01 multiphase flows- fundamental definitions

1

Multiphase Flows

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Dr. Mohammad Jadidi(Ph.D. in Mechanical Engineering)

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Presented by: Mohammad Jadidi 2

Multiphase Flows Choosing a Multiphase Model

The first step in solving any multiphase problem is to determine which of the regimes described in Multiphase Flow Regimes best represents your flow.

As a general guide, there are some parameters that help to identify the appropriate multiphase model as follows:

Particulate Loading Volume Fractions Superficial and Phase Velocities Response Time Stokes Number Dilute and Dense Flows Phase Coupling Other Considerations

Multiphase Models

Euler-Lagrange approach

DPM

Euler-Euler Approach

Eulerian

Model

Mixture Model

VOF Model


Multiphase Flows Fundamental Definitions: Primary & Secondary phases

Multiphase flow is simultaneous flow of: Materials with different states or phases (i.e. gas, liquid or

solid). Materials with different chemical properties but in the

same state or phase (i.e. liquid-liquid systems such as oil droplets in water).

The primary and secondary phases: One of the phases is continuous (primary) while the

other(s) (secondary) are dispersed within the continuous phase.

A diameter has to be assigned for each secondary phase to calculate its interaction (drag) with the primary phase.

particle size distribution is modeled by assigning a separate phase for each particle

diameter

NOTE: A secondary phase with a particle size distribution is modeled by assigning a separate phase for each particle diameter.

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Multiphase Flows Fundamental Definitions: Volume Fractions

The volume fraction of the dispersed phase is defined as:

the volume fraction of continuous phase is:

And by definition, the sum if the volume fractions must be unity


Multiphase Flows Fundamental Definitions: Particulate Loading

Note: that the word “particle” is used in this discussion to refer to a particle, droplet, or bubble

The material density ratio(ϒ):

Particulate loading (β) :

material density ratio is greater than 1000 for gas-solid flows, about 1 for liquid-solid flows, and less than 0.001 for gas-liquid flows.

Particulate loading has a major impact on phase interactions. The particulate loading is defined as the mass density ratio of the dispersed phase (d) to that of the carrier phase (c).


Multiphase Flows

Average distance between the individual particles of the particulate phase can be estimated as follows. (Crowe et al (1998))

For example, for a gas-particle flow with a particulate loading of 1, the interparticle space is about 8; the particle can therefore be treated as isolated (that is, very low particulate loading).

Fundamental Definitions: Average distance between the individual particles

𝜅 =𝛽

𝛾

𝒅𝒅

𝐿


Multiphase Flows Fundamental Definitions: Superficial and Phase Velocities

The superficial velocity of each phase is the mass flow rate of that phase divided by the pipe area A and phase density. The superficial velocity for the dispersed phase is:

The phase velocity is the actual velocity of the phase, and it is related to the superficial velocity by the volume fraction

In other words, superficial velocity is the velocity of the phase if the phase occupied the whole pipe area


Multiphase Flows Fundamental Definitions: Relaxation Time or Particle Response Time

The response time of a particle or droplet is the time required for a particle to be released from rest to achieve 63%, (𝒆−𝟏 / 𝒆), of the free stream velocity

When does the particle follow the flow?

Typical relaxation times in process applications

𝜏𝑝 =𝜌𝑑𝑑𝑑

2

18𝜇𝑐


Multiphase Flows Fundamental Definitions: Stokes Number

Stokes Number (St) is a dimensionless parameter that describes a particle’s flow in a particular fluid. Stokes number is determined by the ratio of the relaxation time of the particle (τp), a characteristic dimension of the obstacle obstructing fluid flow (LF) and the fluid’s velocity (V F):

If St <<1, the particle response time is much less than the characteristic time associated with the flow field. In this case the particles will have ample time to respond to changes in flow velocity and, the particle and fluid velocities will be nearly equal

If St>>1, then the particle will have essentially no time to respond to the fluid velocity changes and the particle velocity will be little affected by fluid velocity change

Normalized particle distribution for varying Stokes number

𝜏𝑝 =𝜌𝑑𝑑𝑑

2

18𝜇𝑐𝜏𝐹 =

𝐿𝐹𝑉𝐹


Multiphase Flows

Answer:Snow particles with a low Stokes Number are carried by the moving fluid. Rain particles with a high Stokes Number settle onto the windshield. If there was no resultant fluid flow, both particles would settle.

Calculation: A value of 20m/s (≈ 45mph) is used as a model velocity for the

car and resultant airflow. The car is assumed to have a characteristic dimension of D = 1 m. Air at 0 degrees Celsius has a dynamic viscosity of 1.71 ∗ 10−5 Ns/m2.

