R. Balasubramaniam, E. Ramé, J. Kizito, and M. Kassemi

National Center for Space Exploration Research, Cleveland, Ohio

Two Phase Flow Modeling: Summary of Flow

Regimes and Pressure Drop Correlations in

Reduced and Partial Gravity

NASA/CR—2006-214085

January 2006
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R. Balasubramaniam, E. Ramé, J. Kizito, and M. Kassemi

National Center for Space Exploration Research, Cleveland, Ohio

Two Phase Flow Modeling: Summary of Flow

Regimes and Pressure Drop Correlations in

Reduced and Partial Gravity

NASA/CR—2006-214085

January 2006

National Aeronautics and

Space Administration

Glenn Research Center

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EXECUTIVE SUMMARY

The purpose of this report is to provide a summary of state-of-the-art predictions

for two-phase flows relevant to Advanced Life Support. The main focus is to identify

gaps in predictive capabilities in partial gravity. The motivation is that the long

duration missions being considered cannot afford to carry all the needed water for

the trip. Consequently, recycling processes for water and other life support resources

must be used. The operation of these processes must be validated for the relevant

gravity conditions (e.g., zero for the trip, and the planetary gravity for the length of

the stay there). Naturally, this is not a comprehensive review. We strive to pick out

the most used and accepted models for pressure drop and flow regime prediction. As

expected, there is a great history of work that applies to terrestrial conditions. More

recently, a good number of works (but very small compared to those for terrestrial

gravity) have addressed two-phase flows in microgravity conditions. A gap is evident

in the lack of information on how to predict flows in partial gravity, especially for

Lunar and Martian applications. Filling this knowledge gap will be the basis for the

recommendations for future work that we will develop following this report.

After a summary of flow regimes and pressure drop correlations for terrestrial and

zero gravity, we analyze the fully developed annular gas-liquid flow in a straight cylindrical tube. This flow is amenable to analytical closed form solutions for the flow field

and heat transfer. In this flow, the gas flows in the center and the liquid forms an

annulus next to the tube wall. No gas bubbles are present in the liquid, nor liquid

drops in the gas. In this respect, our analysis differs from real annular flows where

typically liquid droplets are entrained by the gas core. Therefore, these solutions

may be used as baselines and guides to which realistic measurements in similar flows

may be compared. We have also attached in an appendix the material we submitted

in a previous short report concerning the flow regimes likely to be encountered in

the water recovery equipment currently under consideration for space applications.

The report is a follow-up on recommendation developed in the workshop by Chiaramonte and Joshi, 2004 where issues relevant to (i) air revitalization (ii) water recovery

and (iii) thermal systems, particularly of interest to the Advanced Life Support were

discussed.

NASA/CR—2006-214085

iii 

NASA/CR—2006-214085 v

Table of Contents 

Nomenclature ................................................................................................................. vii 

1 Introduction to two-phase flow............................................................................... 1 

 1.1 Background ..................................................................................................... 1 

 1.2 Description of two phase flow......................................................................... 3 

 1.3 Current knowledge base for reduced gravity .................................................. 3 

 1.3.1 Background on flow regime ............................................................. 3 

 1.3.2 Background on Pressure Drop.......................................................... 5 

 1.3.3 Background on Heat Transfer .......................................................... 6 

 1.3.4 Background on Computational Modeling ........................................ 6 

2 Dimensionless parameters in two-phase flow........................................................ 8 

 2.1 Use of the Buckingham Pi Theorem ............................................................... 8 

 2.2 Scaling Analysis of Two Phase Flow.............................................................. 9 

 2.3 Parameter Range on Current Life Support Hardware ..................................... 10 

3 Flow Regimes in Reduced Gravity ......................................................................... 12 

 3.1 Description of flow regimes ............................................................................ 12 

 3.2 Jayawardena et al. [1] reduced gravity flow regime map................................ 12 

 3.2.1 Bubble-slug transition ...................................................................... 15 

 3.2.2 Slug-annular transition ..................................................................... 15 

 3.3 Other flow regime transition criteria ............................................................... 15 

 3.3.1 Gas void fraction criterion................................................................ 15 

 3.3.2 Weber number criterion.................................................................... 16 

 3.4 Flow regime transition mechanisms................................................................ 16 

 3.4.1 Mechanisms for bubble-slug and slug-annular transition ................ 16 

 3.4.2 Alternative mechanistic approaches for predicting regime 

 transitions ......................................................................................... 18 

4 Flow Regimes in Partial Gravity ............................................................................ 19 

5 Pressure Drop Predictions ...................................................................................... 24 

 5.1 Homogeneous fluid model .............................................................................. 24 

 5.1.1 Pressure drop due to flow acceleration............................................. 24 

 5.1.2 Pressure drop due to gravity............................................................. 24 

 5.1.3 Pressure drop due to wall friction..................................................... 24 

 5.2 Separated flow model...................................................................................... 25 

 5.2.1 Pressure drop due to flow acceleration............................................. 26 

 5.2.2 Pressure drop due to gravity............................................................. 26 

 5.2.3 Pressure drop due to wall friction..................................................... 26 

 5.3 Pressure drop in bubbly flow........................................................................... 27 

 5.4 Pressure drop in slug flow ............................................................................... 28 

 5.5 Pressure drop for fully developed, laminar, annular flow............................... 29 

 5.5.1 Pressure drop due to flow acceleration............................................. 29 
NASA/CR—2006-214085 vi

 5.5.2 Pressure drop due to wall friction in zero gravity ............................ 30 

 5.5.3 Pressure drop due to wall friction in partial gravity......................... 34 

 5.5.4 Pressure drop in turbulent annular flow ........................................... 34 

 5.6 Droplet entrainment in the air stream in annular flow .................................... 34 

 5.7 Experimental measurements of pressure drop in annular flow ....................... 35 

6 Heat Transfer in Fully Developed Annular Flow ................................................. 36 

7 Conclusions............................................................................................................... 39

Appendix A—Fully developed, Laminar, Annular Flow........................................... 41 

 A.1 Core phase center relative to pipe center......................................................... 43 

Appendix B—Heat Transfer in Fully Developed Annular Flow............................... 45 

 B.1 Criterion for phase in contact with wall .......................................................... 47 

Appendix C—Pressure Drop by Integral Momentum Balance with Droplet 

 Entrainment in the Air Stream............................................................................... 51 

Appendix D—Flow regimes in ALS equipment operating in microgravity............. 53 

 D.1 Dimensionless Groups for Current Space ALS Hardware.............................. 53 

 D.2 Flow regime estimation in two-phase ALS technologies operating 

 in Microgravity................................................................................................ 54 

 D.3 Discussion ....................................................................................................... 55 

References....................................................................................................................... 57 
Nomenclature

α denotes the thermal diffusivity m2/s

β thermal conductivity ratio

∆p Pressure drop along the pipe N/m2

m ̇ total rate of mass flow per unit area in the pipe Kg/s

GA area averaged local void fraction of the gas

μG gas viscosity N.s/m2

μH effective viscosity N.s/m2

μL liquid viscosity N.s/m2

φ angle between the vertical the flow direction

ρH effective density Kg/m3

ρef f effective density kg/m3

ρG gas density Kg/m3

ρL liquid density Kg/m3

σ Surface Tension N/m

τW wall shear stress N/m2

θ contact angle of the liquid-gas interface at the solid

h ̃ heat transfer coefficient W/m2.K

Bo static Bond number

cpGgas heat capacity J/kg.K

cpL

liquid heat capacity J/kg.K

Ca Capillary Number

D pipe diameter m

F gravitational parameter

fent fraction of liquid drops entrained

F r modified Froude number

g gravitational acceleration m/s2

gz component of gravity along the axis of the tube m/s2

NASA/CR—2006-214085

vii
H mean curvature of the interface 1/m

h constant gas core thickness m

IntGas Integral of the gas core in eq. 40

IntLiq Integral of the liquid in eq. 40

kL liquid’s thermal conductivity W/m.K

M gas to liquid viscosity ratio

Nu Nusselt number

P

0

G physical pressure in the gas phase N/m2

P

0

L physical pressure in the liquid phase N/m2

P e P ́eclet number

q

∗

constant heat flux per unit wall area W/m2

QGf gas volume flow fraction

QG liquid flow rates m3/s

qLD liquid flow rate that is flowing as drops m3/s

QLF liquid flow rates m3/s

QL liquid flow rates m3/s

R radius of pipe m

ReG gas Reynolds number

ReL liquid Reynolds number

S Slip ratio

Su Suratman number ≡ Re/Ca = Re2/W e

t constant liquid film thickness m

U total superficial velocity m/s

UGS superficial gas velocity m/s

ULS superficial liquid velocity m/s

w

∗ axial velocity field m/s

W e Weber number

x quality (the gas mass flow fraction)

X2ratio of liquid-to-gas frictional pressure gradients with each phase flowing alone

in the pipe

NASA/CR—2006-214085

viii
1 Introduction to two-phase flow

Two-phase flows involving liquids and gases regularly occur in advanced life support

systems operating in space environments. These flows cannot be avoided; therefore

there is a need to understand how these flows can potentially affect the performance

and longevity of life support hardware. Two-phase flows are critical to regenerative

life support technologies, specifically water and atmosphere management. In these

flows, the pressure drop and the rates of transfer of heat and mass are strongly influenced by the flow regimes which in turn are controlled by the relative flow rates and

the dimensions of the component carrying the fluid mixture.

The ultimate goal of this project is to develop predictive analytical and computational tools for two-phase flow with and without phase change in low and partial-g

environments, for applications to Advanced Life Support. The tools will be applicable not only for design but also for predicting performance of space systems from

ground-based test data.

Two-phase flow of gas and liquid, and multiphase flow of gas, liquid and solid have

been identified and given high priority in the final report of a workshop on critical issues in microgravity fluids, transport and reaction processes (Chiaramonte and Joshi,

2004 [4]). This workshop developed several important recommendations for research

in applications to Advanced Life Support.

In this report we have reviewed and cataloged adiabatic two-phase flow behavior

in a pipe. Predictions for the flow regime in normal and reduced gravity, and available correlations for the pressure drop and rate of heat transfer are summarized. To

write this report we have have used information from published knowledge, analysis

of basic equations that describe flow and information gathered from technical reports

by ETH 2005 [5] and Lahey et al. 2003 [6].

1.1 Background

Multiphase technology is likely to become necessary in regenerative life support systems in missions to the Moon or Mars because it will not be possible to carry from

Earth all the resources needed for the trip. Similarly, heat exchange with phase

change will be a must because of its superior efficiency-to-volume ratio. Unfortunately, two-phase technologies have not yet been widely implemented in space due to

lack of reliability from poor prediction tools in non-terrestrial gravity environments.

Nevertheless, two phase flow is often encountered is several space systems that are

planned to be used (Eckart, 1996). Examples of some specific systems are: Vapor Phase Catalytic Ammonia Removal (VPCAR), Vapor Compression Distillation

VCD), Thermoelectric Integrated Membrane Evaporation System (TIMES), Electrolysis Plant (ELEKTRON), Supercritical Water Oxidation (SCWO) and Condensing

Heat Exchanger (CHX). These space systems are used for revitalization of air in the

spacecraft and in the recovery and recycling of water. Some information on the rates

of flow of air and water in these systems and the regimes of two phase flow that occur

are summarized in Appendix D.

NASA/CR—2006-214085 1
The workshop by Chiaramonte and Joshi, 2004 [4] developed several important recommendations for research in applications to (i) air revitalization (ii) water recovery

(iii) solid waste management and (iv) thermal and phase change systems. We summarize below the applications of multiphase flow identified by this workshop in each

of these categories.

Water recovery systems provide a safe supply of potable water critical for human

life for long-term space flight. Resupply is unlikely in long-term flight, and the recovery and use of wastewater is necessary. The workshop report states that multiphase

flow is currently avoided in the design of waste water recovery systems, and that being

able to use multiphase flow systems under microgravity could permit the use of technologies for water recovery that have not been considered before. The understanding

of multiphase flow and separation in microgravity was identified as a critical, high

priority need for design of water recovery systems.

Thermal and phase change systems require enabling technologies for efficient and

reliable generation, storage and transfer of energy. Currently, most of the subsystems involve single-phase fluid and thermal processes. Future subsystems such as

boilers, condensers, evaporators, heat exchangers, fuel cells, heat pipes, etc., will

involve multiphase flow so that the energy-to-mass ratios are improved. The following areas for research in microgravity, among several others, have been identified as

having high priority with critical and severely limiting issues in the workshop report:

phenomenological understanding for multi-phase flow in micro and macro geometries;

boiling, condensation, phase distribution and phase transition; development of empirical correlations, theoretical models, scaling laws and CFD codes for two-phase flow

in complicated geometries; stability criteria for two-phase flow loops; development of

advanced technologies for phase separation.

Air revitalization systems are used for environment conditioning and life support

(ECLS) for the crew. They are necessary for survival of the crew (such as oxygen

supply and carbon dioxide removal) and to maintain crew health and safety (such

as fire detection and suppression, humidity control, and detection and removal of

particulate matter and trace contaminants in air). Control of particulate matter in

the air stream, separation of gases from liquids, and degassing of liquids is a common

thread in many of the systems utilized for air revitalization. The workshop report recommends continued effort to understand and develop techniques for phase separation

and liquid degassing. More importantly, it determined that key information useful for

designing ECLS systems is not available in a format readily useful to designers, but

is scattered in the literature on microgravity research, and in commercial aerospace

and academic communities. Compiling the existing information into design guides

detailing the mechanisms, behaviors, fundamentals and the relevant physics, as well

as providing scaling analysis, correlations, summary of flight experiment data into a

usable format for the ECLS design community was given a high priority.

