This chapter describes the Lagrangian discrete phase capabilities available. Information is organized into the following sections:

Transcription

1 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,

2 Discrete Phase Models 19.1 Overview and Limitations of the Discrete Phase Models Introduction In addition to solving transport equations for the continuous phase, FLU- ENT 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 , and 19-2 c Fluent Inc. December 3, 2001

3 19.1 Overview and Limitations of the Discrete Phase Models instructions for setup, solution, and postprocessing are provided in Sections 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 ). 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 ). The particle cloud model tracks the statistical evolution of a cloud of particles about a mean trajectory (see Section ). 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 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,

4 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

5 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 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,

6 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 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 du p dt = F D(u u p )+ g x(ρ p ρ) ρ p + F x (19.2-1) where F D (u u p ) is the drag force per unit particle mass and F D = 18µ ρ p d 2 p C D Re 24 (19.2-2) Here, u is the fluid phase velocity, u p is the particle velocity, µ is the molecular viscosity of the fluid, ρ is the fluid density, ρ p is the density of 19-6 c Fluent Inc. December 3, 2001

7 19.2 Trajectory Calculations the particle, and d p is the particle diameter. Re is the relative Reynolds number, which is defined as Re ρd p u p u µ (19.2-3) The drag coefficient, C D,canbetakenfromeither C D = a 1 + a 2 Re + a 3 Re 2 (19.2-4) where a 1, a 2,anda 3 are constants that apply for smooth spherical particles over several ranges of Re given by Morsi and Alexander [163], or C D = 24 ( ) 1+b 1 Re b 2 + b 3Re Re b 4 +Re (19.2-5) where b 1 = exp( φ φ 2 ) b 2 = φ b 3 = exp( φ φ φ 3 ) b 4 = exp( φ φ φ 3 )(19.2-6) which is taken from Haider and Levenspiel [85]. The shape factor, φ, is defined as φ = s (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, F D is defined as F D = 18µ d p 2 ρ p C c (19.2-8) c Fluent Inc. December 3,

8 Discrete Phase Models The factor C c is the Cunningham correction to Stokes drag law, which you can compute from C c =1+ 2λ d p ( e (1.1dp/2λ) ) (19.2-9) where λ is the molecular mean free path. A high-mach-number drag law is also available. This drag law is similar to the spherical law (Equation ) 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 for a description of this law. Instructions for selecting the drag law are provided in Section Including the Gravity Term While Equation 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 incorporates additional forces (F x ) 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 F x = 1 ρ d 2 ρ p dt (u u p) ( ) and is important when ρ>ρ p. An additional force arises due to the pressure gradient in the fluid: 19-8 c Fluent Inc. December 3, 2001

9 19.2 Trajectory Calculations F x = ( ) ρ u u p x ρ p ( ) Forces in Rotating Reference Frames The additional force term, F x, in Equation 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 (1 ρρp ) Ω 2 x +2Ω (u y,p ρρp u y ) ( ) where u y,p and u y are the particle and fluid velocities in the Cartesian y direction, and (1 ρρp ) Ω 2 y 2Ω (u x,p ρρp u x ) ( ) where u x,p and u x 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, F x, in Equation : 1 T F x = D T,p m p T x ( ) c Fluent Inc. December 3,

10 Discrete Phase Models where D T,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]: 6πd p µ 2 C s (K + C t Kn) 1 F x = ρ(1 + 3C m Kn)(1 + 2K +2C t Kn) m p T T x ( ) where: Kn = Knudsen number = 2 λ/d p λ = mean free path of the fluid K = k/k p k = fluid thermal conductivity based on translational energy only = (15/4) µr k p = particle thermal conductivity C S =1.17 C t =2.18 C m =1.14 m p = particle mass T = local fluid temperature µ = 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 S n,ij given by [135] where δ ij is the Kronecker delta function, and S n,ij = S 0 δ ij ( ) S 0 = 216νσT ( ) π 2 ρd 5 ρp 2 ( ) p ρ Cc c Fluent Inc. December 3, 2001

11 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 F bi = ζ i πs o t ( ) where ζ i 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 = 2Kν1/2 ρd ij ρ p d p (d lk d kl ) 1/4 ( v v p) ( ) where K =2.594 and d ij 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 ). Optionally, you can include the instantaneous value of the fluctuating gas flow velocity, c Fluent Inc. December 3,

12 Discrete Phase Models u = u + u ( ) 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 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 ). 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 yields the velocity of the particle at each point along the trajectory, with the trajectory itself predicted by dx dt = u p ( ) Equations similar to and 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 du p dt = 1 τ p (u u p ) ( ) c Fluent Inc. December 3, 2001

13 19.2 Trajectory Calculations where τ p is the particle relaxation time. scheme for integrating Equation : FLUENT uses a trapezoidal u n+1 p u n p t = 1 τ (u u n+1 p )+... ( ) where n represents the iteration number and u = 1 2 (un + u n+1 ) ( ) u n+1 = u n + tu n p un ( ) Equations and 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 and ) 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 ) 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 Y d = e (d/ d) n ( ) where d is the size constant and n is the size distribution parameter. Use of the Rosin-Rammler size distribution is detailed in Section c Fluent Inc. December 3,

14 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 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 particles, using the instantaneous fluid velocity, u + u (t), along the particle path during the integration. By computing the trajectory in this c Fluent Inc. December 3, 2001

15 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: T = 0 u p (t)u p (t + s) ds ( ) u p 2 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 u i u j T. For small tracer particles that move with the fluid (zero drift velocity), the integral time becomes the fluid Lagrangian integral time, T L. This time scale can be approximated as T L = C L k ɛ ( ) where C L is to be determined and is not well known. By matching the diffusivity of tracer particles, u i u j T L, to the scalar diffusion rate predicted by the turbulence model, ν t /σ, one can obtain T L 0.15 k ɛ ( ) c Fluent Inc. December 3,

16 Discrete Phase Models for the k-ɛ model and its variants, and T L 0.30 k ɛ ( ) when the Reynolds stress model (RSM) is used [48]. For the k-ω models, substitute ω = ɛ/k into Equation 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, u, v,andw a time scale, τ e The values of u, v,andw that prevail during the lifetime of the turbulent eddy are sampled by assuming that they obey a Gaussian probability distribution, so that u = ζ u 2 ( ) 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 u 2 = v 2 = w 2 = 2k/3 ( ) for the k-ɛ 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: c Fluent Inc. December 3, 2001

17 19.2 Trajectory Calculations u = ζ u 2 ( ) v = ζ v 2 ( ) w = ζ w 2 ( ) 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 for details. The characteristic lifetime of the eddy is defined either as a constant: τ e =2T L ( ) where T L is given by Equation in general (Equation by default), or as a random variation about T L : τ e = T L log(r) ( ) where r is a uniform random number between 0 and 1 and T L is given by Equation The option of random calculation of τ e yields a more realistic description of the correlation function. The particle eddy crossing time is defined as [ ( )] L e t cross = τ ln 1 τ u u p ( ) where τ is the particle relaxation time, L e is the eddy length scale, and u u p 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 c Fluent Inc. December 3,

18 Discrete Phase Models UsingtheDRWModel! The only inputs required for the DRW model are the value for the integral time-scale constant, C L (see Equations and ) 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 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 x i 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 n(x i ) = ṁp (x i,t) ( ) The PDFs for particle position are assumed to be multivariate Gaussian. These are completely described by their mean, µ i, and variance, σ i 2,and are of the form c Fluent Inc. December 3, 2001

19 19.2 Trajectory Calculations P (x i,t)= 1 (8π) 3/2 3 i=1 (s e 2 /2) σ i ( ) where s = 3 i=1 x i µ i σ i ( ) 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: µ i (t) x i (t) = t 0 V i (t 1 ) dt 1 + x i (0) ( ) 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, σi 2 (t), of the PDF can be expressed in terms of two particle turbulence statistical quantities: σ 2 i (t) =2 t 0 t2 u 2 p,i (t 2) 0 R p,ii (t 2,t 1 )dt 1 dt 2 ( ) where u 2 p,i are the mean square velocity fluctuations, and R p,ij (t 2,t 1 ) is the particle velocity correlation function: R p,ij (t 2,t 1 )= u p,i (t 2)u p,j (t 1) [ ] 1/2 ( ) u 2 p,i (t 2)u 2 p,j (t 2) By using the substitution τ = t 2 t 1,andthefactthat R p,ij (t 2,t 1 )=R p,ij (t 4,t 3 ) ( ) c Fluent Inc. December 3,

20 Discrete Phase Models whenever t 2 t 1 = t 4 t 3, we can write t t2 σi 2 (t) =2 u 2 p,i (t 2) R p,ii (τ)dτdt 2 ( ) 0 0 Note that cross correlations in the definition of the variance (R p,ij,i 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, R ij takes the form: where B = particle: R p,11 = u 2 p θ e (τ/τa) + u 2 θ e (τb/t) St T ( ( B 0.5m T γ St2 T B2 +1 θ 1+ m T St 2 T γb θ ) +0.5m T γ τ T ( ) R p,22 = R p,11 ( ) R p,33 = u 2 St T B e (τ/τa) u 2 θ θ e (τb/t) ( ) 1+m 2 T γ2 and τ a is the aerodynamic response time of the ) τ a = ρ pd 2 p 18µ ( ) and T = m T T me m ( ) T fe = C3/4 µ k 3/2 ɛ( 2 3 k)1/2 ( ) c Fluent Inc. December 3, 2001

