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Predicting Multidimensional U

Annular Flows

with a Locally Based

Two-Fluid Model

SP Antal, DP Edwards, and TD Strayer

Lockheed Martin Corporation, Schenectady, NY

Goal of Work

0 Develop a methodology to predict annular flows using a multidimensional four-field, two-fluid Computational Fluid Dynamics (CFD) computer code.

0 Develop closure models which use the CF'D predicted local velocities, phasic volume fractions, etc...

0 Implement a numerical method which allows the discretized equations to have the same characteristics as the differential form.

0 Compare predicted results to local flow field data taken in a R-134a working fluid test section.

Conservation of Phasic Mass

Conservation of Phasic Momentum

a at z z z 1 1 1 1 Z -(a .p *u .) + V.( a *p .u .u .) = - a .vp + v.zi

+ M . + c D. . (U - u . ) + r . . u - r . . u Z ZJ j J Z int ij int j f i

i - Field Indicator, i= cl (continuous liquid), i= dl (dispersed liquid) i = cv (continuous vapor), i = dv (dispersed vapor)

h? i - Non-drag Interfacial Momentum Forces

r D * * - Interfacial Drag Coefficient between phase ‘5” and phase “j”

- Mass Transfer from phase ‘5” to phase “j” ij

ZJ ‘int - Interfacial Velocity

ai7 p i y Mi7 ‘ i - Phasic volume fraction, density, velocity and shear stress

Three Field Model of Annular Flows

Vapor Core Flow (cv field)

V @ ( a c v ~ c v u cv ) = O

CV u u ) = - a Vp+V.z V@(acVPcv cv cv CV

Liquid Film Flow (cl field)

+ M C l + D e l - cv(% - ' e l ) + ' d l - clUdl+ ' c l - dl'cl

Droplet Flow (dl field)

+ Mdl + Ddl - cv('cv - ' d l ) + ' c l - dl'cl + ' d l - cl'dl

Two-Fluid Models Applied on Local Basis

Entrainment and Deposition of Droplets

0 Interfacial Area Density

Interfacial Shear (cv-cl)

0 Dispersive Annular Flow Force

0 Collective Annular Flow Force

Entrainment and Denosition of Droplets

Apply Model from Kataoka and Ishii (1983) on a Local Basis

= G A ' c l - dl E C Z - C V

= G A rdl - cl D C Z - C V

-7 0.74 0.925 = 6.6a 10 Recl W e GEDh

P C l

Local Velocity, Volume Fraction

- PclaclUclDh

P C l -

Interfacial Area Density

e Assume Thin Smooth Films

Interfacial Shear Force

Apply Model from Wallis (1969) on a Local Basis

int 1 z cv-cl = - f o p 2 1 cv IU cv-ucl~(ucv-ucz)

= 0.005 f l

Dispersive Annular Flow Force Thin Film Dynamics Controlled by Collectlte and Dispersive Interfacial Forces

a Apply Simplified Kelvin-Helmholtz Stability Analysis to Thin Film to show form for Dispersive Force

a Unsteady Flow Equations (u = U + u’) for Vapor (i=cv) and Film (i=cl)

n

Dispersive Annular Flow Force

0 Solve in terms of Potential Functions and Wave Function

@ Solution takes the form

-kY k y ikx+st @i' = (Ai" +B.e 1 )e

0 Growth Factor,s = is . + s (Normalized Results)

Evaluated by Substitution Z r s

- S = -ik

- @ cv coth(k( 1 - h ) ) + hclcoth(kh)

fi coth ( i t ( 1 - h ) ) + coth ( i h )

0 pi3 We

- -L I pk coth(i( 1 - h))coth(ki) f ( b coth(i( 1 - h ) ) + coth(ih)) 2 pcoth(k(1-h)) + coth(kh)

Dispersive Annular Flow Force

Film becomes unstable when Radicand is greater than zero Stability Condition Becomes,

- k > We coth ( k ( 1 - h ) ) coth ( k h )

coth( k( 1 - h ) ) + coth ( k h )

Short Wave Length Limit k i < < 1

2 x t h<- We - Short Wave length can be Stable

0 Long Wave Length Limit khs 1

2 x t h<- m - Long Wave Length can be Unstable (Bernoulli Effects)

