass Diffusion Lim-
19-142
c Fluent Inc. December 3, 2001
19.11 Setting Material Properties for the Discrete Phase
ited Rate Constant (C1 in Equation 19.3-66), Kinetics Limited Rate
Pre-exponential Factor (Ai in Equation 19.3-76), Kinetics Limited
Rate Activation Energy (Ei in Equation 19.3-76), Char Porosity (θ
in Equation 19.3-73), Mean Pore Radius (r p 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 in
Equation 19.3-77).
Note that the Intrinsic Combustion Model panel is a modal panel,
which means that you must tend to it immediately before continuing the property definitions.
To use the multiple surface reactions model, select multiple-surfacereactions in the drop-down list. FLUENT will display a dialog box
informing you that you will need to open the Reactions panel, where
you can review or modify the particle surface reactions that you
specified 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 information
about using the multiple surface reactions model.
You will notice that the Burnout Stoichiometric Ratio and Heat of
Reaction for Burnout are no longer available in the Materials panel,
as these parameters are now computed from the particle surface
reactions you defined in the Reactions panel.
Note that the multiple surface reactions model is available only if
the Particle Surface option for Reactions is enabled in the Species
Model panel. See Section 13.3.2 for details.
When the effect of particles on radiation is enabled (for the P-1 or discrete 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, p , used to
compute 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 discrete
ordinates radiation model is active. Note that you must enable
radiation to particles, using the Particle Radiation Interaction option
c Fluent Inc. December 3, 2001
19-143
Discrete Phase Models
in the Discrete Phase Model panel. Recommended values of particle
emissivity are 1.0 for coal particles and 0.5 for ash [143].
Particle Scattering Factor is the scattering factor, fp , due to particles in
the 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 Phase
Model panel. The recommended value of fp for coal combustion
modeling is 0.9 [143]. Note that if the effect of particles on radiation
is enabled, scattering in the continuous phase will be ignored in the
radiation model.
When an atomizer injection model and/or the spray breakup or collision
model is enabled in the Set Injection Properties panel (atomizers) and/or
Discrete Phase Model panel (spray breakup/collision), you will need to
define the following additional parameters:
Viscosity is the droplet viscosity, µl . The viscosity may be defined as
a function of temperature by selecting one of the function types
from the drop-down list to the right of Viscosity. See Section 7.1.3
for details about temperature-dependent properties. You also have
the option of implementing a user-defined function to model the
droplet viscosity. See the separate UDF Manual for information
about user-defined functions.
Droplet Surface Tension is the droplet surface tension, σ. The surface
tension may be defined as a function of temperature by selecting
one of the function types from the drop-down list to the right
of Droplet Surface Tension. See Section 7.1.3 for details about
temperature-dependent properties. You also have the option of
implementing a user-defined function to model the droplet surface
tension. See the separate UDF Manual for information about userdefined functions.
19-144
c Fluent Inc. December 3, 2001
19.12 Calculation Procedures for the Discrete Phase
19.12
Calculation Procedures for the Discrete Phase
Solution of the discrete phase implies integration in time of the force
balance on the particle (Equation 19.2-1) to yield the particle trajectory.
As the particle is moved along its trajectory, heat and mass transfer
between the particle and the continuous phase are also computed via
the heat/mass transfer laws (Section 19.3). The accuracy of the discrete phase calculation thus depends on the time accuracy of the integration and upon the appropriate coupling between the discrete and
continuous phases when required. Numerical controls are described in
Section 19.12.1. Coupling and performing trajectory calculations are
described in Section 19.12.2. Sections 19.12.3 and 19.12.4 provide information about resetting interphase exchange terms and using the parallel
solver 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 particle
trajectory equations:
• the length scale or step length factor, used to set the time step for
integration within each control volume
• the maximum number of time steps, used to abort trajectory calculations when the particle never exits the flow domain
Each of these parameters is set in the Discrete Phase Model panel (Figure 19.12.1) under Tracking Parameters.
Define −→ Models −→Discrete Phase...
