modeling discrete phase

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CFD Modeling Courses

Modeling Discrete Phase

Lecturer: Ehsan.A.SaadatiSharif University of TechnologyOwj Group-Tehran: First Edition Fall 2010- Non [email protected]

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Contents• Introduction • Particle Motion Theory • Multicomponent Particle Theory • Wall-Film Model Theory • Particle Erosion and Accretion Theory • Dynamic Drag Model Theory • Spray Model Theory • Atomizer Model Theory • One-Way and Two-Way Coupling • Discrete Phase Model (DPM) Boundary Conditions • Steps for Using the Discrete Phase Models • Setting Initial Conditions for the Discrete Phase • Setting Boundary Conditions for the Discrete Phase • Setting Material Properties for the Discrete Phase • Solution Strategies for the Discrete Phase • Postprocessing for the Discrete Phase

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Introductory• In addition to solving transport equations for the continuous phase,

FLUENT allows you to simulate a discrete second phase in a Lagrangian frame of reference. This second phase consists of spherical particles (which may be taken to represent droplets or bubbles) dispersed in the continuous phase. FLUENT computes the trajectories of these discrete phase entities, as well as heat and mass transfer to/from them. The coupling between the phases and its impact on both the discrete phase trajectories and the continuous phase flow can be included.

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IntroductoryFLUENT 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

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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 Chapter 23 for information about when you might want to use one of the general multiphase models instead of the discrete phase model.

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Equations of Motion for Particles

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The drag coefficient, CD , can be taken from either

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

The factor CD is the Cunningham correction to Stokes' drag law , which you can compute from:

Where λ is the molecular mean free path.

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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 effect on particles in the additional acceleration (force/unit mass) term, FX , in bellow Equation

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

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Brownian ForceFor sub-micron particles, the effects of Brownian motion can be optionally included in the additional force term.

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

Where ξi 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 nonturbulent models.

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Saffman's Lift ForceThe 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 and is a generalization of the expression provided by Saffman

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

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Turbulent Dispersion of Particles The dispersion of particles due to turbulence in the fluid phase can be predicted using the stochastic tracking model or the particle cloud model.The stochastic tracking (random walk) model includes the effect of instantaneous turbulent velocity fluctuations on the particle trajectories through the use of stochastic methods. The particle cloud model tracks the statistical evolution of a cloud of particles about a mean trajectory. The concentration of particles within the cloud is represented by a Gaussian probability density function (PDF) about the mean trajectory. For stochastic tracking a model is available to account for the generation or dissipation of turbulence in the continuous phase

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Stochastic Tracking When the flow is turbulent, FLUENT will predict the trajectories of particles using the mean fluid phase velocity,ū , in the trajectory equations. Optionally, you can include the instantaneous value of the fluctuating gas flow velocity,

u=ū+u̒�

to predict the dispersion of the particles due to turbulence.

FLUENT uses a stochastic method (random walk model) to determine the instantaneous gas velocity. In the discrete random walk (DRW) 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 nonphysical results in strongly nonhomogeneous diffusion-dominated flows, where small particles should become uniformly distributed. Instead, the DRW will show a tendency for such particles to concentrate in low-turbulence regions of the flow.

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The Integral TimePrediction 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 :

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 :

For small "tracer'' particles that move with the fluid (zero drift velocity), the integral time becomes the fluid Lagrangian integral time, . This time scale can be approximated as

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The Integral Time

for the - model and its variants, and

when the Reynolds stress model (RSM) is used

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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, ,v and w u̒� ̒� ̒�a time scale,τe

The values of ,v and w that prevail during the lifetime of the turbulent eddy are u̒� ̒� ̒�sampled by assuming that they obey a Gaussian probability distribution, so that

Where ς 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 defined (assuming isotropy) as

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The Discrete Random Walk Model for the k-ε model, the k-ω model, and their variants. When the RSM is used, nonisotropy of the stresses is included in the derivation of the velocity fluctuations:

when viewed in a reference frame in which the second moment of the turbulence is diagonal. For the LES model, the velocity fluctuations are equivalent in all directions.