21 19.2 Trajectory Calculations γ = τ ag u ( ) St = τ a ( ) T me St T = τ a ( ) T θ = St 2 T (1 + m2 T γ2 ) 1 ( ) m = ū u ( ) ū T me = T fe u ( ) [ ] G(m) m T = m 1 (1 + St) 0.4(1+0.01St) ( ) G(m) = 2 e y2 dy ( π 0 ( 1+ m2 π π erf(y)y 1+e y 2)) ( ) 5/2 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 W (x i,t) P (x i,t)dv V cell P (x i,t)dv V cloud ( ) 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,

22 Discrete Phase Models! the Set Injection Properties panel for each injection, as described in Section The cloud model is not available for unsteady tracking Particle Erosion and Accretion Particle erosion and accretion rates can be monitored at wall boundaries. The erosion rate is defined as R erosion = N particles p=1 ṁ p C(d p )f(α)v b(v) A face ( ) where C(d p ) is a function of particle diameter, α istheimpactangleof 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,andb =0. Since C, f, andb 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 R accretion = N particles p=1 ṁ p A face ( ) c Fluent Inc. December 3, 2001

23 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 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,andFLUENT 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 Description Laws Activated Inert inert/heating or cooling 1, 6 Droplet heating/evaporation/boiling 1, 2, 3, 6 Combusting heating; evolution of volatiles/swelling; heterogeneous surface reaction 1, 4, 5, 6 In addition to the above laws, you can define your own laws using a 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 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,

24 Discrete Phase Models define, T vap, and after the volatile fraction, f v,0, of a particle has been consumed. These conditions may be written as Law 1: Law 6: T p <T vap (19.3-1) m p (1 f v,0 )m p,0 (19.3-2) where T p is the particle temperature, m p,0 is the initial mass of the particle, and m p 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, T vap, is thus an arbitrary modeling constant used to define the onset of the vaporization/boiling/volatilization laws.) WhenusingLaw1orLaw6,FLUENT uses a simple heat balance to relate the particle temperature, T p (t), to the convective heat transfer and the absorption/emission of radiation at the particle surface: where m p c p dt p dt = ha p(t T p )+ɛ p A p σ(θ 4 R T 4 p ) (19.3-3) m p = mass of the particle (kg) c p = heat capacity of the particle (J/kg-K) A p = surface area of the particle (m 2 ) T = local temperature of the continuous phase (K) h = convective heat transfer coefficient (W/m 2 -K) ɛ p = particle emissivity (dimensionless) σ = Stefan-Boltzmann constant (5.67 x 10 8 W/m 2 -K 4 ) θ R = radiation temperature, ( G 4σ )1/4 Equation assumes that there is negligible internal resistance to heat transfer, i.e., the particle is at uniform temperature throughout c Fluent Inc. December 3, 2001

25 19.3 Heat and Mass Transfer Calculations G is the incident radiation in W/m 2 : G = Ω=4π IdΩ (19.3-4) where I is the radiation intensity and Ω is the solid angle. Radiation heat transfer to the particle is included only if you have 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 is integrated in time using an approximate, linearized form that assumes that the particle temperature changes slowly from one time value to the next: m p c p dt p dt [ ] [ ]} = A p { h + ɛ p σtp 3 T p + ht + ɛ p σθr 4 (19.3-5) As the particle trajectory is computed, FLUENT integrates Equation to obtain the particle temperature at the next time value, yielding T p (t + t) =α p +[T p (t) α p ]e βp t (19.3-6) where t is the integration time step and α p = ht + ɛ p σθ 4 R h + ɛ p σt 3 p (t) (19.3-7) and β p = A p(h + ɛ p σt 3 p (t)) m p c p (19.3-8) FLUENT can also solve Equation in conjunction with the equivalent mass transfer equation using a stiff coupled solver. See Section for details. c Fluent Inc. December 3,

26 Discrete Phase Models The heat transfer coefficient, h, is evaluated using the correlation of Ranz and Marshall [185, 186]: where Nu = hd p k = Re 1/2 d Pr 1/3 (19.3-9) d p = particle diameter (m) k = thermal conductivity of the continuous phase (W/m-K) Re d = Reynolds number based on the particle diameter and the relative velocity (Equation ) Pr = Prandtl number of the continuous phase (c p 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 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, T vap, and continues until the droplet reaches the boiling point, T bp, or until the droplet s volatile fraction is completely consumed: T p <T bp ( ) m p > (1 f v,0 )m p,0 ( ) The onset of the vaporization law is determined by the setting of T vap, 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 T vap c Fluent Inc. December 3, 2001

27 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 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: N i = k c (C i,s C i, ) ( ) where N i = molar flux of vapor (kgmol/m 2 -s) k c = mass transfer coefficient (m/s) C i,s = vapor concentration at the droplet surface (kgmol/m 3 ) C i, = vapor concentration in the bulk gas (kgmol/m 3 ) Note that FLUENT s vaporization law assumes that N i is positive (evaporation). If conditions exist in which N i is negative (i.e., the droplet temperature falls below the dew point and condensation conditions exist), FLUENT treats the droplet as inert (N i =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, p sat, at the particle droplet temperature, T p : where R is the universal gas constant. C i,s = p sat(t p ) RT p ( ) 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,

28 Discrete Phase Models p op C i, = X i ( ) RT where X i is the local bulk mole fraction of species i, p op is the operating pressure, and T is the local bulk temperature in the gas. The mass transfer coefficient in Equation is calculated from a Nusselt correlation [185, 186]: Nu AB = k cd p D i,m = Re 1/2 d Sc 1/3 ( ) where D i,m = diffusion coefficient of vapor in the bulk (m 2 /s) µ Sc = the Schmidt number, ρd i,m d p = particle (droplet) diameter (m) The vapor flux given by Equation 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 m p (t + t) =m p (t) N i A p M w,i t ( ) where M w,i = molecular weight of species i (kg/kgmol) m p = mass of the droplet (kg) A p = surface area of the droplet (m 2 ) FLUENT can also solve Equation in conjunction with the equivalent heat transfer equation using a stiff coupled solver. See Section for details. Defining the Vapor Pressure and Diffusion Coefficient You define the vapor pressure as a polynomial or piecewise linear function of temperature (p sat (T )) during the problem definition. Note that the vapor pressure definition is critical, as p sat is used to obtain the driving c Fluent Inc. December 3, 2001

29 19.3 Heat and Mass Transfer Calculations force for the evaporation process (Equations and ). 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, D i,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: m p c p dt p dt = ha p (T T p )+ dm p dt h fg + A p ɛ p σ(θ R 4 T p 4 ) ( ) where c p = droplet heat capacity (J/kg-K) T p = droplet temperature (K) h = convective heat transfer coefficient (W/m 2 -K) T = temperature of continuous phase (K) dm p dt = rate of evaporation (kg/s) h fg = latent heat (J/kg) ɛ p = particle emissivity (dimensionless) σ = Stefan-Boltzmann constant (5.67 x 10 8 W/m 2 -K 4 ) θ R = radiation temperature, ( I 4σ )1/4,whereI 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,

30 Discrete Phase Models equation 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, T bp, and while the mass of the droplet exceeds the nonvolatile fraction, (1 f v,0 ): and T p T bp ( ) m p > (1 f v,0 )m p,0 ( ) When the droplet temperature reaches the boiling point, a boiling rate equation is applied [120]: d(d p ) dt = 4k ( [ Re d )ln 1+ c ] p, (T T p ) ρ p c p, d p h fg ( ) where c p, = heat capacity of the gas (J/kg-K) ρ p = droplet density (kg/m 3 ) k = thermal conductivity of the gas (W/m-K) Equation has been derived assuming steady flow at constant pressure. Note that the model requires T >T bp in order for boiling to occur and that the droplet remains at fixed temperature (T bp ) throughout the boiling law. When radiation heat transfer is active, FLUENT uses a slight modification of Equation , derived by starting from Equation and assuming that the droplet temperature is constant. This yields dm p dt h fg = ha p (T T p )+A p ɛ p σ(θ R 4 T p 4 ) ( ) c Fluent Inc. December 3, 2001

31 19.3 Heat and Mass Transfer Calculations or d(d p) dt = 2 [ ] k Nu (T T p )+ɛ p σ(θr 4 Tp 4 ) ρ p h fg d p ( ) Using Equation for the Nusselt number correlation and replacing the Prandtl number term with an empirical constant, Equation becomes d(d p) dt = 2 [ 2k [ ] Re d ] (T T p )+ɛ p σ(θr 4 ρ p h fg d T p 4 ) p ( ) In the absence of radiation, this result matches that of Equation in the limit that the argument of the logarithm is close to unity. FLU- ENT uses Equation when radiation is active in your model and Equation 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) Law 4: Devolatilization The devolatilization law is applied to a combusting particle when the temperature of the particle reaches the vaporization temperature, T vap, and remains in effect while the mass of the particle, m p, exceeds the mass of the non-volatiles in the particle: T p T vap and T p T bp ( ) c Fluent Inc. December 3,

32 Discrete Phase Models and m p > (1 f v,0 )(1 f w,0 )m p,0 ( ) where f w,0 is the mass fraction of the evaporating /boiling material if Wet Combustion is selected (otherwise, f w,0 = 0). As implied by Equation , the boiling point T bp and the vaporization temperature T vap should be set equal to each other when Law 4 is to be used. When wet combustion is active, T bp and T vap 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 By default, the constant rate model (Equation ) will be used. The Constant Rate Devolatilization Model The constant rate devolatilization law dictates that volatiles are released at a constant rate [13]: 1 dm p f v,0 (1 f w,0 )m p,0 dt = A 0 ( ) c Fluent Inc. December 3, 2001

