Analysis and Modeling of Annular Liquid-Vapour Flo Summer School 2017 Annular Flow.pdf · Analysis...

Analysis and Modeling of Annular Liquid-Vapour Flow

Andrea Cioncolini

Assistant Professor/Lecturer in Thermal-Hydraulics School of Mechanical, Aerospace and Civil Engineering

University of Manchester-United Kingdom [email protected]

Andrea Cioncolini, University of Manchester UIT Summer School 2017

Aim: Present and discuss recently developed models for annular two-phase flow parameters

prediction (entrained liquid fraction, void fraction, wall shear stress, heat transfer coefficient).

Outline of this lecture: 1. Motivation;

2. Modeling approach;

3. Entrained liquid fraction (e) model;

4. Void Fraction (ε) model;

5. Pressure drop (DP) model;

6. Heat transfer (h) model.

•  Part of the liquid phase flows as a thin film along the tube wall;

•  Part of the liquid phase flows as entrained droplets dispersed in the

gas/vapor phase in the center of the tube;

•  Predominant flow pattern in evaporation and condensation;

•  ‘Friendly’ in applications: smooth operation, large heat transfer

capability,

Annular flow:


1.  Motivation: More accurate/reliable flow-pattern-based two-phase flow models required in several cutting-edge applications:

•  Nuclear Thermal-Hydraulics (power uprates, license extensions, fuel optimization, transient/safety analysis (best estimate in place of older conservative predictions));

•  Refrigeration & Air conditioning;

•  Petrochemical/Chemical Processing;

•  Microscale Heat Sink Design (CPUs, microelectronic components, lasers, particle detectors,…small devices that dissipate a lot of heat…);

More accurate predictions mean more sound and economically feasible designs.


2. Modeling approach: •  Experimental data-driven modeling: the first step was to put together a databank as large

and diversified as possible (currently 15,000+ data points from 100+ literature studies);

•  Adopt a ‘pragmatic’ approach:

•  Analyze trends in collected data and identify and resolve the most important phenomena only, and leave second-order effects and fine structure resolution to future developments;

•  Design minimal models that capture the most important phenomena (from above), and fit to the data: so what you get in the end is a semi-empirical prediction method;

•  Databanks collected from literature always noisy and contain outliers: so use robust statistics in fitting to minimize distortion from noise and outliers.

Advantages:

•  Simplicity (minimal models are simple);

•  Consistency (all models formulated consistently to work well together);

•  Scalability (all models can be extended/improved as new data become available).


Annular flow: closer look:

•  Highly dynamics;

•  Very irregular morphology of liquid-vapor interface;

•  Liquid film sheared and continuously atomized by the gas/vapor flow;

•  Accelerated droplets continuously deposited back on liquid film;

Ø  Tight mass/momentum coupling between liquid and gas/vapor.

‘’Realistic’’ annular flow in tube cross section (from G. Hewitt)

White: liquid; Grey: Gas/Vapor

‘’Idealized’’ annular flow: liquid droplets in gas + smooth liquid film on wall


Annular flow: influencing forces:

•  Inertia (in liquid film and gas core);

•  Viscous dissipation (in liquid film and gas core);

•  Surface tension (at liquid film interface);

•  Gas shear on liquid film;

•  Gravity (horizontal/inclined channels at low flow velocity);

Ø  In principle all strongly interacting (tight coupling between phases).

Additionally:

§  Turbulence modulation in vapor/gas (entrained droplets);

§  Turbulence modulation in liquid film (intermittency/waviness, atomization-deposition, evaporation-condensation).


Ø  Key player: aerodynamic interaction between liquid film-gas core, controlled by the competition between:

q  Disrupting aerodynamic force (gas shear on liquid film);

q  Surface tension retaining force (on liquid film).

Note: similarity with liquid atomization in spray: as a matter of fact, an annular flow can be regarded as an incomplete spray.

Ø  Consequence: Weber number (ratio of disrupting aerodynamic force/surface tension retaining force) controlling dimensionless group in annular flows (as in spray theory);

…in some form: multiplicity of flow parameters in two-phase flow no surprise… [The many densities of two-phase flow, in: Two-Phase Flow in Complex Systems, by S. Levy]


Ø  Consequence: Liquid and gas do not behave as their single-phase counterparts:

0 10 20 30 40 50 60 70 80 90 1000

5

10

15

20

25

30

35

Dimensionless Distance from Wall y+

Dim

ensi

onle

ss V

eloc

ity V

+

V+ = y+

Universal Velocity Profile

•  Data stick with laminar flow line longer than wall-bounded flow (up to y+≈9);

•  In turbulent flow velocity higher than single-phase wall-bounded flow (UVP).

Turbulence intensity in shear-driven annular liquid films is weaker than in single-phase flows, so that:

•  Turbulence does not reach as close to the channel wall as it does in single-phase flows;

•  The turbulent viscosity is lower (and the velocity correspondingly higher) than in single-phase wall-bounded flows.

[Indirect confirmation: single-phase flow theories extrapolated to annular liquid films overpredict the heat transfer coefficient]


1.  Cioncolini, A., Thome, J.R., Lombardi, C., Algebraic turbulence modeling in adiabatic gas-liquid annular two-phase flow, International Journal of Multiphase Flow 35 (2009) 580-596.

2.  Cioncolini, A., Thome, J.R., Lombardi, C., Unified macro-to-microscale method to predict two-phase frictional pressure drops of annular flows, International Journal of Multiphase Flow 35 (2009) 1138-1148.

3.  Cioncolini, A., Thome, J.R., Prediction of the entrained liquid fraction in vertical annular gas-liquid two-phase flow, International Journal of Multiphase Flow 36 (2010) 293-302.

