NUMERICAL MODELING OF NUMERICAL MODELING OF MULTIPHASE PLUMES:MULTIPHASE PLUMES:

A COMPARATIVE STUDY BETWEENA COMPARATIVE STUDY BETWEEN TWO-FLUID AND MIXED-FLUID TWO-FLUID AND MIXED-FLUID

INTEGRAL MODELSINTEGRAL MODELS

presented bypresented by Tirtharaj BhaumikTirtharaj Bhaumik

Masters’ Thesis, Ocean Engineering Program, Masters’ Thesis, Ocean Engineering Program, Department of Civil Engineering, Texas A&M UniversityDepartment of Civil Engineering, Texas A&M University

supervised bysupervised byDr. Scott A. Socolofsky Dr. Scott A. Socolofsky (Chair),(Chair),

Dr. Kuang-An Chang Dr. Kuang-An Chang andand Dr. Yassin A. Hassan Dr. Yassin A. Hassan

OUTLINEOUTLINE

Multiphase Flow TerminologiesMultiphase Flow Terminologies Two-fluid and Mixed-fluid ModelsTwo-fluid and Mixed-fluid Models Governing EquationsGoverning Equations Graphical User InterfaceGraphical User Interface Initial ConditionsInitial Conditions Model Verification with Experimental DataModel Verification with Experimental Data Case StudiesCase Studies ConclusionConclusion

MULTIPHASE FLOW MULTIPHASE FLOW TERMINOLOGYTERMINOLOGY

Multiphase flows are fluid flows involving Multiphase flows are fluid flows involving the kinematics of more than one phase or the kinematics of more than one phase or constituentconstituent

Dispersed & Continuous PhasesDispersed & Continuous PhasesDispersed Phase : Dispersed Phase : Bubbles, Droplets, PowderBubbles, Droplets, Powder

Continuous Phase : Continuous Phase : Water, AirWater, Air

Jets:Jets: Driving force – Momentum flux of dispersed phaseDriving force – Momentum flux of dispersed phase

Plumes:Plumes: Driving force – Buoyancy flux of dispersed phase Driving force – Buoyancy flux of dispersed phase

Two-fluid and Mixed-fluid Two-fluid and Mixed-fluid modelsmodels

The Two-Fluid Model The Mixed-Fluid Model

Dispersed phase Continuous phase

Mixed Phase

Socolofsky & Adams, 2001 McDougall, 1978; Asaeda & Imberger, 1993

WHICH MODEL YIELDS BETTER ESTIMATES ?

Multiphase Flow ApplicationsMultiphase Flow Applications

Bubble BreakwatersBubble Breakwaters Antifreeze measures in HarborsAntifreeze measures in Harbors Bubble curtains for Oil spill containmentBubble curtains for Oil spill containment Reservoir and Reservoir and Lake Destratification Lake Destratification LakeLake and Aquarium and Aquarium AerationAeration COCO22 Sequestration in Ocean Sequestration in Ocean Blood flow modeling in bio-medical engineeringBlood flow modeling in bio-medical engineering Two-phase flow modeling in chemical industriesTwo-phase flow modeling in chemical industries Gas Stirring of molten metals in ladles, nuclear Gas Stirring of molten metals in ladles, nuclear

devices and chemical reactorsdevices and chemical reactors

H

hT

hP

Diffuser Source

Inner Plume

Outer plume

Ambient fluid (water)

Air bubble

SCHEMATIC OF AN AIR-BUBBLE PLUME IN STRATIFIED AMBIENT

DOUBLE PLUME MODEL OF ASAEDA & IMBERGER (1993)

(Mixed-fluid model)

H

hT

hP

Diffuser Source

Inner Plume(Mixed phase)

Outer plume(Single phase)

Ambient fluid (Water)

H

hT

hP

Diffuser Source

Bubble Core

Inner Plume(Dispersed phase + Continuous phase)

Outer plume(Continuous phase)

Ambient fluid

DOUBLE PLUME MODEL OF SOCOLOFSKY & ADAMS (2001)

(Two-fluid model)

MULTIPHASE PLUMES IN STRATIFIED MULTIPHASE PLUMES IN STRATIFIED ENVIRONMENTENVIRONMENT

LIF Image of a Type 3 plume

UN = us / (BN)¼

Top-Hat Distribution

1. SELF – SIMILARITY ASSUMPTION

X(r,z) = X(z), -b r b

= 0 elsewhere

X: variable of interest ( u, C, ∆ρ)

