NHT Asinari Multiphase v1.5

8/12/2019 NHT Asinari Multiphase v1.5

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DEPDEPARTMENT OFARTMENT OF ENERGETICSENERGETICS

Numerical Heat Transfer

Multiphase Flows:Basic Physics and

Engineering Modeling

Pietro Asinari, PhD

Spring 2007, TOP UIC Program: The Master ofScience Degree of the University of Illinois

at the Politecnico di Torino


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NHT: Multiphase Flows

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Outline of this Section

Basic physics of multiphase flows:

flow patterns;

flows around droplet and bubble;

evolution equation for bubble dynamics.

Discrete phase models (DPM): particle based (Lagrangian) formulation.

Continuous phase models (CPM):

singlefluid approach (Mixture models, VolumeofFluid VOF, Cavitation models);

twofluid approach (Eulerian Eulerian models, and

Eulerian granular models).


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Air / Water Mixtures in Vertical Pipe

Basic Physics of Multiphase Flows


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Multiphase Flows

Multiphase flow is used to refer to any fluid flow

consisting of more than one phase (*) (or component)

We exclude those circumstances in which thecomponents are well mixed above the molecular level

Consequently, the flows considered here have somelevel of phase or component separation at a scale wellabove the molecular level This still leaves anenormous spectrum of different multiphase flows

(*) Phase: in thermodynamics, a chemically and physically uniformquantity of matter that can be separated mechanically from a non homogeneous mixture. It may consist of a single substance or of a

mixture of substances.



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Disperse Flows: Trajectory Models

Disperse flows are those consisting of finite particles, drops

or bubbles (the disperse phase) distributed in a connectedvolume of the continuous phase

In trajectory models, the details of the flow around eachof the particles are subsumed into assumed drag, lift and

moment forces acting on and altering the trajectory ofthose particles The following considerations hold: particle-particle interactions and the effects of the particle volume

fraction on the gas phase are negligible (this issue implies that the

discrete phase must be present at a fairly low volume fraction,usually less than 10 %);

the thermal history of the particles can also be tracked if it isappropriate to do so in this case, different mass transferphenomena are affecting the convective and radiative heat transfer.



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Disperse Flows: Twofluid Models

In twofluid models: the disperse phase is treated as a

second continuous phase intermingled and interacting withthe continuous phase The following considerations hold

effective conservation equations (of mass, momentum and energy)are developed for the two fluid flows these included interaction

terms modeling the exchange of mass, momentum and energybetween the two flows;

the two-fluid models neglect the discrete nature of the dispersephase and approximate its effects upon the continuous phase (this

implies a fairly high volume fraction, usually larger than 10 %); inherent in this approach, are averaging processes necessary to

characterize the properties of the disperse phase these involvesignificant difficulties (for example, appropriate boundary

conditions).



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Separated Flows

Separated flows are those where two different streams

can readily travel at different velocities The essentialfeatures of this type of flows are:

the relative motion (and consequently, from the modeling point ofview, the relative velocity);

the separating interfaces (which implies that they must betracked in the computational domain).

These flows can be modeled by means of

singlefluid (or mixture) models, where the main stream (?!) ismodeled first and then the relative drifts;

twofluid models, where the single phase fluid flow equations isapplied in the two streams, coupling them through appropriatekinematic and dynamic conditions at the interface.



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Flow Patterns

From a practical engineering point of view one of the

major design difficulties in dealing with multiphase flowis that the mass, momentum, and energy transfer ratesand processes can be quite sensitive to the geometricdistribution or topology of the phases within the flow

The geometry may strongly effect the interfacial areaavailable for mass, momentum or energy exchangebetween the phases The flow within each phase willclearly depend on that geometric distribution

Hence we recognize that there is a complicated two-way coupling between the flow in each of the phasesand the geometry of the flow

A particular type of geometric distribution of the

components is called a flow regime (or flow pattern)



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Air / Water Mixtures in Horizontal Pipe

Entrance lengthsfor multiphaseflow patterns are

not wellestablished

Several possibleflow patterns

may depend onthe initialconditions



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Flow Regime Map in Horizontal Pipe



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Air / Water Mixtures in Vertical Pipe



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Flow Regime Map in Vertical Pipe