NASA/CR—2006-214085 2
1.2 Description of two phase flow

In two phase flow, two phases, typically gas and liquid flow through a system, such

as a pipe, simultaneously. Because the liquid/gas interface can deform, the location

of the regions occupied by gas and liquid in the pipe is unknown to begin with and

is of great interest in designing the two phase flow system. Some of the common

states of the two phases are: (i) bubbly flow, in which the gas is present within the

liquid in the form of innumerable bubbles of small size (ii) slug flow where individual

gas bubbles merge to form a large gas mass or slug that is often cylindrical in shape

(iii) stratified flow where the gas and liquid regions are separated, with the gas that is

lighter flowing on top of the heavier liquid (iv) annular flow where one fluid (typically

the liquid) occupies the space adjacent to the tube wall and surrounds the other fluid

(gas) that flows in the center of the tube. Where the liquid is, where the gas is, and in

what form, is determined from a flow regime map. Such a map attempts to connect

the observations made of the state of the two phase flow to the tube size, physical

properties of the liquids, and the rates at which the liquid and gas flow. The liquid

and gas flow rates are often specified in terms of superficial velocities, which is the

velocity of the liquid or gas if it alone flows through the pipe. The void fraction refers

to the fraction of the space occupied by the gas in a region within the pipe. Especially of interest to the designer is the pressure drop as the two phases flow through

the pipe, and in the case of non-isothermal systems, the rate at which heat can be

transported in the pipe by the two phase flow.

Gravity is a crucial parameter in two phase flow. The buoyancy force exerted due of

the difference in density between the gas and the liquid has a major impact on the

flow regime, the distribution of void fraction, the pressure drop and the rate of heat

transfer in the pipe.

1.3 Current knowledge base for reduced gravity

The purpose of this section is to review published work on two-phase flow in partial

and microgravity since the pioneering work of Dukler et al. 1988 [7] where models

are proposed to describe gas-liquid flow regimes and their transitions in microgravity

conditions. These models use void fraction as a criterion describing the transition

between flow regimes..

1.3.1 Background on flow regime

Zhao et al. 2004 [8] report a study of experimental data of two-phase flow regimes

and their transitions in a 90-degree bend with inner diameter of 12.7 mm and curvature radius of 76.5 mm in microgravity conditions. In their study, the superficial

gas velocities range from 1.0 to 23.6 m/s and superficial liquid velocities range from

0.09 to 0.5 m/s. The study observes three major flow regimes: slug, slug-annular

transitional, and annular flows.

Takamasa et al. 2004 [9] measured void fraction, interfacial area concentration, bubble

NASA/CR—2006-214085 3
Sauter mean diameter, 1 and bubble number density in bubbly flow in microgravity

and low liquid Reynolds number conditions. These experiments are consistent with

the flow regime map described by Jayawardena et al. 1997 [1]. Nonaka et al. 2002

[10] obtained data on two-phase flow using the drop-shafts of Hokkaido National Industrial Research Institute (HNIRI), Microgravity Laboratory of Japan (MGLAB)

and Japan Microgravity Center (JAMIC). Their data show that when the length of

an annular liquid film in a circular pipe exceeds a critical length under zero gravity,

the liquid film breaks into periodic lobes which result into plugs. The group surmises

that linear stability theory can explain the periodicity observed in the experimental

data.

Choi et al. 2003 [11] have obtained data of flow regimes, void fraction, frictional pressure drop in normal gravity, microgravity and hyper-gravity (2g) aboard the MU-300

aircraft. They conclude that the gravity dependency of flow regimes was more clearly

seen as gas and liquid flow rates decrease. The effect of gravity on two-phase flow

was insignificant for the turbulent flow regions.

Zhao et al. 2001 [12] have performed experimental studies for horizontal two-phase

air-water flows in normal and reduced gravity conditions in a square cross-section

channel aboard the Russian IL-76 reduced gravity airplane. They observe four flow

regimes: bubbly, slug, slug-annular transition and annular flows. In that same year,

Zhao et al. Nov. 2001 [13] report a series of experiments performed aboard the Mir

Space Station on two-phase gas-liquid flow regimes in a test tube with a length of 356

mm and an inside diameter of 10 mm. The working fluids were carbogal and air which

resulted in five types of flow regimes: dispersed bubbly, bubbly, slug, slug-annular

transitional, and annular. Earlier, Zhao et al. May 2001, [14] claim to have discovered

a new region of annular flow with lower liquid superficial velocity. Previously, Zhao et

al. 2000 [15] had developed a model for predicting the transition from slug to annular

flow of adiabatic two-phase gas-liquid flow in microgravity.

W ̈olk et al. 2001 [16] describe a flow regime map for microgravity based on five dimensionless parameters. Observed regimes include: dispersed bubble, slug, slug/annular,

frothy slug/annular and annular flows.

Lowe et al. 1999 [17] identify two-phase flow regimes in microgravity using void

fraction signals on board the NASA DC-9 microgravity aircraft.

Ungar et al. 1998 [18] argue based on analysis of dimensionless parameters that

it should be possible to simulate microgravity conditions on Earth using systems

similitude. It is not clear from this work whether microgravity flow regime maps can

be obtained on Earth under wholly similar conditions.

Colin et al. 1996 [19] suggest that the transition from bubbly to slug flow may be

predicted by introducing a critical void fraction that depends on the fluid properties

1The Sauter mean diameter is the volume to surface area mean diameter of a distribution of

bubbles.

NASA/CR—2006-214085 4
and the pipe diameter. However, the role of coalescence which controls this transition is not clearly understood. Reinarts et al. 1991 [20] present data that indicates

that flow regimes observed under microgravity conditions are primarily annular and

include slug and bubbly-slug regimes.

1.3.2 Background on Pressure Drop

Wang et al. 2004 [21] present data on the interfacial friction factor and relative interfacial roughness on the gas-liquid interface for an air-water annular flow in a tube

with inner diameter of 9.525 mm.2 Their results show that while the roughness in

microgravity is less than half of that in normal gravity, the friction factor was only

about 10% smaller in microgravity than that in normal gravity.

Motil et al. 2003 [22] report experimental data on flow regime transitions, pressure

drop, and flow characteristics for co-current gas-liquid flow through packed columns in

microgravity. They find that two non-dimensional numbers, liquid Reynolds number

and Suratman number, are important in describing flow in a packed bed in microgravity. The flow conditions in the inlet and outlet pipes can be used to determine

the pressure drop across these sections. Earlier, Bousman et al. 1996 [23] presented

data on two-phase gas-liquid flow in the reduced gravity aircraft for void fraction,

liquid film thickness and pressure drop.

Takamasa et al. 2003 [24] present data which they claim can be used for the development of reliable constitutive relations which reflect the true transfer mechanisms in

two-phase flow in microgravity. These relations can be used to determine the pressure

drop across the pipe section. They measure bubble number density, interfacial area

concentration, and Sauter mean diameter of adiabatic nitrogen-water bubbly flows in

a 9-mm diameter pipe using image processing.

Kamp et al. 2001 [25] developed a mechanistic model for bubble coalescence in

turbulent flow. Their model can be used to predict pressure drop in pipes. Their

model was validated by data obtained in a reduced gravity aircraft. They conclude

that, in the absence of gravity, collisions between bubbles smaller than the length

scale of turbulence are primarily due to turbulence.

Iguchi et al. 2001 [26] report wettability of the pipe did not affect the mean rising

velocity of bubbles in microgravity. Taitel et al. 1996 [27] present a model to predict

slug flow in microgravity. Colin et al. 1995 [28] report experiments on gas-liquid flow

in small tubes of 19 mm, 10 mm and 6 mm diameter obtained during reduced gravity

flights, for a range of superficial liquid velocities from 0.1 to 2 m/s and superficial gas

velocities from 0.05 m/s to 10 m/s. Their results show how coalescence and the wall

friction factor affect pressure drop. Also, Zhao et al. 1995 [29] provide experimental

data on pressure drop of air-water two-phase flow in microgravity obtained aboard the

NASA KC-135 aircraft. They compare pressure drop data between μ-g and 1-g vertical flows. They indicate that the magnitude of pressure drop in both cases is similar.

2The interfacial friction factor, fi

, relates to the drag at the liquid-gas interface in the same way

the conventional friction factor applies to the drag at the solid-fluid interface.

NASA/CR—2006-214085 5
Zhao et al. 1993 [30] present data on two-phase air-water flow pattern data obtained aboard NASA KC-135 aircraft. The liquid superficial velocity ranges from

0.09 to 3.73 m/s, and the gas superficial velocity from 0.2 to 29.9 m/s. In this parameter range they observe bubbly, slug, frothy slug-annular and annular flows. In

the same year, Wheeler et al. 1993 [31] present a comparison between two-phase flow

and single phase systems for space applications. The advantages include capability

of achieving high specific power levels. They performed experiments on a reduced

gravity aircraft where annular flow pressure drops were measured through 10.41-mm

ID 1.251-m long glass tubing during periods with acceleration levels in the range 0.05

g. They conclude that the reduced acceleration interfacial friction factor data was

not predicted by the 1-g models.

1.3.3 Background on Heat Transfer

Colin 2002 [32] has raised some important questions regarding two-phase flow in microgravity. These issues are motivated by the technical problems arising in the design

of thermo-hydraulic loops for space applications. Further, she states that most of the

studies are focused on the determination of the flow regime, wall shear stress, heat

transfer and phase fraction. These data have led to many empirical correlations without a clear understanding of some of the basic mechanisms involved. Colin illustrates

this point with an example showing that the transition from bubbly to slug flow is

well predicted by a critical value of the void fraction depending on the Ohnesorge

number (Oh ≡ Su−1/2, see section 2.1). One limitation is that the transition criterion does not take into account the pipe length and the bubble size at the pipe inlet.

In most cases, to improve this criterion, a physical model of bubble coalescence in

turbulent flow is used to predict the bubble size evolution along the pipe in microgravity. Another issue concerns the radial distribution of bubbles in pipe flow which

is very sensitive to gravity and affects heat transfer.

Fore et al. 1997 [33] present measurements on both fluid flow and heat transfer

for two-phase slug flows in microgravity. They use air and two liquids (water and a

50% aqueous glycerine solution) to obtain a range of liquid Reynolds numbers from

1000 to 20,000 in a 25.4 mm inner diameter tube. They show based on a comparison

of microgravity to normal-gravity correlations that the heat transfer coefficients are

smaller in reduced-gravity than in normal-gravity under the same flow conditions.

They surmise that smaller liquid-phase turbulence levels in the absence of buoyancyinduced slip between the gas and liquid can explain this difference.

1.3.4 Background on Computational Modeling

In this report, we do not consider the computational modeling of two-phase flow,

except for the brief introduction to the subject given below. There have been several

studies on the computational modeling of two-phase flow, mostly in normal gravity.

The most popular approach is the multi-field two-fluid model (Lahey, 2002 [6]), and

appears to be the state of the art. In particular, a four-field two-fluid model appears

to be capable of describing the features of different flow regimes in normal gravity. In

NASA/CR—2006-214085 6
what follows, we briefly summarize this approach. The four fields comprise of continuous vapor, continuous liquid, dispersed vapor and dispersed liquid. The unknowns

in this model at a point in the computational domain are (i) the volume fraction of

each of the fields (ii) the velocity vector of these fields (iii) the pressure of each field

(iv) the enthalpy of each field. For each field, equations representing the ensemble

average of conservation of mass, momentum and energy are written. Source and sink

terms are present in these equations to represent the inter-conversions of these quantities among the fields. In order to achieve closure of the system of equations and

unknowns, jump conditions for momentum and energy transport across the interface

of a pair of the fields are written. For bubbly flow, this would involve the jump momentum and energy balances across the interfaces between a continuous liquid and

dispersed vapor phases. Empirical laws using drag coefficients are used for the contribution to the momentum balance arising from drag forces. Typically, the interfacial

area density appears in these empirical laws. An interfacial area transport equation is

usually obtained using kinetic theory methods. Where necessary, potential flow theory is also used to derive many of the interfacial transfer laws. Appropriate models

for turbulence are incorporated in the conservation equations and interfacial transfer

laws.

The results from such simulations for the void fraction distribution and liquid velocity profile are reported to be in good agreement with experiments for up-flow and

down-flow of air and water. The readers should consult Lahey (2002) [6] for more

details and further examples illustrating the use of four-field two-fluid model, such as

pressure drop, prediction of transient phenomena, etc.

Several researchers have used direct numerical simulations to model multi-phase flows.

Some of the approaches used are volume of fluid (VOF) methods [34], level set methods, sharp interface methods and phase field methods [34], and embedded interface

methods [34].

Most of the four-field two-fluid modeling studies as well as the direct numerical simulations appear to have been conducted for two-phase flow under normal gravity, but

in principle they are applicable for modeling flows in microgravity and partial gravity.

NASA/CR—2006-214085 7
2 Dimensionless parameters in two-phase flow

We will consider two-phase flow phenomena when the two phases are being transported via a pipe or a channel from one reservoir to another. In what follows we

only deal with isothermal flow of gas and liquid. A knowledge of the dimensionless

parameters in two phase flow is useful because, as in single phase flow, it can enable theoretical and experimental results for a particular flow situation (fluids, tube

size, velocities, gravity level etc.) to be scaled and extrapolated to a different flow

situation. Quite often, casting results in terms of dimensionless parameters can also

provide a clue to the underlying physical mechanisms.

2.1 Use of the Buckingham Pi Theorem

The independent variables are:

ULS, UGS, μL, μG, ρL, ρG, D, σ, g and φ. (1)

These symbols denote the superficial liquid velocity, superficial gas velocity, liquid

viscosity, gas viscosity, liquid density, gas density, pipe diameter, surface tension,

gravitational acceleration, and angle between the vertical and the flow direction. The

superficial velocity is the average velocity each phase would have if it alone flows

in the pipe. There are ten variables and three fundamental units (mass, length and

time). Using the Buckingham Pi theorem, this yields seven independent dimensionless

parameters. One set of dimensionless parameters is:

ReL =

ρLULSD

μL

, ReG =

ρGUGSD

μG

, F rL =

∆ρ

ρL

gD

U

2

LS

,

W eL =

ρLU

2

LSD

σ

,

μG

μL

,

ρG

ρL

, φ (2)

where ReL is the liquid Reynolds number, ReG is the gas Reynolds number, F rL is

a modified Froude number in the liquid, W eL is the Weber number in the liquid and

∆ρ ≡ ρL − ρG. A commonly used parameter is the slip ratio defined as:

S =

UGS

ULS

(3)

may be derived from the liquid and gas Reynolds numbers and the viscosity ratio.