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The Discrete Random Walk Model The characteristic lifetime of the eddy is defined either as a constant:

τe=2TL

Where TL is given by before mentioned Equation in general, or as a random variation about TL:

τe=-TLlog(r)

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

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The Discrete Random Walk Model The particle eddy crossing time is defined as

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 ς

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Using the DRW Model The only inputs required for the DRW model are the value for the integral time-scale constant, C L 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 22.12.5.

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Stochastic Staggering of ParticlesIn order to obtain a better representation of an injector, the particles can be staggered either spatially or temporally. When particles are staggered spatially, FLUENT randomly samples from the region in which the spray is specified (e.g., the sheet thickness in the pressure-swirl atomizer) so that as the calculation progresses, trajectories will originate from the entire region. This allows the entire geometry specified in the atomizer to be sampled while specifying fewer streams in the input panel, thus decreasing computational expense.

When injecting particles in a transient calculation using relatively large time steps in relation to the spray event, the particles can clump together in discrete bunches. The clumps do not look physically realistic, though FLUENT calculates the trajectory for each particle as it passes through a cell and the coupling to the gas phase is properly accounted for. To obtain a statistically smoother representation of the spray, the particles can be staggered in time. During the first time step, the particle is tracked for a random percentage of its initial step. This results in a sample of the initial volume swept out by the particle during the first time step and a smoother, more uniform spatial distribution at longer time intervals.

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Stochastic Staggering of ParticlesThe menu for staggering is available in the text user interface, under

define/models/dpm/options/particle-staggering

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

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

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Multicomponent Particle Theory A number of industrially important processes, such as distillation, absorption and extraction, bring into contact two phases which are not at equilibrium. The rate at which a specie is transferred from one phase to the other depends on the departure of the system from equilibrium. The quantitative treatment of these rate processes requires knowledge of the equilibrium states of the system. Apart from these cases, vapor-liquid equilibrium (VLE) relationships in multicomponent systems are needed for the solution of many other classes of engineering problems, such as the computation of evaporation rates in spray combustion applications.

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

DPM particles are used to model the wall-film. The wall-film model in FLUENT allows a single component liquid drop to impinge upon a boundary surface and form a thin film. The model can be broken down into four major subtopics: interaction during the initial impact with a wall boundary, subsequent tracking on surfaces, calculation of film variables, and coupling to the gas phase. Next slide figure schematically shows the basic mechanisms considered for the wall-film model.

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Wall-Film Model Theory

Mechanisms of Splashing, Momentum, Heat and Mass Transfer for the Wall-Film

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Wall-Film Model Theory The main assumptions and restrictions for the wall-film model are as follows:

The layer is thin, less than 500 microns in thickness. This limitation is due to the assumption of a linear velocity profile in the film. The temperature in the film particles change relatively slowly so that an analytical integration scheme can be utilized. Film particles are assumed to be in direct contact with the wall surface and the heat transfer from the wall to the film takes place by conduction. The film temperature never exceeds the boiling temperature for the liquid. The simulation is transient. The wall-film model is not available with the Workpile Algorithm shared memory option in parallel processing.

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SplashingIf the particle impinging on the surface has a sufficiently high energy, the particle splashes and several new particles are created. The number of particles created by each impact is explicitly set by the user in the DPM tab in the Boundary Conditions panel, as in Figure 22.4.3. The number of splashed parcels may be set to an integer value between zero and ten. The properties (diameter, magnitude, and direction) of the splashed parcels are randomly sampled from the experimentally obtained distribution functions described in the following sections. Setting the number of splashed parcels to zero turns off the splashing calculation. Bear in mind that each splashed parcel can be considered a discrete sample of the distribution curves and that selecting the number of splashed drops in the Boundary Conditions panel does not limit the number of splashed drops, only the number of parcels representing those drops.

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Splashing

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One-Way and Two-Way CouplingYou can use FLUENT to predict the discrete phase patterns based on a fixed continuous phase flow field (an uncoupled approach or "one-way coupling"), or you can include the effect of the discrete phase on the continuum (a coupled approach or "two-way coupling"). 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 22.9.1 for details. 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.

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Coupling Between the Discrete and Continuous PhasesAs 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 22.9.1.

The End

By: Ehsan SaadatiPardad Petrodanesh [email protected] out more at:

www.petrodanesh.irwww.petrodanesh.com

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Non Completed

January 2010

Ehsan Saadati

modeling discrete phase

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