Max. Number Of Steps is the maximum number of time steps used to
compute a single particle trajectory via integration of Equations
19.2-1 and 19.2-21. When the maximum number of steps is exceeded, FLUENT abandons the trajectory calculation for the current particle injection and reports the trajectory fate as “incomplete”. The limit on the number of integration time steps eliminates the possibility of a particle being caught in a recirculating
c Fluent Inc. December 3, 2001
19-145
Discrete Phase Models
Figure 19.12.1: The Discrete Phase Model Panel
19-146
c Fluent Inc. December 3, 2001
19.12 Calculation Procedures for the Discrete Phase
region of the continuous phase flow field and being tracked infinitely. Note that you may easily create problems in which the
default value of 500 time steps is insufficient for completion of the
trajectory calculation. In this case, when trajectories are reported
as incomplete within the domain and the particles are not recirculating 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 the
equations of motion for the particle. The integration time step is
computed by FLUENT based on a specified length scale L, and the
velocity of the particle (up ) and of the continuous phase (uc ):
∆t =
L
up + u c
(19.12-1)
where L is the Length Scale that you define. As defined by Equation 19.12-1, L is proportional to the integration time step and is
equivalent to the distance that the particle will travel before its
motion equations are solved again and its trajectory is updated. A
smaller value for the Length Scale increases the accuracy of the trajectory and heat/mass transfer calculations for the discrete phase.
(Note that particle positions are always computed when particles
enter/leave a cell; even if you specify a very large length scale, the
time step used for integration will be such that the cell is traversed
in one step.)
Length Scale will appear in the Discrete Phase Model panel when
the Specify Length Scale option is on (the default setting).
Step Length Factor also controls the time step size used to integrate the
equations of motion for the particle. It differs from the Length
Scale in that it allows FLUENT to compute the time step in terms
of the number of time steps required for a particle to traverse a
computational cell. To set this parameter instead of the Length
Scale, turn off the Specify Length Scale option.
The integration time step is computed by FLUENT based on a characteristic time that is related to an estimate of the time required
c Fluent Inc. December 3, 2001
19-147
Discrete Phase Models
for the particle to traverse the current continuous phase control
volume. If this estimated transit time is defined as ∆t∗ , FLUENT
chooses 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 is
roughly equivalent to the number of time steps required to traverse the current continuous phase control volume. A larger value
for the Step Length Factor decreases the discrete phase integration
time step. The default value for the Step Length Factor is 20.
One simple rule of thumb to follow when setting the parameters above
is that if you want the particles to advance through a domain of length
D, the Length Scale times the Max. Number Of Steps should be approximately equal to D.
19.12.2
Performing Trajectory Calculations
The trajectories of your discrete phase injections are computed when
you display the trajectories using graphics or when you perform solution iterations. That is, you can display trajectories without impacting
the continuous phase, or you can include their effect on the continuum
(termed a coupled calculation). In turbulent flows, trajectories can be
based on mean (time-averaged) continuous phase velocities or they can
be impacted by instantaneous velocity fluctuations in the fluid. This section describes the procedures and commands you use to perform coupled
or uncoupled trajectory calculations, with or without stochastic tracking
or cloud tracking.
Uncoupled Calculations
For the uncoupled calculation, you will perform the following two steps:
1. Solve the continuous phase flow field.
19-148
c Fluent Inc. December 3, 2001
19.12 Calculation Procedures for the Discrete Phase
2. Plot (and report) the particle trajectories for discrete phase injections of interest.
In the uncoupled approach, this two-step procedure completes the modeling effort, as illustrated in Figure 19.12.2. The particle trajectories are
computed as they are displayed, based on a fixed continuous-phase flow
field. 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 low
mass and momentum loading, in which case the continuous phase is not
impacted by the presence of the discrete phase.
Coupled Calculations
In a coupled two-phase simulation, FLUENT modifies the two-step procedure above as follows:
1. Solve the continuous phase flow field (prior to introduction of the
discrete phase).
2. Introduce the discrete phase by calculating the particle trajectories
for each discrete phase injection.
3. Recalculate the continuous phase flow, using the interphase exchange of momentum, heat, and mass determined during the previous particle calculation.