Other dimensionless parameters derivable from the above set are the Suratman number:

Su =

Re2

L

W eL

=

ρLDσ

μ

2

L

(4)

that has been used in the literature and is important in demarcation of flow regime

transition boundaries in low gravity, and the static Bond number:

Bo =

(ρL − ρG)gD2

σ

(5)

NASA/CR—2006-214085 8
which is the ratio of the buoyancy to surface tension forces. A question arises as

to what the natural set of seven dimensionless parameters is. To determine these,

we perform a scaling analysis of the governing equations for two phase flow in the

absence of heat transfer.

2.2 Scaling Analysis of Two Phase Flow

The governing equations for two phase flow that we consider are the incompressible

Navier-Stokes equations. They are written separately for the liquid and gas phase

below.

ρL



∂UL

∂t + UL · ∇UL



= −∇PL + μL∇2UL (6)

ρG



∂UG

∂t + UG · ∇UG



= −∇PG + μG∇2UG − (ρL − ρG)g (7)

where ∇PL = −ρLg + ∇p

∗

L

, ∇PG = −ρLg + ∇p

∗

G, ∇p

∗

G − ρGg = ∇PG − (ρL − ρG)g

and p

∗

L

, p∗

G denote the physical pressures in the liquid and gas phases.

We choose the following reference quantities to normalize the variables.

• Gas velocity – UGS = Superficial gas velocity

• Liquid velocity – ULS = Superficial liquid velocity

• Time – D

UL

• Length – D = pipe diameter

• Gas pressure – ρLU

2

LS

• Liquid pressure – ρLU

2

LS

Before we proceed we discuss the choice for the pressure normalization given above.

Both the liquid and gas pressures are referred to the same datum (the hydrostatic

pressure in the liquid at any z plane) and is scaled by ρLU

2

LS. Consider a two phase

flow in which the liquid forms the continuous phase, and contains many gas bubbles

as the dispersed phase. We expect the pressure within a gas bubble to be equal to the

pressure in the liquid surrounding it, except for a contribution arising from surface

tension. Thus the gas and liquid pressures adjacent to the liquid/gas interface are

approximately the same. The normalization scheme we have chosen for the pressures

NASA/CR—2006-214085 9
reflects this idea. If drops of liquid are present in a gas as the continuous phase, the

density of the gas ρG and its superficial velocity UGS must be used in normalizing the

pressure in the liquid and gas. The pressure normalization factors in the liquid and

gas can be different for stratified flow and developing or entrance flows. The scaling

analysis performed here is not valid for such situations and must be suitably modified.

With these choices the governing equations can be written in the following dimensionless form.

∂uL

∂t + uL · ∇uL = −∇pL +

1

ReL

∇2uL (8)

S

∂uG

∂t + uG · ∇uG = −∇pG +

1

ReG

∇2uG − F rG

g

|g|

, (9)

where F rG =

∆ρ

ρG

gD

U2

GS

. We next look at the balance of shear and normal stresses at the

liquid/gas interface. When the surface tension is uniform (i.e., no Marangoni stress

at the interface), the balance of shear stress yields



μLULS

μGUGS 

∂uL

∂s =

∂uG

∂s (10)

where s is distance tangential to the interface. The balance of normal stress may be

written as

W eL (pG − pL) + 2 Ca 

∂uLn

∂n −

μGUGS

μLULS

∂uGn

∂n 

= 2H (11)

where Ca =

μLULS

σ

is the Capillary number, H is the mean curvature of the interface

and n is distance normal to the interface.

Thus the scaling analysis reveals that the dimensionless parameters in two phase

flow are the gas and liquid Reynolds numbers, the slip ratio, the gas phase Froude

number, the viscosity ratio, and the liquid phase Weber number. The orientation

angle φ is an additional parameter which appears implicitly in the components of g

with respect of the direction of flow. The density ratio, Capillary number and Bond

number can be derived from the above dimensionless groups.

When the cross section of the flow duct is not circular, the hydraulic diameter,

Dh ≡

Flow Area

Flow perimeter

should be used instead of D.

2.3 Parameter Range on Current Life Support Hardware

We have performed a comparative study of the dimensionless parameters and flow

regimes likely to occur in life support hardware operating in microgravity. In water

NASA/CR—2006-214085 10
recovery equipment, the Reynolds number of gas and liquid are small to moderate.

The details are shown in Appendix D. We have used the flow regime maps of Jayawardena et al. [1]. We have made educated assumptions about the pipe diameter where

two-phase flow might arise in such hardware, and about the coexistence of water and

air in that pipe. The results are flow regime predictions for typical flow rates quoted

in the literature (Eckart 1996 [35], Eckart 1997 [36], Wieland 1998 [37], Wieland 1998

[38], Wieland 1994 [39]). However, we find that the Jayawardena flow map should be

taken with some skepticism in predicting flow regimes in water recovery equipment.

Critical analysis of the Jayawardena flow map indicates that it may apply best only

when gas and liquid flow at high Reynolds numbers. Therefore, it seems prudent

to undertake an experimental program to establish microgravity and partial gravity

flow regime maps valid in the low-to-moderate Reynolds number flows likely to be

encountered in water recovery equipment.

NASA/CR—2006-214085 11
3 Flow Regimes in Reduced Gravity

3.1 Description of flow regimes

Jayawardena et al. [1] say that in the absence of gravity three major flow patterns

exist – bubbly flow, slug flow and annular flow, with annular flow being obtained for

a wide range of gas and liquid flow rates. For a fixed liquid flow rate, bubbly flow is

obtained for low gas flow rates. This regime transitions to slug flow for higher gas

flow rates. As the flow rate of the gas increases further, annular flow is obtained.

For the operation of two phase flow systems, bubbly and annular flow are preferred.

Because slug flow can cause vibrations, it is usually avoided. Typically, the annular

flow regime yields maximum flow rates for a given two-phase system.

In bubbly flow, bubble sizes are relatively uniform, and the bubbles are distributed

everywhere in the flow. Slug flow results from the merging and coalescence of individual bubbles, and the bubbles are elongated and more cylindrical in shape (Taylor

bubble). The elongated gas bubbles are separated by liquid slugs. Small bubbles are

often present in the liquid slugs.

In typical annular flow, the gas occupies the central or core region of the tube, with

the liquid remaining in a thin layer adjacent to the tube wall. Waves are present on

the liquid/gas interface and the instantaneous shape of the interface fluctuates with

time. Also, small liquid drops are typically entrained by the flowing gas. Figure 1

shows pictures of bubbly, slug and annular flow obtained in low gravity.

3.2 [1] reduced gravity flow regime map

In reduced gravity we can consider the value of the Froude number to be small under

most circumstances, and the influence of the residual gravity on the dynamics of the

two phase flow will be negligible. Therefore F r can be omitted from the list of dimensionless parameters that influences the flow. When gravity has a negligible influence,

the orientation, φ, of the flow relative to g, can be omitted from the list of dimensionless parameters as well because φ only appears in ρg sin φ and ρg cos φ. Thus, instead

of the parameters F r and φ, one could use F r1 ≡ F r sin φ and F r2 ≡ F r cos φ. Further, the gas/liquid density and viscosity ratios typically have small values. One can

therefore surmise that these two property ratios will only have a small influence on

the flow. We are thus left with three dimensionless parameters – the gas and liquid

Reynolds numbers and the Weber number. This is the starting point of Jayawardena

et al. [1] in the representation of the flow regime map in terms of dimensionless parameters.

Figures 2 and 3 show the flow regime map from the work of Jayawardena et al.

They were able to successfully capture the bubbly, slug and annular flow regimes

on a single map, by plotting the Suratman number (= ρLDσ

μ

2

L

) versus the gas/liquid

Reynolds number ratio or the gas Reynolds number. It is remarkable that a flow

regime map for a 3-parameter system was obtained using only two dimensionless parameters (see sec. 2.3 and Appendix D for a discussion). Curiously, Motil et al. 2003

NASA/CR—2006-214085 12

Jayawardena et al.
Figure 1: From top to bottom: photographs of annular, bubbly and slug flow in

reduced gravity (McQuillen et al. [2]).

NASA/CR—2006-214085 13
0.001 0.1 10 1000

104

105

106

107

Slug or Annular

Slug Annular

Bubble

Su

ReG / ReL

Figure 2: Flow map for zero gravity valid at Su < 106.

10 100 1000 104 105 106

107

106

Annular

Bubble or

Su Slug

ReG

Figure 3: Flow map for zero gravity valid at Su > 106.

NASA/CR—2006-214085 14
[22] find that a transition from bubbly flow to pulse (or slug) flow in porous, packed

beds in microgravity is also captured very well on a plot of the Suratman number

versus the ratio of gas to liquid Reynolds number, as in fig. 2.

3.2.1 Bubble-slug transition

Jayawardena et al. [1] report that the transition from bubbly flow to slug flow can

be written as follows.



ReG

ReL



transition

= 464.16 Su−2/3(12)

It may also be written alternatively as

(ReG)

transition = 464.16 Ca2/3 Re1/3

L

(13)

3.2.2 Slug-annular transition

Jayawardena et al. find that the transition from slug flow to annular flow can be

written as follows.



ReG

ReL



transition

= 4641.6 Su−2/3 Su < 106(14)

(ReG)

transition = 2 × 10−9 Su2 Su > 106

(15)

Note that for Su < 106the slug-annular transition occurs at a Reynolds number ratio

that is a factor of 10 greater than the bubble-slug transition Reynolds number ratio.

Jayawardena et al. [1] show that their map agrees well with nine different experiments

in the open literature that used six different fluid pairs (air/water, air/water-glycerin,

air/water-zonyl, R11 vapor/liquid, R12 vapor/liquid, R114 vapor/liquid) and ten

different tube sizes in the range 4.7 mm to 40 mm. Specifically, the accuracy of

bubble-slug transition predictions is 87%, and that for slug-annular transition is 97%.

3.3 Other flow regime transition criteria

Jayawardena et al.’s [1] Suratman number based flow regime map is not the only

map developed for reduced gravity two-phase flow. In this section we summarize

some alternative regime maps.

3.3.1 Gas void fraction criterion

Bousman’s thesis [3] details the use of the Drift-flow model to estimate gas void fraction, GA, knowing the superficial velocities, resulting in a relation GA = F(UGS/ULS).

Since they also measured the void fraction at various flow regimes transitions, GAT r ,

they can plot the regime transition predicted from the Drift-flow model, UGS =

ULS F

−1

(GAT r ) over the measured data. Note that the model only allows a straight

line to be the flow regime boundary. The drift-flow model is adequate for predicting

bubbly-to-slug transition.

NASA/CR—2006-214085 15
Bousman also uses the drift-flow model coupled with a force balance in the annular flow to describe the slug-to-annular boundary. In this approach, expressions for

the void fractions in the slug and in the annular regimes are equated. This yields a

relation for the transition superficial velocities. The evaluation of the annular flow

void fraction requires the calculation of the interfacial shear at the liquid-gas interface

of the annular flow. This is done using a Blasius type friction factor correlation. The

Blasius correlation assumes no-slip of the relevant flow phase; whereas the gas does

slip on the liquid. However, because typically UGS  ULS in annular flow, the error

from assuming that the gas does not slip may be small.

The arguments used to derive these boundaries hinge on a parameter Co that appears in the drift-flow model. This parameter depends on whether the flow in the

liquid is laminar or turbulent. The fact that the values of Co chosen to adjust the

data are not always consistent with the values prescribed based on the turbulence

level of the liquid, weakens the power of this technique.

3.3.2 Weber number criterion

The Weber number criterion postulates that at the slug-to-annular transition inertial

and surface tension forces should both be important. This criterion uses the gas

superficial Weber number. It postulates that at transition W eG = U

2

GS ρG D/σ ≈ 1.

Bousman has tested this criterion with three different fluid systems and two pipe

diameters. The changes in fluid systems test the effects of viscosity and surface

tension. Changes in pipe diameter test, obviously, the dependence on D. A change

in viscosity does not change W eG. If there were a dependency of the transition

threshold on μ, it should appear as a shift in W eG between the two systems with

different viscosities. The data hint at some viscosity dependence of the transition

W eG, although it is difficult to be definite when the scatter is as large as in these

data and no error bars are quoted. When the viscosity is constant and D and σ

change, the transition W eG seems to hold more or less constant. This supports the

Weber-number transition criterion. Comparing the Weber number criterion to the

Suratman number criterion of Jaywardena et al. [1], the latter has a much sharper

demarcation of the flow domains.

3.4 Flow regime transition mechanisms

Mechanistic explanations for regime transitions have been attempted for many years.

Here we summarize the more relevant ones.

3.4.1 Mechanisms for bubble-slug and slug-annular transition

Two interrelated mechanisms that have been proposed [5] are: (i) coalescence of individual bubbles leading to spherical cap and Taylor bubbles. It is believed that

the frequency of bubble collisions, necessary for their eventual coalescence, increases

rapidly when the void fraction is larger than 0.2. This led to the belief that bubbleslug transitions occur at void fractions around 0.25. (ii) Since bubble coalescence is a

NASA/CR—2006-214085 16
slow process that must be preceded by drainage of the intervening liquid film, it is not

very clear that coalescence is the only factor – the coalescing bubbles must be held

in place for a sufficiently long time without relative motion. Therefore it has been

suggested that the existence of “void waves” which creates regions of large bubble

concentration where bubble separation is inhibited is an essential step. Colin et al.

[28] suggest that coalescence happens more readily in reduced gravity because of lack

of relative motion between the gas and the liquid. Thus coalescing bubbles could be

held in place longer in reduced gravity.

Jayawardena et al. [1] do not discuss any mechanisms for the transition of bubbly

to slug flow or from slug to annular flow. It is significant that the data is correlated

by the Suratman number, Su =

ρLDσ

μ

2

L

. Su only depends on the properties of the

liquid and the pipe diameter, and does not depend on the flow rates of the liquid

or gas. Briefly, Su is a modified Reynolds number where the characteristic velocity

is the “capillary” velocity σ/μ. Thus, increasing surface tension should be thought

of as increasing inertia of the capillary-driven flow. There is a likely analogy with

breakup of a liquid filament, where the stability limit is controlled by Su. This should

be so because it is commonly believed that bubbly-slug and slug-annular transitions

are governed by merging and coalescence of bubbles, which are surely influenced by

surface tension (and perhaps viscosity).