4.  Cioncolini, A., Thome, J.R., Algebraic turbulence modeling in adiabatic and evaporating annular two-phase flow, International Journal of Heat and Fluid Flow 32 (2011) 805-817.

5.  Cioncolini, A., Thome, J.R., Void fraction prediction in annular two-phase flow, International Journal of Multiphase Flow 43 (2012) 72-84.

6.  Cioncolini, A., Thome, J.R., Entrained liquid fraction prediction in adiabatic and evaporating annular two-phase flow, Nuclear Engineering and Design 243 (2012) 200-213.

7.  Cioncolini, A., Thome, J.R., Liquid film circumferential asymmetry prediction in horizontal annular two-phase flow, International Journal of Multiphase Flow 51 (2013) 44-54.

8.  Mauro, A.W., Cioncolini, A., Thome, J.R., Mastrullo, R., Asymmetric annular flow in horizontal circular macro-channels: basic modeling of the liquid film distribution and heat transfer around the tube perimeter in convective boiling, International Journal of Heat and Mass Transfer 77 (2014) 897-905.

9.  Cioncolini, A., Del Col, D., and Thome, J.R., An indirect criterion for the laminar to turbulent flow transition in shear-driven annular liquid films, International Journal of Multiphase Flow 75 (2015) 26-38, 2015.

10. Thome, J.R., Cioncolini, A., Unified modelling suite for two-phase flow, convective boiling and condensation in macro- and micro-channels, Heat Transfer Engineering 37 (2016) 1148-1157.

11. Cioncolini, A., Thome, J.R., Pressure drop prediction in annular two-phase flow in macroscale tubes and channels, International Journal of Multiphase Flow 89 (2017) 321-330.

Detailed description of annular flow models:


3. Entrained liquid fraction (ELF):

e = mle

ml

mlf

mg

mle mle: mass flow rate of liquid as entrained droplets; mlf: mass flow rate of liquid in the film; mle+mlf=ml: total mass flow rate of liquid; mg: mass flow rate of gas/vapor;

Dimensionless transport parameter specific of annular flows (not defined for other flow

patterns), bounded between 0 and 1, that tells you what fraction of the liquid mass flow rate is transported as droplets, and what fraction is transported in the film:

e→ 0

e→1

Ideal annular flow: all liquid in film and no entrained droplets;

Mist flow: all liquid travels as entrained droplets;

Entrained Liquid Fraction


Sources of entrained liquid droplets:

Shearing-off of disturbance waves crests induced by faster travelling gas core, similar to wind-induced atomization of sea waves [dominant];

Impingement of a depositing droplet onto the liquid film and secondary atomization [secondary];

Bursting of a gas/vapor bubble at the liquid film interface [secondary];


ELF model: Cioncolini&Thome, International Journal of Multiphase Flow 36 (2010) 293-302 Cioncolini&Thome, Nuclear Engineering and Design 243 (2012) 200-213

State-of-the-art:

•  15-20 correlations available for e: based on limited databanks and/or unnecessarily complicated in their formulation;

•  4-5 mechanistic methods (rate of entrainment and deposition separately correlated, e from balance between the two) (rate of entrainment not accessible experimentally, rate of deposition yes but with compromises…Not enough data to use to full advantage at present).

Collected experimental databank details:

•  2460 data points from 37 literature studies; •  Adiabatic/evaporating annular flow, vertical upflow/downflow and horizontal flows; •  Tube diameters: 5.00-95.3 mm, circular/non circular (annulus and rod bundle); •  8 fluids (H2O, R12, R113, H2O-Air, H2O-Helium, Genklene-Air, Ehtanol-Air, Silicon-Air).


Trends in experimental data:

0.1 0.2 0.3 0.4 0.5 0.6 0.70.4

0.5

0.6

0.7

0.8

0.9

Vapor Quality

Ent

rain

ed L

iqui

d Fr

actio

n

G=1000 kgm-2s-1

G=2000 kgm-2s-1

0.1 0.2 0.3 0.4 0.5 0.6 0.70.4

0.5

0.6

0.7

0.8

0.9

Vapor Quality

Ent

rain

ed L

iqui

d Fr

actio

n

d=10 mmd=20 mm

H2O; 10 mm; 3 MPa [Würtz, 1978] H2O; 1000 kgm-2s-1; 7 MPa [Würtz, 1978]

H2O; 13.3 mm; x = 0.3 [Nigmatulin et al., 1977]

1 2 3 4 5 6 7 8 9 100

0.2

0.4

0.6

0.8

1

Pressure [MPa]

Ent

rain

ed L

iqui

d Fr

actio

n

G=500 kgm-2s-1

G=1000 kgm-2s-1

G=1500 kgm-2s-1

G=2000 kgm-2s-1Note: Effect of P flow conditions dependent.

e∝ x,G,d, 1P;P


Physical interpretation:

Data trends suggest that liquid atomization is driven by the aerodynamics interaction between liquid film and gas core: •  Interfacial shear drives the liquid film atomization; •  Surface tension resists atomization (tells how much energy is required to create interface).

⇒↑x Kinetic energy of core flow increases, so more shear and atomization (e ↑);

⇒↑G Kinetic energy of core flow increases, so more shear and atomization (e ↑);

⇒↑d Surface-to-volume ratio of tube more favorable for droplets to remain entrained longer (e ↑);

)()( ↑↓↓↓⇒↑ eandePl

g σρ

ρ Density ratio decreases means less slip and therefore less interfacial shear (e ↓); Surface tension decreases (saturated flows only) means easier to atomize liquid (e ↑); Two competing effects: so final effect (e ↑ or ↓) fluid and flow conditions dependent.