4. Recalculate the discrete phase trajectories in the modified continuous phase flow field.
c Fluent Inc. December 3, 2001
19-149
Discrete Phase Models
5. Repeat the previous two steps until a converged solution is achieved
in which both the continuous phase flow field and the discrete phase
particle trajectories are unchanged with each additional calculation.
This coupled calculation procedure is illustrated in Figure 19.12.3. When
your FLUENT model includes a high mass and/or momentum loading in
the discrete phase, the coupled procedure must be followed in order to
include the important impact of the discrete phase on the continuous
phase flow field.
continuous phase flow field calculation
particle trajectory calculation
update continuous phase source terms
Figure 19.12.3: Coupled Discrete Phase Calculations
! When you perform coupled calculations, all defined discrete phase injections will be computed. You cannot calculate a subset of the injections
you 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
19-150
c Fluent Inc. December 3, 2001
19.12 Calculation Procedures for the Discrete Phase
flow field. You will then create your injection(s) and set up the coupled
calculation.
For each discrete-phase iteration, FLUENT computes the particle/droplet
trajectories and updates the interphase exchange of momentum, heat,
and mass in each control volume. These interphase exchange terms then
impact the continuous phase when the continuous phase iteration is performed. During the coupled calculation, FLUENT will perform the discrete phase iteration at specified intervals during the continuous-phase
calculation. The coupled calculation continues until the continuous phase
flow field no longer changes with further calculations (i.e., all convergence
criteria are satisfied). When convergence is reached, the discrete phase
trajectories no longer change either, since changes in the discrete phase
trajectories would result in changes in the continuous phase flow field.
The steps for setting up the coupled calculation are as follows:
1. Solve the continuous phase flow field.
2. In the Discrete Phase Model panel (Figure 19.12.1), enable the Interaction with Continuous Phase option.
3. Set the frequency with which the particle trajectory calculations
are introduced in the Number Of Continuous Phase Iterations Per
DPM Iteration field. If you set this parameter to 5, for example,
a discrete phase iteration will be performed every fifth continuous phase iteration. The optimum number of iterations between
trajectory calculations depends upon the physics of your FLUENT
model.
!
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 panel
in the usual manner) you will see the following information in the FLUENT console as the continuous and discrete phase iterations are performed:
c Fluent Inc. December 3, 2001
19-151
Discrete Phase Models
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,
Done.
316 1.9206e-01 1.1860e-01 6.9573e-01
317 2.0729e-01 3.2982e-02 8.3036e-01
318 3.2820e-01 5.5508e-02 6.0900e-01
aborted= 0, trapped= 0, evaporated = 8, i
5.2692e-02 2.3997e-02 2.4532e-03
4.1649e-02 2.2111e-02 2.5369e-01
5.9018e-02 2.6619e-02 4.0394e-02
0:00:02
0:00:01
0:00:00
Note that you can perform a discrete phase calculation at any time by
using the solve/dpm-update text command.
Stochastic Tracking in Coupled Calculations
If you include the stochastic prediction of turbulent dispersion in the
coupled two-phase flow calculations, the number of stochastic tries applied each time the discrete phase trajectories are introduced during
coupled calculations will be equal to the Number of Tries specified in the
Set Injection Properties panel. Input of this parameter is described in
Section 19.9.15.
Note that the number of tries should be set to 0 if you want to perform the coupled calculation based on the mean continuous phase flow
field. An input of n ≥ 1 requests n stochastic trajectory calculations for
each particle in the injection. Note that when the number of stochastic
tracks included is small, you may find that the ensemble average of the
trajectories is quite different each time the trajectories are computed.
These differences may, in turn, impact the convergence of your coupled
solution. For this reason, you should include an adequate number of
stochastic tracks in order to avoid convergence troubles in coupled calculations.
Under-Relaxation of the Interphase Exchange Terms
When you are coupling the discrete and continuous phases for steadystate calculations, using the calculation procedures noted above, FLUENT applies under-relaxation to the momentum, heat, and mass transfer terms. This under-relaxation serves to increase the stability of the
19-152
c Fluent Inc. December 3, 2001
19.12 Calculation Procedures for the Discrete Phase
coupled calculation procedure by letting the impact of the discrete phase
change only gradually:
Enew = Eold + α(Ecalculated − Eold )
(19.12-3)
where Enew is the exchange term, Eold is the previous value, Ecalculated is
the newly computed value, and α is the particle/droplet under-relaxation
factor. FLUENT uses a default value of 0.5 for α. You can modify α
by changing the value in the Discrete Phase Sources field under UnderRelaxation Factors in the Solution Controls panel. You may need to decrease α in order to improve the stability of coupled discrete phase calculations.