0 Model Long Wave Length Dispersive Effects Proportional to a Bernoulli Pressure Force

MDisp 111

= c D isp p c v I 'cv - ~cll~Acl - cv f i int cl- cv

Collective Annular Flow Force 0 Apply Wall Force Model to Vapor Core

to show form for Collective Force

Liquid Flow

The Wall Force in Dispersed Flows has the Form

wa21 l2 f i C a p u - u W dv I cv cl

Rbubble - - Wall Force

c l - dv A4

0 The Collective Force uses Same Form Scaled by Duct Thickness

wall 2 c coll a cl a cv P CV l U c v - U c l

- - Collective c l - cv t

A4

Model Implementation in CFD Code

Finite Volume Discretization Scheme

Segregated, Pressure Based Algorithm

Non-Staggered Grid

0 Subcooled and Saturated Boiling Flows

0 Bubbly, Transitional, and Annular Flows

Numerical Implementation

Improved Pressure-Velocity Coupling for Volume Fraction Terms

W t

Average of Nodal Velocities t coupling

Rhie-Chow Interpolation

t Velocity Volume Fraction

- More Detail in “A Coupled Phase Exchange Algorithm for Three-Dimensional Multi-Field Analysis of Heated Flows with Mass Transfer”, Computers and Fluids, Submitted to Computer and Fluids for publication

Numerical Implementation

Fully Developed without Velocity-Volume Fraction Coupling

U. Y/t 1. Thickness Dimension

Fully Developed with Velocity-Volume Fraction Coupling

Fraction Coupling

f Thickneis Dimension

Numerical Implementation

Discretization of Volume Fraction Gradient Term

e e e W P e

F p a W

F a + P -

F e F a = a + a W - F + F W F + F P

W P W P E F + F e F + F aP

e P e P

n 1 1 F = 1 r = F = e a,(l-ae) P a (1 -a ) aw(l-aw) P P

0 Limiting Conditions for Volume Fraction Gradient Term

I I I I

I I

l>a>O l>a>O l>a>O aa

a Y - < l /Ay

Numerical Implementation

Require Model to Predict Fully Developed Annular Flows

- Under Fully Developed Conditions the Collective and Dispersive Forces Must Balance

= o COll cl- cv + M MDisp

cl- cv

Substitute Collective and Dispersive Models

3

c coll a cl a cv P cv pcv-UclIL t - 'Disp PcvI 'cv

Simplify,

= o aa t - c Dispay a a coll el cv C

Y

May Not Sum to Zero For Fully Developed Flows

Numerical Testing Confirmed Force Imbalance

Numerical Implementation

Require Model to Predict Fully Developed Annular Flows

- Strategy: Develop Sensor,S for Fully Separated Flows

Condition 1: aa aY

Fully Separated Conditions Imply: - = 1/by

1- MDisp

c l - cv #isp

c l - cv aa aY - = l /Ay

< 1

Condition 2: Collective Force Must Be Larger than

Dispersive Force Evaluate with: - = 1/Ay aa aY

J

MColl c l - cv

< 1

Numerical Implementation

Require Both Conditions to be Met

Collective c l - cv M t "waZl

09

0,95

1.0

L

c 0 - c i! is

G

0.0 ill

0.5

1.0

5 Distribution a Distribution el

Results Compare Against Local Volume Fraction Measurements in Lockheed Martin Corporation's R-134a Test Facility - Adiabatic Flow Conditions

- SUVA' working fluid

- Duct - GDS Measurements -170 Hydraulic Diameters Downstream of Inlet

- Homogenous Mixture Entering Test Section - Gamma Densitomer Measurements (GDS)

Front View Port #5 (Fixed TC Rake) - P13

-P12

-P11 Port #4 (Blank) - P10

-P9

-P8 Port #3 (Blank) - P6

- P5

- P4 Port #2 (Blank) - P3

- P2

-P1 Port #1 (Fixed TC Rake) m 4 4 4 3-zone Inlet

Saturated Inlet Flow

1. SUVA@ is a trademark of Dupont Chemicals for R-134a

Results

8 Compare over Range of Mass Flows, Pressures and Met Qualitie,

Table 1 : Test Conditions

Inlet Quality

1 51 2. 2380. 0.5998

Inlet Mass Inlet

(Kg,lm2-s) kPa Case Flux Press u re

2 I 1032. I 1361. I 0.1639 I 3 I 1994. 1 1361. I 0.4360 I 4 I 2040. I 2380. I 0.4905 I

Predicted Phasic Volume Fraction for Case 2

cl Field cv Field

4 4 4 4-b Uniform, Homogenous Thickness

Dimension Inlet Flow

dl Field

4-b Thickness Dimension

Results Predicted Velocities and Volume Fraction (x/L = 170 hydraulic diameters)

