two-phase flow (air-water) characteristics in annulus

CRANFIELD UNIVERSITY

TOBIAS GSCHNAIDTNER

TWO-PHASE FLOW (AIR-WATER) CHARACTERISTICS IN ANNULUS

SCHOOL OF ENGINEERING ENERGY SYSTEMS AND THERMAL PROCESSES

MSc THESIS Academic year: 2013 – 2014

Supervisor: HOI YEUNG September 2014

CRANFIELD UNIVERSITY

SCHOOL OF ENGINEERING ENERGY SYSTEMS AND THERMAL PROCESSES MSc THESIS

Academic year: 2013 – 2014

TOBIAS GSCHNAIDTNER

TWO-PHASE FLOW (AIR-WATER) CHARACTERISTICS IN ANNULUS

Supervisor:

HOI YEUNG

September 2014

This thesis is submitted in partial fulfilment of the requirements for the Degree of Master of Science

© Cranfield University 2014. All rights reserved. No part of this publication may be reproduced without the written permission of the copyright owner.

Abstract

Abstract

The objective of the presented work is to determine the two-phase flow characteristics of water and air in a concentric annulus, consisting of an inner cylinder surrounded by an outer pipe. The outer diameter of the inner pipe is 33.3 mm, the inner diameter of the outer pipe 51.5 mm. The experiments are conducted at the 2’’ test rig in the PSE Lab at Cranfield University. Conductivity probes and differential pressure transducers are used to record the liquid holdup and the pressure drop of the two-phase mixture. The flow pattern is visually recorded by a video camera. The observed flow patterns comprise bubbly flow and intermittent flow. It is observed, that the transition from bubbly flow to intermittent flow takes place at superficial velocities of water at between 2 and 2.5 m/s. The measurements of the liquid holdup show that the liquid holdup decreases with growing air flow rates and increases for higher liquid flow rates. The experimental results also indicate that water is the major contributor, i.e. that the pressure drop tends to raise with increasing flow rates of air and water. In addition to the experimental work, the presented thesis includes adaption of theoretical frameworks from relevant literature to the case of a concentric annular cross-section. Also different diameter approaches are applied. Comparing the literature with the experimental results, it turns out that the Taitel and Dukler model is the most reliable model in terms of flow pattern transition. On the other hand, the Steiner model based on the Crittendon diameter is found to estimate the liquid holdup most accurately. When determining the pressure drop the Beattie and Whalley approach using the Knudsen/Prengle/Rothfus diameter concept performs best.

Key words: Concentric Annulus, Flow Pattern Transition, Liquid Holdup, Pressure Drop, Two-Phase Flow

i

ii

List of Contents

List of Contents

1

2

Introduction ............................................................................................... 1 1.1

Background .......................................................................................... 1

1.2

Aims and Objectives ............................................................................ 3

1.3

Methodology ........................................................................................ 4

Literature Review ...................................................................................... 5 2.1

2.1.1

Single-Phase Flow in Pipes .......................................................... 8

2.1.2

Single-Phase Flow in Annuli ....................................................... 11

2.2

3

4

5

Single-Phase Flow ............................................................................... 5

Two-Phase Flow ................................................................................ 18

2.2.1

Two-Phase Flow in Pipes ........................................................... 22

2.2.2

Two-Phase Flow in Annuli .......................................................... 30

Experimental Setup ................................................................................. 36 3.1

Description of the Test Rig and Test Section ..................................... 36

3.2

Instrumentation and Calibration Process ........................................... 39

3.3

Experimental Procedure and Data Collection .................................... 41

Theoretical Models .................................................................................. 43 4.1

Models for Flow Pattern Determination .............................................. 43

4.2

Models for Liquid Holdup Prediction................................................... 51

4.3

Models for Frictional Pressure Loss Calculation ................................ 53

Experimental Results and Discussions ................................................ 58 5.1

Experimental Results ......................................................................... 58

iii

List of Contents

5.1.1

Single-Phase Flow of Water ....................................................... 58

5.1.2

Air-Water Two-Phase Flow ......................................................... 60

5.2

Comparison of experimental data and theoretical models ................. 63

5.2.1

Flow Pattern Estimation .............................................................. 63

5.2.2

Liquid Holdup Estimation ............................................................ 66

5.2.3

Frictional Pressure Drop Estimation............................................ 70

6

Conclusions............................................................................................. 73

7

Recommendations for future research ................................................. 75

References ...................................................................................................... 77 Appendix A ..................................................................................................... 92 Appendix B ..................................................................................................... 95

iv

List of Figures

List of Figures

Figure 1-1: Flow of the drilling fluid throughout the drilling process in a horizontal wellbore ............................................................................................. 2 Figure 2-1: Flow patterns of the flow of a single-phase fluid .............................. 7 Figure 2-2: Velocity profile of an incompressible, iso-thermal viscous fluid in a pipe (laminar)............................................................................................... 8 Figure 2-3: Velocity profile of an incompressible, iso-thermal viscous fluid in a pipe (turbulent) .......................................................................................... 11 Figure 2-4: Velocity profile of an incompressible, iso-thermal viscous fluid in an annular pipe ............................................................................................. 13 Figure 2-5: Flow patterns of adiabatic air-water two-phase flow through horizontal pipes ................................................................................................ 24 Figure 2-6: Flow pattern map for horizontal two-phase flow (Baker, 1954) ...... 24 Figure 3-1: Schematic structure of the 2’’ test rig ............................................. 38 Figure 3-2: Schematic drawing of the test section ............................................ 37 Figure 3-3: Ring-type conductance probes ...................................................... 40 Figure 3-4: Calibration curve of the conductance probes 2 .............................. 41 Figure 4-1: Geometrical parameters and sections of an annular tube dependent on the liquid height ......................................................................... 45 Figure 5-1: Pressure loss of water single-phase flow in the annular test section over the flow rate of water .................................................................... 59 Figure 5-2: Pressure loss of water single-phase flow in the annular test section .............................................................................................................. 59 Figure 5-3: Flow pattern observation plotted against superficial velocity of air and water..................................................................................................... 61 v

List of Figures

Figure 5-4: Liquid holdup versus superficial velocity of air for different water flow rates ................................................................................................ 62 Figure 5-5: Pressure Drop versus superficial velocity of air for different water flow rates ................................................................................................ 62 Figure 5-6: Flow pattern map using modified transition criterions suggested by Taitel and Dukler ........................................................................ 65 Figure 5-7: Flow pattern map using modified transition criterions suggested by Barnea and based on Equivalent diameter ................................ 65 Figure 5-8: Flow pattern map using modified transition criterions suggested by Beggs and Brill based on Omurlu diameter and Crittendon diameter ........................................................................................................... 66 Figure 5-9: Liquid holdup estimation using homogeneous model..................... 67 Figure 5-10: Liquid holdup estimation using separated model of Zivi, Chisholm, and Lockhart and Martinelli ............................................................. 68 Figure 5-11: Liquid holdup estimation using drift flux model of Steiner, and Danielson and Fan ........................................................................................... 68 Figure 5-12: Liquid holdup estimation using Beggs and Brill model, based on Crittendon diameter ..................................................................................... 69 Figure 5-13: Pressure drop estimations using the homogeneous flow approach based on the Knudsen/Prengle/Rothfus diameter concept ............... 71 Figure 5-14: Pressure drop estimations using the separated flow approach based on the Knudsen/Prengle/Rothfus diameter concept .............................. 72 Figure A-1: Proof of repeatability of measured liquid holdups .......................... 92 Figure A-2: Proof of repeatability of measured pressure drop .......................... 93 Figure A-3: Intermittent flow at 0.4 l/s water flow rate and 0.6 kg/h air flow rate ................................................................................................................... 93 Figure A-4: Intermittent flow at 0.4 l/s water flow rate and 14.6 kg/h air flow rate ................................................................................................................... 93 Figure A-5: Intermittent flow at 1.6 l/s water flow rate and 1.5 kg/h air flow rate ................................................................................................................... 94

vi

List of Figures

Figure A-6: Intermittent flow at 1.6 l/s water flow rate and 6.1 kg/h air flow rate ................................................................................................................... 94 Figure A-7: Bubbly flow at 3.6 l/s water flow rate and 0.3 kg/h air flow rate ..... 94 Figure A-8: Intermittent flow at 3.6 l/s water flow rate and 2.0 kg/h air flow rate ................................................................................................................... 94

vii

List of Tables

List of Tables

Table 2-1: Correlations of various diameter concepts for the flow through annular ducts .................................................................................................... 35 Table 3-1: Test matrix ...................................................................................... 42 Table 5-1: Comparison of the evaluated liquid holdup models ......................... 70 Table 5-2: Comparison of the evaluated pressure loss models based on the Knudsen/Prengle/Rothfus diameter concept .............................................. 71 Table B-1: Comparison of the evaluated diameter concepts used in the Barnea model in order to predict the flow pattern ............................................. 95 Table B-2: Comparison of the evaluated diameter concepts used in the Beggs and Brill model in order to predict the flow pattern ................................ 96 Table B-3: Comparison of the evaluated diameter concepts used in the Beggs and Brill model in order to estimate the liquid holdup ............................ 96 Table B-4: Comparison of the evaluated diameter concepts used in the McAdams model in order to estimate the pressure drop .................................. 97 Table B-5: Comparison of the evaluated diameter concepts used in the Beattie and Whalley model in order to estimate the pressure drop .................. 97 Table B-6: Comparison of the evaluated diameter concepts used in the Sun and Mishima model in order to estimate the pressure drop ...................... 98 Table B-7: Comparison of the evaluated diameter concepts used in the Müller-Steinhagen and Heck model in order to estimate the pressure drop ..... 98 Table B-8: Comparison of the evaluated diameter concepts used in the Beggs and Brill model in order to estimate the pressure drop .......................... 99