33 19.3 Heat and Mass Transfer Calculations where m p = particle mass (kg) f v,0 = fraction of volatiles initially present in the particle m p,0 = initial particle mass (kg) A 0 = rate constant (s 1 ) The rate constant A 0 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]: dm p dt = k[m p (1 f v,0 )(1 f w,0 )m p,0 ] ( ) where m p = particle mass (kg) f v,0 = mass fraction of volatiles initially present in the particle f w,0 = mass fraction of evaporating/boiling material (if wet combustion is modeled) m p,0 = initial particle mass (kg) k = kineticrate(s 1 ) Note that f v,0, the fraction of volatiles in the particle, should be defined using a value slightly in excess of that determined by proximate analysis. The kinetic rate, k, is defined by input of an Arrhenius type pre-exponential factor and an activation energy: c Fluent Inc. December 3,

34 Discrete Phase Models k = A 1 e (E/RT) ( ) FLUENT uses default rate constants, A 1 and E, as given in [5]. Equation has the approximate analytical solution: m p (t + t) =(1 f v,0 )(1 f w,0 )m p,0 + [m p (t) (1 f v,0 )(1 f w,0 )m p,0 ]e k t ( ) which is obtained by assuming that the particle temperature varies only slightly between discrete time integration steps. FLUENT can also solve Equation in conjunction with the equivalent heat transfer equation using a stiff coupled solver. See Section for details. The Two Competing Rates Kobayashi Model FLUENT also provides the kinetic devolatilization rate expressions of the form proposed by Kobayashi [117]: R 1 = A 1 e (E 1/RT p) ( ) R 2 = A 2 e (E 2/RT p) ( ) where R 1 and R 2 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 m v (t) t ( t ) = (α 1 R 1 + α 2 R 2 )exp (R 1 + R 2 ) dt dt (1 f w,0 )m p,0 m a 0 0 ( ) c Fluent Inc. December 3, 2001

35 19.3 Heat and Mass Transfer Calculations where m v (t) = volatile yield up to time t m p,0 = initial particle mass at injection α 1,α 2 = yield factors m a = ash content in the particle The Kobayashi model requires input of the kinetic rate parameters, A 1, E 1, A 2,andE 2, and the yields of the two competing reactions, α 1 and α 2. 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 α 1 be set to the fraction of volatiles determined by proximate analysis, since this rate represents devolatilization at low temperature. The second yield parameter, α 2, should be set close to unity, which is the yield of volatiles at very high temperature. By default, Equation is integrated in time analytically, assuming the particle temperature to be constant over the discrete time integration step. FLUENT can also solve Equation in conjunction with the equivalent heat transfer equation using a stiff coupled solver. See Section 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,

36 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 k g k δ 2 δ 2g 1 k b * k c c+ 2g 2 Figure : 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, g 1.Asbridges 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, g 2. The total population of bridges in the coal lattice matrix can be c Fluent Inc. December 3, 2001

37 19.3 Heat and Mass Transfer Calculations represented by the variable p, wherep = + 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 dg 1 dt dg 2 dt = k b ( ) = k b ρ +1 = [ 2ρk b ρ +1 ( ) ] k g δ ( ) = k g δ ( ) = 2 dc dt ( ) where the rate constants for bridge breaking and gas release steps, k b and k g, are expressed in Arrhenius form with a distributed activation energy: k = Ae (E±Eσ)/RT ( ) where A, E, ande σ 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 = k δ /k c, 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,

38 Discrete Phase Models g = g 1 + g 2 ( ) g 1 = 2f σ ( ) g 2 = 2(c c 0 ) ( ) where f is the fraction of broken bridges (f =1 p). The initial conditions for this system are given by the following: c(0) = c 0 ( ) (0) = 0 = p 0 c 0 ( ) δ(0) = 2f 0 =2(1 c 0 0 ) ( ) g(0) = g 1 (0) = g 2 (0) = 0 ( ) where c 0 is the initial fraction of char bridges, p 0 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 (f gas ), tar precursor fragments (f frag ), and char (f char ). This is accomplished by using the following relationships, which are obtained using percolation lattice statistics: r(g 1 + g 2 )(σ +1) f gas (t) = ( ) 4+2r(1 c 0 )(σ +1) 2 f frag (t) = [ΦF (p)+rωk(p)] ( ) 2+r(1 c 0 )(σ +1) f char (t) = 1 f gas (t) f frag (t) ( ) c Fluent Inc. December 3, 2001

39 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: Φ = [ 1+r p Ω = δ 2(1 p) p F (p) = ( p ) σ+1 σ 1 p K(p) = [ ( ) ]( σ +1 p 1 p 2 p ] (σ 1)δ + 4(1 p) ) σ+1 σ 1 ( ) ( ) ( ) ( ) r is the ratio of bridge mass to site mass, m b /m a,where m b = 2M w,δ ( ) m a = M w,1 (σ +1)M w,δ ( ) where M w,δ and M w,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 p is the root of the following equation in p (the total number of bridges in the coal lattice matrix): p (1 p ) σ 1 = p(1 p) σ 1 ( ) 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,

40 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: dm cross dt = m frag A cross e (Ecross/RT ) ( ) where m cross is the amount of mass reattaching to the matrix, m frag is the amount of mass in the tar precursor fragments (metaplast), and A cross and E cross 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]: A b, E b, E σb, A g, E g,ande σg for the rate constants k b and k g A cross, E cross,andρ 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, p 0 initial fraction of char bridges, c 0 lattice coordination number, σ + 1 cluster molecular weight, M w,1 side chain molecular weight, M w,δ c Fluent Inc. December 3, 2001

41 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 Values for the coal-dependent parameters for a variety of coals are listed in Table Table : Chemical Structure Parameters for 13 CNMRfor13Coals Coal Type σ +1 p 0 M w,1 M w,δ c 0 Zap (AR) Wyodak (AR) Utah (AR) Ill6 (AR) Pitt8 (AR) Stockton (AR) Freeport (AR) Pocahontas (AR) Blue (Sandia) Rose (AFR) (lignite, ACERC) (subbituminous, ACERC) (anthracite, ACERC) 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, C sw, which is defined by you and applied in the following relationship: c Fluent Inc. December 3,

42 Discrete Phase Models d p d p,0 =1+(C sw 1) (1 f w,0)m p,0 m p f v,0 (1 f w,0 )m p,0 ( ) where d p,0 = particle diameter at the start of devolatilization d p = current particle diameter The term (1 f w,0)m p,0 m p f v,0 (1 f w,0 )m p,0 is the ratio of the mass that has been devolatilized to the total volatile mass of the particle. This quantity approaches a value of 1.0 as the 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: m p c p dt p dt = ha p (T T p )+ dm p dt h fg + A p ɛ p σ(θ R 4 T p 4 ) ( ) 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 is solved analytically, by assuming that the temperature and mass of the particle do not change significantly between time steps: where T p (t + t) =α p +[T p (t) α p ]e βpt ( ) c Fluent Inc. December 3, 2001

43 19.3 Heat and Mass Transfer Calculations α p = ha pt + dmp dt h fg + A p ɛ p σθ R 4 ha p + ɛ p A p σt p 3 ( ) and β p = A p(h + ɛ p σt p 3 ) m p c p ( ) FLUENT can also solve Equation in conjunction with the equivalent mass transfer equation using a stiff coupled solver. See Section for details Law 5: Surface Combustion After the volatile component of the particle is completely evolved, a surface reaction begins, which consumes the combustible fraction, f comb, of the particle. Law 5 is thus active (for a combusting particle) after the volatiles are evolved: m p < (1 f v,0 )(1 f w,0 )m p,0 ( ) and until the combustible fraction is consumed: m p > [(1 f v,0 )(1 f w,0 ) f comb ]m p,0 ( ) When the combustible fraction, f comb, has been consumed in Law 5, the combusting particle may contain residual ash that reverts to the inert heating law, Law 6 (see Section ). 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, S b, of the surface burnout reaction: char(s) + S b ox(g) products(g) ( ) c Fluent Inc. December 3,

44 Discrete Phase Models where S b 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 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: dm p dt = 4πd p D i,m Y ox T ρ g S b (T p + T ) ( ) where D i,m = diffusion coefficient for oxidant in the bulk (m 2 /s) Y ox = local mass fraction of oxidant in the gas ρ g = gas density (kg/m 3 ) S b = stoichiometry of Equation Equation 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 c Fluent Inc. December 3, 2001

45 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 and a kinetic rate D 0 = C 1 [(T p + T )/2] 0.75 d p ( ) are weighted to yield a char combustion rate of R = C 2 e (E/RTp) ( ) dm p dt = πd 2 pp ox D 0 R D 0 + R ( ) where p ox 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 is recast in terms of the oxidant mass fraction, Y ox,as dm p dt = πd 2 ρrt Y ox p M w,ox D 0 R D 0 + R ( ) 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 and are entered in the Materials panel, as described in Section c Fluent Inc. December 3,

46 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 ). The intrinsic model uses Equation to compute the diffusion rate coefficient, D 0, but the chemical rate, R, is explicitly expressed in terms of the intrinsic chemical and pore diffusion rates: R = η d p 6 ρ pa g k i ( ) η is the effectiveness factor, or the ratio of the actual combustion rate to the rate attainable if no pore diffusion resistance existed [130]: where φ is the Thiele modulus: η = 3 (φ coth φ 1) ( ) φ2 φ = d p 2 [ ] Sb ρ p A g k i p 1/2 ox ( ) D e C ox C ox is the concentration of oxidant in the bulk gas (kg/m 3 )andd e 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, D e is given by D e = θ [ 1 τ ] 1 ( ) D Kn D 0 where D 0 is the bulk molecular diffusion coefficient and θ is the porosity of the char particle: θ =1 ρ p ρ t ( ) c Fluent Inc. December 3, 2001