Stokes Number for a raindrop Sk = 584 is calculated from a diameter of dp = 0.003m = 3mm and a density of ρ = 1000 kg/m3.

Stokes Number Sk = 58 for snow is calculated using the same diameter and a density of ρ = 100 kg/m3.

Question:“Why is it that I get more snow on my windshield when my car is stopped at a light than when it’s moving, but I get more rain on my windshield when it’s moving than when it’s stopped?”

Fundamental Definitions: Stokes Number-Example


Multiphase Flows Fundamental Definitions: Dilute and Dense Flows

A dilute flow, is one in which the particle motion is controlled by the fluid forces (drag and lift)

A dense flow, on the other hand, is one in which the particle motion is controlled by collisions

In collision-dominated flow the collisions between the particles control the features of the flow, such as in a fluidized bed

In a contact dominated flow, the particle motion is controlled by continuous contact such as in a shear granular flow

There is a further classification of dense flows: collision-and contact-dominated.

Dense flows

Collision-dominated

flow

Contact dominated

flow


Multiphase Flows

One-way-coupled : the fluid carrier influences the particles via drag and turbulence, but the particles have no influence on the fluid carrier

Fundamental Definitions: Phase Coupling

Schematic diagram of coupling

Two-way-coupled: the fluid carrier influences the particulate phase via drag and turbulence, but the particles in turn influence the carrier fluid via reduction in mean momentum and turbulence

Four-way- couple : there is two-way coupling plus particle pressure and viscous stresses due to particles


Multiphase Flows

Dispersed two-phase flow as a function of the particle volume fraction and inter-particle spacing

Fundamental Definitions: Phase Coupling

NOTE: Four-way coupling effects becomeimportant when particle volume fraction exceeds 𝟏𝟎-3


Multiphase Flows Fundamental Definitions: Weber number

Weber number describes the ratio between deforming inertial forces and stabilizing cohesive forces for liquids flowing through a fluid medium. For example, the Weber number characterizes the atomizing quality of a spray and the resulting droplet size.

When a liquid flows through a second fluid phase

(gas or liquid), then the aerodynamic force

FA causes the drops to deform and ultimately disperse.

The cohesion force FK associated with the surface

tension or interfacial tension ,σ, opposes the

increase in surface area which is caused by the

deformation. The drop is therefore held together by the surface or interfacial tension.

If the deforming force increases due to a

higher speed or longer process length, the

drops of a spray disperse more easily and

drops of oil in an aqueous environment are

split apart more easily. A high surface or interfacial tension counteracts this process.

https://www.kruss.de/services/education-theory/glossary/surface-tension/

https://www.kruss.de/services/education-theory/glossary/interfacial-tension/


Multiphase Flows Fundamental Definitions: Weber number

If the deforming force increases due to

a higher speed or longer process

length, the drops of a spray disperse

more easily and drops of oil in an

aqueous environment are split apart

more easily. A high surface or interfacial tension counteracts this process.

VIDEO: Weber number


Choosing a Multiphase Model

Multiphase Flows


Multiphase Flows Choosing a Multiphase Model

Multiphase Models

Euler-Lagrange approach

DPM

Euler-Euler Approach

Eulerian

ModelMixture Model VOF Model

There are two approaches for the numerical calculation of multiphase flows: the Euler-Lagrange approach and the Euler-Euler approach


Multiphase Flows

The VOF model is a surface-tracking technique applied to a fixed Eulerian mesh.

It is designed for two or more immiscible fluids where the position of the interface between the fluids is of interest.

In the VOF model, a single set of momentum equations is shared by the fluids, and the volume fraction of each of the fluids in each computational cell is tracked throughout the domain.

Choosing a Multiphase Model-Euler-Euler approach-Volume of Fluid (VOF)

The VOF models require a proper mesh and numerical advection scheme to approximate the transport of the scalar function in an accurate manner avoiding numerical diffusion

Hydrodynamics and Wave Impact Analysis


Multiphase Flows Choosing a Multiphase Model-Euler-Euler approach-Volume of Fluid (VOF)

Applications of the VOF model include:

Stratified flows Free-surface flows Filling Sloshing Motion of large bubbles in a liquid, Motion of liquid after a dam break, Prediction of jet breakup (surface tension) Steady or transient tracking of any liquid-gas

interface.

Sloshing


Multiphase Flows Choosing a Multiphase Model-Euler-Euler approach-The Mixture Model

The mixture model solves for the mixture momentum equation and prescribes relative velocities to describe the dispersed phases.