P↑⇒ρgρl↓(e↓)


Modeling/fittingModeling/fitting: :

Controlling dimensionless group in aerodynamics atomization of liquids:

Vcore

Vfilm

dcoreσ

ρ

σ

ρ dJdVVWe gcorecorefilmcorecore

c

22)(≈

−=

Vjet

Vgas

Vgas

djetσ

ρ jetjetgasg dVVWe

2)( −=

For example, for a liquid jet atomized by coaxial gas stream:

Extrapolating to annular flow yields the core flow Weber number:

Note: Approximation (gas superficial velocity Jg for relative velocity and tube diameter d for core diameter) generally good and in line with pragmatic approach.


101 102 103 104 1050

0.2

0.4

0.6

0.8

1

Core Flow Weber Number

Ent

rain

ed L

iqui

d Fr

actio

nPlot of data (e is dimensionless) vs. core flow Weber number just defined:

e = (1+ 279.6Wec−0.8395 )−2.209


•  Data points cluster on a sigmoid trend (physically plausible: at low e, an increase of e triggers an increase of Wec and this gives a positive feedback; at high e saturation due to liquid depletion; one of the most frequently observed trends in science (biomass living organisms, learning curves, chemistry,…));

•  Noise/scatter in databank (measuring e very challenging: available techniques quite invasive and of limited accuracy, no better than 20-30%);

•  Outliers in databank (data points that depart from the trend of the bulk of the data, 5-10% typical in large databanks);

•  Generalized logistic function selected to reproduce sigmoid trend (other options available…), and Robust fitting (robust statistics) used to get trend in bulk data and minimize distortion from outliers:


Example of robust fitting:

101 102 103 104 1050

0.2

0.4

0.6

0.8

1


Ent

rain

ed L

iqui

d Fr

actio

n

All DataClean Data

One step further: automatic outlier detection to provide interpretation of outliers: •  Part of the outliers are low quality data points, but part can be fine structure in data, not yet

properly resolved with current databank, so outliers could contain relevant information; •  After a robust fit has been obtained, removing the outliers and refitting gives an idea of how

robust the fitting actually was; •  Automatic outlier detection techniques available to isolate the outliers from the databank

(critical issue in data-mining: credit card fraud detection, athlete performance analysis, network intrusion, voting irregularities, data streaming (CERN: isolating the few collisions good to observe the Higgs boson),…);

•  Outlier detection on present databank (standard symmetric boxplot): reveals that most of the outliers come from a single experiment.


MAPE = 34.1%; 6/10 pts within ± 30% •  Statistically better than existing correlations; •  Good result, for a pragmatic and simple model

based on just one dimensionless group (and considering the high noise in the data).

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

1

Entrained Liquid Fraction: PredictedEnt

rain

ed L

iqui

d Fr

actio

n:E

xper

imen

tal

Cioncolini and Thome, (2012)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

1

Entrained Liquid Fraction: PredictedEnt

rain

ed L

iqui

d Fr

actio

n:E

xper

imen

tal

Ishii and Mishima, (1989)

MAPE = 39.9%; 5/10 pts within ± 30% •  Best correlation among the others considered; •  Overpredicts at high e, not good for dryout

modeling, unless going conservative is preferred (nuclear safety analysys).

Results (entire databank, outliers included):

e = tanh 7.25 ⋅10−7We1.25 Rel0.25( ) We =

ρg Jg2 d

σ

ρl − ρgρg

"

#$$

%

&''

0.33

; Rel =ρl Jl dµl

Ishii & Mishima (1989):


Results-non circular channels:

101 102 103 104 1050

0.2

0.4

0.6

0.8

1


Ent

rain

ed L

iqui

d Fr

actio

n

AdiabaticEvaporating

+ 30 %

- 30 %

102 103 1040

0.2

0.4

0.6


Ent

rain

ed L

iqui

d Fr

actio

n

No Grid SpacersEggCrate-UpstreamEggCrate-DownstreamULTRAFLOW-UpstreamULTRAFLOW-Downstream

+ 30 %

- 30 %

dhyd = 9 mm; drod = 17 mm; dtube = 26 mm P: 3-9 MPa; G: 400-3000 kgm-2s-1

Electrical heating, uniform heat flux (rod only, tube only and both rod/tube); (Data off have void fraction of about 0.7, so contamination from intemittent flow likely)

Rod bundle H2O-air data [Feldhaus et al., 2002]

drod = 9.5 mm; p = 12.6 mm; p/d = 1.326 dhyd = 9.35 mm (11.8 mm infinite bundle, wall effect)

P: 0.15 MPa; G: 300-450 kgm-2s-1

Margin of improvement of de-entraining vanes…


Good results: suggest that cross sectional shape of channel might be of second order importance on the entrained liquid fraction.

Annulus H2O data [Würtz, 1978]

Microscale: No data available yet, only a ‘proof of existence’ from Shedd (2010) with refrigerant R410a:

d = 1.05 mm; G: 600 kgm-2s-1

d = 1.05 mm; G: 200 kgm-2s-1

d = 0.508 mm; G: 400 kgm-2s-1

d = 0.508 mm; G: 800 kgm-2s-1

Note: Droplets size does not scale with tube diameter, so even a few droplets in a small tube can have an effect, and assuming ideal annular flow not appropriate.