19.12.3
Resetting the Interphase Exchange Terms
If you have performed coupled calculations, resulting in non-zero interphase sources/sinks of momentum, heat, and/or mass that you do not
want to include in subsequent calculations, you can reset these sources
to zero.
Solve −→ Initialize −→Reset DPM Sources
When you select the Reset DPM Sources menu item, the sources will
immediately be 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 specify
explicitly that you want to perform the discrete phase calculation in parallel. In the Discrete Phase Model panel, turn on the Workpile Algorithm
option under Parallel and specify the Number of Threads. By default,
the Number of Threads is equal to the number of compute nodes you
specified for the parallel solver. You can modify this value based on the
computational requirements of the particle calculations. If, for example,
the particle calculations require more computation than the flow calculation, you can increase the Number of Threads (up to the number of
available processors) to improve performance.
c Fluent Inc. December 3, 2001
19-153
Discrete Phase Models
Note that the discrete phase model is also available when solving in
parallel on a distributed memory machine or compute cluster. However,
as when running on a shared-memory machine, the particle calculations
will take place entirely within the Host process. Therefore, you will need
to make sure that there is enough memory to store the entire grid on the
machine executing the Host process. In such a situation, the number of
threads 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 coupled
two-phase calculations of interest, you can display and store the particle
trajectory predictions. FLUENT provides both graphical and alphanumeric 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 exchange 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 postprocessing
options.
(Note that plotting or reporting trajectories does not change the source
terms.)
19-154
c Fluent Inc. December 3, 2001
19.13 Postprocessing for the Discrete Phase
19.13.1
Graphical Display of Trajectories
When you have defined discrete phase particle injections, as described in
Section 19.9, you can display the trajectories of these discrete particles
using 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 From
Injections list. (You can choose to track a specific particle, instead,
as described below.)
c Fluent Inc. December 3, 2001
19-155
Discrete Phase Models
2. Set the length scale and the maximum number of steps in the
Discrete Phase Model panel, as described in Section 19.12.1.
Define −→ Models −→Discrete Phase...
If stochastic and/or cloud tracking is desired, set the related parameters in the Set Injection Properties panel, as described in Section 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 the
Pulse button to animate the particle positions. The Pulse button
will become the Stop ! button during the animation, and you must
click on Stop ! to stop the pulsing.
!
For unsteady particle tracking simulations, clicking on Display will
show only the current location of the particles. Typically, you
should select point in the Style drop-down list when displaying transient 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 particle
stream instead of for all the streams in a given injection. To do so, you
will first need to determine which particle is of interest. Use the Injections
panel to list the particle streams in the desired injection, as described in
Section 19.9.4.
Define −→Injections...
Note the ID numbers listed in the first column of the listing printed
in the FLUENT console. Then perform the following steps after step 1
above:
1. Enable the Track Single Particle Stream option in the Particle Tracks
panel.
2. In the Stream ID field, specify the ID number of the particle stream
for which you want to plot the trajectory.
19-156
c Fluent Inc. December 3, 2001
19.13 Postprocessing for the Discrete Phase
Options for Particle Trajectory Plots
The options mentioned above include the following: You can include
the grid in the trajectory display, control the style of the trajectories
(including the twisting of ribbon-style trajectories), and color them by
different scalar fields and control the color scale. You can also choose
node or cell values for display. If you are “pulsing” the trajectories, you
can control the pulse mode. Finally, you can generate an XY plot of the
particle trajectory data (e.g., residence time) as a function of time or
path length and save this XY plot data to a file.
These options are controlled in exactly the same way that pathlineplotting options are controlled. See Section 25.1.4 for details about setting the trajectory plotting options mentioned above.