47 19.3 Heat and Mass Transfer Calculations ρ p and ρ t are, respectively, the apparent and true densities of the pyrolysis char. τ (in Equation ) 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]. D Kn is the Knudsen diffusion coefficient: D Kn =97.0r p Tp M w,ox ( ) where T p is the particle temperature and r p 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 (r p < 10 Å) dominate in high-rank chars [130]. A g (in Equations and ) 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 m 2 /g [33]. k i (in Equations and ) is the intrinsic reactivity, which is of Arrhenius form: k i = A i e (E i/rt p) ( ) where the pre-exponential factor A i and the activation energy E i 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,

48 Discrete Phase Models α, relating the char particle diameter to the fractional degree of burnout U (where U =1 m p /m p,0 ) by [216] d p d p,0 =(1 U) α ( ) where m p 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 , , , , , , and are entered in the Materials panel, as described in Section 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, c Fluent Inc. December 3, 2001

49 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 m p c p dt p dt = ha p (T T p ) f h dm p dt H reac + A p ɛ p σ(θ R 4 T p 4 ) ( ) where H reac is the heat released by the surface reaction. Note that only a portion (1 f h ) of the energy produced by the surface reaction appears as a heat source in the gas-phase energy equation: the particle absorbs a fraction f h of this heat directly. For coal combustion, it is recommended that f h be set at 1.0 if the char burnout product is CO and 0.3 if the char burnout product is CO 2 [24]. c Fluent Inc. December 3,

50 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 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 in conjunction with the equivalent mass transfer equation using a stiff coupled solver. See Section for details 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) + 3Cl 2 (g) 2Al 2 Cl 3 (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 c Fluent Inc. December 3, 2001

51 19.4 Spray Models 19.4 Spray Models In addition to the simple injection types described in Section , 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 : Atomizer Models Section : Droplet Collision Model Section : Spray Breakup Models Section : Dynamic Drag Model 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 Details about the atomizer models are provided below. c Fluent Inc. December 3,

52 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 , , and c Fluent Inc. December 3, 2001

53 19.4 Spray Models r d p p 1 2 liquid jet orifice walls downstream gas L Figure : Single-Phase Nozzle Flow (Liquid completely fills the orifice.) vapor liquid jet vapor orifice walls downstream gas Figure : Cavitating Nozzle Flow (Vapor pockets form just after the inlet corners.) c Fluent Inc. December 3,

54 Discrete Phase Models liquid jet orifice walls downstream gas Figure : 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 Table : List of Governing Parameters for Internal Nozzle Flow nozzle diameter d nozzle length L radius of curvature of the inlet corner r upstream pressure p 1 downstream pressure p 2 viscosity µ liquid density ρ l vapor pressure p v 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 (Re h ) c Fluent Inc. December 3, 2001

55 19.4 Spray Models and a cavitation parameter (K). Re h = dρ l µ 2(p 1 p 2 ) ρ l (19.4-1) K = p 1 p v p 1 p 2 (19.4-2) The liquid flow often contracts in the nozzle, as can be seen in Figures and Nurick [166] found it helpful to use a coefficient of contraction (C c ) 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: C c = 1 1 C ct 11.4r d (19.4-3) C ct 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 (C d ). The coefficient of discharge is a ratio of the mass flow rate through the nozzle, divided by the theoretical maximum mass flow rate: C d = ṁ A 2ρ l (p 1 p 2 ) (19.4-4) The cavitation number (K in Equation ) is an essential parameter for predicting the inception of cavitation. The inception of cavitation is known to occur at a value of K incep 1.9 for short, sharp-edged nozzles. However, to include some of the effects of inlet rounding and viscosity, an empirical relationship is used: ( K incep =1.9 1 r ) (19.4-5) d Re h c Fluent Inc. December 3,

56 Discrete Phase Models Similarly, a critical value of K where flip occurs is defined as K crit : 1 K crit =1+ ( )( ) (19.4-6) 1+ L 4d Re h e 70r/d If r/d is greater than 0.05, then flip is deemed impossible and K crit 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 Depending on the state of the nozzle, a unique closure is chosen for the above equations. For a single-phase nozzle [137], C du = L d (19.4-7) C d = 1 1 C du +20 (1+2.25L/d) Re h (19.4-8) Equation is for the ultimate coefficient of discharge, C du. Equation corrects this ultimate coefficient of discharge for the effects of viscosity. For a cavitating nozzle [166], For a flipped nozzle [166], C d = C c K (19.4-9) C d = C ct ( ) 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 c Fluent Inc. December 3, 2001

57 19.4 Spray Models K K incep K > K incep K < K K K K <K K K crit crit crit crit flipped cavitating flipped single phase Figure : 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 ρ l A ( ) 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 = 2C cp 1 p 2 +(1 2C c )p v C c 2ρl (p 1 p v ) ( ) 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 = ṁ ρ l C ct A ( ) c Fluent Inc. December 3,

58 Discrete Phase Models Spray Angle The correlation for spray angle (θ) comes from the work of Ranz [184]: θ 2 θ 2 [ 4π = tan 1 ρg C A ρ l ] 3 6 ( ) = 0.01 ( ) Equation describes the spray angle for both single-phase and cavitating nozzles. For flipped nozzles, the spray angle has a constant value (Equation ). C A is thought to be a constant for a given nozzle geometry. You must choose the value for C A. The larger the value, the narrower the spray. Reitz [189] suggests the following correlation for C A : C A =3+ L 3.6d ( ) The spray angle is sensitive to the internal flow regime of the nozzle. Hence, you may wish to choose smaller values of C A 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 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: c Fluent Inc. December 3, 2001

59 19.4 Spray Models d 32 = 133.0λWe 0.74 ( ) where d 32 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 lu 2 λ σ ( ) where λ = d/8 andσ is the droplet surface tension. For a more detailed discussion of droplet surface tension and the Weber number, see Section For cavitating nozzles, FLUENT uses a slight modification to Equation 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 = d eff /8, where d eff = 4ṁ πρ l u ( ) For the case of the flipped nozzle, the initial droplet diameter is set to the diameter of the liquid jet: d 0 = d C ct ( ) where d 0 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 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,

60 Discrete Phase Models Table : Values of Spread Parameter for Different Nozzle States State Spread Parameter single phase 3.5 cavitating 1.5 flipped Since the correlations of Wu et al. provide the Sauter mean diameter, d 32, these must be converted to the most probable diameter, d 0. 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: d 0 =1.2726d 32 ( 1 1 s ) 1/s ( ) 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 The interaction between the air and the sheet is not well understood. It is generally accepted that an aerodynamic instability causes the sheet c Fluent Inc. December 3, 2001

61 19.4 Spray Models film formation sheet breakup atomization Figure : Theoretical Progression from the Internal Atomizer Flow to the External Spray to break up. The mathematical analysis below assumes that Kelvin- Helmholtz 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,

62 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(d inj t) ( ) where d inj is the injector exit diameter, and ṁ is the mass flow rate, which must be measured experimentally. The other unknown in Equation 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 U = k v 2 p ρ l ( ) Lefebvre [132] has noted that k v is a function of the injector design and injection pressure. If the swirl ports are treated as nozzles, Equation 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 (C d )forsinglephase 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 C d may be as low as Hence, 0.78 should be a practical upper bound for k v. Reducing k v by 10% to allow for other momentum losses in the injector gives an estimate of 0.7. Physical limits on k v 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 k v : [ ] 4ṁ ρl k v =max 0.7, πd 2 0 ρ l cos θ 2 p ( ) c Fluent Inc. December 3, 2001

63 19.4 Spray Models Assuming that p is known, Equation can be used to find U. Once U is determined, u is found from u = U cos θ ( ) 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 ρ l and ρ g, respectively, and the viscosity of the liquid is µ l. A coordinate system is used that moves with the sheet, and a spectrum of infinitesimal disturbances of the form η = η 0 e ikx+ωt ( ) is imposed on the initially steady, motion-producing fluctuating velocities and pressures for both the liquid and the gas. In Equation η 0 is the initial wave amplitude, k =2π/λ is the wave number, and ω = ω r + iω i is the complex growth rate. The most unstable disturbance has the largest value of ω r, 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,

64 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]+[4ν l k 2 tanh(kh)+2iqku]+ 4ν l k 4 tanh(kh) 4ν 2 l k 3 l tanh(lh) QU 2 k 2 + σk3 ρ l = 0 ( ) where Q = ρ g /ρ l and l 2 = k 2 + ω/ν l. It can be shown that above a critical Weber number of We g = 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 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 ω r = { 1 2ν l k 2 tanh(kh)+ tanh(kh)+q c Fluent Inc. December 3, 2001

65 19.4 Spray Models ] 4νl 2k4 tanh 2 (kh) Q 2 U 2 k 2 [tanh(kh)+q] [ QU 2 k 2 + σk3 ρ l ( ) Two main assumptions have been made to reduce Equation to Equation 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 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 η b at breakup, a breakup time, τ, can be evaluated: η b = η 0 e Ωτ 1 ( ) Ω ln ηb η 0 ( ) where Ω, the maximum growth rate, is found by numerically maximizing Equation 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 L b = Uτ = U ( ) Ω ln ηb η 0 ( ) where the quantity ln( η b η 0 ) 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 d L = 8h K s ( ) c Fluent Inc. December 3,