4. Void fraction:

ε =AgA=

AgAl + Ag

Al

Ag

Al: cross sectional area occupied by the liquid phase; Ag: cross sectional area occupied by the gas/vapor phase; Al+Ag=A: total cross sectional area of the channel;

Void Fraction

Dimensionless geometric flow parameter, defined for any flow pattern (for annular flows is

normally above 0.7-0.8), bounded between 0 and 1, that tells you what fraction of the channel cross section is occupied by the liquid, and what fraction is occupied by the gas/vapor:

ε→ 0

ε→1

Single-phase liquid flow;

Single-phase gas/vapor flow;

ε ≥ 0.7− 0.8 Annular flow;


Void fraction model: Cioncolini&Thome, International Journal of Multiphase Flow 43 (2012) 72-84

State-of-the-art:

•  About ~100+ empirical or semi-empirical correlations available for ε;

•  Best predictions from drift-flux correlations (all flow patterns);

•  Drift-flux not OK for separated flows like annular flow (works best for unseparated flows and relative motion between phases induced by external force field (such as gravity)).


•  2673 data points from 29 literature studies;

•  Segregation: void fraction above 0.7 (contamination from intermittent flow minimal); •  Mostly adiabatic annular flow, vertical upflow; •  Tube diameters: 1.05-45.5 mm, circular/non circular (rectangular, annulus and rod bundle); •  8 fluids (H2O, R410a, H2O-Air, H2O-Argon, H2O-Nitrogen, H2O+Alcohol-Air, Alcohol-Air, kerosene-

air).



0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.70.75

0.8

0.85

0.9

0.95

1

Vapor Quality

Voi

d Fr

actio

n

G=500 kgm-2s-1

G=1000 kgm-2s-1

G=2000 kgm-2s-1

0 0.2 0.4 0.6 0.8 10.2

0.4

0.6

0.8

1

Vapor Quality

Voi

d Fr

actio

n

P=3.1 MPaP=1.9 MPa

R410a; 2.92 mm; 600 kgm-2s-1 [Shedd, 2010] (NB: consider only points above 0.7)

H2O; 20 mm; 7 MPa [Würtz, 1978]

H2O-Argon; 2.1 MPa; 1000 kgm-2s-1 [Alia et al., 1965]

0 0.2 0.4 0.6 0.8 1

0.7

0.8

0.9

1

Vapor Quality

Voi

d Fr

actio

n

d=25 mmd=15 mm Note: Effect of G within 6-7%, so not that

bigger than experimental errors...

ε∝ x, 1P,G


Physical interpretationPhysical interpretation:

⇒↑x More gas/vapor present, so more space needed to accommodate it (ε ↑);

⇒↑P Gas specific volume decreases, so gas occupies less space (ε ↓); For saturated flows surface tension decreases, so easier liquid atomization and behavior closer to homogeneous flow (ε ↑), so two competing effects, first typically dominant;

⇒↑G Higher mass flux means more tight momentum coupling between phases, so finer mixing of phases and behavior closer to homogeneous flow (ε ↑).

0 0.2 0.4 0.6 0.8 10.2

0.4

0.6

0.8

1

Vapor Quality

Voi

d Fr

actio

n

d=2.92 mmd=1.19 mm

R410a; 2.0 MPa; 800 kgm-2s-1 [Shedd, 2010]

Note: Potential effect of tube diameter in small/micro scale: smaller tubes more effective at confining the phases, so more difficult to develop a slip and behavior closer to homogeneous flow (ε ↑):

Result plausible but not confirmed yet (one research only), can delay the onset of annular flow in small channels.


Modeling/fitting:

•  Concentrating on the dominant effects of vapor quality and pressure:

Note: this minimal model contains the same information (vapor quality and density ratio) of the homogeneous model:

•  This yields a minimal model (maximum simplification possible, yet retaining the essential physics) for annular flows;

( )1, −≈ lgxf ρρε

•  The simplest dimensionless parameter used in two-phase flow to capture pressure effects is the density ratio ρg/ρl (OK for saturated and two-component flows), so:

⎟⎠

⎞⎜⎝

⎛≈P

xf 1,ε

εh = 1+1− xx

ρgρl

"

#$

%

&'

−1

=x

x + (1− x)ρg ρl−1


•  Plot of a selection of void fraction data vs. vapor quality and density ratio as parameter (all dimensionless parameters):

0 0.2 0.4 0.6 0.8 10.7

0.8

0.9

1

Vapor Quality

Voi

d Fr

actio

n

Water-air; 45.5mm; 0.1 MPaWater-air; 15.0mm; 2.1 MPaR410a; 1.05mm; 2.6 MPa


ε =h xn

1+ kxn; 0 < h,k 0 < n <1

o  Growing and saturating trend of void fraction with vapor quality, with rate of growth modulated by the density ratio;

o  Hill function selected to reproduce the trend (other options available; note that factorization into two functions NOT appropriate), and Robust fitting to capture the trend of the bulk of the data (wise as a general rule: outliers always present in large databanks).

17.0;110;10;)1(1

13 <<<<<<−+

= −− ερρε lgn

n

xxh

xh

5150.01

2186.01

)(6513.03487.0

)(129.3129.2−

−−

+=

+−=

lg

lg

n

h

ρρ

ρρ

•  Final correlation:

0 0.2 0.4 0.6 0.8 10.7

0.8

0.9

1

Vapor Quality

Voi

d Fr

actio

n

Density Ratio 10-3

Density Ratio 10-2

Density Ratio 10-1

Density Ratio 1

10-3 10-2 10-1 1000

5

10

15

Density Ratio

h10-3 10-2 10-1 1000.2

0.4

0.6

0.8

1

Density Ration


MAPE = 1.8%; 7/10 pts within ±2% •  Statistically better than existing correlations; •  Good result for a minimal model that depends

on vapor quality and density ratio only (minimal models pay off…).