Note that in addition to coloring the trajectories by continuous phase
variables, you can also color them according to the following discrete
phase variables: particle time, particle velocity, particle diameter, particle density, particle mass, particle temperature, particle law number,
particle time step, and particle Reynolds number. These variables are
included in the Particle Variables... category of the Color By list. To display the minimum and maximum values in the domain, click the Update
Min/Max button.
Graphical Display in Axisymmetric Geometries
For axisymmetric problems in which the particle has a non-zero circumferential velocity component, the trajectory of an individual particle is
often a spiral about the centerline of rotation. FLUENT displays the r
and x components of the trajectory (but not the θ component) projected
in the axisymmetric plane.
19.13.2
Reporting of Trajectory Fates
When you perform trajectory calculations by displaying the trajectories (as described in Section 19.13.1), FLUENT will provide information
about the trajectories as they are completed. By default, the number of
trajectories with each possible fate (escaped, aborted, evaporated, etc.)
is reported:
c Fluent Inc. December 3, 2001
19-157
Discrete Phase Models
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 the
trajectories 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 boundary
for which the “escape” condition is set.
• “Incomplete” trajectories are those that were terminated when the
maximum allowed number of time steps—as defined by the Max.
Number Of Steps input in the Discrete Phase Model panel (see Section 19.12.1)—was exceeded.
• “Trapped” trajectories are those that terminate at a flow boundary
where the “trap” condition has been set.
• “Evaporated” trajectories include those trajectories along which
the particles were evaporated within the domain.
• “Aborted” trajectories are those that fail to complete due to roundoff reasons. You may want to retry the calculation with a modified
length scale and/or different initial conditions.
Summary Reports
You can request additional detail about the trajectory fates as the particles exit the domain, including the mass flow rates through each boundary zone, mass flow rate of evaporated droplets, and composition of the
particles.
1. Follow steps 1 and 2 in Section 19.13.1 for displaying trajectories.
19-158
c Fluent Inc. December 3, 2001
19.13 Postprocessing for the Discrete Phase
2. Select Summary as the Report Type and click Display or Track.
A detailed report similar to the following example will appear in the
console window. (You may also choose to write this report to a file by
selecting File as the Report to option, clicking on the Write... button
(which was originally the Display button), and specifying a file name for
the 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
---Incomplete
Escaped - Zone 7
-----2
8
Elapsed Time (s)
Min
Max
Avg
Std Dev
---------- ---------- ---------- ---------- --1.485e+01 2.410e+01 1.947e+01 4.623e+00
4.940e+00 2.196e+01 1.226e+01 4.871e+00
(*)- Mass Transfer Summary -(*)
Fate
---Incomplete
Escaped - Zone 7
Mass Flow (kg/s)
Initial
Final
Change
---------- ---------- ---------1.388e-03 1.943e-04 -1.194e-03
1.502e-03 2.481e-04 -1.254e-03
(*)- Energy Transfer Summary -(*)
Fate
---Incomplete
Escaped - Zone 7
Heat Content (W)
Initial
Final
Change
---------- ---------- ---------4.051e+02 3.088e+02 -9.630e+01
4.383e+02 3.914e+02 -4.696e+01
(*)- Combusting Particles -(*)
Fate
---Incomplete
Escaped - Zone 7
Volatile Content (kg/s)
Initial
Final
%Conv
---------- ---------- ------6.247e-04 0.000e+00 100.00
6.758e-04 0.000e+00 100.00
Char Content (kg/s)
Initial
Final
---------- ---------- -5.691e-04 0.000e+00 1
6.158e-04 3.782e-05
Done.
The report groups together particles with each possible fate, and reports the number of particles, the time elapsed during trajectories, and
c Fluent Inc. December 3, 2001
19-159
Discrete Phase Models
the mass and energy transfer. This information can be very useful for
obtaining information such as where particles are escaping from the domain, where particles are colliding with surfaces, and the extent of heat
and mass transfer to/from the particles within the domain. Additional
information is reported for combusting particles.
Elapsed Time
The number of particles with each fate is listed under the Number heading. (Particles that escape through different zones or are trapped at
different zones are considered to have different fates, and are therefore
listed separately.) The minimum, maximum, and average time elapsed
during the trajectories of these particles, as well as the standard deviation about the average time, are listed in the Min, Max, Avg, and Std
Dev columns. This information indicates how much time the particle(s)
spent in the domain before they escaped, aborted, evaporated, or were
trapped.