66 Discrete Phase Models where K s 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, r 0 : h end = r 0 h 0 r 0 + L b sin ( θ 2 ) ( ) This mechanism is not logical for short waves. For short waves, the determination of the ligament diameter is simpler. The value of d L 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, K s. 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 for more details about Oh): d 0 =1.88d L (1 + 3Oh) 1/6 ( ) 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 c Fluent Inc. December 3, 2001

67 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 ). 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,

68 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 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 : Flat Fan Viewed From Above and From the Side c Fluent Inc. December 3, 2001

69 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 d L = 16h K s ( ) 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 C ct times the nozzle area, where C ct is a constant that you specify during the problem setup: u = ṁ ρ l C ct A ( ) The maximum droplet diameter is set to the effective diameter of the exiting jet: c Fluent Inc. December 3,

70 Discrete Phase Models d max = d C ct ( ) The droplet size is then sampled from a Rosin-Rammler distribution with a spread parameter of 4.0 (see Section ). The most probable droplet size depends on the angle, θ, between the droplet s stochastic trajectory and the injection direction: d 0 = d max e (θ/θs)2 ( ) The dispersion angle multiplier, Θ s, is computed from the quality, x, and the specified value for the dispersion constant, C eff : Θ s = x C eff ( ) 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 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 c Fluent Inc. December 3, 2001

71 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 1 2 N 2. (The factor of 1 2 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 N 2 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 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,

72 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 r 1 +r 2.More c Fluent Inc. December 3, 2001

73 19.4 Spray Models precisely, if the smaller droplet center passes within a flat circle centered around the collector of area π(r 1 + r 2 ) 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(r 1 + r 2 ) 2 v rel 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 P 1 = π(r 1 + r 2 ) 2 v rel t V ( ) Equation can be generalized for parcels, where there are n 1 and n 2 droplets in the collector and smaller droplet parcels, respectively. The collector undergoes a mean expected number of collisions given by n = n 2π(r 1 + r 2 ) 2 v rel t V ( ) 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 n nn P (n) =e n! ( ) where n is the number of collisions between a collector and other droplets. c Fluent Inc. December 3,

74 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 ( b crit =(r 1 + r 2 ) min 1.0, 2.4f ) We ( ) where f is a function of r 1 /r 2, defined as f ( ) r1 r 2 = ( r1 r 2 ) ( r1 r 2 ) ( ) r1 r 2 ( ) The value of the actual collision parameter, b, is(r 1 +r 2 ) Y,whereY is a uniform deviate. The calculated value of b is compared to b crit,andif b<b crit, the result of the collision is coalescence. Equation 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: v 1 = m 1v 1 + m 2 v 2 + m 2 (v 1 v 2 ) m 1 + m 2 ( b b crit r 1 + r 2 b crit ) ( ) This relation is used for each of the components of velocity. No other droplet properties are altered in grazing collisions c Fluent Inc. December 3, 2001

75 19.4 Spray Models 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 illustrates the analogous components. Table : Droplet Comparison of a Spring-Mass System to a Distorting 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 for details. c Fluent Inc. December 3,

76 Discrete Phase Models Use and Limitations The TAB model is best for low-weber-number sprays. Extremely high- Weber-number sprays result in shattering of droplets, which is not described well by the spring-mass analogy. Droplet Distortion The equation governing a damped, forced oscillator is [169] F kx d dx dt = md2 x dt 2 ( ) 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 = C ρ g u 2 F ρ l r ( ) k m = C σ k ρ l r 3 ( ) d m = C µ l d ρ l r 2 ( ) where ρ l and ρ g 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 µ l is the droplet viscosity. The dimensionless constants C F, C k,andc d 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>c b r ( ) where C b 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 c Fluent Inc. December 3, 2001

77 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 is non-dimensionalized by setting y = x/(c b r)and substituting the relationships in Equations : d 2 y dt 2 = C F ρ g u 2 C b ρ l r 2 C kσ ρ l r 3 y C dµ l dy ρ l r 2 dt ( ) where breakup now occurs for y>1. For under-damped droplets, the equation governing y can easily be determined from Equation if the relative velocity is assumed to be constant: [ y(t) =We c +e (t/t d) (y 0 We c )cos(ωt)+ 1 ( dy0 ω dt + y ) ] 0 We c sin(ωt) t d ( ) where We = ρ gu 2 r σ ( ) We c = C F We C k C b ( ) y 0 = y(0) ( ) dy 0 dt = dy (0) ( ) dt 1 = C d t d 2 ω 2 = C k σ ρ l r 3 1 t 2 d µ l ρ l r 2 ( ) ( ) In Equation , 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,

78 Discrete Phase Models C k = 8 C d = 5 C F = 1 3 If Equation is solved for all droplets, those with y>1areassumed 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] E parent =4πr 2 σ + K π [ (dy ) ] 2 5 ρ lr 5 + ω 2 y 2 dt ( ) 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 E child =4πr 2 σ r r 32 + π 6 ρ lr 5 ( ) dy 2 ( ) dt where r 32 is the Sauter mean radius of the droplet size distribution. r 32 can be found by equating the energy of the parent and child droplets (i.e., Equations and ), setting y =1,andω 2 =8σ/ρ l r 3 : r 32 = 1+ 8Ky ρ lr 3 (dy/dt) 2 σ r ( ) ( ) 6K Once the size of the child droplets is determined, the number of child droplets can easily be determined by mass conservation c Fluent Inc. December 3, 2001

79 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 = C b r(dy/dt). Therefore, the child droplets will have a velocity normal to the parent droplet velocity given by v normal = C v C b r dy dt ( ) where C v 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 (t d ) for each droplet at time step n using the following: ( (dy/dt) A = (y n We c ) 2 n ) 2 + ( ) ω According to Equation , breakup is possible only if the following condition is satisfied: We c + A>1 ( ) 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 and its derivative, which are both c Fluent Inc. December 3,

80 Discrete Phase Models based on work done by O Rourke and Amsden [169]: y n+1 =We c + { e ( t/t d) (y n We c )cos(ωt)+ 1 ω ( ) dy n+1 = We c y n+1 + dt t d { [( ωe ( t/t d) 1 dy ω dt ) n + yn We c t d [( dy dt ) n + yn We c t d ] } sin(ωt) ( ) ] } cos(ω t) (y n We c )sin(ω t) ( ) All of the constants in these expressions are assumed to be constant throughout the time step. If the criterion of Equation is met, then breakup is possible. The breakup time, t bu, must be determined to see if breakup occurs within the time step t. Thevalueoft bu 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 t n of an undamped version of Equation : where We c + A cos[ω(t t n )+φ] = 1 ( ) cos φ = yn We c A ( ) and sin φ = (dy/dt)n Aω ( ) c Fluent Inc. December 3, 2001

81 19.4 Spray Models If t bu >t n+1, then breakup will not occur during the current time step, and y and (dy/dt) are updated by Equations and The breakup calculation then continues with the next droplet. Conversely, if t n <t bu <t n+1, then breakup will occur and the child droplet radii are determined by Equation The number of child droplets, N, is determined by mass conservation: N n+1 = N n ( r n r n+1 ) 3 ( ) A velocity component normal to the relative velocity vector, with magnitude computed by Equation , 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 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,

82 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 ρ 2.The liquid has a density, ρ 1,andviscosity,µ 1, and a cylindrical polar coordinate system is used which moves with the jet. An arbitrary infinitesimal axisymmetric surface displacement of the form η = η 0 e ikz+ωt ( ) 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 φ 1 = C 1 I 0 (kr)e ikz+ωt ( ) ψ 1 = C 2 I 1 (Lr)e ikz+ωt ( ) where φ 1 and ψ 1 are the velocity potential and stream function, respectively, C 1 and C 2 are integration constants, I 0 and I 1 are modified Bessel functions of the first kind, L 2 = k 2 +ω/ν 1,andν 1 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: p 21 = ρ 2 (U iωk) 2 kη K 0(ka) K 1 (ka) ( ) c Fluent Inc. December 3, 2001

83 19.4 Spray Models where K 0 and K 1 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 v 1 = η t u 1 r = v 1 z ( ) ( ) and p 1 +2µ 1 σ ( ) a 2 η + a 2 2 η z 2 + p 2 = 0 ( ) which are mathematical statements of the liquid kinematic free surface condition, continuity of shear stress, and continuity of normal stress, respectively. Note that u 1 is the axial perturbation liquid velocity, v 1 is the radial perturbation liquid velocity, and σ is the surface tension. Also note that Equation was obtained under the assumption that v 2 =0. As described by Reitz [188], Equations and can be used to eliminate the integration constants C 1 and C 2 in Equation Thus, when the pressure and velocity solutions are substituted into Equation , the desired dispersion relation is obtained: [ I ω 2 +2ν 1 k 2 ω 1 (ka) I 0 (ka) 2kL I 1 (ka) I 1 (La) ] k 2 + L 2 = I 0 (ka) I 1 (La) σk ρ 1 a 2 (1 k2 a 2 ) ( L 2 a 2 L 2 + a 2 ) ( I1 (ka) I 0 (ka) +ρ 2 U i ω ρ 1 k ) ( 2 L 2 a 2 ) I1 (ka) K 0 (ka) L 2 + a 2 I 0 (ka) K 1 (ka) ( ) As shown by Reitz [188], Equation 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 were generc Fluent Inc. December 3,