Another issue concerning transition boundaries is the question of reversibility. Is

the transition boundary from regime 1 to regime 2 the same as from regime 2 to

regime 1? The experiments we have found do not address this question. Rather, the

gas and liquid flow rates are dialed and the ensuing flow regime is interrogated by the

diagnostic method of choice. In real situations, flow rates might change more or less

gradually, inducing changes in flow regime. It is therefore of interest to determine

whether the regime boundaries are hysteretic or not.

A qualitative or semi-quantitative understanding of the reason why the flow regime

map advocated by Jayawardena et al. [1] is so successful needs to be determined.

One important reason is that, once the mechanism is understood, extensions for nonnegligible values of the Froude number can be contemplated, which will be very useful

for predicting flow regime maps in partial gravity.

The effect of gravity on axial development of bubbly flows for low liquid Reynolds

numbers in small diameter pipes was studied by Takamasa et al. [9, 24] by monitoring key flow parameters such as void fraction, interfacial area concentration, bubble

Sauter mean diameter and bubble number density. The flow measurements were

performed in an air-water system using a non-intrusive image processing method

with three digital cameras placed at 5, 20 and 40 diameters along the pipe section.

The microgravity tests were performed in the underground drop shaft facility at the

Japanese Microgravity Center (JAMIC) providing 10s of reduced gravity environment

with nearly negligible residuals between 10−4 and 10−5 g. The main mechanisms for

transition from bubbly to slug flow were studied by examining bubble coalescence

and breakup. Since for the relatively low Reynolds numbers that prevailed in these

NASA/CR—2006-214085 17
studies, bubble breakup was minimal, the main driving force for flow transition was

due to bubble coalescence. Considerable bubble coalescence was observed in both

the 1g and reduced gravity experiments. Visual records indicate that in 1g the main

mechanism for bubble coalescence is wake entrainment. Wake entrainment refers to

the entrainment of a trailing bubble by the motion of a leading bubble in the liquid

– the motion of the leading bubble sets up a flow in the liquid that tends to entrain

other bubbles. In microgravity, however, there is no mechanism for relative motion of

the liquid and gas; thus wake entrainment is absent. At low liquid Reynolds numbers,

where the flow is basically laminar and the velocity profile is parabolic, Takamasa et

al. report that velocity profile entrainment is the main mechanism for bubble coalescence. In normal as well as microgravity, due to the relatively low Reynolds numbers

encountered, coalescence due to random collisions is minimal. Curiously, it seems

that the relative contributions of velocity profile entrainment, in microgravity, and

wake entrainment, in 1g, to bubble coalescence is the same. This results in nearly

similar flow structure developments for the two environments leading to a decrease in

the interfacial area concentration and an increase in the Sauter mean diameter along

the pipe.

3.4.2 Alternative mechanistic approaches for predicting

regime transitions

Currently, in most of the analyses and codes used for predicting the behavior and

performance of two-phase thermal-hydraulic systems, the effects of the different interfacial structures that evolve as the system transitions between the flow regimes

are incorporated through certain transition criteria dictated by predetermined flow

regime maps such as the one shown in Figs 2 and 3. Unfortunately, in this approach,

no time or length scale is inherently incorporated into the transition criteria. As a

result, such an approach is not well suited for simulation of developing flows. To solve

this problem more mechanistic approaches have been proposed in recent years. One of

the methods with great potential is based on a mechanistic prediction of the changes

that take place in the two-phase flow structure based on the solution of a transport

equation for the interfacial area concentration [9, 24, 40]. A successful formulation

of this methodology is likely to make a drastic improvement in the development of

future two-phase flow models for both 1g and reduced gravity predictions.

NASA/CR—2006-214085 18
4 Flow Regimes in Partial Gravity

Hurlbert et al. 2004 [41] presented data on hydrodynamic measurements for twophase flows in Mars and Moon gravity conditions, obtained aboard the KC-135 aircraft. The data was obtained using dichlorodifluoromethane (R-12) as the working

fluid flowing in a tube with an inner diameter of 11.1 mm. The simulated gravitational

acceleration for Mars was approximately 0.38-g, and that for Moon was approximately

0.17-g. We note that the residual gravity was perpendicular to the direction of flow

in the tube. Also an attempt was made to scale the effects of gravity to predict the

behavior of two-phase flow using Euler (≡ ∆p/(ρU2)) and Froude numbers.

John McQuillen, of NASA Glenn Research Center, obtained data in the KC-135

aircraft in a two-phase flow rig with Lunar gravity. He used air and a 50-50 waterglycerin solution (fig. 4), air-water (fig. 5), and air-water plus Zonyl 0.5% (fig. 6).

The addition of Zonyl to water affected the surface tension without changing the

density or viscosity. Also, we note that the residual gravity was perpendicular to the

direction of flow in the tube.

Figure 4: Flow map for Lunar gravity for Air and a 50-50 Water-Glycerin solution.

We describe next the current issues in the understanding of two-phase flow in normal gravity. Because gravity plays a very important role in two-phase flow, much of

these issues are likely applicable in partial gravity conditions as well. Current state

of knowledge on two-phase flow in normal gravity was discussed recently at a DOE

workshop by members of the Study Group on Flow Regimes in the Workshop on

Scientific Issues in Multiphase Flow Champaign-Urbana, IL [42],[43]. The findings at

this workshop were published in four articles, and can be summarized as: (1) Phase

Distribution in Gas-Liquid Flows where Theofanous et al. [44] identify pattern flow

NASA/CR—2006-214085 19
Figure 5: Flow map for Lunar gravity for Air-Water two-phase flow system.

Figure 6: Flow map for Lunar gravity for Air-(Water-Zonyl mixture) two-phase flow

system.

NASA/CR—2006-214085 20
map as the principal scientific issue permeating essentially every aspect of multifluid

flow. (2) For disperse flow, Sundaresan et al. [45] outlined a number of scientific

challenges which represent building blocks for the comprehensive understanding of

disperse flows encountered in a variety of technologies and in nature. We note that

their report is heavily weighted toward solid-fluid interaction but the findings applied

to fluid-gas interactions. (3) Prosperetti et al. [46] describe the advantages of Computational Physics besides its application to solving actual problems, as (a) a tool

to develop and understand basic physics and as a guide toward asking more penetrating questions about two-phase flow (b) an aid in closing the averaged equations

which are used to describe two-phase flow phenomena. Finally, (4) Delhaye et al.

[47] summarize the role of microscopic phenomena in multiphase flow, specifically,

the need to understand the coupling between molecular scale phenomena and macroscopic phenomena. These issues and those in Prosperetti et al. 2004 [48] are the

current problems that need to be understood in 1-g.

In normal gravity two-phase flow, the inclination of the pipe, φ, becomes an important parameter. The flow regimes in vertical pipes are different from those in

horizontal pipes. The flow regimes in horizontal pipes are (see fig. 7):

• Dispersed bubbly flow

• Plug flow

• Stratified flow

• Semi-slug flow

• Stratified wavy flow

• Slug flow

• Annular dispersed flow

The flow regimes in vertical pipes are (see fig. 8):

• Bubbly flow

• Slug or Plug flow

• Churn flow

• Annular flow

• Wispy annular flow

Thus gravity and its orientation are important parameters in two-phase flow. In the

absence of gravity, principally only bubbly, slug and annular flow have been observed

to occur. The flow regime map in microgravity is therefore relatively simplified.

NASA/CR—2006-214085 21
Figure 7: Flow patterns observed under normal gravity two-phase flows in a horizontal

pipe.

NASA/CR—2006-214085 22
Figure 8: Flow patterns observed under normal gravity two-phase flows in a vertical

pipe. Gravity direction is downwards.

NASA/CR—2006-214085 23
5 Pressure Drop Predictions

The knowledge of pressure drop in a two-phase flow system is important for its design.

It enables the designer to size the pump required for the operation of the flow system.

We present below the most common approaches in prediction of the pressure drop in

two-phase flows.

5.1 Homogeneous fluid model

The pressure gradient in a two-phase flow can be thought of as arising from three

additive contributions: (i) frictional (ii) flow acceleration and (iii) hydrostatic head.

Thus,

dp∗

dz∗

=



dp∗

dz∗



fr

+



dp∗

dz∗



ac

+



dp∗

dz∗



gr

(16)

In the homogeneous fluid model, the fluid is characterized by an effective fluid that

has suitably averaged properties of the liquid and gas phases.

5.1.1 Pressure drop due to flow acceleration

The pressure gradient due to flow acceleration can be written as



dp∗

dz∗



ac

=

d

dz∗



1

A

Z

A

ρu2dA=  ̇m2

d

dz∗



1

ρH



(17)

where  ̇m is the total rate of mass flow per unit area in the pipe, and the effective

density ρH is defined by:

ρH = GAρG + (1 − GA)ρL (18)

where GA is the area averaged local void fraction of the gas, ρG and ρL are the

densities of the gas and the liquid.

5.1.2 Pressure drop due to gravity

The pressure gradient due to gravity may be written as



dp∗

dz∗



gr

= gzρH = gz[GAρG + (1 − GA)ρL] (19)

where gz is the component of gravity in the z-direction.

5.1.3 Pressure drop due to wall friction

The frictional pressure drop due to the shear stress exerted by the tube wall is considered the most problematic term in two-phase pressure drop. The frictional pressure

drop may be related to the wall shear stress τW by a force balance:



dp∗

dz∗



fr

=

2τ

∗

W

R

(20)

NASA/CR—2006-214085 24
We present various approaches used in the literature to determine the frictional pressure drop.

Friction factor

Analogous to single phase flow, the use of friction factors in determining the twophase frictional pressure drop is common. A typical expression for the friction factor

f is given below.



dp∗

dz∗



fr

=

fm ̇

2

ρHR

f = 0.079Re−0.25 (21)

where the Reynolds number is Re =

2  ̇mR

μH

with the effective viscosity approximated

as [49]:

μH = μL or μH = xμG + (1 − x)μL or (22)

1

μH

=

x

μG

+

(1 − x)

μL

or (23)

μH = βμG + (1 − β)μL or (24)

μH = μL(1 − β)(1 + 2.5β) + μGβ or (25)

μH =

μLμG

μG + x

1.4

(μL − μG)

(26)

where x is the quality (the gas mass flow fraction).

Two-phase friction factor multiplier

Here the frictional pressure gradient for two-phase flow is determined by multiplying

the frictional pressure gradient for flow of the liquid alone by a two-phase multiplication factor. The mass flow rate of the single phase liquid flow is the same as that of

the liquid in the two-phase flow.



dp∗

dz∗



fr,T P

= φ

2

LO 

dp∗

dz∗



fr,LO

(27)

where subscript TP denotes two-phase and LO liquid only. A typical correlation for

the two-phase multiplier is

φ

2

LO =

ρL

ρH



1 + x

(ρL − ρG)

ρG

 1 + x

(μL − μG)

μG

−1/4

(28)

5.2 Separated flow model

In this model the flow of the gas and vapor are analyzed as though the gas and liquid

streams flow through separate tubes, with cross-section area proportional to the void

fraction. To account for the interaction of the gas and the liquid, appropriate rates

of transfer (e.g. mass, momentum or energy) are considered in the balance equations

in a control volume formulation.

NASA/CR—2006-214085 25
5.2.1 Pressure drop due to flow acceleration

The pressure gradient due to flow acceleration may be written as follows.



dp∗

dz∗



ac

=

d

dz∗



1

A

Z

A

ρu2dA=  ̇m2

d

dz∗



x

2

ρG

2

GA

+

(1 − x)

2

ρL(1 − GA)

2



(29)

5.2.2 Pressure drop due to gravity

The expression for the pressure drop due to gravity is the same as that give for the

homogeneous flow model.

5.2.3 Pressure drop due to wall friction

Lockhart-Martinelli correlation

The Lockhart-Martinelli correlation for the two-phase pressure gradient is similar in

idea to the two-phase multiplier. A parameter X is defined as

X

2 =

(dp∗/dz∗)LS

(dp∗/dz∗)GS

(30)

which is the ratio of the frictional pressure gradients for the liquid and gas alone,

flowing at their respective superficial velocities. The single phase pressure drops are

obtained using the Fanning equation that uses a friction factor:

dp∗

dz∗

=

f ρ U2

R

, (31)

where:

f =

16

Re

for laminar flow, (32)

and for turbulent flow, two correlations are used:

f =

0.079

Re0.25 Re ≤ 2 × 104

, or (33)

f =

0.046

Re0.2

Re ≥ 2 × 104(34)

where Re is based on pipe diameter i.e. Re =

ρ UD

μ

[50].

The two-phase frictional pressure gradient is then obtained by a multiplier defined as



dp∗

dz∗



fr

= φ

2

G



dp∗

dz∗



GS

= φ

2

L



dp∗

dz∗



LS

(35)

φ

2

G = 1 + CX + X

2 φ2

L = 1 + CX−1 + X

−2

(36)

NASA/CR—2006-214085 26
where C is empirically determined and is in the range 5 < C < 20. The value of

C is 5 when the flow of liquid and gas is laminar and 20 when both flows are turbulent.

Friedel correlation

In the Friedel correlation, the two-phase multiplier φLO is calculated differently and

the expression for it is given below.

φ

2

LO = E +

3.24F H

F r0.045W e0.035 (37)

where

E = (1 − x)

2 + x2



ρLfGO

ρGfLO 

F = x

0.78(1 − x)0.24

H =



ρL

ρG

0.91 

μG

μL

0.19 

1 −

μG

μL

0.7

F r =

m ̇

2

2gRρ2

H

W e =

2Rm ̇

2

σρH

ρH ≡



x

ρG

+

1 − x

ρL

−1

,

and fLO, fGO are the friction factors calculated as if the liquid and the gas respectively

were flowing alone in the pipe (see M. M. Awad and Y. S. Muzychka, [51] where these

parameters are defined).