viii

Abbreviations

Abbreviations

Abbreviations

English wording

ABS

Acrylonitrile Butadiene Styrene

OBD

Overbalanced drilling

PSE

Process Systems Engineering

UBD

Underbalanced drilling

UBHD

Underbalanced horizontal drilling

ix

Nomenclature

Nomenclature

Latin alphabet Symbol

Quantity

Unit

a

Radius ratio

-

A

Area

m²

Ag

Area covered by air

m²

Al

Area covered by water

m²

C

Constant

-

C0

Distribution parameter

-

d1

Outer diameter of inner tube

m

d2

Inner diameter of outer tube

m

dc

Stable diameter

m

d cb

Critical bubble size

m

d cd

Critical bubble size

m

dH 2

Equivalent diameter

m

dh

Equivalent diameter

m

D

Diameter

m

DH

Hydraulic diameter

m

x

Nomenclature

f

Friction factor

-

fg

Friction factor of air

-

f go

Friction factor of air

-

f hom

Friction factor of homogeneous mixture

-

fl

Friction factor of water

-

f lo

Friction factor of water

-

Frhom

Froude number of homogeneous mixture

-

g

Gravitational constant

m/s²

G

Mass flux

kg/m²/s

hg

Gas level

m

hl

Liquid level

m

l

Length

m

L1

Correlation boundary

-

L2


-

L3


-

L4


-

La

Laplace number

-

Lentry

Entrance length

m

Mg

Mass flow rate of air

kg/s

Ml

Mass flow rate of water

kg/s

p

Pressure drop

Pa

xi

Nomenclature

pa

Pressure drop due to acceleration

Pa

p f

Frictional pressure drop

Pa

pg

Gravitational pressure drop

Pa

P

Wetted perimeter

m

Q

Volumetric flowrate

m³/s

Qg

Volumetric flowrate of air

m³/s

Ql

Volumetric flowrate of water

m³/s

r

Radius

m

r1

Outer radius of inner tube

m

r2

Inner radius of outer tube

m

rmax

Locus of maximum velocity

m

rw

Tube radius

m

Re

Reynolds number

-

Re*

Modified Reynolds number

-

Reh

Modified Reynolds number

-

Reh ,g

Modified Reynolds number of air

-

Reh ,l

Modified Reynolds number of water

-

Rehom

Reynolds number of homogeneous mixture

-

Re s

Superficial Reynolds number

-

s

Sheltering coefficient

-

S

Correction factor

-

S

Velocity ratio

-

Sg

Wetted perimeter of air

m

xii

Nomenclature

Si

Wetted perimeter between water and air

m

Sl

Wetted perimeter of water

m

u

Velocity

m/s

u

Average velocity

m/s

ud

Drift velocity

m/s

ug

Velocity of air

m/s

ul

Velocity of water

m/s

uhom

Velocity of homogeneous mixture

m/s

umax

Maximum velocity

m/s

us

Superficial velocity

m/s

us , g

Superficial velocity of air

m/s

us ,l

Superficial velocity of water

m/s

us , m

Superficial mixture velocity

m/s

V

Normalised voltage

-

Vg

Volume covered by air

m³

Vl

Volume covered by water

m³

Vtot

Total volume

m³

x

Vapour/mass quality

-

X

Lockhart-Martinelli parameter

-

xiii

Nomenclature

Greek alphabet Symbol

Quantity

Unit

g

Void fraction

-

 g ,DFM

Void fraction of drift flux model

-

 g ,hom

Void fraction of homogeneous mixture

-

 g ,sep

Void fraction of separated model

-

l

Liquid holdup

-



Volumetric quality

-



Input liquid content

-



Dynamic viscosity

kg/s/m

g

Dynamic viscosity of air

kg/s/m

l

Dynamic viscosity of air

kg/s/m

hom

Dynamic viscosity of homogeneous mixture

kg/s/m

l

Kinematic viscosity of water

m²/s

g

Kinematic viscosity of air

m²/s



Density

kg/m³

g

Density of air

kg/m³

hom

Density of homogeneous mixture

kg/m³

l

Density of water

kg/m³

xiv

Nomenclature



Surface tension of water

N/m

g

Shear stress of air

N/m²

i

Interfacial shear stress

N/m²

l

Shear stress of water

N/m²



Function

-

g

Two-phase friction multiplier

-

go


-

l


-

lo


-

xv

xvi

Introduction

1 Introduction 1.1 Background Two-phase flow is the result when two fluids of different matter move concurrently through a canal. The exact form of two-phase flow is determined according to the phases appearing in the mixture, namely solid-liquid, gas-solid (e.g. particles in a gas or liquid) and gas-liquid (droplets in gas and gas bubbles in a liquid). Due to its omnipresence in many industrial applications, the simultaneous flow of gas and liquid is of major interest to the chemical, nuclear and petroleum industry. It has been the subject of intense research in the last decades. Typical examples of two-phase gas-liquid flow are boilers and condensers, nuclear reactors, various types of chemical reactors and the production/transportation of gas and oil. Thereby, the gas-liquid mixture mainly flows in conduits with a circular, near circular or rectangular geometry. Double pipe heat exchangers in terms of evaporation and condensation, which are consisting of two concentric pipes, are a typical example for the occurrence of multiphase flow. With one fluid moving in the inner pipe and the other fluid flowing in the annular section of the two pipes, two-phase flow can appear either in an annular or in a circular conduit. Since the parameters (heat transfer and pressure drop) to design the heat exchanger depend on the flow pattern, indicating the visible distribution or structure of the phase, a detailed understanding of the different types of flow pattern is of great importance. Annular geometries can also appear in other types of applications: Heat exchangers such as the heating of fluids by an electrical heating-rod placed in the middle of a pipe or drilling operations in the petroleum industry are just two more examples among many more. In the petroleum industry drilling of wells is required to explore and exploit reservoirs containing oil or gas. A typical well bore drilling system consists of the drilling equipment (e.g. drill bit and drill pipe) and a circulation system. The drill bit is used to drill a variety of boreholes, by cutting and crushing the sedimentary rock. Simultaneously, the circulation system pumps a drilling fluid through the drill pipe and the drill bit down to the bottom of the borehole, where the fluid enters the space of the annular conduit between the drill string and the borehole, as shown in Figure 1-1 (Lyons, 2008).

1

Introduction

Figure 1-1: Flow of the drilling fluid throughout the drilling process in a horizontal wellbore

Together with the drill cuttings, the drilling fluid flows through the annular space back to the surface, where the cuttings are removed from the fluid. In conventional drilling techniques (overbalanced drilling, OBD) the hydrostatic pressure of the drilling fluid in the wellbore exceeds the pore pressure of the sedimentary formation. However, in various cases, one faces the possibility of fluid loss under extreme overbalanced conditions, where underbalanced drilling (UBD) techniques offer several advantages (Bennion and Thomas, 1994). This applies in particular to fractured sandstone or carbonate formations, where the permeability is mainly embodied in the fracture system. Also for carbonates with a high vugular porosity and a high permeability as well as for sands and carbonates in pressure depleted formations (Bennion and Thomas, 1994) underbalanced drilling may lead to better results. In contrast to OBD, the pressure of the drilling fluids in the borehole of UBD operations is less than the pore pressure of the sedimentary rock. Thus, the rate of penetration can be increased, the risk of lost circulation can be minimised, the lifetime of the drilling bit can be prolonged and in connection with a missing filter cake during the drilling, no problems with the differential pipe-sticking occur (Guo and Ghalambor, 2002). Additionally, due to the lower pressure of the drilling fluid, the invasion of the drilling fluid into the permeable formation zone is avoided resulting in reduced damage to the formation. Given the substantial advantages, the demand for UBD techniques steadily increases among gas and oil producers. In recent years, the exploitation of unconventional petroleum reservoirs including shale and tight-sandstone gas has become an alternative to natural gas. In this context, UBD of horizontal wellbores (UBHD) is the key for the successful exploitation of such unconventional reservoirs (Zou, 2013). The primary advantages of horizontal wells over vertical wells include a larger drainage area, an increased production per well and hence, a reduction of number of wells needed, higher producing rates and the ability to turn non-

2

Introduction

accessible reservoirs into productive petroleum sources by avoiding barriers and protected landscape (Zou, 2013; Rehm et al., 2012). The underbalanced condition during UBD operations is generally achieved by the injection of gas into the drilling fluid. In this way, compressed gas can be used to create a lightweight fluid by reducing its density. Yet, by using a gasliquid mixture as the drilling fluid in UBD operations, the estimation of design parameters such as the pressure loss of the gas-liquid flow, the choice of the appropriate flow and the performance of the hole cleaning becomes more complex than for single-phase drilling fluids due to the interaction of the two phases. Additionally, poorly designed and performed UBHD operations could lead to deeper sedimentary formation damage in comparison to a well prepared and executed OBD operation (Lage et al., 2000). In order to reduce damage throughout the drilling process, to calculate the pressure drop accurately, to choose the optimal flow rates of the gas and liquid phase and to make UBHD as economical as possible, it is necessary to better understand the hydraulics of aerated drilling liquids. A considerable amount of theoretical as well as experimental studies dealing with multiphase flow in horizontal pipes and annuli were conducted using the hydraulic diameter. However, the hydraulic diameter, commonly used in applications with a single-phase flow, has to the author's knowledge not yet been proven to be accurate enough for multiphase flows. Given the lack of research about the hydraulic behaviour of two-phase fluids in annular geometries, the focus of the presented thesis lies on the study of twophase flow of gasified drilling fluids in horizontal wellbores.

1.2 Aims and Objectives The main aim of this research project is to determine the two-phase flow hydraulics and characteristics of water and air in an annulus, consisting of an inner cylinder surrounded by an outer pipe. The outer diameter of the inner cylinder is 33.3 mm and the inner diameter of the outer pipe is 51.5 mm respectively. The main aim of the thesis can be divided into the following objectives:  design and set up of the test rig and the adaption of appropriate measurement devices  calibration of a prototype liquid hold up measurement device for annuli (conductivity rings)

3

Introduction  observation and recording of flow regimes using photography and video, measurement of liquid hold up values, determination of the characteristic velocities and recording the pressure drop respectively  verification of the data by comparing the results with other relevant scientific work

1.3 Methodology First, a thorough literature review about single- and two-phase flow is given in Chapter 2 to provide a detailed overview of previous work, of the different flow regimes and of the instrumentations, used to measure two-phase flows. The set up and design of the test rig is carried out using existing facilities and measurement devices, which have been proven in former test rigs to be adequate for two-phase flow. The accuracy of the measurement devices with special regards to liquid hold up measurement depends on the calibration of the devices and has to be adapted to the annulus. The measurements will be accomplished using devices such as conductivity rings, photography and video. Once the experiments are carried out, the acquired data are compared with theoretical frameworks using different diameter correlations. Finally, all correlations for estimating the liquid holdup, the flow pattern transition and the pressure drop are compared to each other.

4

Literature Review

2 Literature Review Single-phase and two-phase flows are the most common flow conditions of drilling fluids occurring during a drilling process. Depending on the drilling technique, in OBD operations single-phase fluids are widely used to transport the cuttings from the drilling to the surface, while UBD operations generally rely on two-phase fluids. Since the flow of two-phases is strongly related to the flow of fluids of one phase, this chapter provides an overview of both, two-phases as well as single-phase flows. When applying single-phase and two-phase flows to conduits with a noncircular geometry (e.g. annuli), the common procedure is to first analyse the hydraulics of the particular flow in the simplest possible type of a conduit which is a circular conduit/pipe. The information gained in the first step is then adjusted to the flow through non-circular conduits. Usually, a satisfying description of the flow behaviour can be achieved by modifying the obtained correlations for circular conduits with new concepts based on non-circular geometry. In the following, it is distinguished between single-phase and twophase flow through circular and annular pipe sections.

2.1 Single-Phase Flow A sound understanding of the basic physics underlying the flow of singlephase fluids, e.g. air and water flowing separately, is necessary for studying the hydraulics of two-phase flow. Many models describing the behaviour of twophase flow are based on equations used for single-phase flow. For this reason, the aim of this section is to provide a basic understanding of the flow of singlephase fluids. In fluid mechanics, the flow of a fluid is illustrated using the concept of a particle, which is called fluid particle (Yamaguchi, 2008). In continuum mechanics, there are three conservation equations governing the physical behaviour of the particles. They are the conservation of mass, momentum and energy. For a detailed discussion of these equations it is referred to (Yamaguchi, 2008; Falkovich, 2011; Kambe, 2007). In practice, the most difficult task is to characterise the flow of the single fluid. Fluids are normally represented by a multitude of single particles flowing 5

Literature Review

side by side in the fluid. They can also be seen as viscous mass. The particles in viscous fluids are considered as sensitive to the internal forces of neighbouring particles, also known as viscous stress. For example in closed conduits, such as pipes, viscous stresses cause the fluid particles close to the pipe wall to move slower than particles located in the centre of the pipe. Under non-slip condition, the velocity of a moving fluid at the surface of a wall is assumed to be zero. In the flow of a viscous fluid the layer from the surface (velocity is zero) to the location, where the fluid reaches its maximum velocity is known as the boundary layer. According to Kleinstreuer (2003) viscous fluids are grouped into Newtonian and non-Newtonian fluids, incompressible and compressible fluids, internal and external flows, laminar and turbulent flows and isothermal and non-isothermal flows. For the present study of air-water two-phase flow through an annular geometry, the viscous fluid flow of air, respectively water, can be limited to  Newtonian fluids, i.e. the viscous stresses are proportional to the local strain rate and hence the viscosity at a given temperature and velocity of the fluid can be seen as constant,  internal flows, i.e. the fluid flows in a closed conduit with a constant mass flow,  isothermal flows, i.e. the properties of the fluid are evaluated at a specific temperature. The two fluids encountered in this study, air and water, are assumed to be effectively incompressible (i.e. a fluid with a constant density) as long as the flow velocities are far below the speed of sound (Lautrup, 2011). Depending on the motion of the particles within the fluid, the three common terms to describe a single-phase flow are: laminar, turbulent and transitional. In his well-known experiment, Reynolds (Reynolds, 1883) showed that under certain conditions the flow of a fluid is either laminar or turbulent. According to Reynolds (Reynolds, 1883) the flow of a fluid is laminar when the particles within the fluid move directly (i.e. in parallel lines (streamlines)) with no disturbance between the different lines or layers. However, when the flow becomes turbulent, the well-organised structure of a laminar flow regime, is destructed by eddies or swirls. The state in which the laminar flow regime is increasingly disrupted before becoming fully turbulent is called laminar-turbulent transition or onset of turbulence. Figure 2-1 illustrates the different flow patterns of a single-phase fluid.

6

Literature Review

Figure 2-1: Flow patterns of the flow of a single-phase fluid

To predict the particular flow regime, Reynolds (Reynolds, 1883) used a dimensionless value in his study, commonly known as Reynolds number. It is defined as the ratio of the inertial forces to the viscous forces of a fluid flow passing a body (Falkovich, 2011). The Reynolds number, crucial for many models describing the single-phase flow, can be expressed in one dimensional coordinates as given in Equation (2-1).

Re 

 u  D 

(2-1)

In Equation (2-1), ρ is the density, ū the average velocity and μ the dynamic viscosity of the fluid. D is the diameter of a circular conduit. In the literature (Moody, 1944; Lautrup, 2011; Schlichting and Gersten, 2000) it is widely accepted, that the Reynolds number categorises the type of flow of a single fluid as follows:

Re  2000

Laminar :

Transitional : 2000  Re  4000 4000  Re

Turbulent :

Especially in terms of velocity distribution and pressure drop, the Reynolds number is of great importance. It is used for estimating the friction factor f, which is a dimensionless factor determining the pressure drop Δp of a fluid over the length l (see Equation (2-2)).

p  l   f 

l  u 2  D 2

(2-2)

A simple method to estimate the friction factor and pressure drop in clean new pipes and in closed conduits under steady state flow conditions was presented 7

Literature Review

by Moody (Moody, 1944). Moody provided the so called Moody diagram, which is commonly used to estimate the friction factor for laminar and turbulent flow. Following, with special regard to the limitations stated above and the categorisation of the condition of the fluid flow (laminar, turbulent or transition area), the velocity distribution and pressure drop of a single-phase fluid in circular and annular pipes are discussed in more detail.

2.1.1 Single-Phase Flow in Pipes The flow of an isothermal, viscous Newtonian fluid with a single phase trough a circular pipe is discussed in this section of Chapter 2. The flow of single-phase fluids through circular pipes can generally be categorized into three types: laminar flow, transitional flow and turbulent flow. Following this categorization, the velocity distribution as well as the friction factor is outlined for each flow pattern in the upcoming passages. Laminar Flow: Hagen (Hagen, 1839) and Poiseuille (Poiseuille, 1840) first quantitatively examined laminar flow. Neglecting gravity, the flow of a fluid through a circular conduit is symmetrical around the central line of the pipe. Assuming a no-slip condition at the wall and a velocity gradient of zero at the centre of the pipe, the velocity profile of the fluid can be expressed in cylindrical coordinates as a parabolic shape with the maximum velocity at the centre of the pipe (see Figure 2-2) (Lautrup, 2011).