Fate
Number
---Incomplete
Escaped - Zone 7
-----2
8
Elapsed Time (s)
Min
Max
Avg
Std Dev
---------- ---------- ---------- ---------- --1.485e+01 2.410e+01 1.947e+01 4.623e+00
4.940e+00 2.196e+01 1.226e+01 4.871e+00
Also, on the right side of the report are listed the injection name and
index of 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 initial
and final mass flow rates and the change in mass flow rate are reported
19-160
c Fluent Inc. December 3, 2001
19.13 Postprocessing for the Discrete Phase
in the Initial, Final, and Change columns. With this information, you
can determine how much mass was transferred to the continuous phase
from the particles.
(*)- Mass Transfer Summary -(*)
Fate
---Incomplete
Escaped - Zone 7
Mass Flow (kg/s)
Initial
Final
Change
---------- ---------- ---------1.388e-03 1.943e-04 -1.194e-03
1.502e-03 2.481e-04 -1.254e-03
Energy Transfer Summary
For all particles with each fate, the total initial and final heat content
and the change in heat content are reported in the Initial, Final, and
Change columns. This report tells you how much heat was transferred
from the continuous phase to the particles.
(*)- Energy Transfer Summary -(*)
Fate
---Incomplete
Escaped - Zone 7
Heat Content (W)
Initial
Final
Change
---------- ---------- ---------4.051e+02 3.088e+02 -9.630e+01
4.383e+02 3.914e+02 -4.696e+01
Combusting Particles
If combusting particles are present, FLUENT will include additional reporting on the volatiles and char converted. These reports are intended
to help you identify the composition of the combusting particles as they
exit the computational domain.
(*)- Combusting Particles -(*)
Fate
---Incomplete
Escaped - Zone 7
Volatile Content (kg/s)
Initial
Final
%Conv
---------- ---------- ------6.247e-04 0.000e+00 100.00
6.758e-04 0.000e+00 100.00
c Fluent Inc. December 3, 2001
Char Content (kg/s)
Initial
Final
%Conv
---------- ---------- ------5.691e-04 0.000e+00 100.00
6.158e-04 3.782e-05
93.86
19-161
Discrete Phase Models
The total volatile content at the start and end of the trajectory is reported 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 is
reported in the Initial and Final columns under Char Content. The
percentage 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 of
the particle trajectory/trajectories. Such reports can be obtained in
alphanumeric format. This capability allows you to monitor the particle
position, velocity, temperature, or diameter as the trajectory proceeds.
The procedure for generating files containing step-by-step reports is
listed 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 Track
Single Particle Stream option.
2. Select Step By Step as the Report Type.
3. Select File as the Report to option. (The Display button will become
the Write... button.)
4. In the Significant Figures field, enter the number of significant figures to be used in the step-by-step report.
5. Click on the Write... button and specify a file name for the stepby-step report file in the resulting Select File dialog box.
A detailed report similar to the following example will be saved to the
specified file before the trajectories are plotted. (You may also choose
to print the report in the console by choosing Console as the Report to
option and clicking on Display or Track, but the report is so long that it
is unlikely to be of use to you in that form.)
19-162
c Fluent Inc. December 3, 2001
19.13 Postprocessing for the Discrete Phase
The step-by-step report lists the particle position and velocity of the
particle at selected time steps along the trajectory:
Time
0.000e+00
3.773e-05
5.403e-05
9.181e-05
1.296e-04
1.608e-04
.
.
.
X-Position
1.411e-03
2.411e-03
2.822e-03
3.822e-03
4.821e-03
5.644e-03
.
.
.
Y-Position
3.200e-03
3.200e-03
3.192e-03
3.192e-03
3.192e-03
3.192e-03
.
.
.
Z-Velocity
0.000e+00
0.000e+00
0.000e+00
0.000e+00
0.000e+00
0.000e+00
.
.
.
X-Velocity
2.650e+01
2.648e+01
2.647e+01
2.644e+01
2.642e+01
2.639e+01
.
.