5.3 Pressure drop in bubbly flow

The homogeneous and separated flow models do not consider the regime of the flow in

order to predict the pressure drop. One would expect the pressure drop correlations

to be tailored to the relevant flow regimes, but such correlations are not generally

available.

In the case of bubbly flow in reduced gravity, Colin, Fabre and McQuillen (1996)

[19] have determined that a friction factor approach correlates the experimentally

NASA/CR—2006-214085 27
determined pressure drop quite well. Basically they find that a single-phase Blasius

relationship for the friction factor works well when the liquid flow is turbulent.



dp∗

dz∗



bub

=

fLρLU

2

L

R

, fL = 0.079Re−0.25

L

, UL =

ULS

1 − 

, ReL =

2ρLULSR

μL

(38)

The predicted pressure drop is quite accurate for ReL in the range 20,000 to 80,000.

For ReL less than 20,000 the experimentally obtained friction factor is larger than

the above prediction. Further, in the case of single phase flow, a sharp increase in

the friction factor is observed at a Reynolds number of around 8,000, where there is a

transition from a laminar-like behavior to a turbulent-like behavior. Such a behavior

is not observed for two phase flow.

An alternative approach that deserves consideration is to use a homogeneous fluid

formulation with an effective viscosity μeff given by Einstein’s viscosity for dispersions:

μeff = μL



1 + φ

μL +

5

2

μG

μL + μG

+ c2 φ

2



,

where φ is the volume fraction of the gas bubbles and c2 is a coefficient that must be

determined experimentally [52]. We are not aware of the use of Einstein’s viscosity

for dispersions in the modeling of two-phase flows.

5.4 Pressure drop in slug flow

Colin, Fabre and McQuillen (1996) [19] comment that for slug flow, the above prediction for bubbly flow shows similar trends when compared to experimental results

for slug flow. The data, however display much more scatter. There does not appear

to be any other reliable prediction for the pressure drop in slug flow.

Sang Young Son and Jeff Allen [53] have studied pressure drop in two-phase flow

through a square microchannel of 330μm side. The pressure drop found for slug flow

increases as the gas flow rate increases and the flow transitions to “slug-annular”.

However, a further increase in gas flow rate causes a drastic decrease in pressure

drop as the flow transitions to all-annular. This indicates that the extra pressure

drop caused in slug flows by the presence of contact lines and/or thin lubricating

films around gas bubbles is significant. Alternatively, one could think of this as the

effective viscosity of the all-slug flow being higher than that of the all-annular flow

for similar flow rates. This is analogous to the well-known fact that the effective

viscosity of a foam (i.e., a collection of bubbles separated by very thin liquid layers)

is much larger than the viscosity of either of its gas or liquid constituents. In circular channels, there are two types of slug flow: one type has continuous liquid phase

through the presence of a lubricating liquid layer around each gas bubble (type 1);

the other type has completely separate gas-liquid sections down the tube (type 2).

The pressure drop in type 1 is less than in type 2, for the same flow rates and pipe

size. In square channels, type 2 slug flow cannot exist because there is always liquid

in the corners that connects the liquid all along the pipe. In this case, we can have

NASA/CR—2006-214085 28
either type 1 when the liquid wholly surrounds each gas bubble, or a hybrid of types

1 and 2 when liquid exists around the bubble only in the corners while the rest of the

bubble contacts dry channel wall.

Other works in pressure drop for two-phase flow in microchannels include Kawahara

et al. [49] and Triplett et al. [54]. Microchannel flow work is important because it

may provide information relevant to microgravity. However, one should keep in mind

that just scaling the size down does not make a system similar to microgravity. Case

in point, bubbly flow is widely documented in microgravity, but it is very difficult

to observe bubbly flow (in the sense of having many bubbles across the tube cross

section) in microchannels [54].

5.5 Pressure drop for fully developed, laminar, annular flow

For two-phase flow applications involved in advanced life support, the flow rates are

expected to be somewhat low. Therefore, predictions of pressure drop etc., in laminar

flow may be an appropriate baseline for such applications. Our analysis of fully developed annular flow when the flow is laminar is presented in Appendix A. The analysis

reported there also includes the effect of gravity when g is parallel or antiparallel to

the flow.

In fully developed annular flow there are two issues that need to be discussed concerning the structure and stability of the flow. First, there is the issue of which fluid

flows next to the wall and which one flows in the center core. In all reduced gravity

air-water flows that we are familiar with, water has been reported to be the fluid that

is adjacent to the tube wall, and the gas is in the core. It has been suggested that in

some two-phase flows a minimum dissipation principle is obeyed, which states that

the less viscous fluid occupies the region where the shear stresses are the greatest

[55]. In air-water flows, this principle would imply that air is in contact with the tube

wall, which appears contrary to experimental observations. In Appendix B, we have

proposed an argument based on the interfacial energy to develop criteria to determine

which phase is in contact with the wall. For most gas-liquid annular flows, we find

that the gas is indeed predicted to occupy the core and the liquid is in contact with

the wall.

The second question is the position of the core center relative to the pipe center.

We have pointed out in Appendix A that when the gas core is concentric with the

axis of the tube, the total flow rate in the tube is a maximum for a fixed value of the

pressure gradient.

5.5.1 Pressure drop due to flow acceleration

Flow acceleration occurs when the flow is developing, such as in entrance flows, or if

there is a change in the flow geometry, such as when the flow encounters an expansion,

contraction, or a bend. Flow acceleration can also be caused by phase change. In the

case of fully developed flow, there is no flow acceleration. Therefore, strictly speaking,

the pressure drop due to flow acceleration is zero. One might, however, approximate

NASA/CR—2006-214085 29
the flow to be locally fully developed, with variations in z occurring over a much

longer length scale than the length of pipe required to achieve fully developed flow.

Using such an assumption, the pressure drop due to flow acceleration may be written

as follows.



dp∗

dz∗



ac

=

d

dz∗



1

A

Z

A

ρu2dA(39)

In eq. 39 all variables are dimensional. Using the pseudo velocity scale wps = Q/πR2,

and the pseudo length scale R,

3 we find



dp∗

dz∗



ac

=

−8Q2

π

2μLR5

dR

dz∗



ρL

Z 1

h

w

2

L

rdr + ρG

Z h

0

w

2

Grdr

. (40)

All quantities in eq. 40 are dimensional except those in the integrals, where wL(r),

wG(r) and h are the dimensionless quantities found in Appendix A. The integrals

are independent of z under the assumption of locally fully developed flow (i.e., the

wavelength of variations in z is  R). In figs. 9 and 10 we show the two integrals in

eq. 40 for M = 0.02 (the viscosity ratio of air to water) as a function of h for various

positive and negative values of F. The line F = 0 corresponds to zero-gravity. In

fig. 11 we take advantage that, for large F, both IntLiq and IntGas are ∼ F

2

. Thus,

a single line independent of F represents accurately IntLiqF

−2 and IntGasF−2 uniformly in 0 < h < 1.

Figures 9 to 11 show the integrals that appear in eq. 40 for virtually any F. From

these graphs one can estimate pressure gradients due to acceleration in microgravity

and in partial gravity. A negative F indicates that the direction of the vector g is

opposite the axial coordinate direction in the pipe. For reference, in normal gravity,

and assuming a 1 cm diameter pipe with characteristic liquid velocity of 1 cm/s,

F ∼ 105. Thus, Martian gravity corresponds to F ∼ 3 × 104 and Lunar gravity to

F ∼ 1.5 × 104. These three cases are therefore covered by the large-F asymptotic

behavior displayed in fig. 11. The microgravity case (|F| ∼ 0.1) is covered in figs. 9

and 10.

5.5.2 Pressure drop due to wall friction in zero gravity

The results from Appendix A are specialized below when the effect of gravity is

negligible by setting F = 0. The dimensional pressure gradient is



dp∗

dz∗



fr

=

8μLU

R2

M

h

4

(M − 1) − M

(41)

where U is the total superficial velocity, M =

μG

μL

is the viscosity ratio, and h (h must

be less than 1) is obtained from

QGf =

h

4

(2M − 1) − 2Mh2

h

4

(M − 1) − M

⇒ h =

"

[M(QGf − 1){M(QGf − 1) − QGf }]

1/2 − M

1 + M(QGf − 2) − QGf #1/2

(42)

3We use the term “pseudo scale” to indicate that the scale is not a constant quantity but is a

function of z through R = R(z).

NASA/CR—2006-214085 30
0.1 0.15 0.2 0.3 0.5 0.7 1

1 -h

0.0001

0.01

1

100

IntLiq

0.001 0.01 0.1 1

h

0.0001

0.01

1

100

IntGas

Figure 9: The integrals in eq. 40 as a function of h, for M = 0.02 and various values

of F = 0, 10, 100, 300, 500. F = 0 corresponds to zero gravity. In IntLiq, the lowest

line is F = 0 and the highest is F = 500. In IntGas, the lines correspond to 10, 0,

100, 300, 500, in the order of increasing maxima. The asymptote of the gas integral

is IntGas ∼ 2h

2

for h  1.

NASA/CR—2006-214085 31
0.15 0.2 0.3 0.5 0.7 1

1 -h

0.0001

0.01

1

100

IntLiq

0.001 0.01 0.1 1

h

0.0001

0.01

1

100

IntGas

Figure 10: The integrals in eq. 40 as a function of h, for M = 0.02 and various values

of F = 0, −10, −100, −300, −500. F = 0 corresponds to zero gravity. In IntGas, the

lowest line is F = 0 and the highest is F = −500. In IntLiq, the lines correspond

to 0, -10, -100, 300, 500, from the lowest to the highest vertical axis intercepts. The

asymptote of the gas integral IntGas ∼ 2h

2

for h  1.

NASA/CR—2006-214085 32
0.2 0.4 0.6 0.8 1

h

0.0002

0.0004

0.0006

0.0008

0.001

0.0012

IntLiq êF2

Good for »F»>200

0.2 0.4 0.6 0.8 1

h

0.001

0.002

0.003

0.004

0.005

0.006

0.007

0.008

IntGas êF2 Good for »F»>500

Figure 11: The integrals in eq. 40 as a function of h, for M = 0.02, valid for large

|F| as marked.

NASA/CR—2006-214085 33
where QGf is the gas volume flow fraction. The physical thickness of the liquid film

adjacent to the tube wall is R(1 − h).

5.5.3 Pressure drop due to wall friction in partial gravity

In Appendix A we have plotted the pressure gradient in two plots (see figs. 16)

according to the decomposition, obvious from eq. 58, dp/dz = (dp/dz)0 +F (dp/dz)1.

From these graphs, the pressure gradient may be estimated for arbitrary values of the

gravity parameter F.

5.5.4 Pressure drop in turbulent annular flow

The analysis we have performed in Appendix A can in principle be used for turbulent

flow as well. Specifically, the relationship between the flow rate in the liquid film, the

pressure gradient, and the liquid film thickness (often called the “triangular relationship”, [56]) can be extended to the case of turbulent flow. Turbulent eddy-viscosity

models are used for the transport of momentum in the thin liquid film. Numerical

integration of the governing equations need to be performed to arrive at the triangular relationship. The reader is referred to literature by Hewitt, [56, 57] and [5] for

further details.

5.6 Droplet entrainment in the air stream in annular flow

A key issue in annular flow is the entrainment of liquid droplets in the air stream.

For the two-phase flow of air and water, annular flow naturally occurs when the air

flow rate is large, with the velocity of the air in the core being considerably larger

than the velocity of the liquid in the layer adjacent to the tube wall. When the relative phase velocity is large, interfacial waves with rather large amplitudes develop.

As a consequence, especially when the air flow is turbulent, these waves break up

portions of the interface into small droplets that are entrained and accelerated by the

air stream. These droplets can also collide with the interface and be deposited back

on the interface (see Hewitt and Govan, 1990, [58]). Because the air phase velocity

is much larger than the liquid’s, the total momentum when entrained liquid droplets

flow in the air core is larger than otherwise. Therefore, the pressure drop with drop

entrainment must be higher than the pressure drop without entrainment.

For flow under reduced gravity, Bousman et al. [23] suggest that droplet entrainment has an important and measurable role in the pressure drop. We have performed

an integral momentum balance (reported in Appendix C) to obtain the effect of entrainment on the pressure drop. In this analysis, we have assumed that no droplets

are present in the inlet gas stream, and that a fraction, fent, of the liquid film flow

rate is entrained as droplets in the exiting gas flow. Hewitt and Govan (1990) [58]

provide an empirical correlation for the entrainment fraction, but find considerable

scatter when compared to experimental data. The pressure drop from Appendix C is

provided below.

NASA/CR—2006-214085 34
∆P =

2τ

∗

W L

R

+ ρLU

2

LS "

fent

UGS

ULS

+ (1 − fent)

3/2



UGS

2ULS 1/2 

μG

μL

1/2

− 1

#

(43)

Using our notation, the correlation given by Hewitt and Govan, 1990, [58] for the

entrainment rate is

fent = 5.75 × 10−5

ρGUGS

ρLULS 

ρ

3

LD

ρ

2

Gσ

(ULS − ULSF )

2

0.316

where

ULSF =

μL

ρLD

exp 5.8504 + 0.4299

μG

μL

rρL

ρG



5.7 Experimental measurements of pressure drop in annular

flow

Bousman (1995) [3] performed experiments in reduced gravity aboard the NASA

Learjet. Bousman used air-water, air-water/glycerin and air-water/zonyl as the fluid

systems in the experiments. Water/glycerin mixture was used to study the effect of

increase in viscosity and the surfactant zonyl was used to study the effect of reduction

of surface tension.4 The experiments were performed in a tube of internal diameter

of 12.7 mm. Bousman compared the measured pressure drop in annular flow to

predictions from the Lockhart-Martinelli correlation, and reports that this model is

in agreement for all three fluids with an average error of ± 20% (see fig. 12). Bousman

also recorded comparisons between the friction factor calculated from his measured

pressure drops and the friction factor calculated using the standard relations 32-

34, see figs. 13 and 14. He notes that, despite the “homogeneous” label of the

model, he used μL to evaluate f (shown as fT P in the figures’ axes) because mostly

liquid is in contact with the wall. Given the large scatter in the data it is hard

to assess the goodness of the comparison besides the apparent observation that the

calculated friction factor falls in the data cloud for slug flows but falls outside for

bubbly and bubbly-slug flows. Bousman also indicates that separated flow models

like the Lockhart-Martinelli predict poorly the pressure drops in bubbly and slug

flows where the relative phase velocity is small. Earlier microgravity experiments by

Miller et al. (1993) [59] found that in the case of freon R-12, the Lockhart-Martinelli

model has an average error of 22% for flow in a tube of diameter 10.5 mm, but the

error for a 4.6 mm tube is 56%.