Figure 2-2: Velocity profile of an incompressible, isothermal viscous fluid in a pipe (laminar)

Equation (2-3) implies that the point velocity u depends on the maximum velocity umax and on the ratio of the radius r and the tube radius rw (Knudsen and Katz, 1958).

8

Literature Review

  r 2  u  r   umax  1       rw  

(2-3)

Experimental investigations carried out by Ferrell, Richardson et al. (Ferrell et al., 1955) and Senecal and Rothfus (Senecal and Rothfus, 1953) are consistent with the parabolic velocity distribution for laminar flow. The average velocity ū of the parabolic velocity profile is, according to analytical (Knudsen and Katz, 1958) and experimental studies (Senecal and Rothfus, 1953; Stanton and Pannell, JR, 1914) one half of the maximum velocity at the centre of the pipe:

u

umax 2

(2-4)

The average velocity is not only used to compute the mass discharge, but also to calculate the Reynolds number and the pressure drop. For calculating the pressure drop, the friction factor is required in addition to the average velocity. Moody suggested, to calculate the pressure drop with the Darcy friction factor for laminar flow (shown in Equation (2-5)), which is derived from the Poiseuille law.

f 

64 Re

(2-5)

However, Equations (2-3) to (2-5) are only valid for fully developed velocity profiles in a circular tube. Since the velocity distribution (i.e. the boundary layer) is, due to the non-slip condition, not established immediately at the entrance region of the tube, an entrance length is considered (Lautrup, 2011). According to Lautrup (2011), the distance before the flow is fully developed, depends on the Reynolds number, a factor k (k=0.063) and the hydraulic diameter of the pipe (which in terms of a pipe is the diameter of the pipe):

Lentry  0.063  D  Re

(2-6)

Transitional Flow: Generally, with Reynolds numbers higher than 2000, the streamlines of the laminar flow are destructed. The formation of eddies and swirls was first made visible by Reynolds (Reynolds, 1883) using coloured filaments injected in the flow of a fluid through a pipe. Whereas in laminar flow the coloured filaments 9

Literature Review

move as filaments with sharp boundaries, in the onset of turbulence a strong mixing effect occurs and the fluid seems to be evenly coloured (Schlichting and Gersten, 2000). Reynolds associated the appearance of turbulences with a critical Reynolds number, which he identified as Recrit = 2300. Additionally, Rothfus and Prengle (Rothfus and Prengle, 1952) carried out experiments on the appearance of turbulences in 1 and 2 inches pipes by using dye filament injections. They observed eddies in form of tight spirals for a Reynold number of 1500 and higher. Above a Reynolds number of 2100, Rothfus and Prengle witnessed the formation of large spiral eddies. However, the actual mechanism when migrating from laminar to turbulent flow is not well known. As of today, the transition from laminar to turbulent flow is understood as a statistical phase transition (Moxey and Barkley, 2010; Avila et al., 2011; Manneville, 2009). In the Moody chart the appearance of turbulences is considered as a critical zone without definite lines, i.e. the friction factor is not exactly predictable for this zone. The friction factor line for laminar flow can be interpreted as the minimum friction factor for this zone. The maximum friction factor for the transitional zone is the Colebrook function (Colebrook, 1939), which corresponds to the friction factor of a turbulent flow. Turbulent Flow: In case the Reynolds number exceeds 4000, the flow becomes fully turbulent. In comparison to a laminar fluid flow in a pipe, which is streamlined, the turbulent flow is of chaotic nature. As a result of the irregularities and the continuous mixing during turbulent flow, a quantitative description is not possible (Knudsen and Katz, 1958). Nevertheless, an empirical theory and a statistical theory were developed to model the turbulent flow. Major contributions in this area were provided by Prandtl (1925) and Taylor (1921). Since the turbulent flow varies rapidly with respect to time and highly depends on the location within the pipe, there are no universally accepted theories for turbulent pipe flow (Lautrup, 2011). Due to the high complexity of turbulent flow, this review focuses on the average velocity profile in a pipe. Similar to the laminar flow, the velocity profile of a turbulent flow consists of the boundary layers close to the wall and takes a parabolic shape. Yet, with the exception of the thin boundary layers, the velocity field is, due to the occurring mixture effect, assumed to be approximately constant across the pipe (Lautrup, 2011). This is shown in Figure 2-3. Beyond a Reynolds number of 4000 in the Moody chart, the Colebrook function (Colebrook, 1939) is used to determine the friction factor for turbulent flow. As shown in Equation (2-7), the Colebrook function for smooth tubes is

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Literature Review

Figure 2-3: Velocity profile of an incompressible, isothermal viscous fluid in a pipe (turbulent)

implicitly given in the friction factor f, which depends only on the Reynolds number.





1  2.0  log10 Re f  0.8 f

(2-7)

Since the use of implicit functions is a major challenge for the implementation in practical applications, Fang et al. (2011) recently provided an explicit correlation for the Reynolds-number range from 3000 to 108 to determine the friction factor in smooth tubes with a mean absolute error of 0.022 % (see Equation (2-8)).   150.39 152.66   f  0.25  log  0.98865   Re     Re

2

(2-8)

Equivalent to the case of laminar flow, the average velocity profile of a turbulent flow in a pipe requires an entrance length in order to be fully developed. Schiller and Kirsten (Schiller and Kirsten H., 1929) worked out experimentally that an entrance length of more than 50 tube diameters is required for the turbulent velocity profile to be fully developed:

Lentry  50  D

(2-9)

2.1.2 Single-Phase Flow in Annuli Compared to the flow of a single-phase fluid through a circular pipe which is considered as the simplest geometry, the flow through pipes with an annular geometry is much more complex. In particular, the Reynolds number and the pressure drop, defined in equations (2-1) and (2-2), are not adaptable to an 11

Literature Review

annular pipe section, because the definition uses the diameter of the pipe. Instead, the diameter has to be replaced by a modified concept applicable for non-circular conduits. The most common and widely accepted concept for single-phase flow is to replace the diameter by an equivalent diameter, called the hydraulic diameter DH. The hydraulic diameter is defined as a ratio of the cross sectional area A to the wetted perimeter of the cross-sectional area P (see Equation (2-10)).

DH 

4 A P

(2-10)

According to Schiller (1923) and Fromm (1923), the hydraulic diameter concept is applicable to geometries of any given cross-sectional area and suitable for practical applications. Furthermore, Davies and White (1929) noted that the hydraulic diameter is “a convenient substitute for (the characteristic dimension) in the Reynolds number”. For an annular geometry it can be shown, that the hydraulic diameter can be reduced to the difference between the inner diameter of the outer pipe d2 and the outer diameter of the inner cylinder d1: DH  d2  d1

(2-11)

Even though, the hydraulic diameter is the most common method for predicting pressure drop in annular geometries, its use is questionable. Therefore, in the following the adaptability of the hydraulic radius is also discussed. Similar to the characterisation of single-phase flow through circular conduits, the flow through annular conduits can be subdivided into laminar, transitional and turbulent flow characteristics. Laminar Flow: For the description of the flow of a single-phase fluid through a concentric annular pipe section, it is assumed that the flow is symmetrical around the central line of the inner and outer pipe, as shown in Figure 2-4. Considering noslip conditions at the surface of the outer and inner pipe and a velocity gradient of zero at the locus of maximum velocity, the theoretical position of the locus of the maximum velocity rmax can, in accordance with Knudsen and Katz (1958), be expressed as

12

Literature Review

Figure 2-4: Velocity profile of an incompressible, isothermal viscous fluid in an annular pipe

rmax 

r22  r12 r  2  ln  2   r1 

(2-12)

where r2 is the inner radius of the outer and r1 the outer radius of the inner pipe. Using the radius of maximum velocity Knudsen and Katz (1958) proposed an analytical solution to calculate the point velocity: r  2 r22  r 2  rmax  ln  2  r u  2u  2 2 2 r2  r1  2  rmax

(2-13)

Thereby the average velocity ū is given in Equation (2-14), as the ratio of the volumetric flow rate Q and of the cross sectional area of the annular geometry.

u

Q  r  r12    2 2

(2-14)

Prengle’s and Rothfus’ dye experiments (1955) indicated that Equations (2-12) and (2-13) only hold up to Reynolds numbers of around 700. Exceeding this value, Rothfus et al. (1955) showed that for Reynolds numbers of up to 1,500 the proposed radius of the maximum velocity in Equation (2-12) underestimates the radius. For Reynolds numbers in the range from 1,500 to 2,000 the maximum radius shifts more toward the outer radius of the core pipe than predicted by Equation (2-12). As a result, Prengle and Rothfus as well as Rothfus et al. suggest using Equation (2-15) rather than Equation (2-12) for a laminar flow and for Reynolds numbers just above 700.

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rmax  r1  r2

(2-15)

For estimating the friction factor of laminar flow through concentric annular pipes, the first approach would be to apply the hydraulic diameter concept as suggested by Schiller (1923), Moody (1944) and Fromm (1923). In other words, when replacing the diameter by the hydraulic diameter given in Equation (2-11), Equations (2-1), (2-2) and (2-5) can be used to predict the pressure drop in an annular geometry. However, since the locus of the maximum velocity in an annular geometry is shifted towards the core pipe, the shear stress is larger on the surface of the core pipe than it is on the surface of the outer pipe (Gnielinski, 2007). For this reason, the hydraulic diameter is considered as insufficient for estimating the frictional pressure drop for laminar flow in an annular geometry (Jones and Leung, 1981; White, 2006; Mironer, 1979). Therefore, amongst others, Knudsen and Katz (1958), Koch and Feind (1958), Jones and Leung (1981), Lohrenz and Kurata (1960), Gnielinski (2007) as well as Reed and Pilehvari (1993) suggest, to modify the hydraulic diameter according to a function first derived by Wien (1900) and Lamb (1907). They propose a function, which depends on the ratio a=r1/r2 (see Equations (2-16) and (2-17)).

f 

64   a  Re

(2-16)

1  a   ln  a   a  1  a   ln  a   1  a  2

2

2

(2-17)

Comparing Equation (2-16) to experimental studies (Tiedt, 1966), (Koch and Feind, 1958), Gnielinski found the mean square error of Equation (2-16) to be within a range of 2 %. In addition to the correlation given in Equation (2-17), Bourgoyne et al. (1991) and Crittendon (1959) proposed equivalent diameters for annular cross sections. These are, however, according to Sorgun and Ozbayoglu (2010) and Scheid et al. (2011), not accurate enough to determine the pressure drop through annular geometries. Besides a concentric arrangement of the outer and inner pipe, some researchers also dealt with an eccentric arrangement (Snyder and Goldstein, 1965; Özgen and Tosun, 1987; Tosun, 1984; Jonsson and Sparrow, 1965; James F. Heyda, 1959; Shah and London, 1978). Following, a short overview of the effects of eccentricity is provided by recounting the findings of the just

14

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mentioned research articles. In addition, a general method for estimating the friction factor for eccentric annuli is presented. Snyder and Goldstein (1965) as well as Heyda (1959) investigated on the laminar velocity distribution in an eccentric annulus and developed a formula describing the velocity distribution. Using an inversion technique, Özgen and Tosun (1987) developed a procedure to determine the maximum velocity locus in an eccentric annular cross section. An example for calculating the flow rate is given by Tosun (1984). He provides an equation to calculate the flow rate through an eccentric annulus as a function of eccentricity ratio and radius ratio, which is able to reproduce the exact values. Additionally, Snyder and Goldstein studied the friction factors in eccentric annular pipes and found the friction factor to change significantly with eccentricity by a factor of 2. In this context, Shah and London (1978) showed analytically that the flow resistance of a laminar flow through an annular geometry decreased with the eccentricity. Likewise, Jonsson and Sparrow (1965) observed that eccentricity leads to a decreasing friction factor. Since the hydraulic diameter concept is insensitive to the eccentricity of the core pipe of an annular duct, White (2006) concludes that in case of eccentric annular cross sections exact solutions are needed taking into account the ratio of eccentricity. A common method for estimating the friction factor of an eccentric annular arrangement is the use of the modified friction factor defined in Equation (2-16). Typical values for the factor function φ, which is now dependent on the eccentricity, can be identified in the VDI Heat Atlas (2010). Transitional Flow: As mentioned above, the transition from laminar to turbulent in circular tubes usually takes place in between a Reynolds number range from 2000 to 4000. On the contrary, in single-phase flow through an annular geometry several investigators (Monrad and Pelton, 1942; Carpenter et al., 1946) found the transition zone to occur over an extended Reynolds number range than in a circular tube. In this context, Rothfus and Prengle (1952) observed the first sign of turbulence at a Reynolds number of about 900. Additionally, the extend of the transition region was found by Rothfus et al. (1950) to be a function of the radius ratio r1/r2, with a longer range for smaller values of the radius ratio. Regarding the velocity profile, Prengle and Rothfus (1955) introduced a Reynolds number with an equivalent diameter based on the radius of maximum velocity as in Equation (2-15) (see Equation (2-18). dH 2 

2 2   r22  rmax 

r2

15

(2-18)