84 Discrete Phase Models ated for the maximum growth rate, Ω, and the corresponding wavelength, Λ, and are given by Reitz [188]: Λ a ( ρ1 a 3 ) Ω σ = 9.02 ( Oh0.5 )( Ta 0.7 ) ( We ) 0.6 ( ) ( We = ) (1 + Oh)( Ta 0.6 ( ) ) where Oh = We 1 /Re 1 is the Ohnesorge number and Ta = Oh We 2 is the Taylor number. Furthermore, We 1 = ρ 1 U 2 a/σ and We 2 = ρ 2 U 2 a/σ are the liquid and gas Weber numbers, respectively, and Re 1 = Ua/ν 1 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 In other words, r = B 0 Λ ( ) where B 0 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 dt where the breakup time, τ, isgivenby r) = (a, r a ( ) τ τ = 3.726B 1a ΛΩ ( ) c Fluent Inc. December 3, 2001

85 19.4 Spray Models and Λ and Ω are obtained from Equations and , respectively. The breakup time constant, B 1, is related to the initial disturbance level on the liquid jet and has been found to vary from one nozzle to another [118] 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]: C d,sphere = Re > 1000 ( 24 Re Re2/3) 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 ) and a value of 1.52 corresponding to a disk [142]. The drag coefficient is given by c Fluent Inc. December 3,

86 Discrete Phase Models C d = C d,sphere ( y) ( ) where y is the droplet distortion, as determined by the solution of d 2 y dt 2 = C F ρ g u 2 C b ρ l r 2 C kσ ρ l r 3 y C dµ l dy ρ l r 2 dt ( ) 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 is obtained from the TAB model for spray breakup, described in Section , but the dynamic drag model can be used with either of the breakup models 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 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 c Fluent Inc. December 3, 2001

87 19.5 Coupling Between the Discrete and Continuous Phases mass-exchange heat-exchange momentum-exchange typical particle trajectory typical continuous phase control volume Figure : Heat, Mass, and Momentum Transfer Between the Discrete and Continuous Phases F = ( ) 18µC D Re ρ p d 2 p 24 (u p u)+f other ṁ p t (19.5-1) where µ = viscosity of the fluid ρ p = density of the particle d p = diameter of the particle Re = relative Reynolds number u p = velocity of the particle u = velocity of the fluid C D = drag coefficient ṁ p = mass flow rate of the particles t = time step F other = 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,

88 Discrete Phase Models Section 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 Q = where [ mp c p T p + m ( )] Tp p h fg + h pyrol + c p,i dt ṁ p,0 m p,0 m p,0 T ref (19.5-2) m p = average mass of the particle in the control volume (kg) m p,0 = initial mass of the particle (kg) c p = heat capacity of the particle (J/kg-K) T p = temperature change of the particle in the control volume (K) m p = change in the mass of the particle in the control volume (kg) h fg = latent heat of volatiles evolved (J/kg) h pyrol = heat of pyrolysis as volatiles are evolved (J/kg) c p,i = heat capacity of the volatiles evolved (J/kg-K) T p = temperature of the particle upon exit of the control volume (K) T ref = reference temperature for enthalpy (K) ṁ p,0 = initial mass flow rate of the particle injection tracked (kg/s) This heat exchange appears as a source or sink of energy in the continuous phase energy balance during any subsequent calculations of the continuous phase flow field and is reported by FLUENT as described in Section A similar equation governs heat exchange under Law 5, in which the heat of surface combustion is incorporated c Fluent Inc. December 3, 2001

89 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 = m p m p,0 ṁ p,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 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 F new = F old + α(f calculated F old ) (19.5-4) Q new = Q old + α(q calculated Q old ) (19.5-5) M new = M old + α(m calculated M old ) (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,

90 Discrete Phase Models Interphase Exchange During Stochastic Tracking When stochastic tracking is performed, the interphase exchange terms, computed via Equations to , are computed for each stochastic trajectory with the particle mass flow rate, ṁ p0, 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 to 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 OverviewofUsingtheDiscretePhaseModels The procedure for setting up and solving a problem involving a discrete phase is outlined below, and described in detail in Sections 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 If you are using unsteady particle tracking, define the unsteady parameters as described in Section Specify the initial conditions, as described in Section Define the boundary conditions, as described in Section Define the material properties, as described in Section Set the solution parameters and solve the problem, as described in Section Examine the results, as described in Section c Fluent Inc. December 3, 2001

91 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 ). Define Models Discrete Phase Including Radiation Heat Transfer to the Particles If you want to include the effect of radiation heat transfer to the particles (Equation ), 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 This option is available only when the P-1 or discrete ordinates radiation model is used Including the Thermophoretic Force on the Particles If you want to include the effect of the thermophoretic force on the particle trajectories (Equation ), 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 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 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 ) 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,

92 Discrete Phase Models Figure : The Discrete Phase Model Panel c Fluent Inc. December 3, 2001

93 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 (C c in Equation ) 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 ) in the particle trajectory. To do this, turn on the Saffman Lift Force option in the Discrete Phase Model panel 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 ), Diameter Function (C(d p ) in Equation ), and Velocity Exponent Function (b(v) in Equation ) in the Wall boundary conditions panel for each wall zone (as described in Section ) 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, andhigh-mach-number laws described in Section are always available, and the dynamicdrag law described in Section is available only when one of the droplet breakup models is used in conjunction with unsteady tracking. See Section 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 ) must be specified. c Fluent Inc. December 3,

94 Discrete Phase Models For the Stokes-Cunningham law, the Cunningham Correction factor (C c in Equation ) must be specified 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, andscalar 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 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 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) c Fluent Inc. December 3, 2001

95 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 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 For the TAB model, you will need to specify a value for y 0 (the initial distortion at time equal to zero in Equation ) in the y0 field. For the wave model, you will need to specify values for C 0 and C 1, which are the integration constants of the velocity potential and stream function models represented in Equation , 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,

96 Discrete Phase Models constant B 0 in Equation , 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 for information about choosing the drag law. Modeling Droplet Collisions To include the effect of droplet collisions, as described in Section , select the Droplet Collision option under Spray Models. There are no further inputs for this model 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 ). For moving reference frames, relative velocities should be specified. Diameter of the particle, d p. Temperature of the particle, T p. Mass flow rate of the particle stream that will follow the trajectory of the individual particle/droplet, ṁ p (required only for coupled calculations) c Fluent Inc. December 3, 2001

97 19.9 Setting Initial Conditions for the Discrete Phase! Additional parameters if one of the atomizer models described in Section 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 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,

98 Discrete Phase Models For each non-atomizer injection type, you will specify each of the initial conditions listed in Section , 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 ). Create a group injection (Figure ) 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 ). 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 Particle Types When you define a set of initial conditions (as described in Section ), 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 ) and is c Fluent Inc. December 3, 2001

99 19.9 Setting Initial Conditions for the Discrete Phase Figure : Particle Injection Defining a Single Particle Stream Figure : Particle Injection Defining an Initial Spatial Distribution of the Particle Streams Figure : Particle Injection Defining an Initial Spray Distribution of the Particle Velocity c Fluent Inc. December 3,

100 Discrete Phase Models subject to heating or cooling via Law 1 (Section ). 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 ) and that experiences heating/cooling via Law 1 followed by vaporization and boiling via Laws 2 and 3 (Sections and ). 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 ) and experiences heating/cooling via Law 1 followed by devolatilization via Law 4 (Section ), and a heterogeneous surface reaction via Law 5 (Section ). 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 and ) 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 Creating, Copying, Deleting, and Listing Injections You will use the Injections panel (Figure ) 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.) c Fluent Inc. December 3, 2001

101 19.9 Setting Initial Conditions for the Discrete Phase Figure : 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 ). 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,

102 Discrete Phase Models about using this panel, see Section 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, andz position in the columns headed (X), (Y), and(z) x, y, andz 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) c Fluent Inc. December 3, 2001

103 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) Defining Injection Properties Once you have created an injection (using the Injections panel, as described in Section ), you will use the Set Injection Properties panel (Figure ) 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 ! 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, andfile) are described in Section 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,

104 Discrete Phase Models Figure : The Set Injection Properties Panel c Fluent Inc. December 3, 2001

105 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 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 and described in detail in Section 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 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 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,

106 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 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 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 c Fluent Inc. December 3, 2001

107 19.9 Setting Initial Conditions for the Discrete Phase 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-, andz-velocity fields. (Z-Velocity will appear only for 3D problems.) Diameter: Set the initial diameter of the injected particle stream in the Diameter field. Temperature: Set the initial (absolute) temperature of the injected particle stream in the 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 Point Properties for Group Injections For group injections, you will define the properties described in Section for single injections for the First Point and Last Point in the group. That is, you will define a range of values, φ 1 through φ N,for each initial condition φ by setting values for φ 1 and φ N. 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,

108 Discrete Phase Models φ i = φ 1 + φ N φ 1 (i 1) (19.9-1) N 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, c Fluent Inc. December 3, 2001

109 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 Mass Fraction Range (µm ) in Range 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, Y d : Y d = e (d/d)n (19.9-2) FLUENT refers to the quantity d in Equation 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 Y d : Mass Fraction with Diameter, d (µm) Diameter Greater than d, Y d (0.00) c Fluent Inc. December 3,

110 Discrete Phase Models Mass Fraction > d, Y d Diameter, d ( µm) Figure : Example of Cumulative Size Distribution of Particles c Fluent Inc. December 3, 2001

111 19.9 Setting Initial Conditions for the Discrete Phase AplotofY d vs. d is shown in Figure Next, derive values of d and n such that the data in Figure fit Equation The value for d is obtained by noting that this is the value of d at which Y d = e From Figure , you can estimate that this occurs for d 131 µm. The numerical value for n is given by n = ln( ( ln Y ) d) ln d/d By substituting the given data pairs for Y d 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 Rosin- Rammler curve fit is compared to the example data in Figure You can input values for Y d 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 Rosin- Rammler 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,