4

It is known that soluble surfactants –such as Zonyl– have dynamic effects besides lowering the

surface tension. Dynamic effects arise from surfactant transport in the bulk and the free surface, and

their coupling via adsorption kinetics. Dynamic effects will likely introduce deviations in interfacial

deformation dynamics from the dynamics present if the surface tension were uniform.

NASA/CR—2006-214085 35
Figure 12: Bousman’s [3] comparison between pressure drop in microgravity annular

flow and the Lockhart-Martinelli prediction.

6 Heat Transfer in Fully Developed Annular Flow

In single fluid flow analyses, the relationship between wall heat transfer and wall

friction has been exploited routinely to develop expressions for heat transfer coefficients from the friction factor data. For fully developed turbulent flow in a pipe with

molecular and turbulent Prandtl numbers close to unity (P r =

ν

α

, where ν = μ/ρ is

kinematic viscosity, and α = k/ρ cp is thermal diffusivity), the assumption that heat

and momentum are transported at the same rate produces the following relationship

between the thermal and flow fields:

q

∗

cpτ

∗

du = dT, (44)

where q

∗

is heat flux normal to the wall and τ is shear stress. If, in addition, it

is further assumed that the ratio of heat flux to shear stress is constant across the

domain such that

q

∗

τ

∗

=

q

∗

W

τ

∗

W

,

where the subscript W denotes the quantity evaluated at the tube wall, eq. 44 can

be integrated between the wall and mean bulk conditions to provide the following

relationship between wall heat flux and shear stress:

q

∗

W um

τ

∗

W cp

= TW − Tb. (45)

NASA/CR—2006-214085 36
Figure 13: Bousman’s [3] comparison between Fanning friction factor in microgravity

bubbly and bubbly-slug flows and the homogeneous fluid prediction (eqs. 32-34 where

the liquid viscosity μL is used.

Figure 14: Bousman’s [3] comparison between Fanning friction factor in microgravity

slug flow and the homogeneous fluid prediction (eqs. 32-34 where the liquid viscosity

μL is used.

NASA/CR—2006-214085 37
Here um and Tb are the flow velocity and temperature outside the boundary layer

respectively. When the definitions of heat transfer coefficient, q

∗ = h ̃(TW − Tb), and

Fanning friction factor, τ

∗

W =

f

2

ρu2

m are substituted into eq. 45, a relationship between

wall heat transfer and Fanning friction factor is derived:

St =

f

2

. (46)

Here the Stanton number, St, represents the ratio of wall to convective heat transfer

given by:

St ≡

h ̃

ρcpum

.

Equation 46 is called the Reynolds analogy for pipe flow and strictly speaking is only

valid for fully developed turbulent pipe flows with molecular and turbulent Prandtl

numbers equal to one [60]. A similar relationship between wall friction factor and heat

transfer coefficient can be derived for turbulent and laminar boundary layer flows over

a flat plate. This relationship is also known as the Colburn analogy [60]:

St P r2/3 =

f

2

. (47)

Colburn’s analogy suggests that there is a Prandtl number dependence of P r2/3for

the flat plate problem. Curiously, it turns out this dependence also works well for

turbulent pipe flows. Thus for turbulent pipe flows, a Reynolds analogy can be written that includes the Prandtl number dependence of the fluid in the form of eq. 47.

Even for single phase flows, one has to be cautioned that, while the friction-heattransfer analogies can be applied to both laminar and turbulent flows over a flat plate

and to turbulent pipe flows, they are not valid for laminar pipe flows and that, in general, more rigorous treatments of the governing equations and boundary conditions

are needed when seeking new applications for these analogies. Specifically, Knudsen

and Katz (1958) [61] have shown that for more generalized applications of the heattransfer-fluid-friction analogies, the resulting expressions often do not take the simple

forms represented by eq. 47.

Two-phase heat transfer in the fully developed laminar annular flow in pipes is

amenable to analytical solution and an expression for the heat transfer coefficient

can be derived as shown and described in Appendix B. Therefore, in two-phase flow,

the heat-transfer fluid-friction analogy has been most widely applied to developed

and developing turbulent annular flows. If the turbulent annular flow is fully developed and the thermal boundary layer is totally contained with in the liquid layer,

then eq. 47 can be employed to estimate the heat transfer coefficient. For developing turbulent annular flows, application of Reynolds analogy becomes more involved.

Chitti and Anand (1995) [62] have come up with an analytical procedure to derive

the heat transfer coefficient through an iteration procedure for calculating film thickness. This procedure eliminates the need to calculate void fraction, pressure drop and

NASA/CR—2006-214085 38
shear stress distribution. Instead, a distribution of heat flux in the radial direction is

obtained by performing a steady state radial energy balance in the fluid and neglecting any axial contributions to heat transfer. The results indicate that the predicted

values of the local heat transfer coefficient are within 25% of experimental data with

a mean deviation of 15%.

The Reynolds analogy has been also been used rather loosely in some two-phase

flow codes for piping systems. For example, the transient two-phase flow pipeline

analysis code, TREMOLO, (Eliscon et al, 1999, [63]) uses Eq. 46 to calculate the

heat transfer coefficient for annular flow, under any operating conditions. For plug

flow, the model assumes that the phases contact the pipe wall in proportion to their

volume fractions. The liquid phase heat transfer coefficient is then calculated according to eq. 46, while a constant condensation heat transfer coefficient is assumed for

the vapor phase. The adequacy of such an approach to serve as a general procedure

for calculating heat transfer coefficients for two-phase pipe flows is at best questionable.

Finally Ghajar (2004) [64] has pointed out the need for more systematic research

in this area whereupon accurate isothermal pressure drop measurements are taken

in the same regions where measured heat transfer data are available. The pressure

drop data can then be used through a modified Reynolds analogy to back out heat

transfer coefficients. By comparing the predicted heat transfer results against the

experimentally measured heat transfer data, the correct form of the corresponding

modified Reynolds analogy can be derived semi-empirically. In reality, this task needs

to be undertaken for all the different two-phase flow regimes in 1g. Once the correct

Reynolds analogy relationships are established for the two-phase pipe flows, they can

then be conceivably used to obtain two-phase heat transfer data for regions and environments where, due to our experimental limitations, obtaining heat transfer data

is not easily done.

7 Conclusions

In this report, we have presented a summary of flow regimes and pressure drop correlations that exist in the literature for two phase flow in a pipe, during operation in

normal gravity, microgravity and partial gravity. While a lot of data exists for two

phase flow in normal gravity, data for microgravity and especially partial gravity are

scarce. Several applications of two phase flow in advanced life support systems for

space applications suggest that the flow rates of gas and liquid are quite low compared

to typical values in previous experiments. There is therefore a compelling need to

revisit microgravity and partial gravity experiments and validate existing flow regime

maps for the reduced flow rate conditions. In the case of laminar, fully-developed

annular flow in a pipe, with the direction of flow and gravity being collinear, we find

that the problem is amenable to rigorous mathematical treatment. Such a treatment

has been extended to the heat transfer as well, when the wall heat flux is a constant.

The effect of varying gravity level (Moon, Mars, microgravity) on the pressure drop

and rate of heat transfer can be baselined from these analyses. There is a need to valNASA/CR—2006-214085 39
idate these predictions, and the scalings (especially of the gravitational level) implied

by these predictions, by appropriately designed experiments.

NASA/CR—2006-214085 40
Appendix

A Fully developed, Laminar, Annular Flow

The goal is to predict the liquid film thickness, R − h

∗

, pressure gradient, dp∗

dz∗ and

wall shear stress, τW , for given liquid and gas flow rates. Note that

• Liquid is adjacent to the wall, air is in the core of the flow, and g is oriented

along the tube axis (z

∗ direction).

• The liquid and gas flow rates are specified to be QL and QG.

• Assume incompressible, laminar, axisymmetric, fully developed flow, and a constant liquid film thickness.

UG

UL

CL

h

R

Figure 15: Schematic of fully developed annular flow with liquid annulus surrounding

a gas core.

The governing equation for the flow in the liquid and gas is written below. A superscript ∗ denotes a physical (dimensional) quantity.

0 = −

dp∗

dz∗

+ μ

1

r

∗

d

dr∗



r

∗

dw∗

dr∗



+ ρ g (48)

The boundary conditions are

dw∗

dr∗

= 0 at r

∗ = 0 (49)

w

∗ = 0 at R∗ = R (50)

w

∗

and μ

dw∗

dr∗

are continuous at r

∗ = h∗

(51)

Dimensionless variables are defined as follows.

w =

w

∗

WR

, p =

p

∗ − ρLgz∗

PR

, r =

r

∗

R

, z =

z

∗

R

, h =

h

∗

R

, WR =

Q

πR2

NASA/CR—2006-214085 41
Q = QL + QG, PR =

μLWR

R

, M =

μG

μL

, F =

(ρL − ρG)gR2

μLWR

(52)

For fully developed flow, the difference in pressure between the liquid and gas at any

z location must be independent of z. Therefore, dpL

dz =

dpG

dz which will be denoted by

dp

dz . The governing equation for the flow in the liquid and gas become

1

r

d

dr 

r

dwL

dr 

=

dp

dz h ≤ r ≤ 1 (53)

1

r

d

dr 

r

dwG

dr 

=

1

M

dp

dz +

F

M

0 ≤ r ≤ h (54)

The solution is

wL =

1

4

dp

dz (r

2 − 1) + 1

2

F h2ln r (55)

wG =

1

4

dp

dz 

h

2



1 −

1

M



− 1 +

r

2

M



+

F

4 M

[r

2 + h2

(2M ln h − 1)] (56)

The liquid film thickness and the pressure gradient are determined from the following

equations:

QGf =

h

4

(2M − 1) − 2Mh2

h

4

(M − 1) − M

+

1

2

F h4ln h −

F

8

h

4

(h

2 − 1)[h2

(4M − 3) − 4M + 1]

h

4

(M − 1) − M

(57)

dp

dz =

8M − h

2F [h2

(2M − 1) − 2M]

h

4

(M − 1) − M

(58)

where QGf =

QG

Q

and QG = 2π

R h

0 wG r dr. The shear stress exerted by the wall onto

the fluid, τ

∗

W = τ

∗

rz is 5

τ

∗

W = μL



dw∗

L

dr∗



W

=

μLWR

R



dwL

dr 

r=1

=

1

2

μLWR

R



dp

dz + F h2



(59)

5

In order to get the signs correct, ∆P

L

should be interpreted as dp

dz and τW is positive when it

points along the negative z direction.

NASA/CR—2006-214085 42
Note that the physical pressure gradient is

dp∗

dz∗

= ρLg +

PR

R

dp

dz (60)

Using dp/dz given in eq. 59, and inserting the definition of PR ≡ μLWR/R, we find

dp∗

dz∗

= ρLg +

μLWR

R2



2τ

∗

W R

μLWR

− F h2



= ρLg +

2τ

∗

W

R

−

μLWR

R2

F h2 =

ρLg +

2τ

∗

W

R

− ∆ρgh2 = g



ρL(1 − h

2

) + ρGh

2



+

2τ

∗

W

R

(61)

In other words, the physical pressure gradient is composed of the wall shear and a

gravity component with an effective density

ρef f ≡ ρL(1 − h

2

) + ρGh

2

(62)

The area fractions of gas and liquid are h

2 and 1 − h2

respectively. Eq. 62 is an

exact result for fully developed annular flow. It shows that the averaging that yields

the effective density must be weighted with the phase area fractions given above. As

expected, the result

dp∗

dz∗

= g



ρL(1 − h

2

) + ρGh

2



+

2τ

∗

W

R

(63)

agrees with the force balance around a control volume of the annular flow. The pressure gradient is balanced by the weight and the drag that the wall exerts on the

liquid. The results for the gravitational and frictional pressure gradient given above

are in accord with those given in Section 5.1.

In figs. 16 we plot the two components of the pressure gradient given in eq. 58.

(dp/dz)0 shows the purely viscous part, obtained for F = 0; whereas (dp/dz)1 shows

the gravitational part, given by the coefficient of F; refer to eq. 58. With these

two figures we can calculate pressure gradients for arbitrary F. Similarly, in figs.

17 we show Qgf0 = Qgf (F = 0) and the coefficient of F, Qgf1. For arbitrary F,

Qgf = Qgf0 + F Qgf1.

For most parameter ranges, there is a unique h for each Qgf and M. However,

we note that, in certain ranges of F for given M, there exist more than one h for

given Qgf . This is illustrated in fig. 18, where we show Qgf for M = 0.02 and F = 18.

Depending on Qgf , there may be one, two or three values of h that satisfy eq. 57.

The implications of this multiplicity in film thickness for fixed gas flow fraction are

not clear at this time and will be addressed later during the course of this project.

Nevertheless, for Qgf & 0.5 for which annular flow can be expected to occur, the film

thickness is uniquely determined.

A.1 Core phase center relative to pipe center

It may be shown that, for a given pressure gradient, dp/dz, the maximum total flow

rate QL + QG is obtained when the center fluid core is concentric with the pipe. Any

eccentricity causes the total flow rate to fall below the concentric case maximum.

NASA/CR—2006-214085 43
0.2 0.4 0.6 0.8 1

h

2

4

6

8

-J

dP



dz N

0

F=0

0.2 0.4 0.6 0.8 1

h

0.2

0.4

0.6

0.8

1

-J

dP



dz N

1

F-Comp.