Literature Review

Considering the point of maximum velocity, Prengle and Rothfus (1955) noticed a shift of the locus of the maximum velocity towards the core pipe of the annulus at Reynolds numbers between 700 and 2300. Moreover, Rothfus et al. (1955) showed that Equation (2-12) is not applicable for the transition zone. For Reynolds numbers ranging from 700 to 10,000, Rothfus et al. also detected an inwards shift of the point of maximum velocity contrary to the results of Equation (2-12). Hence, in order to approach the position of maximum velocity for the transition from laminar to turbulent Rothfus et al. suggest using Equation (2-15). Similar to single-phase flow through circular tubes, there is no particular equation given for calculating the friction factor of transitional flow in annular cross sections. According to Gnielinski (2007), neither the approaches for laminar flow nor for turbulent flow are accurate enough to estimate the friction factor for the transition zone. However, for practical purposes Gnielinski (2007) recommends using the friction factor for turbulent flow, which will be presented in the following. Turbulent Flow: Several researchers studied the turbulent flow of single-phase fluids through annuli. Under the hydraulic diameter concept, Turbulent flow in annuli takes place at Reynolds numbers above 4000. Using the equivalent diameter defined in Equation (2-18), Prengle and Rothfus (1955) found turbulent flow to be existent at Reynolds numbers above 2200. The change in the velocity profile from laminar to turbulent annular flow can, to a certain extent, be described corresponding to single-phase flow through a circular tube. According to experimental investigations by Knudsen and Katz (1950), Rothfus et al. (1950), Quaramby (1967), Nouri et al. (1993) and Brighton and Jones (1964), the turbulent velocity profile in an annular cross section turns out to be flatter at the point of maximum velocity as shown in Figure 2-3. This is similar to the findings for the pipe flow. However, regarding the maximum point of velocity, opinions differ on its exact position. Rothfus et al.(Rothfus et al., 1950; 1955) and Knudsen and Katz (1950; 1958) observed, when allowing for an experimental error, that the locus of maximum velocity for fully turbulent flow is the same as for laminar flow. Thus, the point of maximum velocity can be expressed according to Equation (2-12). Yet, investigations carried out several years later by Quarmby (1967) and Brighton and Jones (1964) showed that the results of Rothfus et al. and Knudsen and Katz are incorrect. Brighton and Jones found the location of the point of maximum velocity at high Reynolds numbers nearer the core tube than for laminar flow. Following the experimental results of Brighton and Jones (1964), Rothfus et al. (1966) introduced a new formula for calculating the maximum radius: 16

Literature Review 2   rmax  r1   r1    r2  r1   r2 

0.20

(2-19)

Clump and Kwasnoski (1968), who also proposed a method to predict the location of maximum velocity, found Equation (2-19) to perfectly agree with the values predicted by their own method. In order to describe the pressure drop in concentric annuli, a number of investigations (Knudsen and Katz, 1958; Davis, 1943; Rothfus et al., 1950; Meter and Bird, 1961; Quarmby, 1967; Brighton and Jones, 1964; Nouri et al., 1993; Rehme, 1973; Deissler et al., 1955; Gnielinski, 2007; Brighton and Jones, 1964; Koch and Feind, 1958; Sorgun and Ozbayoglu, 2010; Jones and Leung, 1981) were carried out to propose correlations, which estimate the friction factor accurately. While some of the approaches were identified as incorrect, others are too complicated. Hence, in the following only the most common correlations will be introduced. Similar to the laminar flow through an annular geometry, the use of the hydraulic diameter seems to be not an adequate concept for describing the pressure drop under turbulent flow conditions. According to Mironer (1979), “the hydraulic diameter significantly underestimates the pressure drop of turbulent flows through annular passages”. To identify the most accurate correlation, Jones and Leung (1981) compared different correlations from relevant literature. They demonstrated that the so-called laminar Equivalent diameter, based on the factor given in Equation (2-17), is also applicable for the use in turbulent flow through annuli. Moreover, Jones and Leung conclude with the proposition of using the Colebrook-function for smooth tubes (the modified Reynolds number Re*=Re/φ(a) applied to Equation (2-7)):





1  2.0  log10 Re*  f  0.8 f

(2-20)

Comparing the results of Equation (2-20) with experimental data from Koch und Feind (1958), Gnielinski (2007) found the mean deviation to be of less than 1 %. Besides the method suggested by Jones and Leung (1981), Gnielinski also refers to a simpler correlation developed by Rehme (1973), which can also be modified using Equation (2-17): f  1.8  log10  Re*  1.5

2

(2-21)

Since the method of Rehme is not implicit and its accuracy, compared to experimental investigations, lies in between a range of 5 %, Equation (2-21) is 17

Literature Review

more appropriate for practical applications than Equation (2-20) (Gnielinski, 2007). Following the same argument, one could also use Equation (2-8) and modifying it with Equation (2-17). Regarding the hydrodynamic entrance length, Quarmby (1967) found the entrance length to be in the same order of magnitude as for circular tubes. The effect of eccentricity on the velocity profile and the friction factor was studied only by a few, including (Rehme, 1973; Nouri et al., 1993; Wolffe and Clump, 1963; Jonsson and Sparrow, 1965, 1966; Kacker, 1973; Deissler et al., 1955). Deissler and Taylor (1955) were the first to develop an analytical model to describe the velocity profile of turbulent flow through annular geometries. Wollfe and Clump (1963) studied the locus of maximum velocity and came to the conclusion that the point of maximum velocity can be sufficiently described by Heyda's solution (James F. Heyda, 1959) of the Navier-Stokes equation for laminar flow. In their experimental investigations Jonsson and Sparrow (1965; 1966) found the frictional factors to decrease and the hydrodynamic development length to increase with increasing eccentricity. Additionally, Jonsson and Sparrow provided a description of the velocity field in terms of contour diagrams. Besides Jonsson and Sparrow, Kacker (1973) and Nouri et al. (1993) also examined the turbulent flow through eccentric annular geometries. Thereby, Kacker proposed a correlation using the hydraulic diameter (Equation (2-11) to estimate the friction factor. It was found to be less than for a concentric annular section. Nouri et al. derived from their experiments also the phenomenon of a decreasing friction factor with an increasing eccentricity by 22.5 %. In this context, Rehme (1973) proposed a correlation based on the geometry factor for laminar flow. It shows however, high accuracy only for low diameter ratios, when compared to experimental data.

2.2 Two-Phase Flow Whereas in a single-phase flow only one fluid of one phase is flowing through a conduit, two-phase flow comprises the simultaneous flow of two different fluids with distinct fluid properties and phases, for example air (gas) and water (liquid) which are the focus of this study on two-phase flow. The major difference, which makes two-phase flow by far more complicated than single-phase flow, is the interface between the two different fluids (here: between air and water). Generally, the flow pattern of a single fluid through a conduit is affected by the shape and the size of the conduit, by the velocity and by the properties of the fluid. In case of air-water two-phase flow, the specific flow pattern depends on even more conditions, such as the fluid-fluid combination, the rate and direction of the flow, the geometry of the conduit as 18

Literature Review

well as its alignment (Kleinstreuer, 2003). As a result, additionally to laminar, transitional and turbulent flow, many other flow patterns can be found in the airwater two-phase flow. These are described in the following sections. In the literature, the common method for presenting the particular flow regimes is based on the use of two-dimensional maps, the so-called flow pattern maps (see for example Figure 2-6). They are generally plotted using data gained from experimental investigations (Hand and Spedding, 1993). These maps can be categorised into maps, which are based on physical coordinates, or into dimensionless groups (Taitel et al., 1978). Physical coordinates include values, such as superficial and mixture velocities, mass flow rates as well as liquid holdup and void fraction, while dimensionless coordinates including the Reynolds number. Since these quantities are important to characterise the air-water two-phase flow, i.e. the flow pattern, the most important ones are briefly outlined in the following: Superficial velocities us are, according to Equation (2-22), defined as the ratio of the volumetric flow rate of air and water and the cross sectional area of the conduit. us , g / l 

Qg / l A

(2-22)

Summing up the superficial gas and liquid velocities yields the superficial mixture velocity us,m, as represented by Equation (2-23). us , m 

Qg  Ql A

(2-23)

Furthermore, the mass flux G is defined as the summation of the superficial gas and liquid velocity multiplied with the density of each phase:

G  us , g   g  us ,l  l

(2-24)

Since in multiphase flow the fraction of the total mass flow is, in addition to the velocities, of great importance, one defines the vapour/mass quality x as the ratio of the gas mass flow rate Ṁg to the total mass flow rate. Mg x (2-25) M g  Ml Furthermore, the volumetric quality β is defined as the ratio of the volumetric flow rate of the gas phase to the total volumetric flow rate:

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Literature Review



Qg

(2-26)

Qg  Ql

Equivalent to the Reynolds number of a single-phase fluid, the superficial Reynolds number for a fluid of a two-phase combination is determined by the use of the superficial velocity: Re s 

  us  D 

(2-27)

Using the flow area of each phase instead of the cross sectional area another velocity can be introduced. It is called the mass velocity: ug /l 

Qg / l

(2-28)

Ag / l

In order to calculate the mass velocity, the fractional area or volume of each phase in the conduit is required. The fractional volume of gas, i.e. air, in the tube is an important quantity in the flow of twp-phases, because it is used in almost all modelling approaches (VDI e.V., 2010). It is named void fraction α g and can be defined as the ratio of the volume of gas V g to the total volume Vtot which includes air and water (see Equation (2-29)). Thereby, the total volume can be expressed in terms of the volume of the tube. In this context, liquid holdup αl represents the volumetric fraction of water. It can be expressed using Equation (2-30), or as the remaining fraction of the void fraction (1 – αg).

g 

l 

Vg



Vg



Ag Ag  Al

(2-29)

Vl Vl Al   Vtot Vg  Vl Ag  Al

(2-30)

Vtot

Vg  Vl

However, due to discontinuities in the fluctuation of the flow, the void fraction or liquid holdup is determined by time-averaging over a particular time interval (VDI e.V., 2010). Under such discontinuous flow conditions, it is evident that in practical applications simplified models have to be used to describe the two-phase flow as a continuous matter. In the past, mechanistic as well as empirical and semi-empirical models were developed, either to determine the flow pattern or to estimate the pressure drop. These models can be seen as extensions of the models developed for single-phase flow. According to Collier 20

Literature Review

(1994) and VDI Heat Atlas (2010) the methods used to analyse the flow of two phases can be distinguished into four groups:  Homogeneous flow model: the two-phase flow is treated as a singlephase flow with pseudo-properties depending on the individual phases and no slip between the phases.  Separated flow model: The two phases are artificially separated into two flows each flowing in its own pipe with divergent mean velocities (see Equation (2-28)), which are assumed to be constant over the cross section.  Drift flux model: the two phases are treated as a homogeneous mixture but still separated, using averaged values of the velocity and the void fraction, which is specified in terms of a drift velocity.  The flow pattern model: the two phases are categorised in different flow regimes according to their flow pattern and for each particular regime the basic equations are solved separately. Depending on the method, theses models are more or less accurate in estimating the pressure drop during the two-phase flow through closed conduits. The total change in pressure of two-phase flow along a conduit can be written as the pressure loss due to acceleration Δpa, gravitation Δpg and friction Δpf:

p  pa  pg  p f

(2-31)

In the case of a horizontal conduit, by definition the gravitational pressure loss disappears and Equation (2-31) is simplified to Equation (2-32).

p  pa  p f

(2-32)

Furthermore, on the basis of an unheated two-phase flow the pressure drop due to acceleration is often neglected and hence, the loss due to friction is the major contributor (Friedel, 1978; Orell and Rembrand, 1986; Hasan and Kabir, 1992). As a result, in an adiabatic two-phase flow the total pressure drop can solely be derived from the frictional pressure loss:

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Literature Review

p  p f

(2-33)

To estimate the frictional pressure drop several authors derived approaches from one of the four basic models mentioned above. In the following, besides the approaches to calculate the frictional pressure drop, the different flow patterns, which were observed, as well as correlations to determine the transition between the flow regimes for the case of horizontal circular and annular cross sections are presented.