112 Discrete Phase Models Mass Fraction > d, Y d Diameter, d ( µm) Figure : Rosin-Rammler Curve Fit for the Example Particle Size Data c Fluent Inc. December 3, 2001

113 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 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-, andz-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, andz-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 c Fluent Inc. December 3,

114 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 ) in the Radius field. The particles will be distributed about the axis with the specified radius. θ r origin axis Figure : 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 Point Properties for Surface Injections For surface injections, you will define all the properties described in Section for single injections except for the initial position of the particle streams. The initial positions of the particles will be the location c Fluent Inc. December 3, 2001

115 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 In 2D, you can create a rake using the Line/Rake Surface panel, as described in Section 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 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 ). Spread Parameter: the exponential parameter, n, in Equation 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,

116 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 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, Y- Position, andz-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, andz-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 (p v in Table ) in the Vapor Pressure field. Diameter: Set the diameter of the orifice in the Injector Inner Diam. field (d in Table ). Orifice length: Set the length of the orifice in the Orifice Length field (L in Table ) c Fluent Inc. December 3, 2001

117 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 ). Nozzle parameter: Set the constant for the spray angle correlation in the Constant A field (C A in Equation ). Azimuthal angles: For 3D sectors, set the Azimuthal Start Angle and Azimuthal Stop Angle. See Section for details about how these inputs are used 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 , 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 ). Pressure: Set the pressure upstream of the injection in the Upstream Pressure field (p 1 in Table ). 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( η b η 0 ) in Equation ). Ligament diameter: For short waves, set the proportionality constant that linearly relates the ligament diameter, d L,tothewavelength that breaks up the sheet in the Ligament Constant field (see Equations ). See Section for details about how these inputs are used. c Fluent Inc. December 3,

118 Discrete Phase Models 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 , 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 ). 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 ). 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( η b η 0 ) in Equation ). Ligament diameter: For short waves, set the proportionality constant that linearly relates the ligament diameter, d L,tothewavelength that breaks up the sheet in the Ligament Constant field (see Equations ). See Section for details about how these inputs are used Point Properties for Flat-Fan Atomizer Injections The flat-fan atomizer model is available only for 3D models. For this type of injection, you will define the following initial conditions under Point Properties: c Fluent Inc. December 3, 2001

119 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, andz-center fields (see Figure ). Virtual position: Set the coordinates of the virtual origin of the fan in the X-Virtual Origin, Y-Virtual Origin, andz-virtual Origin fields. This point is the intersection of the lines that mark the sides of the fan (see Figure ). Normal vector: Set the direction that is normal to the fan in the X- Fan Normal Vector, Y-Fan Normal Vector, andz-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 ). See Section for details about how these inputs are used 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,

120 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 , 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 ). 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 (C eff in Equation ). 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 for details about how these inputs are used Modeling Turbulent Dispersion of Particles As mentioned in Section , 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 Click the Turbulent Dispersion tab in the Set Injection Properties panel. 2. Enable stochastic tracking by turning on the Stochastic Model under Stochastic Tracking c Fluent Inc. December 3, 2001

121 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 ), 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 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 issetto1if Stochastic Tracking is enabled. If you want the characteristic lifetime of the eddy to be random (Equation ), enable the Random Eddy Lifetime option. You will generally not need to change the Time Scale Constant (C L in Equation ) 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 illustrates a discrete phase trajectory calculation computed via the mean tracking (number of tries = 0) and Figure 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,

122 Discrete Phase Models 3.04e e e e e e e e e e e e e e e e+00 Particle Traces Colored by Particle Time (s) Figure : Mean Trajectory in a Turbulent Flow 3.00e e e e e e e e e e e e e e e e+00 Particle Traces Colored by Particle Time (s) Figure : Stochastic Trajectories in a Turbulent Flow c Fluent Inc. December 3, 2001

123 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 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 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,

124 Discrete Phase Models Figure : 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 c Fluent Inc. December 3, 2001

125 19.9 Setting Initial Conditions for the Discrete Phase 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 , if you select more than one injection in the Injections panel, clicking the Set... button will open the Set Multiple Injection Properties panel (Figure ) 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,

126 Discrete Phase Models Figure : The Set Multiple Injection Properties Panel c Fluent Inc. December 3, 2001

127 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 ). 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,

128 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 c Fluent Inc. December 3, 2001

129 19.10 Setting Boundary Conditions for the Discrete Phase 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 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 ) coefficient of restitution = V 2,n V 1,n θ 1 θ 2 Figure : 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]: e n = v 2,n ( ) v 1,n where v n is the particle velocity normal to the wall and the subscripts 1 and 2 refer to before and after collision, respectively. Simic Fluent Inc. December 3,

130 Discrete Phase Models larly, the tangential coefficient of restitution, e t, 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, θ 1, in Figure 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 In the case of combusting particles, the remaining volatile mass is passed into the vapor phase. volatile fraction flashes to vapor θ 1 Figure : 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 c Fluent Inc. December 3, 2001

131 19.10 Setting Boundary Conditions for the Discrete Phase particle vanishes Figure : 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,

132 Discrete Phase Models 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 ). Figure : Discrete Phase Boundary Conditions in the Wall Panel Select reflect, trap, orescape 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 You can also select a user-defined function in this list. For internal boundary c Fluent Inc. December 3, 2001

133 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, orpiecewise-polynomial function for the Normal and Tangent coefficients of restitution underdiscrete Phase Reflection Coefficients. See Section 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 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 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, andvelocity Exponent Function (f(α), C(d p ), and b(v) in Equation ). See Section for a detailed description of these functions and Section for details about using the panels for defining polynomial, piecewise-linear, and piecewise-polynomial functions 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 Summary of Property Inputs Tables 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 c Fluent Inc. December 3,

134 Discrete Phase Models Table : Property Inputs for Inert Particles Property Symbol density ρ p in Eq specific heat c p in Eq particle emissivity ɛ p in Eq particle scattering factor f in Eq thermophoretic coefficient D T,p in Eq Table : Property Inputs for Droplet Particles Properties Symbol density ρ p in Eq specific heat c p in Eq thermal conductivity k p in Eq viscosity µ in Eq latent heat h fg in Eq vaporization temperature T vap in Eq boiling point T bp in Eq , volatile component fraction f v0 in Eq , binary diffusivity D i,m in Eq saturation vapor pressure p sat (T ) in Eq heat of pyrolysis h pyrol in Eq droplet surface tension σ in Eq , particle emissivity ɛ p in Eq , particle scattering factor f in Eq thermophoretic coefficient D T,p in Eq c Fluent Inc. December 3, 2001

135 19.11 Setting Material Properties for the Discrete Phase Table : Property Inputs for Combusting Particles (Laws 1 4) Properties Symbol density ρ p in Eq specific heat c p in Eq latent heat h fg in Eq vaporization temperature T vap = T bp in Eq volatile component fraction f v0 in Eq swelling coefficient C sw in Eq burnout stoichiometric ratio S b in Eq combustible fraction f comb in Eq heat of reaction for burnout H reac in Eq fraction of reaction heat given to solid f h in Eq particle emissivity ɛ p in Eq , particle scattering factor f in Eq thermophoretic coefficient D T,p in Eq devolatilization model law 4, constant rate constant A 0 in Eq law 4, single rate pre-exponential factor A 1 in Eq activation energy E in Eq law 4, two rates pre-exponential factors A 1,A 2 in Eq , activation energies E 1,E 2 in Eq , weighting factors α 1,α 2 in Eq law 4, CPD initial fraction of bridges in coal lattice p 0 in Eq initial fraction of char bridges c 0 in Eq lattice coordination number σ + 1 in Eq cluster molecular weight M w,1 in Eq side chain molecular weight M w,δ in Eq c Fluent Inc. December 3,

136 Discrete Phase Models Table : Property Inputs for Combusting Particles (Law 5) Properties Symbol combustion model law 5, diffusion rate binary diffusivity D i,m in Eq law 5, diffusion/kinetic rate mass diffusion limited rate constant C 1 in Eq kinetics limited rate pre-exp. factor C 2 in Eq kinetics limited rate activ. energy E in Eq law 5, intrinsic rate mass diffusion limited rate constant C 1 in Eq kinetics limited rate pre-exp. factor A i in Eq kinetics limited rate activ. energy E i in Eq char porosity θ in Eq mean pore radius r p in Eq specific internal surface area A g in Eq , tortuosity τ in Eq burning mode α in Eq law 5, multiple surface reaction binary diffusivity D i,m in Eq c Fluent Inc. December 3, 2001

137 19.11 Setting Material Properties for the Discrete Phase 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, andcombusting-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 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,

138 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 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 and ) 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 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 c Fluent Inc. December 3, 2001

139 19.11 Setting Material Properties for the Discrete Phase Cp is the specific heat, c p, 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 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- Fin British units and is treated as a constant by FLUENT. Latent Heat is the latent heat of vaporization, h fg, required for phase change from an evaporating liquid droplet (Equation ) or for the evolution of volatiles from a combusting particle (Equation ). This input is supplied in units of J/kg in SI units or of Btu/lb m in British units and is treated as a constant by FLU- ENT. Thermophoretic Coefficient is the coefficient D T,p in Equation , and appears when the thermophoretic force (which is described in Section ) 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 for details about temperature-dependent properties. Vaporization Temperature is the temperature, T vap,atwhichthecalculation of vaporization from a liquid droplet or devolatilization from a combusting particle is initiated by FLUENT. Until the particle temperature reaches T vap, the particle is heated via Law 1, Equation This temperature input represents a modeling decision rather than any physical characteristic of the discrete phase. Boiling Point is the temperature, T bp, at which the calculation of the boiling rate equation ( ) 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 T bp.the boiling point should be defined as the saturated vapor temperature c Fluent Inc. December 3,