2. ENTRAINMENT HYPOTHESIS

e cu u

3. DILUTE PLUME ASSUMPTION

(1 ) 1c

Assumptions in Integral Models:Assumptions in Integral Models:

Control Volume

3-D to 1-D

1. Conservation of Volume flux 1. Conservation of Volume flux 2. Conservation of Momentum flux 2. Conservation of Momentum flux 3. Conservation of Buoyancy flux 3. Conservation of Buoyancy flux 4. Conservation of Temperature flux 4. Conservation of Temperature flux 5. Conservation of Concentration flux 5. Conservation of Concentration flux 6. Conservation of Salinity flux 6. Conservation of Salinity flux 7. Conservation of Mass flux of dispersed phase 7. Conservation of Mass flux of dispersed phase

Governing Differential equations:

2Q b u

2 2J b ua w

r

F Q g

H QT

C Qc

S Qs

2 ( )b s bW b c u u

General Form of the model equations:

Numerical Scheme : 4th order Runga-Kutta

…….. Coupled, non-linear ODEs

Primary variables (10 total): , , , , , , , , ,s w b bb u u c T c s d

Closure Equations (3 additional):

8. ( , , )w f T s p ……….. Sea-Water Equation of State

10. ( , , , , , )s b w b w bu f d ……….. Clift et. al (1978)

9. ( , , , )b f T s p Z ……….. Air/CO2 Equation of State

Flux variables (7 total): , , , , , , bQ J F H C S W

Equation Balance:

Two-Fluid Model : Buoyant forces on two distinct phases

Mixed-Fluid Model : Buoyant force only on a single mixed phase

1 2

1 2

1 2

min( , , )

0

min( , , )

0

min( , , )

0

( )( ) 2 ,

( )(1 )2 ,

( )2 ,

(1 )

b b b

a bdispersed s

r

b b b

a wcontinuous

r

b b b

a mixmixture

r

mix b w

B g u u c rdr

B gu c rdr

B gu rdr

c c

DIFFERENCE BETWEEN MIXED-FLUID AND TWO-FLUID MODELS

Control Volume(Axisymmetric)dA = 2πrdr

= Void Fractionc

DIFFERENCE BETWEEN MIXED-FLUID AND TWO-FLUID MODELS

Conservation of Momentum Conservation of Momentum fluxflux

…………… (Two-fluid model)

……………………………. (Mixed-fluid model)

1 20 iff 0 and su

Conservation of Buoyancy fluxConservation of Buoyancy flux

DIFFERENCE BETWEEN MIXED-FLUID AND TWO-FLUID MODELS

Mixture phase is transported at the velocity of the continuous phase !Slip velocity of the dispersed phase not properly accounted for !

…………… (Two-fluid model)

……………………………. (Mixed-fluid model)

DESIGNED GRAPHICAL USER INTERFACE

- (needed for each of the 7 flux variables in the model equations)- (needed for each of the 7 flux variables in the model equations) - Of these, Q and J must be non-zero, otherwise there is a divide by zero - Of these, Q and J must be non-zero, otherwise there is a divide by zero errorerror - However, u = 0 initially because a plume by definition has a zero initial - However, u = 0 initially because a plume by definition has a zero initial velocityvelocity - Initial values of other flux variables can be obtained from the ambient - Initial values of other flux variables can be obtained from the ambient propertiesproperties

First Inner First Inner PlumePlume

At release point At release point (diffuser level)(diffuser level)

Top of ZFETop of ZFE

Outer PlumeOuter Plume At the peeling At the peeling

locationslocations

Subsequent Subsequent Inner Inner plumesplumes

Just above the Just above the peeling locationspeeling locations

INITIAL CONDITIONS

H

hT

hP

Diffuser Source

Ambient fluid

Peel Height

Trap Height

FIRST INNER PLUME INITIAL CONDITIONS

OUTER PLUME INITIAL CONDITIONS

SUBSEQUENT INNER PLUME INITIAL CONDITIONS

1. VIRTUAL POINT SOURCE

CONCEPTCedarwall & Ditmars(1974),

McDougall(1978)