Figure 16: The purely viscous ((dp/dz)0 ≡ dp/dz for F = 0) and gravitational (the

coefficient of F in eq. 58) components of the pressure gradient. M = 0.02. To get

dp/dz for arbitrary F one should add (dp/dz)0 + F (dp/dz)1.

0.2 0.4 0.6 0.8 1

h

0.2

0.4

0.6

0.8

1

Qgf0

F=0

0.2 0.4 0.6 0.8 1

h

-0.04

-0.03

-0.02

-0.01

Qgf1

Figure 17: The purely viscous (Qgf0 ≡ Qgf for F = 0) and gravitational (the coefficient of F in eq. 57) components of Qgf . M = 0.02. To get Qgf for arbitrary F one

should add Qgf0 + F Qgf1.

0.2 0.4 0.6 0.8 1

h

0.2

0.4

0.6

0.8

1

Qgf

F=18 Figure 18: Qgf for M = 0.02 and F = 18.

NASA/CR—2006-214085 44
B Heat Transfer in Fully Developed Annular Flow

The results given in the Appendix A can be used to determine the heat transfer in

fully developed annular flow. We assume that a constant heat flux per unit wall area,

q

∗

, is imposed at the tube wall. The goal is to obtain the Nusselt number, defined as

Nu ≡ hR/k  ̃

L, where h ̃ is the heat transfer coefficient, R the pipe radius and kL the

liquid’s thermal conductivity.

We begin by writing the steady state energy equation:

w

∗

∂T∗

∂z∗

= α



∂

2T∗

∂z∗2

+

1

r

∗

∂

∂r∗



r

∗

∂T∗

∂r∗

 (64)

where α ≡ k/ρ cp denotes the thermal diffusivity, with cp being the fluid’s heat capacity. The axial velocity field w

∗ depends only on the radial coordinate, r∗

, and is

given in Appendix A.

Because the rate of heat input along the wall is constant with z

∗

, we anticipate

that the temperature field T

∗

(r

∗

, z∗) will be the sum of a function of r

∗ plus a linear

function of z

∗

. We will show that a constant axial temperature gradient G is established by the imposed wall heat flux. G will be determined as a part of the solution.

Based on these ideas, we seek a solution where the temperature field is written as

follows:

T(r) ≡

T

∗

(r

∗

, z∗) − Gz∗

GR (65)

where T(r) is dimensionless. With this substitution, the energy equation in each

phase becomes:

P eL wL(r) = 1

r

d

dr 

r

dTL

dr 

(66)

P eG wG(r) = 1

r

d

dr 

r

dTG

dr 

(67)

where P e(G,L) =

wRR

α(G,L)

is the P ́eclet number of the gas or the liquid. The boundary

conditions are: a constant heat flux, q ≡ q

∗/kLG is imposed at the tube wall, the

temperature field is symmetric about the axis of the tube, and the temperature and

heat flux are continuous across the liquid/gas interface.

dT

dr = q at r = 1 (68)

dT

dr = 0 at r = 0 (69)

TL = TG at r = h (70)

NASA/CR—2006-214085 45
β

dTG

dr =

dTL

dr at r = h (71)

where q ≡

q

∗

kLG

and β ≡

kG

kL

is the thermal conductivity ratio. In this analysis the

temperature field depends only on temperature differences and/or gradients. Thus,

T(G,L)(r) can only be determined up to an additive constant. To choose one value for

this constant, we set TL(1) to be zero. With these boundary conditions, the solution

is

TL =

P eL

8



FMh2



1 − r

2 + ln r



r

2 + h2

(1 − 2 ln h)

 +

dp

dz 

h

2

ln r(1 −

h

2

2

) + (r

2 − 1)(r2 − 3) +

P eG β

8



2F h4ln r



M ln h −

1

4



+

dp

dz h

2

ln r



h

2



1 −

1

2M



− 1

 (72)

TG =

P eL

8



FMh2



1 + h

2

(−1 + 2 ln h(1 − ln h))+

1

2

dp

dz 

3

4

+ h

2



h

4

4

− 1



+ h

2

ln h(2 − h

2

)

 +

P eG

8



F



h

2M ln h(r2 − h2 + 2βh2

ln h) + 1

8



(h

2 − r2

)(3h

2 − r2

) − 4βh4ln h





+

1

2

dp

dz 

(h

2 − 1)(r2 − h2 + 2βh2

ln h) + (h

2 − r2

)(3h

3 − r2

) − 4βh4ln h

4M

 , (73)

where dp

dz is given by eq. 58. By applying the heat flux boundary condition, −q =

dT/dr at r = 1, we determine the unknown axial temperature gradient G from:

q

∗

kLG

= q =

1

2

[P eL(1 − QGF ) + βP eGQGF ] . (74)

Noting that βP eG = P eL

ρG cpG

ρL cpL

, we conclude that both TG and TL are proportional to

P eL, and depend on the heat capacity ratio ρG cpG

ρL cpL

and β in a more complex manner.

This is useful, because then P eL is the only parameter that contains flow rate in the

expressions for temperature.

To compute the Nusselt number we start from the expression h ̃(T

∗

W − T

∗

AV ) = −q

∗

.

The average temperature in the fluid is defined as

T

∗

AV =

R

A

ρ cPw

∗T∗dA

R

A

ρ cPw∗dA .

The weighting by ρcp defines an average temperature that closely reflects the actual

average temperature in an instantaneous sample of a cross sectional slice of the flow.

In other words, a representative average temperature cannot come from an arithmetic

average over the cross sectional area. This is because the change in fluid temperature

by the heat input q

∗ must depend on heat capacity: for example, if ρGcpG = 0, the

NASA/CR—2006-214085 46
0.2 0.4 0.6 0.8 1

h

0.25

0.5

0.75

1

1.25

1.5

1.75

2

-Nu0

8 H1-hL ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ 15 F=105

0.2 0.4 0.6 0.8 1

h

0.25

0.5

0.75

1

1.25

1.5

1.75

2

-Nu0

8 H1-hL ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ 15

F=3 104

0.2 0.4 0.6 0.8 1

h

0.25

0.5

0.75

1

1.25

1.5

1.75

2

-Nu0

8 H1-hL ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ 15

F=1.5 104

0.2 0.4 0.6 0.8 1

h

-0.2

0.2

0.4

0.6

Nu1

64 M H1 - hL

3

ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ 15 F=105

0.2 0.4 0.6 0.8 1

h

-0.2

0.2

0.4

0.6

Nu1

64 M H1 - hL

3

ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ 15

F=3 104

0.2 0.4 0.6 0.8 1

h

-0.2

0.2

0.4

0.6

Nu1

64 M H1 - hL

3

ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ 15

F=1.5 104

Figure 19: Coefficients Nu0 and Nu1 as defined in eq. 76, for F values as marked,

and M = 0.02. The horizontal axis, h, is the radius of the gas core. The values of F

represent, in decreasing order, Earth’s, Mars’ and Moon’s gravity levels.

gas cannot accept heat and therefore TG does not change in the streamwise direction;

however, the liquid temperature increases more rapidly than if ρGcpG6= 0 because

then all the heat input q

∗ goes to the liquid.

Making the preceding expression dimensionless, and evaluating q

∗ = −k(∂T/∂r)|W ,

Nu (TW − TAV ) = ∂T

∂r









W

(75)

Thus, the Nusselt number can be calculated after solving for T(r). In general, Nu =

f(h, F, M, β, ρG cpG

ρL cpL

). The expression was worked out in Mathematica and it’s too

complicated to be written here. However, we have analyzed one relevant limiting

case,

ρG cpG

ρL cpL

→ 0 up to order O 

ρG cpG

ρL cpL



. We assume the following Taylor series:

Nu(h, F, M, β, ρG cpG

ρL cpL

) = Nu0(h, F, M) + ρG cpG

ρL cpL

Nu1(h, F, M) + ... (76)

Interestingly, up to this order of approximation, the dependence on β disappears exactly.

For illustration purposes, we show in figs. 19 through 22 the two coefficients of

eq. 76 for M = 0.02 (the viscosity ratio of air to water), for various F positive and

negative, and 0 ≤ h ≤ 1. Since Nu0 ∼ (1 − h)

−1 and Nu1 ∼ (1 − h)−3 as h → 1, we

plot each term multiplied by its divergent behavior to suppress the singularity and

thus make the plot bounded. Note that the coefficients are quite insensitive to the

gravitational parameter F in a broad range of F  1:

B.1 Criterion for phase in contact with wall

Assume that liquid and gas flow rates QL, QG are injected into a pipe of radius R.

Assume, further, that these parameters together with the fluid properties are such

NASA/CR—2006-214085 47
0.2 0.4 0.6 0.8 1

0.95 h

1

1.05

1.1

1.15

-Nu0

8 H1-hL ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ 15

F=0

0.2 0.4 0.6 0.8 1

h

0.2

0.4

0.6

0.8

1

Nu1

64 M H1 - hL

3

ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ 15

F=0

Figure 20: Coefficients Nu0 and Nu1 as defined in eq. 76, for F = 0 (zero gravity).

2 4 6 8 10 12 14 h

è!!!! F

0.5

1

1.5

2

2.5

3

3.5

4

-Nu0

F=105

2 4 6 8 10 12 14 h

è!!!! F

0.5

1

1.5

2

2.5

3

3.5

4

-Nu0

F=3 104

2 4 6 8 10 12 14 h

è!!!! F

0.5

1

1.5

2

2.5

3

3.5

4

-Nu0

F=1.5 104

Figure 21: Coefficient Nu0 near h = 0 for F values as in fig. 19.

0.2 0.4 0.6 0.8 1

h

0.25

0.5

0.75

1

1.25

1.5

1.75

2

-Nu0

8 H1-hL ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ 15 F=-105

0.2 0.4 0.6 0.8 1

h

0.25

0.5

0.75

1

1.25

1.5

1.75

2

-Nu0

8 H1-hL ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ 15

F=-3 104

0.2 0.4 0.6 0.8 1

h

0.25

0.5

0.75

1

1.25

1.5

1.75

2

-Nu0

8 H1-hL ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ 15

F=-1.5 104

2 4 6 8 10 12 14 h

è!!!!!!! -F

0.5

1

1.5

2

2.5

-Nu0

F=-105

2 4 6 8 10 12 14 h

è!!!!!!! -F

0.5

1

1.5

2

2.5

-Nu0

F=-3 104

2 4 6 8 10 12 14 h

è!!!!!!! -F

0.5

1

1.5

2

2.5

-Nu0

F=-1.5 104

Figure 22: Coefficient Nu0for negative F values as in fig. 19. The plot on the right

shows a detail near h = 0; note the scaled h variable.

NASA/CR—2006-214085 48
hG R

gas

liquid

CL

Figure 23: Schematic of a fully developed annular flow with gas center core of radius

hG. If liquid flows in the center, then its core radius would be hL.

that conditions for annular flow are met. Even in this case, the operator has no

control over the phase distribution within the pipe. Which fluid flows next to the

wall –and which one in the core– is selected by the system without the operator’s

intervention. Here we propose an argument based on the interfacial energy of the

system that begins to address this question.

Refer to fig. 23. hL and hG refer respectively to the core radii of the liquid or the

gas, depending on whether liquid or gas flows in the core. We evaluate the interfacial

energies of the two possible configurations. The quantities, EGC and ELC refer to the

interfacial energies per unit pipe length of the system with gas core and liquid core,

respectively:

1) Gas core: EGC = 2π (R σSL + hG σ).

2) Liquid core: ELC = 2π (R σSG + hL σ).

Subtracting the two expressions, EGC − ELC = 2π [R (σSL − σSG) + (hG − hL)σ].

Here, σSL and σSG are the interfacial energies per unit area of the solid-liquid and the

solid-gas interfaces, respectively; and σ is the liquid-gas interfacial tension. Young’s

equation relates these three quantities to the contact angle, θ: σ cos θ = σSG − σSL.

Inserting this into the previous expression we obtain:

EGC − ELC = 2πσ [−R cos θ + (hG − hL)] (77)

The energy principle states that the lowest energy state is most favorable. This implies that gas core should be expected when EGC − ELC < 0, and liquid core when

EGC − ELC > 0. In terms of the contact angle,

Gas Core : cos θ > hG − hL

R

. (78)

Liquid Core : cos θ < hG − hL

R

. (79)

In fig. 24, we can see that, for air-water (M =

μG

μL

= 0.02), the gas core is thicker

NASA/CR—2006-214085 49
0.2 0.4 0.6 0.8 1

QGF

-1

-0.75

-0.5

-0.25

0.25

0.5

0.75

1

hG -hL

Figure 24: Difference if gas and liquid core radii normalized by pipe radius R, as a

function of gas flow rate fraction, QGF for M = μG/μL = 0.02. hG is calculated with

eq. 42; hL is obtained from the same equation, by replacing M by M−1 and QGF by

1 − QGF .

than the liquid core (i.e., hG > hL) if QGF & 0.71. Then the right-hand side of eq.

78 is positive. This condition is met in most gas-liquid annular flows.

From these expressions we conclude that, for example, in order to have liquid next

to the wall (i.e., gas core) with QGF & 0.71, having a wetting liquid (i.e., cos θ > 0,

or θ < π/2) is necessary but not sufficient; in fact, the liquid must be more wetting

than given in equation 78, i.e., the contact angle must be less than an upper bound

given by:

1 > cos θ > hG − hL

R

> 0, or (80)

0 < θ < arccos

hG − hL

R

< π/2. (81)

In order to have liquid core when QGF & 0.71, eq. 79 shows that the contact angle

may be < 90◦ as long as the inequality 79 is met. If, on the other hand, QGF . 0.71,

the gas core is thinner than the liquid core for the same QGF and a gas core exists

when

1 > cos θ > hG − hL

R

, or (82)

0 < θ < arccos

hG − hL

R

. (83)

In this case, because hG−hL

R < 0, it might be possible to have liquid flowing next to

the wall even for “non-wetting” liquids with contact angles π/2 < θ < arccos hG−hL

R

.