Multiphase Transition


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Instabilities of Disperse Flows

actually lead to multiphase transitions (D S):

the turbulent mixing (and the consequent coalescence)is larger than the separation processes driven by thedensity difference causing phase separation (by gravityor, in a non-uniform or turbulent flow, by the

Lagrangian fluid accelerations); the wave perturbations may destroy a homogeneous,

quiescent multiphase mixture because it may beinternally unstable as a result of gravitationally-induced

relative motion

As they grow in amplitude thesewave-like volume fraction perturbations seem to evolvein several ways depending on the type of flow and themanner in which it is initiated (Jackson instability)



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Instability of Separated Flows

actually lead to multiphase transitions (S D):

1. the interface may become unstable when waves form onthe interface between the two fluid streams these arecalled KelvinHelmoltz instabilities and they depend onthe considered driving force:

a) buoyancy force due to gravity (Rayleigh-Taylor instability);

b) Bernoulli effect that implies a change in the pressure acting on theinterface caused by a change in velocity (Bernoulli instability).



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Multiphase Heat Transfer



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Boiling (Main) Phenomena

Horizontal Surfaces

Pool Boiling: it is when a pool of liquid is heated frombelow through a horizontal surface

Nucleate Boiling: it is when temperature differenceactivates increasingly smaller (and therefore more

numerous) sites

Film Boiling: it is when at or near boiling crisis a film of

vapor is formed that coats the surface and substantiallyimpedes heat transfer

Vertical Surfaces

Film Boiling: qualitatively similar to that on a horizontalsurface except for the upward liquid and vapor velocities

caused by natural convection



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Pool Boiling

Buoyancy

Surface Tension

Heat transfer enhancement

cold liquid is pushedtowards the heating wall

movement of bubblesenhances mixing

heat pipe effect

Boiling Crisis !!



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Nucleate Boiling

Nucleation: increased temperature difference activates

increasingly smaller (and therefore more numerous) sites The ebullition cycle consists of

the bubble growth period, which is directly related to the rate ofheat supply per single bubble

the moment of detachment, when the upward buoyancy forcesexceed the surface tension forces at the bubble-wall contact line

the waiting period, due to the temporary local cooling of the wall



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Film Boiling

At or near boiling crisis a film of vapor is formed that coats

the surface and substantially impedes heat transfer

This vapor layer presents the primary resistance to heattransfer

These flows are usually quite unsteady since the vapor /

liquid interface is unstable (Rayleigh-Taylor instability)Vapor bubbles are introduced into the liquid and travelupwards while liquid droplets are also formed and falldown through the vapor toward the hot surface (thedroplets do not normally touch the hot surface becausethe vapor created on the droplet surface nearest the wallcreates a lubrication layer that suspends the droplet the

Leidenfrost effect)



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Film Boiling in Vertical Surfaces


Boiling on a heated

vertical surface isqualitatively similar tothat on a horizontalsurface except for theupward liquid andvapor velocitiescaused by natural

convection For a well-defined

elevation, boilingcrisis can appear


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Film Condensation in Vertical Surfaces


The spectrum of flow processes

associated with condensation on asolid surface are almost a mirrorimage of those involved in boiling

Drop condensation on the underside

of a cooled on a vertical surface isvery analogous to nucleate boiling When the population of dropletsbecomes large they run together toform condensation films

The phenomenon is most apparent asthe misting up of windows or mirrors.


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Discrete Phase Models (DPMs)


In addition to solving transport equations for the

continuous phase, this approach consists of simulating adiscrete second phase in a Lagrangian frame ofreference, made of discrete particles (which may betaken to represent droplets or bubbles) dispersed in the

continuous phase Essentially the key idea is to compute the trajectories of

these discrete phase entities, as well as heat and masstransfer to/from them The coupling between thephases and its impact on both the discrete phasetrajectories and the continuous phase flow can beincluded by means of proper closure models

Acceptable for low volume fraction (less than 10 %)


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Very Dilute Multiphase Flows


One-way coupling: quite accurate solutions are then

obtained by solving a single phase flow for thecontinuous phase and inputting those fluid velocitiesinto equations of motion for the particles.

1. It is sufficient to solve for the velocity and pressure

of the continuous suspending fluid while ignoring theparticles or disperse phase.