Single-phase plume equations are used up to

the top of ZFEin order to predict non-zero values of b and Um at the

start of computation (top of ZFE)

b

Z0 = 10D

z = 0 D

5D

5D

Location of virtual point source

Location of diffuser unit

Um

Initial values of Um and b are calculated here

Zone of Flow Establishment (ZFE)

Zone of Established Flow

ZFE

2. DENSIMETRIC FROUDE NUMBER CONCEPT

Wuest(1992)

Multiphase plume formulation in terms of the Froude number can be used to predict non-zero values of b and Um

at the start of computation

(diffuser level)

u = 0, b = 0

0.025*H

First Inner plume Initial conditions:

Fr = 0.8

Fr = 1.7

H

hT

hP

Diffuser Source

Ambient fluid

Peel Height

Trap Height

FIRST INNER PLUME INITIAL CONDITIONS

OUTER PLUME INITIAL CONDITIONS

SUBSEQUENT INNER PLUME INITIAL CONDITIONS

Extra Entrainment

OUTER PLUME INITIAL CONDITIONS

Fractional Peeling

EEXPERIMENTALXPERIMENTAL S SETUPETUP

U is measured using PIV and Ub is measured using PTV

Q0= 0.5, 1.0, 1.5 l/minBubble diameter = 3 mm

40 cm40 cm

70 cm

532 nm

4 ms

10-bit

10-bit

12-bit

PIV/PTV

4 mm thick

DETERMINATION OF MODEL PARAMETERS FROM EXPERIMENTAL DATA

= 0.09 = 1.17 = 0.2 m/s

MODEL RUNS WITH INITIAL CONDITIONS OBTAINED EXPERIMENTALLY:

Mixed-fluid model over-estimates momentum flux and continuous phase velocity

MODEL RUNS WITH MCDOUGALL’S (1978) INITIAL CONDITIONS

Not very useful when the depth H is large

MODEL RUNS WITH WUEST’S (1992) INITIAL CONDITIONS

No restrictions on water depth or diffuser diameter, hence more useful in general

MODEL RUNS WITH WUEST’S (1992) INITIAL CONDITIONS

The Two-fluid model with Wuest’s initial conditions matches Froude number best

Correlation of plume trap height to UN. Right-pointing triangles are datafrom Lemckert and Imberger (1993), circles are from Asaeda and Imberger(1993) and squares are from Socolofsky and Adams (2005). Open symbols areair-bubble experiments; closed symbols are glass bead experiments. Typicalerror bars are shown for one data point.

MODEL VERIFICATION WITH EXPERIMENTAL DATA IN A STRATIFIED AMBIENT

SENSITIVITY ANALYSIS FOR alpha

Model results are sensitive to alpha, alpha = 0.09 gives best match

SENSITIVITY ANALYSIS FOR gamma

Model results are insensitive to gamma, gamma = 1.2 is chosen

Entrainment Coefficient (alpha)Entrainment Coefficient (alpha) 0.090.09

Entrainment ratio (kappa)Entrainment ratio (kappa) 0.50.5

Momentum amplification factor Momentum amplification factor (gamma)(gamma)

1.21.2

Bubble spreading ratio Bubble spreading ratio (lambda_1)(lambda_1)

0.80.8

Initial conditions (First inner Initial conditions (First inner plume)plume)

WuestWuest

Selected Model Parameters

Case StudiesCase Studies

CASE 1:CASE 1: Lake DestratificationLake Destratification

CASE 2:CASE 2: Lake AerationLake Aeration

CASE 3:CASE 3: COCO22 Sequestration in Sequestration in oceanocean

BASE CASE PARAMETERS FOR THE THREE CASE STUDIES

ParameterParameter UniUnitt

CASE - 1CASE - 1 CASE - 2CASE - 2 CASE - 3CASE - 3

Release DepthRelease Depth mm 50.050.0 50.050.0 800.0800.0

Dispersed phase Dispersed phase diameterdiameter

mmmm 1010 22 55

Diffuser DiameterDiffuser Diameter mm 7.07.0 7.07.0 1.01.0

Diffuser Flowrate at the Diffuser Flowrate at the release depthrelease depth l/sl/s 12.012.0 6.06.0 1.11.1

Mass Transfer Mass Transfer Coefficient RatioCoefficient Ratio 1.01.0 1.01.0 0.50.5