2.2.1 Two-Phase Flow in Pipes

The flow of two phases through circular tubes has been subject to many studies in the last century. Theoretical as well as experimental investigations were carried out with the aim of fully understanding the physics of two-phase flow in pipes. In these studies correlations were derived to calculate the frictional pressure loss, the variety of flow patterns were examined and transition models were developed in order to distinguish between the different flow patterns. Experiments on the different flow patterns of two-phase flows were carried out in a number of studies. The early work of Bergelin and Gazley (1949) as well as the paper of Kosterin (1949) were one of the first to propose flow-pattern maps for horizontal two-phase flow. Based on the common practice of classifying the diverse flow patterns by characteristic visual observations using photography and video or by theoretical approaches and of plotting them in flow-pattern maps (Corradini, 1998), several other maps were published following the examples of Bergelin and Gazley, and Kosterin. These maps are either of empirical and of theoretical or of semi-theoretical nature (Cheng, 2012). Johnson and Abou-Sabe (1952), Alves (1954), Baker (1954), Hoogendoorn (1959), Mandhane et al. (1974), Spedding and co-workers (1980; 1993), Wong and Yau (1997), Yang and Shieh (2001), Kim and Ghajar (2002) and Abduvayt et al. (2003) amongst others published several empirical flow pattern maps by plotting experimental data. Throughout the numerous studies a variety of flow patterns as well as technical terms for them were established. The most widely accepted flow patterns and terms, which were introduced by Alves (1954), are described in detail by Collier and Thome (1994) and are shown in Figure 2-5:  Bubbly flow: The gas phase moves in form of discrete bubbles, consisting of different sizes, in a continuous liquid phase. Due to 22

Literature Review

gravity, in horizontal pipes the gas bubbles are located in the upper half of the cross section.  Plug flow: Gas bubbles, being, in comparison to the pipe diameter, rather small for bubbly flow, occupy almost the entire pipe section in plug flow. Similar to the bubbles in bubbly flow, the plugs also tend to move in the upper half of the pipe.  Stratified flow: In stratified flow, the two phases appear to flow separately in the pipe, with a relatively smooth interface between the two phases. Due to density differences, the gas phase travels at the upper half and the liquid phase at the bottom of the pipe.  Wavy flow: Compare to the smooth interface between the two phases in stratified flow, the interface in wavy flow is disturbed by waves of the liquid phase.  Slug flow: In slug flow, liquid splashes or slugs move, separated by gas plugs, through the pipe.  Annular flow: In annular flow, a gas core is built up surrounded by a liquid film at the periphery of the circular tube. The film is not necessarily continuous and the annular flow may be interrupted by a drop flow region consisting of liquid droplets. Due to gravity, the liquid film is thicker at the bottom than at the top of the pipe. The different flow patterns are formed depending on the flow rates or velocities of the liquid and gas phase. Stratified flow usually takes place at low velocities and at low flow rates of the gas and liquid phase. With increasing flow rates, stratified flow changes to wavy flow and ends up in slug flow. Raising the liquid velocity whilst keeping the gas velocity constant, bubbly flow and plug flow tend to occur. When the gas flow is much higher than the liquid flow, annular flow is the predominant flow pattern. Chisholm (1973) observed that, under transient conditions, the flow pattern transitions take place at different flow rates than it is the case for steady-state conditions. Even though the afore mentioned flow patterns can be seen as the major categories, many other hybrid flow patterns, mainly combined or subcategorised flow patterns were identified by Hand and Spedding (1993), Wong and Yau (1997), Spedding and Spence (1993), and Spedding and van Nguyen (1993). The common way to provide flow pattern maps is under steady-state conditions. One of the best-known flow pattern map for horizontal two-phase flow is the map from Baker (1954), which is widely used in the petroleum in23

Literature Review

Figure 2-5: Flow patterns of adiabatic air-water two-phase flow through horizontal pipes

Figure 2-6: Flow pattern map for horizontal two-phase flow (Baker, 1954)

24

Literature Review

dustry (Mandhane et al., 1974) and illustrated in Figure 2-6. Baker used the gas (Gg) and liquid mass velocities (Gl) divided, respectively multiplied by the fluidproperty-correction-factors (λ, ψ) as coordinates. He also added two additional sub flow patterns, namely Dispersed and Froth. Although Baker’s flow pattern map is the most widely used of the proposed flow pattern maps, later researchers (Mandhane et al., 1974; Spedding and van Nguyen, 1980; Spedding and Spence, 1993) question its performance when plotted against different pipe sizes. Moreover, this matter also reveals the endemic problem of empirical flow pattern maps. Consequently, theoretical as well as semi-theoretical unified models were developed in order to determine the transition boundaries between the different flow patterns. Taitel and Dukler (1976) and Taitel et al. (1978; 1980) were pioneers in this field introducing a solution based on physical concepts of the transition mechanism for steady flow as well as transient flow. They used dimensionless groups to classify each flow pattern. The result of a comparison with experimental data (Barnea et al., 1980) supports the consistency of the proposed model with the theory. Following Taitel and Dukler’s work, Barnea and co-workers (Barnea et al., 1982; Barnea, 1986, 1987) adopted their approach, modified it and provided a unified model for the whole range of pipe inclinations. Based on Taitel’s and Dukler’s and on Barnea’s models, in recent years Xiao et al. (1990) and Petalas and Aziz (2000) developed comprehensive methods to detect the flow pattern and to predict liquid hold up and pressure drop. As mentioned before, there are four different methods used in the literature to predict the pressure drop: homogeneous model, separated model, drift flux model and flow pattern depending model. Starting with the homogeneous model, Wallis (1969) provided the most complete mechanistic model to determine the pressure drop and liquid holdup for all flow patterns in two-phase flow (Osgouei, 2010). In his “homogeneous no-slip flow model” Wallis treated the two phases as a single fluid with no slip between the two phases. Thereby, Wallis determined the averaged physical properties of the fluid by assuming the liquid holdup to be equal to the gas-liquid volume fraction. In this model the two-phase frictional pressure drop can be estimated using Equation (2-34), which is derived from Equation (2-2) assuming a constant friction factor and the same properties.

l G2 p f  l   f   D 2  hom

(2-34)

The mixture density of the two phases ρhom in Equation (2-34) is commonly calculated with Equation (2-35) (Xu et al., 2012).

25

Literature Review

hom

 x 1 x       g l  

1

(2-35)

The homogeneous void fraction can be written as:

 g ,hom 

1 1 x g 1  x l

(2-36)

And the homogeneous velocity is defined according to Equation (2-37).

uhom  ug  ul

(2-37)

The friction factor of the homogeneous model can be estimated using the correlations given for single-phase flow, where the Reynolds number is rewritten for two-phase flow according to Equation (2-38)

Rehom 

GD

hom

(2-38)

To calculate the mean two-phase flow viscosity of the homogeneous model μhom, a variety of different correlations is given. They include the ones provided by McAdams et al. (1942), Cicchitti et al. (1960), Dukler et al. (1964a; 1964b), Beattie and Whalley (1982), Lin et al. (1991), Chen et al. (2001), Awad and Muzychka (2008) and Shannak (2008). All these models are summarised and evaluated by Xu et al. (Xu et al., 2012). Xu et al. found that the correlation models of Beattie and Whalley and McAdams et al. perform best for the range of data, which was used. Based on the homogeneous approach (no-slip), Beggs and Brill (1973) provided their famous flow pattern dependent model. Using the Froude number, in this approach first, the flow pattern was determined and then, dependent on the flow pattern, the liquid holdup and the pressure drop was estimated by developing a new friction factor correlation. Due to its simplicity and applicability to the whole range of pipe inclination the correlation of Beggs and Brill is widely used in two-phase flow engineering. Even though the homogeneous model is easy to use, it is limited to a number of applications only. According to Whalley (1990), the homogeneous model gives adequate results for ρl/ρg < 10 and G > 2000 kg/m²/s.

26

Literature Review

This fact leads to the second concept of two separated fluids, each flowing in its own pipe with constant velocities, i.e. the separated flow model. A first version of it was proposed by Lockhart and Martinelli (1949). Based on a twophase friction multiplier for the liquid phase φl and gas phase φg, which is defined as the ratio of the two-phase frictional pressure gradient to the frictional pressure gradient which would result from one of the fluids flowing alone, the pressure drop can be estimated using single-phase friction factors for each phase. The frictional pressure drop of two-phase flow in case of the separated model can be written as

l G  x  2 p f  l   f g    g D 2  g 2

(2-39)

for the gas phase, and as l G  1  x   p f  l   fl     l2 D 2  l 2

(2-40)

for the liquid phase flowing alone in the pipe, where the friction factor is calculated on the basis of the correlations for the friction factor of a single-phase fluid using the mass flux and the fluid properties of the particular single-phase. In order to calculate the two-phase multiplier Lockhart und Martinelli defined the well-known Lockhart-Martinelli parameter X as the ratio of the liquid pressure drop to the gas pressure drop (see Equation (2-41)) and provided a graph for determining the two-phase multiplier.

 dp    dl X 2   l  dp     dl  g

(2-41)

Since graphs are inconvenient for calculations (Xu et al., 2012) many studies (see for example Chisholm, 1967; Sun and Mishima, 2009; Zhang et al., 2010) carried out investigations to find an appropriate correlation for the friction multipliers. All proposed correlations are collected and compared in a comprehensive survey by Xu et al. (2012). Xu et al. conclude that the correlation of Sun and Mishima performs best. Besides the latter approach, in the separated flow model another method exists to determine the pressure drop. Similar to the first approach, the friction

27

Literature Review

pressure drop can be calculated in accordance with Equation (2-42) and Equation (2-43).

l G  x  2 p f  l   f go    go D 2  g

(2-42)

l G  x  2 p f  l   flo    lo D 2  l

(2-43)

2

2

In comparison to the multipliers from the first method, this approach defines the multiplier as the ratio of the two-phase pressure gradient to the pressure gradient of the total mixture being either gas or liquid. Following the second method of the separated flow model, Chisholm (1973), Friedel (1978) and Müller-Steinhagen and Heck (1986) proposed various correlations to calculate the two-phase multipliers. Again, it is referred to Xu et al. (2012) who summarised ten different correlations. The correlation given by MüllerSteinhagen and Heck (1986) was found to perform best (Xu et al., 2012; Didi et al., 2002). In the separated model a velocity ratio S is introduced, which is the ratio of the mass velocities of the gas and liquid phase:

S

ug

(2-44)

ul

Using the velocity ratio defined in Equation (2-44), the homogeneous void fraction can be extended to the separated void fraction:

 g ,sep 

1 1  x g 1  S x l

(2-45)

In addition to the void fraction from Equation (2-45), the relevant literature (see for example Zivi 1964 and Smith 1969) proposes modified analytical as well as empirical correlations. These are summarised by (2004). In this context, a third model, namely drift flux model, was introduced by Wallis (1969) and enhanced by Zuber and Findlay (1965) and Ishii (1977). Similar to the homogeneous model, the two phases are also treated as a mixture. Additionally, the drift flux model includes the relative motion between the phases (Ishii, 1977). Averaging the cross-sectional area, the drift flux model 28

Literature Review

is simplified to a one dimensional problem. Since the drift flux is, due to its complexity difficult to implement, it is generally used only in complex program systems (VDI e.V., 2010) and will not be described in detail. Nevertheless, the drift flux model is, due to its physical basis of great importance for practical applications in estimating the void fraction (Hibiki and Ishii, 2003). The void fraction of the drift flux model is outlined in Equation (2-46), where the distribution parameter C0 and the drift velocity ud represent the drift flux parameters (Choi et al., 2012).

 g ,DFM 

us , g

C0   us ,l  us , g   ud

(2-46)

Different correlations for estimating the drift flux parameters were proposed by Zuber and Findlay (1965), Wallis (1969) and Ishii (1977). A summary of the different values can, for example, be found in Thome (2004) and Choi et al. (2012). However, the drift flux parameters can be calculated with or without reference to the particular flow pattern (Thome, 2004). For estimating the pressure drop and void fraction based on flow patterns, the flow pattern dependent model is used. The flow pattern model, also called mechanistic modelling approach, emerged in the early 80’s (Gomez et al., 2000). These models are based on fundamental laws and hence, are able to model geometric and fluid property variations more accurately than empirical models (Petalas and Aziz, 2000). The common procedure of these models is first to determine the flow pattern and then to solve the basic equations within each framework in order to predict the liquid holdup, the pressure drop and other two-phase flow model parameters (Collier and Thome, 1994; Petalas and Aziz, 2000; Mokhatab and Poe, 2012). Earlier mechanistic models presented in the literature can be seen as either incomplete, as they only provide correlations for flow pattern determination (Taitel and Dukler, 1976; Barnea, 1987) or as limited in terms of pipe inclination (1994; Xiao et al., 1990). However, in recent years new and more robust mechanistic approaches were introduced (Petalas and Aziz, 2000; Gomez et al., 2000; Zhang et al., 200 Garc a et al., 2003). Petalas and Aziz (2000) proposed a mechanistic model, which is applicable under all conditions in the petroleum industry in terms of pipe geometries and fluid properties. The flow pattern determination was based on Taitel’s and Dukler’s (1976) as well as on Barnea’s (1987) flow transition criteria. Additionally, empirical correlations for interfacial friction were developed, based on a large amount of experimental data. Gomez et al. (2000) developed a unified steady-state two-phase flow mechanistic model in order to predict flow pattern, liquid hold up and pressure

29

Literature Review

drop for horizontal to upward vertical flow. In this work, correlations from Taitel and Dukler (1976) and Barnea (1987) were used to determinate the flow pattern and empirical correlations to estimate the friction factor. Garcia et al. (2003) presented a unified mechanistic model for the same range of inclination as Gomez et al. (2000). A universal composite correlation for the gas-liquid friction factor, valid for all flow patterns, and composite friction factor correlations sorted by flow pattern were developed. The friction factor was based on the mixture density, mixture velocity and mixture Reynolds number. Although these new mechanistic systems tend to be more robust, further investigations and testing of these models are required (Mokhatab and Poe, 2012). Whereas in single-phase flow the entrance length is well established (see Equations (2-6) and (2-9)), there is no generally accepted corresponding entrance length for two-phase flow (Brennen, 2005).