MAPE = 2.3%; 5/10 pts within ±2 % •  Probably the best general purpose drift-flux

correlation currently available; •  Good clustering but a bit distorted at the top.

Results-entire databank:

0.65 0.7 0.75 0.8 0.85 0.9 0.95 1

0.7

0.8

0.9

1

Void Fraction: Predicted

Voi

d Fr

actio

n: M

easu

red Cioncolini & Thome (2012)

- 5 %+ 5 %

0.65 0.7 0.75 0.8 0.85 0.9 0.95 1

0.7

0.8

0.9

1


Voi

d Fr

actio

n: M

easu

red Woldesemayat & Ghajar (2007)

- 5 %+ 5 %

ε =Jg

C0 (Jl + Jg )+VdriftC0 =

JgJl + Jg

1+ JlJg

!

"##

$

%&&

n'()

*)

+,)

-); n =

ρgρl

!

"#

$

%&

0.1

Vdrift = 2.9gσ d (1+ cosϑ )(ρl − ρg )

ρl2

"#$

%&'

0.25

(1.22+1.22sinϑ )PatmP


Results-non circular channels:

0.65 0.7 0.75 0.8 0.85 0.9 0.95 1

0.7

0.8

0.9

1


Voi

d Fr

actio

n: M

easu

red

Jones&Zuber (1975)Hori et al. (1995)Takenaka&Asano (2005)Morooka et al. (1989)Das et al. (2002)

- 5 %

+ 5 % Fluids: H2O and H2O-air Cross sections: High aspect-ratio rectangular (1); Annulus (2); Inner subchannel of PWR (3); Rod bundle (4).

dhyd = 4.65-25.4 mm

P: 0.1-9.8 MPa; G: 140-3057 kgm-2s-1

Good results: suggest that cross sectional shape of channel might be of second order importance on the void fraction in annular flow.

1 2 3 4

0.65 0.7 0.75 0.8 0.85 0.9 0.95 1

0.7

0.8

0.9

1


Voi

d Fr

actio

n: M

easu

red Shedd(2010)

+ 5 %

- 5 %

Results-available microscale data:

R410a data [Shedd, 2010], the first reliable and verifiable microscale data that appear in the literature (capacitance measurement);

d = 1.05 mm; 2.96 mm

P: 1.9-3.1 MPa; G: 400-800 kgm-2s-1

Good results: suggest no major microscale effects on void fraction in annular flow down to about 1 mm.


5. Pressure Drop (DP):


In the study of channel two-phase flows it is customary to use a one-dimensional approximation for all space variables: all flow parameters are assumed to be constant in the channel cross section, and to be functions of only the axial coordinate along the channel axis z:

Z1 Z2

For the pressure in particular:

Pressure drop [Pa]: difference between the pressures at two arbitrary channel cross sections z1 and z2;

Specific pressure drop per unit channel length [Pa/m], informally called the pressure gradient: difference between the pressures at two arbitrary channel cross sections z1 and z2, divided by the distance between the cross sections.

DP model: Cioncolini et al., International Journal of Multiphase Flow 35 (2009) 1138-1148 Cioncolini&Thome, International Journal of Multiphase Flow 89 (2017) 321-330 (*)

Note: discussion here limited to most recent macroscale model (*); State-of-the-art:

•  Tens of empirical correlations available for frictional pressure gradient (τw);

•  Almost all correlations are flow-pattern independent (neglect peculiarities of flow regimes);

•  Many correlations based on ‘old ideas’, and look obscure and unnecessarily complicated in their formulation.


•  6291 data points from 13 literature studies; •  Adiabatic/evaporating flow, vertical upflow/horizontal flow; •  Tubes and annuli with diameters of 3-25 mm (macroscale); •  13 fluids (H2O, CO2, R12, R22, R134a, R245fa, R410a, R1234ze, H2O-Air, H2O-Ar, H2O-N2, Alcohol-

Ar, H2O+alcohol-Ar).



0 0.1 0.2 0.3 0.4 0.5 0.6 0.710

20

30

40

50

60

70

80

90

Vapor Quality

Pre

ssur

e G

radi

ent

[kP

am-1

]

P=3.0 MPaP=5.0 MPaP=7.0 MPa

H2O; 10 mm; 2000 kg/m2s [Würtz, 1978]

0 0.1 0.2 0.3 0.4 0.5 0.6 0.70

10

20

30

40

50

60

70

Vapor Quality

Pre

ssur

e G

radi

ent

[kP

am-1

]

G=1000 kgm-2s-1

G=2000 kgm-2s-1

G=3000 kgm-2s-1

H2O; 10 mm; 7.0 MPa [Würtz, 1978]

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.84

6

8

10

12

14

16

18

20

22

Vapor Quality

Pre

ssur

e G

radi

ent [

kPam

-1]

d=10 mmd=20 mm

H2O; 7.0MPa, 1000 kg/m2s [Würtz, 1978]

dPdz

∝ x, 1P, 1d,G



⇒↑x Flow accelerates, so steeper velocity gradient at wall (DP ↑);

⇒↑G

⇒↑d If mass flow rate kept constant means decrease in mass flux (DP ↓);

Density ratio decreases means flow decelerates, so milder velocity gradient at wall (DP ↓)

P↑⇒ρgρl↓

Flow accelerates, so steeper velocity gradient at wall (DP ↑);

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

20

40

60

80

100

120

140

Vapor Quality

Pre

ssur

e G

radi

ent

[kP

am-1

]

d=2.96 mmd=1.19 mmd=0.508 mm

(R410a; 1.9 MPa; 600 kg/m2s [Shedd, 2010])

Note: DP trend with vapour quality not always monotonic: sometimes max appears; possible explanations: •  Laminarization of liquid film flow; •  Suppression of atomization from the liquid

film; •  Liquid film looses continuity (dryout);

Note: turbulence likely to have an effect on DP trends, but turbulence structure in two-phase flow still largely unexplored.