140 Discrete Phase Models at the system pressure that you defined in the Operating Conditions panel. Volatile Component Fraction (f v0 ) is the fraction of a droplet particle that may vaporize via Laws 2 and/or 3 (Sections and ). For combusting particles, it is the fraction of volatiles that may be evolved via Law 4 (Section ). Binary Diffusivity is the mass diffusion coefficient, D i,m,usedinthevaporization law, Law 2 (Equation ). This input is also used to define the mass diffusion of the oxidizing species to the surface of a combusting particle, D i,m, as given in Equation (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, p sat, defined as a function of temperature, which is used in the vaporization law, Law 2 (Equation ). 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 for details about temperature-dependent properties.) In the case of unrealistic inputs, FLUENT restricts the range of P sat 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. HeatofPyrolysis is the heat of the instantaneous pyrolysis reaction, h pyrol, 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 Swelling Coefficient is the coefficient C sw in Equation , which governs the swelling of the coal particle during the devolatilization law, Law 4 (Section ). A swelling coefficient of unity (the default) c Fluent Inc. December 3, 2001

141 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, S b, for the burnout reaction, Equation , in terms of mass of oxidant per mass of char in the particle. Combustible Fraction is the mass fraction of char, f comb, in the coal particle, i.e., the fraction of the initial combusting particle that will react in the surface reaction, Law 5 (Equation ). Heat of Reaction for Burnout is the heat released by the surface char combustion reaction, Law 5 (Equation ). 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 f h (Equation ), 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 , retain the selection of constant in the drop-down list to the right of Devolatilization Model and input the rate constant A 0 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 ) 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, A 1,andtheActivation Energy, E, to be used in Equation for the computation of the kinetic rate. When the two competing rates model (two-competing-rates) isse- lected, the Two Competing Rates Model panel will appear and c Fluent Inc. December 3,

142 Discrete Phase Models you will enter, for the First Rate and the Second Rate, thepreexponential Factor (A 1 in Equation and A 2 in Equation ), Activation Energy (E 1 in Equation and E 2 in Equation ), and Weighting Factor (α 1 and α 2 in Equation ). The constants you input are used in Equations through 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 (p 0 in Equation ), Initial Fraction of Char Bridges (c 0 in Equation ), Lattice Coordination Number (σ + 1inEqua- tion ), Cluster Molecular Weight (M w,1 in Equation ), and Side Chain Molecular Weight (M w,δ in Equation ). Note that the Single Rate Devolatilization Model, Two Competing Rates Model,andCPD 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 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 (C 1 in Equation ), Kinetics Limited Rate Pre-exponential Factor (C 2 in Equation ), and Kinetics Limited Rate Activation Energy (E in Equation ). 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 c Fluent Inc. December 3, 2001

143 19.11 Setting Material Properties for the Discrete Phase! ited Rate Constant (C 1 in Equation ), Kinetics Limited Rate Pre-exponential Factor (A i in Equation ), Kinetics Limited Rate Activation Energy (E i in Equation ), Char Porosity (θ in Equation ), Mean Pore Radius (r p in Equation ), Specific Internal Surface Area (A g in Equations and ), Tortuosity (τ in Equation ), and Burning Mode, alpha (α in Equation ). 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 If you have not yet defined any particle surface reactions, you must be sure to define them now. See Section 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 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, ɛ p,usedto compute radiation heat transfer to the particles (Equations , , , , and ) 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,

144 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, f p, due to particles in the P-1 or discrete ordinates radiation model (Equation ). 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 f p 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, µ l. 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 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. Thesurface 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 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 c Fluent Inc. December 3, 2001

145 19.12 Calculation Procedures for the Discrete Phase Calculation Procedures for the Discrete Phase Solution of the discrete phase implies integration in time of the force balance on the particle (Equation ) 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 Coupling and performing trajectory calculations are described in Section Sections and provide information about resetting interphase exchange terms and using the parallel solver for a discrete phase calculation 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 ) 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 and 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,

146 Discrete Phase Models Figure : The Discrete Phase Model Panel c Fluent Inc. December 3, 2001

147 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 10 9 ). 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 (u p ) and of the continuous phase (u c ): t = L u p + u c ( ) where L is the Length Scale that you define. As defined by Equation , 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,

148 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 where λ is the Step Length Factor. As defined by Equation , 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 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 c Fluent Inc. December 3, 2001

149 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 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 continuous phase flow field calculation particle trajectory calculation Figure : 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,

150 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 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 : 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 c Fluent Inc. December 3, 2001

151 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 ), 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 FLU- ENT console as the continuous and discrete phase iterations are performed: c Fluent Inc. December 3,

152 Discrete Phase Models iter continuity x-velocity y-velocity k epsilon energy time/ite e e e e e e-03 0:00: e e e e e e-03 0:00:03 DPM Iteration... number tracked= 9, number escaped= 1, aborted= 0, trapped= 0, evaporated = 8, i Done e e e e e e-03 0:00: e e e e e e-01 0:00: e e e e e e-02 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 specifiedinthe Set Injection Properties panel. Input of this parameter is described in Section 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 1requestsn 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, FLU- ENT applies under-relaxation to the momentum, heat, and mass transfer terms. This under-relaxation serves to increase the stability of the c Fluent Inc. December 3, 2001

153 19.12 Calculation Procedures for the Discrete Phase coupled calculation procedure by letting the impact of the discrete phase change only gradually: E new = E old + α(e calculated E old ) ( ) where E new is the exchange term, E old is the previous value, E calculated 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 Under- Relaxation Factors in the Solution Controls panel. You may need to decrease α in order to improve the stability of coupled discrete phase calculations 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 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. Bydefault, 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,

154 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 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.) c Fluent Inc. December 3, 2001

155 19.13 Postprocessing for the Discrete Phase 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 ). Display Particle Tracks... Figure : 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,

156 Discrete Phase Models 2. Set the length scale and the maximum number of steps in the Discrete Phase Model panel, as described in Section 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 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 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 c Fluent Inc. December 3, 2001

157 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 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 Reporting of Trajectory Fates When you perform trajectory calculations by displaying the trajectories (as described in Section ), 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,

158 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 ) 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 for displaying trajectories c Fluent Inc. December 3, 2001

159 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 Elapsed Time (s) Min Max Avg Std Dev Incomplete e e e e+00 Escaped - Zone e e e e+00 (*)- Mass Transfer Summary -(*) Fate Mass Flow (kg/s) Initial Final Change Incomplete 1.388e e e-03 Escaped - Zone e e e-03 (*)- Energy Transfer Summary -(*) Fate Heat Content (W) Initial Final Change Incomplete 4.051e e e+01 Escaped - Zone e e e+01 (*)- Combusting Particles -(*) Fate Volatile Content (kg/s) Char Content (kg/s) Initial Final %Conv Initial Final Incomplete 6.247e e e e+00 1 Escaped - Zone e e e e-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,

160 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, andstd 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 Elapsed Time (s) Min Max Avg Std Dev Incomplete e e e e+00 Escaped - Zone e e e e+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 e e e+00 injection-0 1 injection e e e+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 c Fluent Inc. December 3, 2001

161 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 Mass Flow (kg/s) Initial Final Change Incomplete 1.388e e e-03 Escaped - Zone e e e-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 Heat Content (W) Initial Final Change Incomplete 4.051e e e+01 Escaped - Zone e e e+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 Volatile Content (kg/s) Char Content (kg/s) Initial Final %Conv Initial Final %Conv Incomplete 6.247e e e e Escaped - Zone e e e e c Fluent Inc. December 3,

162 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 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 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.) c Fluent Inc. December 3, 2001

163 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 X-Position Y-Position Z-Velocity X-Velocity Y-Velocity Z-Veloc 0.000e e e e e e e 3.773e e e e e e e 5.403e e e e e e e 9.181e e e e e e e 1.296e e e e e e e 1.608e e e e e e e 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 Y-Velocity Z-Velocity Diameter Temperature Density Mass 650e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e 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 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,

164 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 X-Position Y-Position Z-Position X-Velocity Y-Velocity Z-Veloc 0.000e e e e e e e 1.672e e e e e e e 3.342e e e e e e e 5.010e e e e e e e 6.675e e e e e e e 8.338e e e e e e e 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 Y-Velocity Z-Velocity Diameter Temperature Density Mass 000e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e c Fluent Inc. December 3, 2001

165 19.13 Postprocessing for the Discrete Phase 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 and ) will not affect the values of these exchange terms. c Fluent Inc. December 3,

166 Discrete Phase Models 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 ). Report Discrete Phase Sample... Figure : 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 c Fluent Inc. December 3, 2001

167 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 ) 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 Histogram Reporting of Samples Histograms can be plotted from sample files created in the Sample Trajectories panel (as described in Section ) using the Trajectory Sample Histograms panel (Figure ). Report Discrete Phase Histogram... The procedure for plotting histograms from data in a sample file is listed below: c Fluent Inc. December 3,

168 Discrete Phase Models Figure : 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. Thenumberof bins orintervalsintheplotcanbesetinthedivisions field. You can delete samples from the list with the Delete button and update the Min/Max values with the Compute button c Fluent Inc. December 3, 2001