2.2.2 Two-Phase Flow in Annuli In the literature, many studies focusing on two-phase flow in circular cross sections are available (see Chapter 2.1.1). However, one only encounters a limited number of investigations, theoretical as well as experimental, discussing the flow of two phases in annular pipe geometries. Whereas in single-phase flow the implementation of the hydraulic diameter tends to yield reasonable results, the description of two-phase flow in concentric and eccentric annuli is more complicated (Ettehadi Osgouei et al., 2010; Metin and Ozbayoglu, 2009). The common method of using the hydraulic diameter and adapting it to annular geometries in order to describe the behaviour of twophase flow is questionable (Omurlu, 2006; Osgouei, 2013; Ettehadi Osgouei et al., 2010; Omurlu, 2006; Sunthankar et al., 2003). Moreover, existing models for two-phase flow through circular geometries perform poorly for annular geometries (Osgouei, 2013; Sunthankar et al., 2003). Nonetheless, several studies were carried out for the case of two-phase flow through vertical annular cross sections (see for example Kelessedis and Dukler 1989, Das et al. 2000, Ozar et al. 2008, Julia et al. Julia et al., 2011 or Mendes et al. Mendes et al., 2013). In the following, only experimental and theoretical investigations performed in horizontal and near horizontal annular pipes are outlined. Salcudean et al. (1983a; 1983b) investigated the influence of flowobstruction geometry on pressure drops and void fraction in horizontal air-water

30

Literature Review

flow. They used a variety of central and peripheral segments, including an annular geometry, and found the obstructions to have a significant impact on the phase distribution of slug and bubbly flow. Furthermore, Salcudean et al. observed larger pressure drops compared to the case where an obstruction is absent. They also noted that the size of the pressure drop depends on the location of the obstruction and that there exists a discrepancy between theoretical models and their experimental counterparts. Osamusali (1988) and Osamusali and Chang (1988) studied two-phase flow pattern and flow pattern transition theoretically as well as experimentally in horizontal pipes, annuli and rod bundles. Taitel’s and Dukler’s (1976) transition model was extended to the case of eccentric annuli based on geometric parameters. In his experiments, Osamusali observed similar flow patterns for circular and annular pipe geometries by direct visual observation and by ring type capacitance transducers. Additionally, the different flow regimes were presented in a flow pattern map. In comparison to pipe flow, Osamusali noticed a significant shift of the flow pattern transitions of two-phase flow through annuli and found that the diameter ratio of the inner to the outer diameter exerts a strong influence. In addition to Osamusali (1988), Ekberg et al. (1999), Sunthankar (2003), Wongwises and Pipathatatakul (2006) and Mendes et al. (2011) also presented flow pattern maps for various annular pipe sizes and found discrepancies when compared to those of circular pipe geometries and modified theoretical correlations. Ekberg et al. (1999) experimentally studied gas-liquid flow regimes, void fraction and pressure drop in narrow horizontal concentric annuli. Sunthankar (2003) performed extensive experiments with and without drill pipe rotation in large-scale annulus. In most of the experiments Sunthankar observed plug and slug flow, which are, in contrast to the homogeneous model, used in simulation practices. Moreover, compared to modified pipe flow models, Sunthakar noticed significant errors and concluded that there is a need for new mechanistic models. Wongwises and Pipathatatakul (2006) carried out experiments in horizontal and inclined narrow concentric annular pipes. No satisfying agreement was found for pressure drop and void fraction when compared to theoretical models. Mendes et al. (2011) presented flow pattern maps for horizontal annular pipes and angels of inclinations of up to 45° for pipe size dimensions close to offshore applications. A simple pressure-signature technique was used to objectively identify the flow pattern transitions. Additionally, they concluded that new phenomenological models are required. Afore mentioned studies clearly show, that new models are required to describe horizontal two-phase flow in annular geometries with sufficient 31

Literature Review

accuracy. As a result, in the past decade, several theoretical investigations were carried out and a number of mechanistic models were developed. Sunthankar (2000) modified the model of Xiao et al. (1990) and the model of Taitel and Dukler (1976) by implementing the hydraulic diameter concept. However, comparing to experimental results, Sunthankar found the extrapolation from circular pipe flow to annular pipe flow based on the hydraulic diameter to be incorrect in predicting flow pattern and pressure drop (Sunthankar et al., 2003). Lage et al. (2000) performed small-scale experiments and formulated a mechanistic model for two-phase flow in horizontal and slightly inclined fully eccentric annuli. In this model, the flow pattern determination relied on the transition models form Taitel and Dukler (1976) and from Barnea (1987) and the pressure loss prediction on different approaches, including the homogeneous model. When validated against experimental data, a satisfying consensus between pressure drop and flow pattern data was found. Akbar and Ghiaasiaan (2003) analysed the flow regime transition of stratified wavy flow pattern in annular geometries. They inherited the model proposed by Taitel and Dukler (1976) and extended it to the case of two-phase flow through annular ducts using experimental data form Osamusali and Chang (1988) and Ekberg et al. (1999). Zhou et al. (2004) developed a mechanistic model based on conservation equations and two-phase pipe flow correlations. Moreover, experiments were carried out in order to validate the proposed model. The model achieved a good result with an absolute average error of 12.2% against the experimental data. Furthermore, throughout the experiments an effect of the temperature on the frictional pressure loss was observed. Omurlu Metin and co-workers (Omurlu, 2006; Omurlu and Ozbayoglu, 2006a, 2006b; Metin and Ozbayoglu, 2007a; Metin and Ozbayoglu, 2007b; Metin and Ozbayoglu, 2009) extensively studied two-phase flow through fully eccentric annuli. They proposed a mechanistic model for flow pattern determination and pressure drop estimation by introducing a new areal representative diameter and a weighting factor. Moreover, friction factor correlations are developed both, dependent on and independent of flow pattern. They concluded that the hydraulic diameter approach is not applicable to fully eccentric annuli due to remarkable shifts at the flow pattern maps. In addition to these investigations, Osgouei et al. (2012) formulated a mechanistic model which was extended to two-phase flow in fully eccentric and inclined annular ducts using conformal mapping technique. Experimental and theoretical studies about two-phase flow through concentric annuli were carried out by Ozbayoglu and Omurlu (2007). They introduced a new mathematical model to predict flow pattern and frictional 32

Literature Review

pressure drop. The model was based on a new equivalent diameter concept adapted from Kozicki’s and Tiu’s (1971) work. Compared to experimental data, the model was found to be reasonably accurate. It was also observed, that the geometry and the viscosity of the liquid phase influence the flow pattern transition and the frictional pressure losses significantly. Ozbayoglu and Ozbayoglu (2007; 2009) and Ozbayoglu and Yuksel (2012) presented a new method for predicting flow pattern and pressure loss. They suggested using artificial neural networks based on superficial Reynolds numbers rather than mechanistic models. When compared to empirical and mechanistic correlations, it was found that the new models could improve the performance in predicting flow pattern and frictional pressure loss. Later on, Osgouei (2010) and Osgouei et al. (2010) proposed another classification of the two-phase flow pattern. They first presented a new flow pattern map and then used diaquadratic discriminant analysis in order to theoretically describe the flow pattern transitions. They also applied Beggs and Brill (1973) and Taitel and Dukler (1976) flow pattern models by implementing the hydraulic diameter concept and concluded that these models cannot predict the flow pattern when based on hydraulic diameter. Throughout their experiments, liquid holdup was measured by using high speed camera and image processing techniques. Using the same method, experiments were conducted by Ozbayoglu and Yuksel (2012), Sorgun et al. (2011; 2013) and Ozbayoglu et al. (2012). Sorgun et al. (2011; 2013) compared Eulerian-Eulerian computational fluid dynamics model for two-phase flow patterns in annulus with experimental studies. Their studies showed that their computational fluid dynamics model can predict flow patterns and pressure losses with reasonable accuracy. In addition, Sorgun et al. noticed a slightly higher pressure loss for concentric annular geometries compared to fully eccentric annular cross sections. Recently, Osgouei et al. (2013) modified traditional correlations by Lockhart und Martinelli (1949) and Beggs and Brill (1973) for circular pipes. The modified correlations showed good agreement when compared to experimental data. Overall, the results from previous work show that there is still considerable requirement for further investigation in horizontal and near horizontal two-phase flow in annular ducts. Additional experimental work performed at the 2’’ test rig in the Process Systems Engineering (PSE) Lab at Cranfield University might help to expand the existing experimental database and to fully understand the hydraulics of two-phase flow in annular cross sections. Moreover, empirical and mechanistic models provided in the literature require further validation using experimental data. Although the hydraulic diameter concept is possibly the most widely used concept in the petroleum industry (Bourgoyne et al., 1991), an experimental validation, respectively a theoretical proof is still required to show 33

Literature Review

that the pipe flow equations can truly be extended to annular geometries using the hydraulic diameter concept or similar concepts proposed in the literature. A summary of different diameter concepts used in the literature is given in Table 2-1 (Scheid et al., 2011).

34

Literature Review

Table 2-1: Correlations of various diameter concepts for the flow through annular ducts Author

(Wien, 1900) (Lamb, 1907)

DH  d2  d1

Hydraulic diameter concept, ratio of the cross sectional area to the wetted perimeter of the cross-sectional area

DH 

Developed for single-phase flow of Newtonian fluids in annular ducts (Lamb/Wien diameter)

d 22  d12 d  ln  2   d1 

2 2   r22  rmax 

r2

where

rmax

 r22  r12 ; laminar   2  ln  r2      r1      r1  r2 ; transitional  0.20   r 1  1 r1  2   r2  r1    r  ; turbulent  2   

  2   d 22  d12   1  4 4 2 2  DH   d 2  d1   d 2  d1  2 4 d  ln  2     d1   

(Crittendon, 1959)

(Knudsen and Katz, 1958) (Koch and Feind, 1958; Jones and Leung, 1981) (Lohrenz and Kurata, 1960) (Gnielinski, 2007) (Reed and Pilehvari, 1993) (Langlinais et al., 1985) (Sorgun and Ozbayoglu, 2010)

(Ozbayoglu and Omurlu, 2007)

Comments

DH  d 22  d12 

(Prengle and Rothfus, 1955) (Rothfus et al., 1966) (Rothfus et al., 1955) (Knudsen and Katz, 1958)

(Omurlu, 2006)

Correlation

   d 2    1   1    2  d 2  d1   1   d1     d 2    DH     2  d    d1     d 2  ln  1   1      d  2     d 2    

DH  0.816   d2  d1 

35

Empirical relationship based on data obtained from a large number of annular treatments (Crittendon diameter) Developed for single-phase flow of Newtonian fluids in annular ducts, based on diameter given by Wien (1900) and Lamb (1907) (Equivalent diameter) Based on the gap of a slot (Slot diameter) Based on area between outer and inner diameter of annular geometry (Omurlu diameter)

DH  d 22  d12

  d   d 2  d1   0.9983  1.4472   1   d2     2 3 d  d    7.2649   1   16.833   1   d d DH    2  2  4  d1    +13.564       d 2    0.816   d  d  2 1 

Developed for single-phase flow of Newtonian fluids in annular ducts dependent on flow pattern (Knudsen /Prengle/Rothfus diameter)

; d1 d 2  0.5

; d1 d 2  0.5

Based on geometrical shape factors by Kozicki and Tiu (1971) and the gap of a slot (Ozbayoglu diameter)

Experimental Setup

3 Experimental Setup This chapter provides an overview of the experimental work carried out at the 2’’ test rig in the PSE Lab at Cranfield University. First, the general structure of the test rig and the test section are described. In Section 3.2, all employed measurement devices are listed and the calibration procedure is outlined. In Section 3.3, the focus lies on the experimental method and data collection.

3.1 Description of the Test Rig and Test Section The 2’’ test rig in the PSE Lab at Cranfield University consists of a closed loop Acrylonitrile Butadiene Styrene (ABS) pipe system with an inner diameter of 2’’ or 51.5 mm, respectively. The schematic drawing of the test section in Figure 3-1 and the drawing of the test rig is shown in Figure 3-2. The air is supplied through an air compressor and can be fed into the ABS pipeline through two inlets. One inlet is located in front of the test section. The second inlet is mounted at the beginning of the test section in order to avoid disturbances on the two-phase flow induced by the abrupt change from the circular to the annular pipe geometry in the entrance of the test section. A domestic pool pump pumps water from a 2 m³ capacity storage tank into the test rig. Before the air-water mixture flows back to the water tank, it passes the annular test section. The outer pipe of the annular section is constructed from a clear acryl pipe, which allows visual observation of the air-water two-phase flow patterns. An ABS pipe acts as the inner cylinder of the annular geometry. The inner diameter of the outer pipe is 51.5 mm and the outer diameter of the inner pipe is 33.3 mm, which results in a hydraulic diameter of the annular cross section of 18.2 mm. The measurement and observation area of the test section is located downstream from the two mixing points of air and water. An entrance length of more than 6 m (> 329 x hydraulic diameter dh) and an exit length of 1 m (> 54 x dh) ensures the flow (or the flow pattern) to be fully developed and avoids entrance and exit effects on the measurements. Assuming Equation (2-9) to be valid for both, the entrance and the exit length of the two-phase flow, the arrangement results in a total length of more than 9 m for the flow pattern to be fully developed and the measurements to be examined. The measurement section includes an absolute and three differential pressure transducers in order 36

Experimental Setup

to measure frictional pressure losses and the average pressure inside the loop. Additionally, two pairs of conductivity rings are mounted at the measurement section in order to determine the liquid holdup. They also can be used to measure the velocity of the gas and liquid phase via cross correlation. Detailed information about the different measurement devices is given in Section 3.2.