2. Once this solution is given, one could then solve an

equation of motion for the particle to determine itstrajectory.

Two-way coupling: as the concentration of the dispersephase is increased, complications can effect both the

continuous phase flow and the disperse phase motions


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Flows around a Sphere (Low Re)


Suspending Flow


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Flows around a Sphere (High Re)


Suspending Flow

Drag Crisis


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Deformation due to Translation



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Particle Force Balance in Fluent



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Particle Growth and Collapse


Unlike solid particles or liquid droplets, gas / vapor

bubbles can grow or collapse in a flow and in doingso manifest a host of phenomena with technologicalimportance (in particular for combustion)

Let us consider the fundamental dynamics of a

growing or collapsing bubble in an infinite domain ofliquid that is at rest far from the bubble

The assumption of spherical symmetry is violated in

several important processes, but it is necessary tofirst develop the baseline dynamics of clouds ofbubbles or of bubbly flows


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Rayleigh Plesset Equation


(1) Instantaneous driving term

(2) Thermal term

(3) Non condensable gases

(4) Inertial term

(5) Viscous resistance

(6) Surface tension


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(1) Inertially Controlled Dynamics


First we consider some of the characteristics of bubble

dynamics in the absence of any significant thermaleffects This kind of bubble dynamic behavior istermed inertially controlled to distinguish it from thethermally controlled behavior discussed later Under

these circumstances the temperature in the liquid isassumed uniform


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Isothermal Oscillating Bubbles


Cavitation ?!

NHT M l i h Fl


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(2) Thermally Controlled Dynamics


When thermal deriver in Rayleigh Plesset equation is

dominant, then an additional relationship between thetemperature and radius dynamics must be provided

In some cases, the effects due to decompression in thebubble growth can be neglected and the practical

formula are highly simplified Several other factors can complicate and alter the

dynamics of thermally controlled bubbles:

the presence of a nearby wall;

nonequilibrium effects (high evaporation rates);

surface instabilities (bubble surfaces are usually rough andirregular);

convective heat transfer;...

NHT M lti h Fl


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Complete System of Equations


We need to correlate T(t) with R(t) the generalized

conduction equation in the liquid phase is highly nonlinear no exact analytical solution exists

Also the relationship between p(t) and R(t) is complicatedby the fact that the actual temperature is changing in time

NHT Multiphase Flows


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(3) Diffusion Controlled Dynamics


In most of the circumstances, it is assumed that the

events occur too rapidly for significant mass transfer ofnoncondensable gas to occur between the bubble andthe liquid Actually all of the gasfilled microscopicbubbles that are present in a subsaturated liquid (and

particularly in water) should dissolve away if theambient pressure is sufficiently high (Henrys law)

The process of mass transfer can be modeled bymeans of a proper concentration equation



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Cavitation


Cavitation occurs in flowing liquid systems when the

pressure falls sufficiently low in some region of the flowso that vapor bubbles are formed.

Parsons (1906) recognized the role played byvaporization The phenomenon has been a subject of

intensive research ever since because of the adverse effects it has on performance,

because of the noise it creates,

and, most surprisingly, the damage it can do to nearby solid

surfaces.

Cavitation number:



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Heat & Mass Transfer in Fluent


Inert heating/cooling

Droplet vaporization

Droplet boiling

Devolatilization

Surface combustion



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Back Coupling in Fluent


Tracking of heat, mass, and momentum gained or lost

by the particles with known trajectories allows one totake into account the effects with regards to thecontinuous (back coupling or two way coupling)

The changes of the particles which pass through each

control volume are summed up



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Continuous Phase Models (CPMs)

The dispersed and the continuous phases are combinedtogether and modeled as a new, continuous phases




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Definitions: Volumetric Quantities




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Definitions: Mass Weighted Quantities




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Definitions: Drift Quantities




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Equations of Motion

We need to assume that there exists an infinitesimalvolume of dimension, such that

it was not only very much smaller than the typical distanceover which the flow properties varied significantly (necessaryin order to define derivatives of the flow properties within theflow field);

but also very much larger than the size of the individual phaseelements, i.e. the disperse phase particles, drops or bubbles(each averaging volume contain representative samples ofeach of the components or phases).