Modeling/fitting:

Pressure drop predicted from integrating the one-dimensional linear momentum equation along the channel:


−dPdz

= 2 ftpρmVm

2

dhyd+G2 d

dz1ρm

⎛

⎝⎜

⎞

⎠⎟+ ρavg gsin(ϑ )

ρavg = 1−ε( )ρl +ερg

ρm =(1− e)2 (1− x)2 x

(1−ε)xρl − eε (1− x)ρg+e x (1− x)+ x2

ε ρg

⎡

⎣⎢⎢

⎤

⎦⎥⎥

−1

ftp =2τ wρmVm

2

Vm =Gρm

Two-phase Fanning friction factor

Momentum velocity

Average (cross-sectional) density

Momentum density for annular flow with 3 phases: liquid film, vapor/gas core, and entrained liquid droplets.

Note: void fraction and entrained liquid fraction models already discussed, so the only thing missing here is a closure relation for the friction factor.


In modeling the frictional DP, we assumed that the dominant effect was the aerodynamics interaction between the liquid film and the gas core (evaluated a posteriori with goodness of fit) this corresponds to the following minimal model for the wall shear stress:

τw = τw (ρm,Vm,dhyd,σ ) Vertical upflow

τw = τw (ρm,Vm,dhyd,σ ,g) Horizontal flow

The aerodynamics interaction between liquid film and gas core is controlled by the competition between:

•  Disrupting aerodynamics force that tends to atomize the liquid film: accounted for by a characteristic density and a characteristic velocity;

•  Surface tension retaining force that contrasts the liquid atomization: accounted for by surface tension and a characteristic length scale;

Turns out that the momentum density, momentum velocity and hydraulic diameter work best as characteristic scales (based on how well the models fit the available databank); these scales are suggested by the momentum equation written in dimensionless form (other tried as well…);

•  Acceleration of gravity included to account for possible gravity effects in horizontal flows.

Applying dimensional analysis to the minimal models for the wall shear stress yields:


ftp = ftp Wem( ) Vertical upflow

ftp = ftp Wem,Frm( ) Horizontal flow

Wem =ρmVm

2dhydσ

Frm =Vm2

gdhyd

where:

Momentum Weber number: captures the aerodynamics interaction between liquid film and gas core.

Momentum Froude number: takes into account the effect of gravity in horizontal channels:

Left: measured liquid film distribution in horizontal annular flow (H2O-air; d = 9.85 mm; G=412 kg/m2s)

Note: viscous dissipation in the liquid film (which would be captured by a Reynolds number) not critical in annular flow DP, at least with conventional scale channels (work in progress with microchannels).


Check/fit minimal models with experimental data:

ftp = 0.2140Wem−0.3884

ftp = 0.2140Wem−0.3884 0.1009Frm

0.6425

1+ 0.1009Frm0.6425

•  Vertical upflow:

•  Horizontal flow:

Results (entire databank):

Vertical upflow (top-left); horizontal flow (bottom-left); independent data on vertical evaporating flow (bottom-right) MAPE = 12.9%; 7/10 pts within ±15%


The key flow parameter in convective boiling/condensation is the heat transfer coefficient h:

h = qwallTwall −Tsat

•  qwall: heat flux exchanged at the channel wall;

•  Twall: temperature of the channel wall in contact with the flow;

•  Tsat: local saturation temperature of the flow

6. Heat Transfer Coefficient (HTC):


HTC model: Cioncolini&Thome, International Journal of Heat and Fluid Flow 32 (2011) 805-817

•  Heat and linear momentum transfer (have to be considered together) through the annular liquid film depend on the turbulence structure in the liquid film;

•  Sound approach to predict heat transfer in annular flow is therefore turbulence modeling in the liquid film;

•  Study limited to algebraic turbulence modeling (level ‘zero’ of turbulence models): the effect of turbulence is limited to increasing the fluid viscosity and thermal conductivity (position dependent in the flow).

State-of-the-art: •  Tens of empirical correlations available for the heat transfer coefficient (h);

•  Almost all correlations are flow-pattern independent (neglect peculiarities of flow regimes);

•  Many correlations based on ‘old ideas’, and look obscure and unnecessarily complicated in their formulation;

•  Most turbulence models for annular flow actually extrapolate single-phase boundary layer theory (with limited success, so turbulence structure similar but not the same).


0.2 0.3 0.4 0.5 0.6 0.7 0.83.5

4

4.5

5

5.5

6

6.5

7

7.5

Vapor Quality

Hea

t T

rans

fer

Coe

ffic

ient

[kW

m-2

K-1

]

G=424 kgm-2s-1

G=583 kgm-2s-1

G=742 kgm-2s-1

0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.6535

40

45

50

55

60

65

70

75

80

Vapor Quality

Hea

t Tra

nsfe

r C

oeff

icie

nt [k

Wm

-2K

-1]

d=4.99 mmd=9.18 mm


R22; 7.7 mm; 0.7 MPa; 25 kW/m2 [Shin et al. 1997] H2O; 1500 kg/m2s; 7 MPa; 210 kW/m2 [Bertoletti et al. 1964]

0.2 0.25 0.3 0.35 0.4 0.45 0.545

50

55

60

65

70

75

80

85

90

Vapor Quality

Hea

t Tra

nsfe

r C

oeff

icie

nt [k

Wm

-2K

-1]

qw=210 kWm-2

qw=860 kWm-2

h∝ x,G, 1d,qwall


H2O; 1500 kg/m2s; 7 MPa; 4.99 mm [Bertoletti et al. 1964]

Note: effect of heat flux qwall only if nucleation at wall is active, effect of pressure not large with water.