UUNN (Non-dimensional (Non-dimensional slip velocity)slip velocity) 1.841.84 2.252.25 2.412.41

FIELD DATA FOR THE CASE STUDIES

LAKE (Cases 1 and 2) OCEAN (Case 3)

Source: Nepf (1995) Source: Teng et. al (1996)

Density and Compressibility are calculated from the above data using EQUATION OF STATE

Case 1.Case 1. Lake Lake DestratificationDestratificationMaintain water quality by artificial mixing,Maintain water quality by artificial mixing, Prevent surface ice formation Prevent surface ice formation in winterin winter

50m

N = f (z) N = 0

Initial state Final state

OBJECTIVE: Find optimal flow rate to achieve a well-mixed system

Q0 = 15.0 l/s (Two-Fluid model)

Q0 = 6.0 l/s (Mixed-Fluid model)

COMPARISON OF OPTIMAL FLOWRATE AS PREDICTED BY THE TWO MODELS

db = 10 mm

Case 2.Case 2. Lake Aeration Lake AerationOxygenation below hypolimnion to prevent lake eutrophication in Oxygenation below hypolimnion to prevent lake eutrophication in summersummer

Hypolimnion

Air-bubble plume

50m

Diffuser port

OBJECTIVE: Find optimal bubble diameter to dissolve all bubbles below hypolimnion

COMPARISON OF COMPUTED FLUX VARIABLES AS PREDICTED BY THE TWO MODELS

Mixed-fluid model predict higher values by about 30%

db = 2 mm, Q0 = 6 l/s

ESTIMATION OF DMPR BY THE TWO MODELS

Smaller the bubbles, faster they dissolveSmaller the bubbles, lower is the slip velocity and hence less is the difference in the two model results

Case 3.Case 3. CO CO22 SequestrationSequestrationReduce greenhouse gas concentration in atmosphere by dumping Reduce greenhouse gas concentration in atmosphere by dumping carbon di-oxide in deep ocean in liquid form, Monitor pH changecarbon di-oxide in deep ocean in liquid form, Monitor pH change

Phase change depth

Diffuser port

800 m

Multiphase plume of liquid CO2 droplets and

water

Ocean

Atmosphere

Pipeline

350 m

OBJECTIVE: Find best combination of Q0 and db to dissolve all droplets below phase change depth

COMPARISON OF COMPUTED FLUX VARIABLES AS PREDICTED BY THE TWO MODELS

Mixed-fluid model predict higher values by about 40%

db = 5 mm, Q0 = 1.1 l/s

ESTIMATION OF DMPR BY THE TWO MODELS

More the hydrate formation, lesser is the dissolution and hence higher is the DMPRMinimum Depth for hydrate formation: 450m in West Pacific, 820m in North Atlantic

CONCLUSIONSCONCLUSIONS The main difference between the mixed-fluid and the two-fluid The main difference between the mixed-fluid and the two-fluid

models lies in the formulation of the conservation equation for models lies in the formulation of the conservation equation for momentum fluxmomentum flux

In the existence of non-zero slip velocity and equal spreading In the existence of non-zero slip velocity and equal spreading ratios, the mixed-fluid model differs from the two-fluid model in ratios, the mixed-fluid model differs from the two-fluid model in the calculation of buoyant forces the calculation of buoyant forces

The model results are sensitive to the entrainment coefficient The model results are sensitive to the entrainment coefficient (alpha) and the entrainment ratio (kappa), but insensitive to the (alpha) and the entrainment ratio (kappa), but insensitive to the value of the momentum amplification factor (gamma)value of the momentum amplification factor (gamma)

Wuest’s method of obtaining initial conditions for the first inner Wuest’s method of obtaining initial conditions for the first inner plume using the concept of bubble Froude number is the most plume using the concept of bubble Froude number is the most reasonable and matches experimental data most closelyreasonable and matches experimental data most closely

Typically, the mixed-fluid model is found to predict higher values Typically, the mixed-fluid model is found to predict higher values of the peel height and the DMPR by about 30% as compared to of the peel height and the DMPR by about 30% as compared to the two-fluid modelthe two-fluid model

Numerical simulations done using the two-fluid model gives Numerical simulations done using the two-fluid model gives more physically realistic, economic and more accurate designsmore physically realistic, economic and more accurate designs

Thank you Thank you