Above two conditions are rarely both satisfied Theaveraging volumes contain a finite number of finite-sized particles and therefore flow properties varysignificantly from point to point




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Individual Phase Continuity Equation




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Individual Phase Momentum Equation




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Difficulties for Energy Equation


In single-phase flows it is usually adequate to assumethat the fluid is in an equilibrium thermodynamic state atall points in the flow and that an appropriatethermodynamic constraint (for example, constant andlocally uniform entropy or temperature) may be used to

relate the pressure, density, temperature, entropy, In many multiphase flows the different phases and/or

components are often not in equilibrium andconsequently thermodynamic equilibrium arguments

that might be appropriate for single phase flows are nolonger valid Under those circumstances it isimportant to evaluate the heat and mass transferoccurring between the phases and/or components



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Relaxation Times

Velocity relaxation time tU: it is a characteristic time thattypifies the natural attenuation of velocity differencesbetween the phases Downstream of some disturbancethat creates a relative velocity, the drag will tend toreduce that difference

Temperatures relaxation time tT: it is characteristic timeassociated with the equilibration of temperatures throughthe process of heat transfer between the phases It canbe obtained by equating the rate of heat transfer from the

continuous phase to the particle with the rate of increaseof heat stored in the particle: the heat transfer to theparticle can occur as a result of conduction, convection,radiation,




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Individual Phase Energy Equation




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Homogeneous Flows




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Condensation Shocks




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Non Ideal Phenomena



C i Ph M d l (CPM )


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Mixture Sound Speed


Air (G)Water (L)


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Volume of Fluid (VOF) Model


The VOF model can model two or more immisciblefluids by solving a single set of momentum equationsand tracking the volume fraction of each of the fluidsthroughout the domain

The VOF formulation relies on the fact that two or more

fluids (or phases) are not interpenetrating For eachadditional phase that you add to your model, a variableis introduced: the volume fraction of the phase in thecomputational cell (in each control volume, the volume

fractions of all phases sum to unity)




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VOF: Interpolation near the Interface





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Mixture Model


The mixture model, like the VOF model, uses a singlefluid approach. It differs from the VOF model in tworespects:

the mixture model allows the phases to be interpenetratingthe volume fractions q and p for a control volume can thereforebe equal to any value between 0 and 1;

the mixture model allows the phases to move at differentvelocities, using the concept of slip (or drift) velocities (note thatthe phases can also be assumed to move at the same velocity,and the mixture model is then reduced to a homogeneous

multiphase model).




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Drift Velocity





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Eulerian Model


The Eulerian multiphase model allows for the modeling

of multiple separate, yet interacting phases. Theessential features of this model are:

a set of conservation equations for momentum, continuity and(optionally) energy is individually solved for each phase;

the single phase volume fractions are introduced and someproper scalar equations are considered (individual phasecontinuity equations);

some proper coupling terms are introduced in the previous

equations among the phases the mechanisms for theexchange of momentum, heat, and mass between the phasesare modeled by means of phenomenological correlations(interphase exchange coefficients)




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Interphase Exchange Laws




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Summary

A fully satisfactory description and understanding of

multiphase flows is still missing and this explains thelarge set of phenomenological correlations used tocharacterize different flows in different geometricalconfigurations

The key problems are the following: different phases and/or components are often not in equilibrium

(non negligible relaxation times), and this rises difficulties inevaluating the heat and mass transfer occurring among the

phases (usual thermodynamic assumptions may not apply);

moreover, flow properties, such as the continuous phase velocity,vary significantly from point to point and averaged flow propertiesare not suitable to compute the mean of the gradients



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Further Readings

V.P. Carey, Liquid Vapor Phase Change

Phenomena, Taylor & Francis, 1984. C.E. Brennen, Fundamentals of Multiphase

Flow, Cambridge University Press, 2005.

http://caltechbook.library.caltech.edu/51/01/content.htm

M.L. Corradini, Fundamentals of MultiphaseFlow, Department of Engineering Physics

University of Wisconsin, USA.http://wins.engr.wisc.edu/teaching/mpfBook



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Acknowledgements

Prof. C. E. Brennen Department of

Mechanical Engineering, CaliforniaInstitute of Technology, USA.

Documents

NHT Asinari Multiphase v1.5