- 0 - 0.2 0.8 a8

Dgtance ylt 1

- 0 ' 0.2 - a6 0.8

Di;ance yh 1

Case 3 0 0.2 0.8

D iiia nccy/t 1

Case 4 J

0 0.2 06

D igfanccy/t 1

3.5 cv Field 7

0 0.2 0.4 0.6 0.8

Distance ylt

4 -

h

g5: x 3 c. .- 0 0 2 5 - -

cl Field Case 2 2 .

1.5 0 0.2 0.4 0.6 0.8 1

Distance y/t

cv Field A.

- 0 0.2 0.4 06 0.8 1

Distance y/t

-0 0.2 0.4 0 6 .-OS I

Distance y/t

Results Predicted Film Thickness and Average Volume Fraction

0 . 6 - 0.045 . Case 2 - aw. E

so.035 . am.

B 0.025 case 1 ’ 0

0 0 a2 a4 0.6 OB 1

Axial Distance WU

- t i . . . . . . * . .

Measured Average Void Fraction

Collective and Dispersive Lateral Force for Case 2

E 0 F e! 6 t G

I Collective Force

Dispersive Foree

Results Predicted Void Distribution and Film Thickness for Inclined Flow (Inclined 450 from Vertical). (Inlet Flow Conditions are same as Case 2)

Distance ylt

Lower Surface

5 0.036.

8 0.025 . I Upper Surface g 0 . 0 3 . d

0.02.

- E O.M6

0.M

0.006

I

I 1 01 0.2 0.4 0.8 0.8 0

Axial Distance W U

Uniform, Homogen Inlet Flow

Liquid Volume Fraction Phase Distribution for 450 Inclined Annular Flow

Results Predicted Void Distribution and Film Thickness for Horizontal Flow. (Met Flow Conditions are same as Case 2)

Lower Surface

Upper Surface

0.5 0.7s 1 “ ~ “ ‘ * ‘ “ ‘ ‘ D a ’

Distance y/t

1 .o

0.5

0.0

Gravity

I Uniform, Homogenous *

Inlet Flow -w *

Liquid Volume Fraction Phase Distribution for Horizontal Annular Flow

Conclusions

With Proper Modeling and Numerical Treatment, Annular Flows can be Predicted with a Multidimensional, Two-Fluid Model

Annular Flow Models Developed From Simplified Analysis of Identified Force Mechanisms

0 Numerical Treatment of Annular Flow Models - Void Gradient Treatment - Improved Pressure-Velocity Treatment - Requirements for Fully Developed Annular Flows

0 Comparisons against Local Volume Fraction Data - Good Predictions for Local Volume Fraction - Good Predictions for Average Volume Fraction

Model Predictions for Inclined Orientations - 45' Inclined Annular Flows - Horizontal Orientation Annular Flows

Additional supporting information on what will be discussed

Topic: Dispersive Annular Flow Force

The approved paper states, “Simple inviscid (Kelvin-Helmoltz) analysis of vertical separated flows indicate that while surface tension is able to stabilize small wavelength disturbances, Ber- noulii pressure effects tend to make the long wavelength disturbances unstable by increasing the pressure along the wave trough and decreasing the pressure along the wave crest.” The presenta- tion will include three overheads summarizing the supporting analysis. A summary of the planned discussion is:

Apply Simplified Kelvin-Helmholtz Stability Analysis to Thin Film to Show Form for Disper- sive Force

- Discuss wavy thin film flowing on a wall. - Amplitude of thin film may grow (unstable) or decay (stable) for various wave lengths.

- Discuss writing single phase Navier-Stokes equation on each side of the thin film. - Expand equations by dividing the velocity and pressure into steady and unsteady terms. - Crank through math and arrive at unsteady flow equations (as done in many text books).