.
Y-Velocity
0.000e+00
0.000e+00
0.000e+00
0.000e+00
0.000e+00
0.000e+00
.
.
.
Z-Veloc
0.000e
0.000e
0.000e
0.000e
0.000e
0.000e
.
.
.
Also listed are the diameter, temperature, density, and mass of the particle. (You may need to use the scroll bar to view this information.)
elocity
650e+01
648e+01
647e+01
644e+01
642e+01
639e+01
.
.
.
19.13.4
Y-Velocity
0.000e+00
0.000e+00
0.000e+00
0.000e+00
0.000e+00
0.000e+00
.
.
.
Z-Velocity
0.000e+00
0.000e+00
0.000e+00
0.000e+00
0.000e+00
0.000e+00
.
.
.
Diameter
Temperature
2.000e-04
3.000e+02
2.000e-04
3.006e+02
2.000e-04
3.009e+02
2.000e-04
3.015e+02
2.000e-04
3.022e+02
2.000e-04
3.027e+02
.
.
.
.
.
.
Density
1.300e+03
1.300e+03
1.300e+03
1.300e+03
1.300e+03
1.300e+03
.
.
.
Mass
5.445e-09
5.445e-09
5.445e-09
5.445e-09
5.445e-09
5.445e-09
.
.
.
Reporting Current Positions for Unsteady Tracking
When using unsteady tracking, you may want to obtain a report of the
particle trajectory/trajectories showing the current positions of the particles. Selecting Current Positions under Report Type in the ParticleTracks
panel enables the display of the current positions of the particles.
The procedure for generating files containing current position reports is
listed 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, using
the Track Single Particle Stream option.
c Fluent Inc. December 3, 2001
19-163
Discrete Phase Models
2. Select Current Position as the Report Type.
3. Select File as the Report to option. (The Display button will become
the Write... button.)
4. In the Significant Figures field, enter the number of significant figures to be used in the step-by-step report.
5. Click on the Write... button and specify a file name for the current
position report file in the resulting Select File dialog box.
The current position report lists the positions and velocities of all particles that are currently in the domain:
Time
0.000e+00
1.672e-05
3.342e-05
5.010e-05
6.675e-05
8.338e-05
.
.
.
X-Position
1.000e-03
1.168e-03
1.337e-03
1.508e-03
1.680e-03
1.854e-03
.
.
.
Y-Position
3.120e-02
3.128e-02
3.137e-02
3.145e-02
3.153e-02
3.161e-02
.
.
.
Z-Position
0.000e+00
0.000e+00
0.000e+00
0.000e+00
0.000e+00
0.000e+00
.
.
.
X-Velocity
1.000e+01
1.010e+01
1.019e+01
1.028e+01
1.038e+01
1.047e+01
.
.
.
Y-Velocity
5.000e+00
4.988e+00
4.977e+00
4.965e+00
4.954e+00
4.942e+00
.
.
.
Z-Veloc
0.000e
0.000e
0.000e
0.000e
0.000e
0.000e
.
.
.
Also listed are the diameter, temperature, density, and mass of the particles. (You may need to use the scroll bar to view this information.)
elocity
000e+01
010e+01
019e+01
028e+01
038e+01
047e+01
.
.
.
19-164
Y-Velocity
5.000e+00
4.988e+00
4.977e+00
4.965e+00
4.954e+00
4.942e+00
.
.
.
Z-Velocity
0.000e+00
0.000e+00
0.000e+00
0.000e+00
0.000e+00
0.000e+00
.
.
.
Diameter
Temperature
7.000e-05 3.000e+02
7.000e-05 3.009e+02
7.000e-05 3.019e+02
7.000e-05 3.028e+02
7.000e-05 3.037e+02
7.000e-05 3.046e+02
.
.
.
.
.
.
Density
Mass
1.300e+03 2.335e-10
1.300e+03 2.335e-10
1.300e+03 2.335e-10
1.300e+03 2.335e-10
1.300e+03 2.335e-10
1.300e+03 2.335e-10
.
.
.
.
.