Figure 3-1: Schematic drawing of the test section

37

Figure 3-2: Schematic structure of the 2’’ test rig

Experimental Setup

38

Experimental Setup

3.2 Instrumentation and Calibration Process Along the 2” test rig a variety of measurement devices are implemented in order to collect the required data. Two flow meters, a thermal flow meter (Proline t-mass 65, Endress and Hausser) and an electromagnetic flow meter (MagMaster MS E500, ABB Limited), are installed to measure the flow rate of air and water, respectively. Both meters are located upstream of the test section. Additionally, a pressure indicator and a pressure regulator are mounted behind the thermal flow meter. They are used to keep the pressure of the compressed air at a constant level before entering the test section. The test section itself is comprised of 3 differential pressure transducers (PMP 4170), and 1 absolute pressure transducer (PMP 1400). The pressure transmitters are supposed to measure the frictional pressure drop and to indicate the system pressure of the air-water mixture at different locations along the test section. The differential pressure is measured between the first and the second, the first and the third, and the first and the fourth pressure tapping illustrated in Figure 3-1. Besides the pressure of the air-water flow, the temperature of the mixture is required to determine the characteristic parameters of air and water. Therefore, a type T thermocouple is added to the test section (not shown in Figure 3-1). Throughout the process of selecting the appropriate measurement devices, particular focus is placed on the determination of the void fraction. The void fraction is one of the most important parameters in two-phase flow applications and is a key physical value in terms of density, viscosity, flow pattern transition and pressure drop (Thome, 2004). In general, three methods are commonly used to measure the void fraction (see Thome 2004) in experiments: chordal, cross-sectional and volumetric. The chordal void fraction is based on the intensity of a radioactive beam, which is differently influenced by the water and air phase. To determine the volumetric void fraction along the test section, a pair of quick-closing valves is installed. However, when it comes to measuring the void fraction over a cross-section, an optical device or the impedance method is used. Comparing the above mentioned methods, the impedance method tends to be the best solution because quick-closing valves cannot measure the liquid holdup in real time, the radiation method is dangerous and complicated in its construction and the optical devices cannot easily be applied to different flow patterns (Shi, 2010). The impedance method is a simple device in terms of requirements and can indicate in real time the void fraction of fluids with sufficiently different electrical properties (Shi, 2010; Devia and Fossa, 2003). The two classifications of the impedance method are either capacitance or conductance method. Since water is the continuous phase in most of the examined flow patterns, according to Shi et al. (2010) the conductance method is preferred. Different arrangements of 39

Experimental Setup

the electrodes of the conductance technique are presented in the literature, such as ring electrodes or plate-shaped electrodes. However, electrodes in form of rings are more suitable for practical applications due to less care during calibration and a better measurement uncertainty when compared to plateshaped electrodes (Fossa, 1998). As a result, throughout the experimental work two homemade ring-type conductance probes comprising two stainless steel ring electrodes are used. Figure 3-3 shows the configuration of the ring-type conductance probes installed at the test rig. The electrodes are flush mounted to the outer pipe of the annular test section and separated by a distance of 66 mm. Following the recommendations suggested by Devia and Fossa (2003) and Andreussi et al. (1988), the gap between the ring electrodes is 13 mm and each ring electrode has a width of 8 mm. The maximum output voltage of this design of the conductance probes is just below 5 V, representing a pipe section fully filled with water, and the minimum output around 0 V, representing an empty pipe. In general, the voltage response of the conductance probes is typically of a non linear nature and depends on the electrode geometry (Devia and Fossa, 2003). As a consequence, the conductance probes require an off-line calibration rig. The calibration process of the conductance rings was carried out following the procedure proposed by Fan and Yan (2014). The normalised calibration curve of each probe is illustrated in Figure 3-4, where “1” represents a full pipe and “0” an empty pipe.

Figure 3-3: Ring-type conductance probes

40

Experimental Setup

Figure 3-4: Calibration curve of the conductance probes 2

The graph is approached by a polynomial function and hence, the liquid holdup can be determined using Equation (3-1).

 l  2.8517 V 6  15.033 V 5  26.382 V 4  20.651V 3  7.4935 V 2  0.0515 V  0.0019

(3-1)

3.3 Experimental Procedure and Data Collection Throughout the experimental work air and water are used as the working fluids for the gas and liquid phase, respectively. For different gas and liquid flow rates flow patterns and pressure losses are measured and recorded using a homemade program and a National Instruments LabVIEW data collection system. The test procedure includes the following steps:  The connection pipes between the differential pressure transducers are flushed with water in order to prevent trapping of gas bubbles in these lines.  Before water is pumped through the annular pipe section offset readings of the pressure transducers and the conductivity rings are taken.

41

Experimental Setup  The water is pumped from the storage tank into the test loop and the water flow rate is adjusted.  Once the water flow rate is regulated, the control valve of the air input line is opened and the air flow rate is adjusted to the desired value.  As soon as the air-water two-phase flow is stabilised, the temperature and the static pressure of the two-phase mixture as well as the flow rates, the frictional pressure losses and the liquid holdup are recorded. The recording time of each flow condition point is set to 2 minutes after steady state condition is established.  Throughout the 2 minutes recording time a digital video camera is used to record the observed flow pattern.  When the experiments are finished experimental data is analysed by averaging the recorded data of the 2 minutes recording time. The test matrix, shown in Table 3-1 is based on the range of the maximum and minimum capacities of the air and water supply and the limits set by the differential pressure transducers (maximum differential pressure of 230 mbar) and the construction of the rig (avoidance of too much vibration). Whilst the water flow rate is kept constant, the flow rate of air is increased gradually. Once the whole range of the air flow rates for the particular water flow rate is covered, the water flow rate is increased. This procedure is repeated for the whole range of the water and air flow rates. Table 3-1: Test matrix Water Flow Rate (kg/s)

Range of Air Flow Rate (kg/h)

0.3

0.1 to 23.8

0.4

0.6 to 14.6

0.8

0.4 to 8.3

1.2

0.4 to 6.0

1.6

0.4 to 6.1

2.0

0.3 to 4.9

2.4

0.3 to 3.9

2.8

0.4 to 3.4

3.2

0.3 to 2.5

3.6

0.3 to 2.0

42

Theoretical Models

4 Theoretical Models In the literature, several models can be found for describing the simultaneous flow of two-phases. The basic idea of some of these models is outlined in Chapter 2. However, with regard to the flow pattern, the liquid holdup and the pressure loss of air-water two-phase flow in this chapter, two-phase flow models are presented which are validated against the experimental results of this study. Since all used models are derived from the flow of two phases through circular geometries, in this study, these models are extrapolated to annular cross-sections using the hydraulic diameter approach and the diameter concepts gathered in Table 2-1. To avoid a misunderstanding, in the following the hydraulic diameter is expressed as do - di and the use of any of the diameter concepts is indicated in terms of dh.

4.1 Models for Flow Pattern Determination As mentioned in Chapter 2, the flow pattern determination is one of the central issues in two-phase flow applications. In this thesis, the use of three different models, namely Taitel and Dukler model (Taitel and Dukler, 1976), Barnea model (Barnea et al., 1982; Barnea, 1986, 1987) and Beggs and Brill model (Beggs and Brill, 1973; Brill, 1991), is made in order to identify the flow pattern. Beggs and Brill model (Beggs and Brill, 1973; Brill, 1991): The flow pattern recognition in the Beggs and Brill model is based on the Froude number of the homogeneous mixture Frhom and the input liquid content λ, which are defined in Equations (4-1) and (4-2).

Frhom

2 uhom  g  dh

  1 

43

(4-1)

(4-2)

Theoretical Models

Using the input liquid content in accordance with Equations (4-3) to (4-6), the correlation boundaries can be calculated.

L1  316   0.302

(4-3)

L2  0.0009252   2.4684

(4-4)

L3  0.10   1.4516

(4-5)

L4  0.5   6.738

(4-6)

Hence, the flow pattern is determined as follows: Frhom  L1 and   0.01    segregated Frhom  L2 and   0.01 L2  Frhom  L3 and   0.01  transition L3  Frhom  L1 and 0.01 1187) for each phase depending on the hydraulic diameter concept. Therefore, using Equations (4-64) and (4-65), the frictional pressure loss for each phase can be calculated and hence, the frictional pressure loss of the two-phase flow can be estimated. The frictional pressure loss correlation suggested by Müller-Steinhagen and Heck is outlined in Equation (4-66). 1   dp   dp    dp   dp     dp  3        2  x           1  x  3     x  dl  f   dl lo  dl  go   dl  go  dl lo  

(4-66)

Beggs and Brill model (Beggs and Brill, 1973): As mentioned in Chapter 2.2.1 the Beggs and Brill model is a flow pattern dependent homogeneous model. As a result, the flow pattern determination of a single fluid (laminar or turbulent) and the calculation of the friction factor are conducted in the same manner as in the homogeneous model. However, since the Beggs and Brill model also takes into account diverse flow patterns of the two-phase flow, different values for the liquid holdup can be derived (see 56

Theoretical Models

Equations (4-50) to (4-53)). Depending on the value of the liquid holdup Beggs and Brill introduced a correction factor for the friction factor based on the homogeneous mixture. The correction factor S is given as      ln  2    l     S  exp  2 4   0.0523  3.182      0.8725      0.01853     2 2 2   l   l    l  

In order to determine the liquid holdup and the flow pattern it is referred to Chapter 4.1 and 4.2.

57

(4-67)

Experimental Results and Discussions

5 Experimental Results and Discussions The results of the experimental work carried out at the 2’’ test rig are outlined in the following sections. In Section 5.1 an overview of the recorded flow patterns as well as of the collected data with respect to liquid holdup and pressure drop is given. It includes single- and two-phase flow. Using the obtained data from the experiments in Section 5.2, a variety of correlations is verified by using different approaches for the diameter concept and the estimation of the flow pattern transition, the liquid holdup and the pressure drop.

5.1 Experimental Results The experimental results are split into two categories, namely single-phase flow of water and air water two-phase flow.

5.1.1 Single-Phase Flow of Water Before conducting a number of two-phase flow experiments with a wide range of water and air input flow rates, first, single-phase flow experiments with water are examined. Thereby, pressure losses and flow rates of the water flow through the annular test section are recorded to ensure the accuracy of the collected data. Moreover, comparing the experimental data to theoretical models, the applicability of frictional pressure loss correlations and of the various diameter concepts (see Chapter 2.2.2) are inspected. The values for the pressure losses are taken from the differential pressure loss readings between the first and the last pressure indicator separated by a distance of 4.03 m (see Figure 3-1). Figure 5-1 and Figure 5-2 show the results of the experimental pressure losses as well as the accuracy of the different diameter concepts. Equation (2-5) and Equation (2-8) were used to determine the friction factors for the pressure loss estimation. According to Figure 5-1 the experimental pressure drop of the single-phase flow increases as expected with an increasing flow rate of water. When compared to theoretical models depending on the diameter concept, the diameter suggested by Knudsen, Prengle and Rothfus (indicated

58


as Knudsen/Prengle) tend to perform best.

Figure 5-1: Pressure loss of water single-phase flow in the annular test section over the flow rate of water

Figure 5-2: Pressure loss of water single-phase flow in the annular test section

59


Similar results are obtained when applying the hydraulic diameter concept. The Slot diameter, the diameter suggested by Lamb and Wien (Lamb/Wien) and the Ozbayoglu diameter, which slightly overpredicts the pressure drop, also lead to reasonable results. However, pressure drop correlations depending on the Equivalent diameter highly overpredict the pressure drop of water, and the diameters introduced by Crittendon and Omurlu highly underestimate the pressure loss exceeding the ±20 % error margin. Overall, when compared to most of the diameter concepts found in the literature, the readings taken for single-phase flow of water can be considered as accurate and reasonable.

5.1.2 Air-Water Two-Phase Flow After the measurement devices are calibrated and after the accuracy of the pressure drop readings is confirmed by the experimental data of water singlephase flow, air-water two-phase flow experiments are examined. In total, 84 data points depending on the water and air flow rate are measured following the procedure mentioned in Chapter 3.3. Throughout the experiments, the temperature of the mixture as well as its pressure is found to vary in a range between 20 to 30 °C and between an absolute pressure of 1 to 1.5 bar. To proof repeatability of the measured data points, measurements are examined twice for randomly chosen data points. The graphs, which proof repeatability of the measured data points regarding the liquid holdup and the pressure drop, are attached in Appendix A. The observed flow patterns comprise bubbly flow, plug flow and slug flow. In the following, for sake of simplicity, the latter two are summarised as intermittent flow. Exemplary pictures of the observed flow pattern for different flow rates of water and air are given in Appendix A. In Figure 5-3 the flow pattern of the 84 data points are plotted against the superficial velocities of air and water. Considering the arrangement of the plotted flow patterns, bubbly flow (blue dots) tends to occur at high flow rates of water. The transition from bubbly flow to intermittent flow takes place at superficial velocities of water in between 2 and 2.5 m/s, depending on the flow rate of air. By increasing the superficial velocity of air (red stars) the transition from intermittent to bubbly flow occurs at higher superficial velocities of water. However, as mentioned before, the experimental work is restricted by the capacities and the design of the test rig. Therefore, no further statements regarding the transition boundaries of the flow regimes are possible. The liquid holdup measurements are illustrated in Figure 5-4. The figure shows, that for constant water flow rates the liquid holdup decreases with rising air flow rates. This is due to the increase of the volume of air whilst the volume

60


of water remains constant. It can also be concluded from Figure 5-4 that the liquid holdup increases with a higher liquid flow rate. The effect on the pressure drop is represented in Figure 5-5. In general, with increasing water and air flow rates the pressure loss becomes larger. At low water velocities the effect of an increase in the air flow rate on the pressure drop is insignificant, whereas at higher water flow rates the influence of air becomes substantial. This fact indicates that water is the major contributor to the pressure drop in air-water two-phase flow. The readings of the two-phase pressure drop are taken from the same differential pressure transducer as used for single-phase flow of water.