NASA/CR—2006-214085 50
C Pressure Drop by Integral Momentum Balance

with Droplet Entrainment in the Air Stream

The goal is to predict the pressure drop, ∆p, given the wall shear stress, τW and the

flow rates QL and QG, when drops of liquid break off the liquid film and are entrained

in the core gas flow. Clearly this event must alter the balance between τW and ∆P

derived in Appendix A because the liquid drops, moving at the average gas phase velocity, will carry a higher momentum than a case where all the liquid flows in the film.

UG UG

UL

Figure 25: Schematic of annular flow with liquid drops entrained into the gas stream.

The gas velocity UG is  UL, the liquid velocity. Once the entrained liquid drops have

attained UG in the gas stream, a fraction fent of the liquid moves with a momentum

significantly higher than when that fraction was in the liquid annulus. As a result,

the pressure has to reduce to compensate for the momentum needed to accelerate

these drops.

We assume that

• Liquid is adjacent to the wall and air is in the core of the flow.

• Gravity is negligible.

• Some liquid droplets are entrained in the air core of the flow and no gas is

entrained in the liquid film.

• The flow is incompressible, laminar, axisymmetric, fully developed, and the

liquid film thickness, t ≡ R − h, is constant.

The liquid QL and gas flow QG rates are related to the superficial velocities by:

QL = πR2ULS (84)

QG = πR2UGS (85)

If qLD is the liquid flow rate that is flowing as drops entrained in the gas phase, then

conservation of liquid flow rate gives

QLF = QL − qLD (86)

NASA/CR—2006-214085 51
If ULF is the average liquid velocity in the film,

ULF ≡

QLF

2πRt

, (87)

and substituting in eq. 86

ULF 2πRt = QL − qLD (88)

Since R ≈ R − t, the shear stress balance at the interface is approximately

μLULF

t

=

μGUGS

R

(89)

Replacing ULF from eq. 88 we solve for t

t

R

=



1

2

QL − qLD

QG

μL

μG

1/2

(90)

To write the momentum in, we assume that the two phases are well mixed. To write

the momentum out, we assume that droplets, with total flow rate qLD, are convected

with UGS and that they do not get deposited in the liquid film. Then:

M|

out = 2πRhρLU

2

LF + ρGU

2

GSπR2 + ρLqLDUGS + PoutπR2

(91)

M|

in = ρLU

2

LSπR2 + ρGU

2

GSπR2 + PinπR2

(92)

The change in momentum flux is balanced by the shear stress at the wall, M|

out −

M|

in = τ

∗

W . Therefore,

∆p ≡ pin − pout =

2τW L

R

+

qLDρLUGS

πR2

+

2ρLtU2

LF

R

− ρLU

2

LS (93)

Using eqs. 88 and 90, we may write

tU2

LF =



QL − qLD

2πR 2

1



1

2π

QL−qLD

UGS

μL

μG

1/2

=

(2QG)

1/2

4π

2

(QL − qLD)

3/2

R3



μG

μL

1/2

(94)

Substituting into eq. 93,

∆p =

2τW L

R

+ ρLU

2

LS "

qLD UGS

πR2U

2

LS

+



QG

2QL

1/2 

1 −

qLD

QL

3/2 

μG

μL

1/2

− 1

#

(95)

The only undetermined parameter in the above expression is qLD. If we define the

fraction of liquid drops entrained as fent ≡ qLD/QL, we may write:

∆p =

2τW L

R

+ ρLU

2

LS "

fent

UGS

ULS

+ (1 − fent)

3/2



UGS

2ULS 1/2 

μG

μL

1/2

− 1

#

(96)

NASA/CR—2006-214085 52
D Flow regimes in ALS equipment operating in

microgravity

We describe an estimation tool for flow regimes that might develop in two-phase flows

occurring in water recovery equipment in zero gravity. The calculation is based on

the flow regime map discovered by Jayawardena, Balakotaiah and Witte (1997) [1].

We have had to make reasonable assumptions about pipe diameters and flow rates

for lack of concrete information in the available space systems literature (Eckart 1996

[35], Eckart 1997 [36], Wieland 1998 [37], Wieland 1998 [38], Wieland 1994 [39]). The

only experiments on oxygen generation in microgravity are by Ohira et al., 2001 [65]

who performed experiments that used cultivated microalgae.

First we took nominal gas and liquid flow rates input into a few of the systems as

shown in Eckart (1996)[35]. We assume that the gas input shown in the table refers

to gas dissolved in the urine feed. We also assume that, as the pressure decreases

downstream, this dissolved gas comes out of solution completely, thus generating a

two-phase flow with the flow rates indicated.

We have developed an Excel worksheet that returns the flow regime and the values of the liquid and gas Reynolds numbers as the answer, for given liquid and gas

mass flow rate, and pipe diameter inputs. The worksheet has fixed liquid and gas

properties (density and viscosity), as well as water-air surface tension. We have assumed a surface tension value of σ = 70 dyn/cm, a good estimate for pure water but

maybe as much as twice that of urine.

The liquid and gas Reynolds numbers should give a rough indication of whether

the flow is laminar or turbulent. We discuss some ideas about transition mechanisms,

and the usefulness of the Reynolds number to decide whether the flow is laminar.

D.1 Dimensionless Groups for Current Space ALS Hardware

In zero gravity, and because μG  μL and ρG  ρL, of the seven dimensionless groups

identified in section 2.1, only three independent groups remain:

Su ≡

ρLDσ

μ

2

L

, ReL ≡

ULS ρLD

μL

, and ReG ≡

UGS ρGD

μG

,

where UGS ≡ 4QG/πD2 and ULS ≡ 4QL/πD2 are the gas and liquid “superficial” velocities. It is clear that these are not the actual average phase velocities, but instead

the velocities of the gas and liquid if each phase were flowing in the pipe alone.

Su is the Suratman number. It is a “Reynolds” number where the characteristic

velocity is the capillary velocity scale σ/μ. For this reason, it does not involve any

externally imposed velocity scale; it only depends on pipe diameter and material

properties. Of course, {ReG, ReL, Su} is not the only choice of groups; but it is

the one that best (i.e., most compactly) correlates the data, at least in the range of

NASA/CR—2006-214085 53
parameters probed in Jayawardena et al. (1997). Any other choice of groups must

comprise three groups and only three.

D.2 Flow regime estimation in two-phase ALS technologies

operating in Microgravity

Below is an example of a list of two-phase systems that are critical to regenerative life

support technologies for water and atmosphere management. We use the flow regime

maps (and regime boundaries) discussed in section 3.1.

Example: Vapor Phase Catalytic Ammonia Removal (VPCAR): From Eckart (1996,

p. 224), we find M ̇

L = 13 Kg/day, M ̇G = 0.02 Kg/day. From these, using fluid densities, we get QG = 0.23 cm3/s and QL = 0.15 cm3/s, ReG/ReL = 7.67 10−2

. For an

assumed 1/4 inch diameter straight cylindrical pipe, D = 0.64 cm and Su = 4.5 105.

Since ReG/ReL = 7.67 10−2is less than 464.16 Su−2/3 = 7.9 10−2(the location of

the left-most boundary in Fig. 2), we conclude that the flow is bubbly (although

the point is so close to the boundary that one should be cautious about flow type

without experimental verification). Further, since ReG and ReL are not large, we may

surmise that both phases flow in the laminar regime. In the table below we compare

three systems as found in Eckart (1996). We assume D = 0.64 cm, ρL = 1 g/cm

3

,

ρG = 10−3 g/cm

3

, μL = 10−2 Poise, μG = 2 10−4 Poise, σ = 70 dyn/cm. Note that

these units are all in the CGS system so they may be entered into the dimensionless

groups as is. The decision about laminar flow is based on the values of ReG and ReL;

see more in Discussion section.

Other systems include:

Vapor Compression Distillation (VCD): From Eckart (1996, p. 221), we find M ̇

L = 33

Kg/day, M ̇G = 0.03 Kg/day.

Thermoelectric Integrated Membrane Evaporation System (TIMES): From Eckart

(1996, p. 225), we find M ̇

L = 22 Kg/day, M ̇G = 0.7 Kg/day.

Electrolysis Plant (ELEKTRON): From Wieland (1998, p. 57, Vol. 2), we find

M ̇

L = 8 Kg/day, M ̇G = 6 Kg/day.

Supercritical Water Oxidation (SCWO): From Eckart (1996, p. 241), we find M ̇

L =

25.6 Kg/day, M ̇G = 2.59 Kg/day.

Condensing Heat Exchanger (CHX): From Wieland (1998, p. 104, Vol. 1 and Scull

et al. 1998), we find M ̇

L = 24 Kg/day, M ̇G = 11132 Kg/day.

NASA/CR—2006-214085 54
Table 1: Summary comparison of three water recovery systems for D = 0.64 cm.

Mass flow data from Eckart (1996) and Wieland (1998)

M ̇ G M ̇

L D Flow

SYSTEM Kg/day Kg/day cm ReG ReL Regime Laminar?

VPCAR 0.02 13 0.64 2.3 30 Bubble Yes

VCD 0.03 33 0.64 3.5 76 Bubble Yes

TIMES 0.7 22 0.64 80 50 Annular Yes

ELEKTRON 6 8 0.64 690 18 Annular Inconclusive

SCWO 2.59 25.6 0.64 299 58 Annular Yes

CHX 11132 24 0.64 1.2 106 55 Annular No

D.3 Discussion

First we address some ideas about flow regime transitions that can be gleaned from

the map. For fixed Su, transition from bubble to slug happens as the ratio of superficial Reynolds numbers approaches a certain value. In our case, with a pipe

diameter D = 0.64cm (1/4 inch), Su = 4.5 × 105. And at transition, UGS/ULS =

464.16 Su−2/3 = 0.08. When the gas superficial velocity is much less than the liquid’s,

we get bubble flow. In this flow, bubbles (of a certain size) at relatively low concentration (so that the gas volume fraction is  1), move essentially at the liquid’s average

velocity. For fixed ratio of superficial Reynolds numbers, the transition occurs when

Su exceeds a certain value. For a given material system, increasing Su is the same

as increasing pipe diameter. As D increases, for fixed UG and UL, Reynolds number

increases and we may expect inertia forces to stimulate coalescence. As enough bubbles coalesce, the bubble spans the whole pipe diameter, and a slug forms. However,

a fixed ratio ReG/ReL may also obtain for fixed flow rates, QG and QL. If instead

of UG and UL being fixed, the flow rates QG and QL are held fixed, an increase in D

actually lowers the Reynolds number Re = 4 Q ρ/π D μ.

We believe that this case cannot be fully explained by the 2-dimensional map of

Jayawardena et al. (1997) [1]. In principle, since the system is described by three

parameters, any regime map should be sensitive to three parameters. The fact that

the authors find a 2-dimensional map sufficient, may indicate that the dependence on

the third parameter has been lost due to its either high or low value. Unfortunately,

we do not have information on individual values of ReG and ReL in the map shown in

Fig. 2. But we suspect that both are quite large, so that variations in a high value do

not introduce additional dependencies to the map that would require the drawing of

a third parametric axis. However, one should warn users of the Jayawardena

et al. map that, when ReG and/or ReL are not large, the map may not

be valid. This is potentially very important for water recovery systems,

where the Reynolds number for neither phase is large.

Next, we offer some discussion on whether we can expect laminar flow. The values of Re for gas or liquid are based on each phase superficial velocity. Superficial

velocities do not, in general, reflect the actual characteristic velocity of the phase in

NASA/CR—2006-214085 55
question. Consequently, this definition of Re cannot be used to assess with certainty

whether a flow is laminar or turbulent. For example, in bubbly flow, the bubbles

are likely flowing at a velocity close to the liquid phase velocity; the relative phase

velocity is therefore small. There is no clear guideline to assess whether a bubbly flow

is laminar or turbulent.

In annular flow the situation is a bit different. According to the regime map, a

condition for annular flow is ReG > 10 ReL. Inputing typical values of fluid properties, this implies QG > 10 QL. Since each phase is flowing separately in its own

region of space, and typically the gas core radius h ≈ D/2 with (D − 2h)/D  1,

we may estimate more “realistic” gas and liquid velocities as: UG ≡ QG/πh2 and

UL ≡ QG/πD(D/2 − h). Inserting these velocities in the definition of gas and liquid

Reynolds numbers, we get the following alternative definitions: ReL ≡ ULρL(D/2 −

h)/μL = QL ρL/πD = ReL/4 and ReG ≡ UGρGD/μG = QG ρG/πD + O 

D/2−h

h



=

ReG + O 

D/2−h

h



. Although we have no criterion for the transition to turbulence in

these annular flows, we may surmise that the flow should be laminar if ReL and ReG

are obviously “too small” for turbulence.

In slug flow, assuming that the liquid slug and gas bubble move at the same velocity UG = UL = U (i.e., there is no liquid film surrounding the gas bubble, or the

film thickness t  D/2), then we may use similar arguments as those used for annular

flow, but using the definition ReL ≡ U ρLD/μL.

NASA/CR—2006-214085 56
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E–15422

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NCC3–975

72

Two Phase Flow Modeling: Summary of Flow Regimes and Pressure Drop

Correlations in Reduced and Partial Gravity

R. Balasubramaniam, E. Ramé, J. Kizito, and M. Kassemi

Fluid dynamics

Unclassified -Unlimited

Subject Category: 34

National Center for Space Exploration Research

Glenn Research Center

21000 Brookpark Road

Cleveland, Ohio 44135

Project Manger, Brian Quigley, Exploration Systems Division, NASA Glenn Research Center, organization

code PTO, 216–433–8672.

The purpose of this report is to provide a summary of state-of-the-art predictions for two-phase flows relevant to Advanced

Life Support. We strive to pick out the most used and accepted models for pressure drop and flow regime predictions. The

main focus is to identify gaps in predictive capabilities in partial gravity for Lunar and Martian applications. Following a

summary of flow regimes and pressure drop correlations for terrestrial and zero gravity, we analyze the fully developed

annular gas-liquid flow in a straight cylindrical tube. This flow is amenable to analytical closed form solutions for the flow

field and heat transfer. These solutions, valid for partial gravity as well, may be used as baselines and guides to compare

experimental measurements. The flow regimes likely to be encountered in the water recovery equipment currently under

consideration for space applications are provided in an appendix.