0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.650

50

100

150

200

250

300

Vapor Quality

Hea

t Tra

nsfe

r C

oeff

icie

nt [k

Wm

-2K

-1]

P=4.1 MPaP=5.5 MPaP=6.9 MPaP=8.3 MPa

H2O; 1100 kg/m2s; 5.2 mm; 1000 kW/m2 [Silvestri et al. 1963]


R22, P=0.7 MPa, G=742 kg/m2s, d=7.7 mm

0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.654.5

5

5.5

6

6.5

7

7.5

8

8.5

Vapor Quality

Hea

t Tra

nsfe

r C

oeff

icie

nt [k

Wm

-2K

-1]

qw=18 kWm-2

qw=25 kWm-2

qw=30 kWm-2


⇒↑x Flow accelerates, so steeper temperature gradient at wall (h ↑);

⇒↑G

⇒↑d If mass flow rate kept constant means decrease in mass flux (h ↓);

Bubbling activity at wall more intense, so more heat absorbed by flow (creating a bubble requires energy), and more intense local mixing (h ↑)

Flow accelerates, so steeper temperature gradient at wall (h ↑);

qwall ↑⇒

Note: turbulence likely to have an effect on HTC trends (similarly to pressure drop), but turbulence structure in two-phase flow still largely unexplored. Note: effects of all influencing parameters strongly interdependent, so interpretation of experimental trends particularly challenging.


Turbulence modeling in liquid film:

Starting from Navier-Stokes Eqs. and making ‘standard’ assumptions for RANS and algebraic turbulence modeling (incompressible flow, constant thermo-physical properties, Newtonian behavior, steady-state, time-smoothed, cylindrical symmetry, Boussinesq assumption,…) yields:

( )Rr

drdV

wtl τµµ −=+

( )rRq

drdTkk wtl =+

τw

qw

rT(r)V(r)

y

Two 1° order ODE for the velocity V(r) and temperature T(r) profiles, where:

•  µl and kl are the liquid viscosity and thermal conductivity;

•  µt and kt are the turbulent viscosity and thermal conductivity;

•  τw and qw are the wall shear stress and the heat flux;

•  R is tube radius and r the radial coordinate;

Note: nucleation at the tube wall NOT considered, only convective evaporation.


Switching r → y (radius → wall distance), going dimensionless and adding IC (no-slip and imposed wall temperature Tw ):

( ) ( ) 00;0;11 =≤≤−=+ ++++

+

+

++ Vty

Ry

dydV

tν

( ) ( ) ++++

−

+

+

+

++ =≤≤⎟⎟

⎠

⎞⎜⎜⎝

⎛−−=+ wt TTtyRy

dydT 0;0;11

1

α

Two Cauchy problems for the (dimensionless) velocity and temperature profiles, where:

Turbulent eddy diffusivity for momentum; lower bounded at 1 (laminar flow)

Turbulent eddy diffusivity for heat; lower bounded at 1 (laminar flow)

l

tt µ

µν =+

l

tt k

k=+α

(t is average liquid film thickness)

To solve the Cauchy problems, two closure models for these eddy diffusivities are required (substitute in above eqs and then integrate).


In algebraic turbulence modeling, the eddy diffusivities are normally modeled as:

( )++ = lftν

( )lt lg Pr,++ =α

where:

•  l+ is the length scale characteristic of the problem (ideally, it should capture the size of turbulent eddies typical of the flow of interest) ;

•  Prl is the liquid Prandtl number.

So, turbulence modeling (in this approximation) is finally reduced to two operations:

•  a) Fix the length scale of the problem (l+);

•  b) Define the relations f and g that define the eddy diffusivities (by empirically fitting experimental data).


Vcore

Region close to the tube wall: •  Flow is mostly affected by the tube wall (wall-bounded flow); •  l+ ~ y+ (distance from wall, analogous to single-phase wall-bounded flow

theory)

a) Length scale: we assumed that:

Region away from tube wall: •  Flow is mostly affected by the interaction with shearing

gas core flow (fluid-bounded flow); •  l+ ~ t+ (liquid film thickness, analogous to jets and wakes)

Buffer region: •  To allow smooth transition between the two other zones: •  l+ ~ (y+, t+) (intermediate between two previous cases)

Physically, this means that the liquid film is assumed to be mostly affected by the interaction with the gas core, except very close to the tube wall (this is the main assumption of model).


b) Functional relations f and g:

•  The ‘standard’ way to define f and g is through comparison with experimental data: measured velocity profiles and measured temperature profiles;

•  These data are currently not available (few researches available on velocity profile, more qualitative than quantitative; nothing on temperature profiles);

Available experimental information:

•  Adiabatic flows: t (average liquid film thickness), τw (wall shear stress, from dP/dz);

•  Evaporating/condensing flows: Tw (wall temperature), Tb (fluid bulk temperature, typically derived from pressure profile reconstruction), qw (wall heat flux).

Crude approximation to advance using available data: neglect wall and buffer regions:

( )++ = tftν

( )lt tg Pr,++ =α

The eddy diffusivities are thus independent of variable of integration y+, and ODEs can be solved analytically.