- Discuss use of potential function and wave function to solve for film amplitude

Unsteady Flow Equations for Vapor (i=cv) and Film (i=cl)

Solve in terms of Potential Functions and Wave Function

growth factor. Film becomes unstable when Radicand is greater than zero

- If growth factor has positive real value then amplitude will grow with time. - If growth factor has negative real value then amplitude will decay (stable interface).

-Discuss looking at limiting conditions, short and long wave lengths. Analysis shows that short wave length will be stable but long wave lengths can be unstable (i.e., can grow with time)

Model Long Wave Length Dispersive Effects Proportional to a Bernoulli Pressure Force - Discuss long wave length growth are related to pressure difference on each side of the film interface. This effect can be modeled as a Bermoulli pressure force times the interfacial area density.

Short Wave and Long Wave Length Limit

Topic: Collective Annular Flow Force

The approved paper states, “This effect, which is generalized to take the form of an aerodynamic lift force applied to the interface can be written as:” The presentation will include some support- ing information. A summary of the planned discussion is:

Apply Wall Force Model to Vapor Core to show form for Collective Force - Discuss wall having significant influence on location of vapor between parallel walls - Simplify process to vapor slug traveling between parallel walls

- Discuss modeling vapor slug as large bubble having the influence of each wall - Use wall force model (available in open literature) to model effects of walls on the vapor

The Wall Force in Dispersed Flows has the Form

field

2

The Collective Force uses Same Form Scaled by Duct Thickness - Discuss use of wall force model to prescribe form of collective force model

Topic: Model Implementation in CFD Code:

The paper states, “The multifield, two-fluid models have been implemented into a segregated, pressure based algorithm to solve the system of equations and to predict the phase distribution and velocity profiles (see Siebert et al., 1994).” To better describe the code the following will be sum- marized.

Finite Volume Discretization Scheme Segregated, Pressure Based Algorithm Non-Staggered Grid Subcooled and Saturated Boiling Flows Bubbly, Transitional, and Annular Flows

- These are general attributes of the code and will better convey the results to the audience. - The code uses a standard finite volume discretization scheme on a non-staggered grid. - The code has capability to predict flows from single phase heating, through high

volume fraction annular flows. The code has capability to model subcooled and saturated boiling flows and can model bubbly flows coalescing into annular flows.

Topic: Improved Pressure-Velocity Coupling for Volume Fraction Terms

The paper states, “The requirement to conform to the realizable limits (Oaolume fractiond) and their restrictions on implementation into a discretized computational model is a process by which any model(s) must endure. However, most other models do not produce sharp void fraction gradi- ent; therefore, a more general numerical treatment is adequate.” Included in the discretized com- putational model is the need to improve the pressure-velocity coupling. This is an important ingredient to adequately predicting annular flows. The discussion will summarize the need to include numerical treatment to improve annular flow predictions. A example of fully developed adiabatic annular flows with and without the treatment is included in the presentation. In addition, an approved paper (A Coupled Phase Exchange Algorithm for Three-Dimensional Multi-Field Analysis of Heated Flows with Mass Transfer, RF Kunz et al.) has been submitted to the Journal of Computer and Fluids for publication which will also present and discuss improved pressure- velocity coupling. A reference to this submitted paper will be included in the presentation.

Topic: Results

The results were limited in the paper by the conference restriction of a maximum length of eight pages. The presentation will include results presented in the paper with additional supporting information. All results are taken from the analysis of the four cases included in the approved paper. The planned presentation of the results will include:

Predicted Phasic Volume Fraction for Case 2 - color contours (included in paper) Predicted Velocities and Volume Fraction - line plots for each of 4 cases (included in paper Predicted Film Thickness and Average Volume Fraction - line plots for each of 4 cases (included in paper) Collective and Dispersive Lateral Force for Case 2 - color contour plot (included in paper)

Predicted Void Distribution and Film Thickness for inclined Flow (Inclined 45' from Verti- cal). - line plots of film thickness (included in paper) and color contour plot of liquid fraction distribution. Predicted Void Distribution and Film Thickness for Horizontal Flow. - line plot of film thick- ness and color contour plot of liquid film fraction