Applications of the mixture model include:

particle-laden flows with low loading bubbly flows sedimentation and cyclone separators

NOTE: The mixture model can also be used without relative velocities for the dispersed phases to model homogeneous multiphase flow.


Multiphase Flows Choosing a Multiphase Model-Euler-Euler approach-The Eulerian Model

The Eulerian model is the most complex of the multiphase models in ANSYS Fluent. It solves a set of n momentum and continuity equations for each phase. In the Eulerianapproach both the dispersed particle phase and continuous fluid phase are solved using the NS equations. Coupling is achieved through the pressure and interphase exchange coefficients.

Applications of the Eulerian multiphase model include:

bubble columns Risers particle suspension fluidized beds

NOTE: It can be used to compute any multiphase flow regime, provided that an adequate closure relation for the interfacial coupling terms are provided


Multiphase Flows

The Lagrangian Discrete Phase Model (DPM) in ANSYS Fluent follows the Euler-Lagrange approach.

The fluid phase is treated as a continuum by solving the Navier-Stokes equations

The dispersed phase is solved(Using: the Newton’s second law) by tracking a large number of particles, bubbles, or droplets through the calculated flow field.

In DPM individual particles are treated as rigid spheres (i.e., neglecting particle deformation and internal flows)

The dispersed phase can exchange momentum, mass, and energy with the fluid phase.

Choosing a Multiphase Model-Euler-Lagrange Approach-The DPM Model

spray dryers coal and liquid fuel combustion some particle-laden flows

Applications of the DPM model include:


Multiphase Flows

The discrete phase formulation used by ANSYS 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.

Choosing a Multiphase Model-Euler-Lagrange Approach-The DPM Model

Representation of the particle streams at the end of the injection (t=0.11 s), image shows the particles

coloured by its velocity magnitude. The particle streams are draw as spheres with proportional size

scaled 50 times more than the real diameter


Multiphase Flows

For bubbly, droplet, and particle-laden flows in which the phases mix and/or dispersed-phase volume fractions exceed 10% mixture or the Eulerian model

For slug flows & stratified/free-surface flows VOF model For pneumatic transport the mixture model for homogeneous

flow or the Eulerian model for granular flow For fluidized beds Eulerian model for granular flow For slurry flows and Hydrotransport the mixture or Eulerian For sedimentation the Eulerian model

Choosing a Multiphase Model based on the flow regime

The use of the DPM is limited to low volume fractions (less than or equal to 10% ), unless you are using the dense discrete phase model (DDPM) formulation. In addition, for the discrete phase model simulation, you can choose many more advanced combustion models compared to the Eulerian models.


Multiphase Flows Choosing a Multiphase Model - Mixture Or Eulerian model?

If accuracy is more important than computational effort, the Eulerian model is a better choice. However, the complexity of the Eulerian model can make it less computationally stable than the mixture model.

If there is a wide distribution of the dispersed phases (that is, if the particles vary in size and the largest particles do not separate from the primary flow field), the mixture model may be preferable (that is, less computationally expensive).

If the dispersed phases are concentrated just in portions of the domain, you should use the Eulerian model instead.

If interphase drag laws that are applicable to your system are available the Eulerian model can usually provide more accurate results than the mixture model.

if the interphase drag laws are unknown or their applicability to your system is questionable, the mixture model may be a better choice.

If you want to solve a simpler problem, which requires less computational effort, the mixture model may be a better option, since it solves a smaller number of equations than the Eulerian model.


Multiphase Flows

For very low loading, the coupling between the phases is one-way. The DPM , mixture, and Eulerian models can all handle this type of problem correctly. Since the Eulerian model is the most expensive, the discrete phase or mixture model is recommended.


Choosing a Multiphase Model based on Loading and St.

For high loading, there is two-way coupling plus particle pressure and viscous stresses due to particles (four-way coupling). Only the Eulerian model will handle this type of problem correctly


Multiphase Flows

For intermediate loading, the coupling is two-way. The DPM, mixture, and Eulerian models are all applicable in this case


Which one is better?

Example: For a coal classifier with a characteristic length of 1 m and a characteristic velocity of 10 m/s, the Stokes number is 0.04 for particles with a diameter of 30 microns, but 4.0 for particles with a diameter of 300 microns. Clearly the mixture model will not be applicable to the latter case.

Choosing a Multiphase Model based on Loading and St

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Multiphase Flows

Thanks

https://ir.linkedin.com/in/moammad-jadidi-03ab8399

[email protected]

Dr. Mohammad Jadidi(Ph.D. in Mechanical Engineering)

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Date post:	14-Feb-2017
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