+++

+

+

++

++ ≤≤

+≈⎟⎟

⎠

⎞⎜⎜⎝

⎛−

+= tyy

RyyV

tt

0;12

)(11 2

νν

Solution for the velocity profile:

( )++ = tftνand:

Linear profile in y+ in thin liquid film approximation t << d (typically OK).

Imposing mass conservation for the liquid film:

2**

2

)(2)1)(1(;

2)(

yVxetR

llf

lft πρ

νΓ−−

=ΓΓ

≈ ++

+++

The right-hand side can be calculated using:

•  Available adiabatic data for (t , τw);

•  Entrained liquid fraction prediction method already discussed (for e).

This yields the eddy diffusivity (in crude approximation) as function of liquid film thickness, and allows calibrating the turbulence model for momentum transfer.

(Γlf+: dimensionless liquid film mass flow rate)


101 10210-1

100

101

Dimensionless Liquid Film Thickness

1+E

ddy

Diff

usiv

ity

30030;1033.0

30;0

<<−=

<=+++

++

ttt

t

t

ν

ν

Results: •  Good clustering for a crude approximation; •  Noise at low t+ due to near-wall effects that

have been neglected;

•  Linear trend typical of shear-driven flows (jet and wakes), so length scale selected looks promising (far from wall).

•  Rearranging yields a prediction method for the liquid film thickness t+:

Collected experimental adiabatic databank details (t , τw):

•  1146 pts from 11 literature studies; •  Adiabatic annular flow, vertical upflow; •  Tube diameters: 15.1-31.8 mm, circular tubes only; •  5 fluids (H2O, H2O-Air, H2O-Argon, H2O-Nitrogen, H2O+Alcohol-Argon).

t+ =maxRelf2; 0.0165 Relf

!

"##

$

%&&


Solution for temperature profile:

+++

++

+

+

+

+++ ≤≤

+−≈⎟⎟

⎠

⎞⎜⎜⎝

⎛−

++= tyyT

RyRTT

tw

tw 0;

11ln

1 αα

Linear profile in y+ in thin liquid film approximation t << d (typically OK). Assuming the liquid film interface is at saturated temperature yields:

1−−

≈++

++

bwt TT

tα

The right-hand side can be calculated using:

•  Available diabatic data (Tw, Tb, qw);

•  Estimating (t+) from velocity profile just discussed;

•  Entrained liquid fraction model already discussed for (e);

•  Friction model just discussed for (τw).

This yields the eddy diffusivity (in crude approximation) as function of liquid film thickness and Prandtl , and allows calibrating the turbulence model for heat transfer.


Collected experimental diabatic databank details (Tw, Tb, qw):

•  1311 pts from 8 literature studies; •  Evaporative only annular flow, vertical upflow and horizontal flow; •  Tube diameters: 1.03-14.4 mm, circular tubes only; •  9 fluids (H2O, R12, R22, R32, R134a, R290, R600a, R245fa, R236fa).

Results:

101 102 103

100

101

Dimensionless Liquid Film Thickness

(1+E

ddy

Diff

usiv

ity)/P

r l0.52

•  Good clustering for crude approximation; •  Rearranging yields a prediction method for

the heat transfer coefficient:

80010;1Pr)(0776.0

10;052.090.0 <<−=

<=+++

++

ttt

lt

t

α

α

Nu = h tkl= 0.0776(t+ )0.9 Prl

0.52


R22 d = 7.7 mm; Tsat: 285 K; qw: 25 kWm-2;

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 14

5

6

7

8

Vapor Quality

Hea

t Tr.

Coe

ff. [

kW/m

2 K]

G=424 kg/m2sG=583 kg/m2sG=742 kg/m2s

0.19 0.2 0.21 0.22 0.23 0.2413.5

14

14.5

15

15.5

16

Vapor Quality

Hea

t Tr.

Coe

ff. [

kW/m

2 K]

qw=46 kW/m2

qw=71 kW/m2

qw=100 kW/m2

H2O d = 14.4 mm; Tsat: 417 K; G: 135 kgm-2s-1

Note: no effect of heat flux, consistent with wall nucleation suppressed.

Results-selected evaporation data:

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

2000

4000

6000

8000

Vapor Quality

Hea

t Tr.

Coe

ff. [

W/m

2 K]

G=200 kg/m2sG=300 kg/m2sG=400 kg/m2sG=750 kg/m2s

R134a d = 8.0 mm; Tsat: 313 K;

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

2000

4000

6000

8000

Vapor Quality

Hea

t Tr.

Coe

ff. [

W/m

2 K]

G=200 kg/m2sG=400 kg/m2sG=600 kg/m2sG=750 kg/m2s

R22 d = 8.0 mm; Tsat: 313 K;

Results-selected condensation data:


Conclusions: •  Annular flows controlled by the aerodynamics interaction between the liquid film and the

gas/vapor core, which is the key player; •  Significant need for more experiments to fill-in gaps and better resolve little covered

parametric regions; •  Great need of new experiments and new experimental techniques for small scale

applications (non-invasive and meaningful measurements challenging);

Suggestions for Modeling: •  Start as simple as possible (but not simpler…), add complexity on the go as needed; •  Use large and diversified experimental databanks for fitting; •  Try not to have ‘rigid’ expectations (the eyes see (and the ears hear) what the brain wants to

see (or hear)); •  If possible, use the knowledge from similar problems to understand your problem; •  Do use modern techniques borrowed from data-mining and robust statistics to handle data;


Analysis and Modeling of Annular Liquid-Vapour Flo Summer School 2017 Annular Flow.pdf · Analysis...

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