.
c Fluent Inc. December 3, 2001
19.13 Postprocessing for the Discrete Phase
19.13.5
Reporting of Interphase Exchange Terms and Discrete
Phase Concentration
FLUENT reports the magnitudes of the interphase exchange of momentum, heat, and mass in each control volume in your FLUENT model. It
can also report the total concentration of the discrete phase. You can
display these variables graphically, by drawing contours, profiles, etc.
They are all contained in the Discrete Phase Model... category of the
variable selection drop-down list that appears in postprocessing panels:
• DPM Concentration
• DPM Mass Source
• DPM X,Y,Z Momentum Source
• DPM Swirl Momentum Source
• DPM Sensible Enthalpy Source
• DPM Enthalpy Source
• DPM Absorption Coefficient
• DPM Emission
• DPM Scattering
• DPM Burnout
• DPM Evaporation/Devolatilization
• DPM (species) Source
• DPM Erosion
• DPM Accretion
See Chapter 27 for definitions of these variables.
Note that these exchange terms are updated and displayed only when
coupled calculations are performed. Displaying and reporting particle
trajectories (as described in Sections 19.13.1 and 19.13.2) will not affect
the values of these exchange terms.
c Fluent Inc. December 3, 2001
19-165
Discrete Phase Models
19.13.6
Trajectory Sampling
Particle states (position, velocity, diameter, temperature, and mass flow
rate) can be written to files at various boundaries and planes (lines in
2D) 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 listed
below:
1. Select the injections to be tracked in the Release From Injections
list.
2. Select the surfaces at which samples will be written. These can be
19-166
c Fluent Inc. December 3, 2001
19.13 Postprocessing for the Discrete Phase
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 particle
tracking, the Compute button will become the Start button (to
initiate sampling) or a Stop button (to stop sampling).
Clicking on the Compute button will cause the particles to be tracked and
their 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 open
the files and write the file header sections. If the solution is advanced
in time by computing some time steps, the particle trajectories will be
updated and the particle states will be written to the files as they cross
the selected planes or boundaries. Clicking on the Stop button will close
the files and end the sampling.
For stochastic tracking, it may be useful to repeat this process multiple
times and append the results to the same file, while monitoring the
sample statistics at each update. To do this, enable the Append Files
option before repeating the calculation (clicking on Compute). Similarly,
you can cause erosion and accretion rates to be accumulated for repeated
trajectory calculations by turning on the Accumulate Erosion/Accretion
Rates option. (See also Section 19.13.8.) The format and the information
written for the sample output can also be controlled through a userdefined function, which can be selected in the Output drop-down list. See
the 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 Trajectories panel (as described in Section 19.13.6) using the Trajectory Sample
Histograms panel (Figure 19.13.3).
Report −→ Discrete Phase −→Histogram...
The procedure for plotting histograms from data in a sample file is listed
below:
c Fluent Inc. December 3, 2001
19-167
Discrete Phase Models
Figure 19.13.3: The Trajectory Sample Histograms Panel
1. Select a file to be read by clicking on the Read... button. After
you read in the sample file, the boundary name will appear in the
Sample list.
2. Select the data sample in the Sample list, and then select the data
to be plotted from the Fields list.
3. Click on the Plot button at the bottom of the panel to display the
histogram.
By default, the percent of particles will be plotted on the y axis. You can
plot the actual number of particles by deselecting Percent under Options.
The number of “bins” or intervals in the plot can be set in the Divisions
field. You can delete samples from the list with the Delete button and
update the Min/Max values with the Compute button.
19-168
c Fluent Inc. December 3, 2001
19.13 Postprocessing for the Discrete Phase
19.13.8
Postprocessing of Erosion/Accretion Rates
You can calculate the erosion and accretion rates in a cumulative manner (over a series of injections) by using the Sample Trajectories panel.
First select an injection in the Release From Injections list and compute
its trajectory. Then turn on the Accumulate Erosion/Accretion Rates option, select the next injection (after deselecting the first one), and click
Compute again. The rates will accumulate at the surfaces each time you
click Compute.
! Both the erosion rate and the accretion rate are defined at wall face
surfaces only, so they cannot be displayed at node values.
c Fluent Inc. December 3, 2001
19-169
Discrete Phase Models
19-170
c Fluent Inc. December 3, 2001