Figure 5-3: Flow pattern observation plotted against superficial velocity of air and water

61


Figure 5-4: Liquid holdup versus superficial velocity of air for different water flow rates

Figure 5-5: Pressure Drop versus superficial velocity of air for different water flow rates

62


5.2 Comparison of experimental data and theoretical models In this section, the experimental results are compared with the predictions in terms of flow pattern estimation, liquid holdup determination and pressure drop prediction. Different models and correlations including various diameter concepts found in the literature are validated with the measured data points. All models used in the comparison were presented in Chapter 2 and Chapter 4.

5.2.1 Flow Pattern Estimation Three different flow pattern transition models are considered, namely Taitel and Dukler model (Taitel and Dukler, 1976), Barnea model (Barnea et al., 1982; Barnea, 1986, 1987) and Beggs and Brill model (Beggs and Brill, 1973; Brill, 1991). The Barnea and Beggs and Brill model also take into account the different diameter concepts defined in Chapter 2.2.2. The accuracy of these models is validated with the observed flow pattern of the 84 measured data points. To illustrate the findings, general flow pattern maps are generated for each model. Thereby, the temperature and the static pressure of the two-phase flow are averaged across the 84 data points and used to calculate the fluid properties. The average temperature is 28.2 °C and the average static pressure sums up to 1.26 bar. The modified model suggested by Taitel and Dukler is the most accurate among the three evaluated models predicting 84 out of 84 data points correctly. Hence, regarding the measured data points the modified Taitel and Dukler model can be seen as highly accurate. The flow pattern map generated by using the transition criterions by Taitel and Dukler is represented in Figure 5-6. It clearly shows that the predicted transition from intermittent to bubbly flow is supported by the experimental results. The modified Barnea model is, in terms of accuracy, the second best model. Depending on the chosen diameter concept the modified Barnea model is able to predict up to 80 out of 84 data points correctly (Equivalent diameter concept). In this context, the Crittendon diameter concept and the Omurlu diameter concept performs worst by predicting only 73 out of 84 data points correctly. The results of all diameter concepts are shown in Appendix B. The Barnea flow pattern based on the Equivalent diameter concept is illustrated in Figure 5-7. In Figure 5-7, the comparison of the predicted transition boundary and the experimental results show that at lower air flow rates the transition from intermittent to bubbly flow takes place at lower water flow rates than predicted by the Barnea model. Nevertheless, similar to the modified Taitel and Dukler model, the Barnea model predicts the flow pattern of air-water two-phase flow with a high accuracy. 63


The third model, the modified Beggs and Brill model, turns out to be the third best model, with a prediction accuracy of less than 60 %. Using the Crittendon diameter concept or the Omurlu diameter concept, a maximum of 48 out of 84 data points can be estimated correctly, whereas the Equivalent diameter is able to make only 39 correct predictions (results see Appendix B). The flow pattern map generated by using the transition criterions suggested by Beggs and Brill and by applying the Crittendon diameter and the Omurlu diameter is shown in Figure 5-8. From Figure 5-8, it is evident that the transition from intermittent to bubbly flow takes place at higher superficial velocities of water than predicted by the Beggs and Brill model. As a result, a higher accuracy may be achieved by changing the transition boundaries to higher liquid rates. Nevertheless, the modified Beggs and Brill model, as used in this study, is questionable in its applicability to concentric annular cross-sections. Overall, the Taitel and Dukler model as well as the Barnea model can predict the flow pattern, in terms of intermittent and bubbly flow, with a high degree of accuracy, whereas the Beggs and Brill model fails to accurately predict the flow pattern. However, it has to be mentioned that, due to limitations set by the test rig, only data for intermittent and bubbly flow are collected. As a result, no statement can be made regarding stratified (smooth) flow pattern, wavy flow pattern and annular flow pattern. Therefore, to proof the applicability of the afore mentioned models in terms of all flow regimes, more experimental data is required.

64


Figure 5-6: Flow pattern map using modified transition criterions suggested by Taitel and Dukler

Figure 5-7: Flow pattern map using modified transition criterions suggested by Barnea and based on Equivalent diameter

65


Figure 5-8: Flow pattern map using modified transition criterions suggested by Beggs and Brill based on Omurlu diameter and Crittendon diameter

5.2.2 Liquid Holdup Estimation To estimate the liquid holdup of the air-water two-phase flow, the presented models in Chapter 4.2 are applied and evaluated. The experimental liquid holdup from the conductivity probes and the theoretical estimations are compared in Figure 5-9 to Figure 5-12. In Figure 5-9 the results of the homogeneous model are represented. According to Figure 5-9, the homogeneous model highly underestimates the liquid holdup for smaller values of the liquid holdup. However, when the liquid holdup exceeds a value of 80 % the homogeneous model is able to predict the liquid holdup with a high accuracy and a deviation of less than 20 %. Figure 5-10 illustrates the separated models of Zivi (1964), Chisholm (1973) and Lockhart and Martinelli (1949). The Zivi model tends to overestimate the flow pattern by more than ±20 % (even up to a value of liquid holdup of 80 %). The Chisholm model, in contrast, underestimates the liquid holdup. However, good results can be obtained for liquid holdup values of more than 60 % using the Chisholm model. The Lockhart and Martinelli concept seems to be the most accurate separated model, because the predicted liquid holdup lies within the error margin of ±20 %. Nevertheless, for the range from 30 to 50 % the Lockhart 66


and Martinelli model tends to slightly underestimate the liquid holdup and slightly overestimate the holdup by 50 to 80 %. The drift flux models of Steiner (2010), and Danielson and Fan (2009) are compared in Figure 5-11. Both models follow the same trend. For lower liquid holdup values, the two models highly underestimate the liquid holdup. At moderate and higher liquid holdup values the Steiner model and the Danielson and Fan model provide promising results with a deviation of less than ±20 %. The Beggs and Brill model based on the Crittendon diameter concept is shown in Figure 5-12. The Crittendon diameter gives the best results of all diameter concepts evaluated in this study. A similar trend as described for the drift flux models of Steiner, and of Danielson and Fan can be observed, whereby the accuracy of the Beggs and Brill model inclines to more deviation than the drift flux models.

Figure 5-9: Liquid holdup estimation using homogeneous model

67


Figure 5-10: Liquid holdup estimation using separated model of Zivi, Chisholm, and Lockhart and Martinelli

Figure 5-11: Liquid holdup estimation using drift flux model of Steiner, and Danielson and Fan

68


Figure 5-12: Liquid holdup estimation using Beggs and Brill model, based on Crittendon diameter

In general, all evaluated liquid holdup models show good accuracy for higher values of the liquid holdup, whereas for decreasing values the accuracy of the estimation declines. Taking all measured data into account, the separated model by Lockhart and Martinelli is the most accurate model in estimating the liquid holdup with a mean absolute error of 5.3 % followed by the drift flux model by Steiner (6.5 %) and the separated model by Chisholm (7.0 %). The results of the liquid holdup models are summarised in Table 5-1 outlining the mean average error and the maximum and minimum error. All the results regarding the various diameter concepts implemented in the Beggs and Brill model can be found in Appendix B.

69


Table 5-1: Comparison of the evaluated liquid holdup models Model

Mean Average Error (-)

Maximum Error (-)

Minimum Error (-)

Separated Lockhart and Martinelli Model

0.053

0.296

0.00071

Drift Flux Steiner Model

0.065

0.436

0.00007

Separated Chisholm Model

0.070

0.405

0.00043

Drift Flux Danielson and Fan Model

0.084

0.635

0.00057

Beggs and Brill

0.117

0.465

0.00773

Homogeneous Model

0.150

0.791

0.00004

Separated Zivi Model

0.193

0.634

0.00224

(Crittendon Diameter)

5.2.3 Frictional Pressure Drop Estimation In this section the experimental pressure drop of the 2’’ annular test rig is compared to five different models. These include the homogeneous model by McAdams (1942) and by Beattie and Whalley (1982), the separated model by Müller-Steinhagen and Heck (1986) and by Sun and Mishima (2009), and the Beggs and Brill model (Beggs and Brill, 1973). All of these models depend on the diameter of the conduit. Hence, the models are evaluated using the different diameter concepts introduced in Chapter 2.2.2. Here, it is only referred to the best performing diameter concept for each model. For all models, the Knudsen/Prengle/Rothfus diameter best fits the experimental data in terms of mean average error. The comparison between the pressure drop estimation models and the experimental pressure drop are illustrated in Figure 5-13 and Figure 5-14. Whereas in Figure 5-13 the pressure drop models based on the homogeneous flow approach are represented, Figure 5-14 shows the pressure drop calculated using the separated flow approach. In general, all models evaluated in this study show reasonable results in estimating the pressure drop. Most of the predicted pressure losses are within a deviation of ±20 %. For lower values of the pressure loss, the models tend to overestimate the pressure drop. At higher values, the models slightly underestimate the pressure loss. Comparing the 5 different models with each other, the correlations suggested by Beattie and Whalley with a mean average error of 8.0 % perform best, followed by Beggs 70


and Brill (11.9%) and Müller-Steinhagen and Heck (11.9 %). The results of the Knudsen/Prengle/Rothfus diameter in terms of mean average error, maximum and minimum error are outlined in Table 5-2. The results of all diameter concepts can be found in Appendix B. Table 5-2: Comparison of the evaluated pressure loss models based on the Knudsen/Prengle/Rothfus diameter concept Model


Maximum Error (-)

Minimum Error (-)

Beattie and Whalley

0.080

0.628

0.00041

Beggs and Brill

0.119

0.370

0.01315

Müller-Steinhagen and Heck

0.119

1.051

0.00277

McAdams

0.122

1.057

0.00472

Sun and Mishima

0.129

0.374

0.00449

Figure 5-13: Pressure drop estimations using the homogeneous flow approach based on the Knudsen/Prengle/Rothfus diameter concept

71


Figure 5-14: Pressure drop estimations using the separated flow approach based on the Knudsen/Prengle/Rothfus diameter concept

72

Conclusions

6 Conclusions In this study, experiments are carried out at the 2’’ test rig in the PSE Lab at Cranfield University in order to study air-water two-phase flow through annular cross-sections. Conductivity probes and differential pressure transducers are used to record the liquid holdup and the pressure drop of the two-phase mixture. In the theoretical part of this study, various models from the literature are adopted to the case of an annular cross-section using different diameter concepts. The evaluated models comprise flow pattern transition models, liquid holdup estimation models, and pressure drop determination models. Evaluating the experimental data and comparing them with the different models of this study, the following remarks can be concluded:  The flow patterns observed at the annular test section of the 2’’ test rig in the PSE Lab at Cranfield University are intermittent flow and bubbly flow.  The transition from bubbly flow to intermittent flow takes place at superficial velocities of water in between 2 and 2.5 m/s. With increasing air flow rates the transition line tends to move to higher water flow rates.  The liquid holdup decreases with increasing air flow rates and increases with a higher liquid flow rate.  The pressure drop depends mainly on the liquid phase of the airwater two-phase flow. With increasing air and water flow rates the pressure drop increases.  An accurate flow pattern prediction can be achieved using the modified transition models of Taitel and Dukler or of Barnea, whereas the transition model suggested by Beggs and Brill is not recommended for annular cross-sections. Applying the Barnea model, the Equivalent diameter yields the best results.

73

Conclusions  Regarding the liquid holdup estimation, the best outcomes tend to be achieved by the Lockhart and Martinelli model, closely followed by drift flux model by Steiner, the separated model by Chisholm and the drift flux model by Danielson and Fan. Applying the Beggs and Brill model, the Crittendon diameter yields the best results.  The pressure loss can be estimated best using the homogeneous approach by Beattie and Whalley, while all other models still yield reasonable results. For all models, the Knudsen/Prengle/Rothfus diameter concept performs best.

74

Recommendations for future research

7 Recommendations for future research Experiments are carried out in terms of air-water two-phase flow through a concentric annulus with an inner diameter of the outer tube of 51.5 mm and an outer diameter of the inner tube of 33.3 mm. First results agree with the expected results from modified theoretical models which were derived from pipe flow. However, further studies are needed to extend the data base of the measured data points and to proof the accuracy of the evaluated models. Additionally, new correlations might be examined in order to achieve a better performance when compared to experimental data. The recommendations for future academic studies comprise the following aspects:  The capacity of the water pump should be expanded in order to achieve lower and higher water flow rates. Additionally, improving the design of the test rig may allow using higher air flow rates without damaging the construction of the test rig.  The inner tube is observed to be moved by the two-phase flow at high air flow rates. A better design of the inner tube should prevent any movements and avoid possible influences on the measurements.  The flow patterns observed throughout the experiments include bubbly flow and intermittent flow. In order to proof the applicability of the evaluated models for the whole range of flow patterns data for annular, stratified and wavy flow is required.  At higher flow rates of water and air the use of a video camera is inappropriate in order to record the flow pattern. High speed cameras may solve this problem and accurately illustrate the observed flow pattern.  Only the case of a concentric annular cross-section is investigated in this study. The effect of eccentricity might lead to other conclusions than it is the case for a concentric annulus. Moreover, the influence varying the radius ratio of outer diameter of inner pipe to inner diameter of outer pipe is not examined in this study. 75

Recommendations for future research

 In this study, only water and air are used as the working fluids. So far, the effect of the viscosity of the liquid phase on flow pattern transition and the accuracy of the evaluated models in annular crosssection has not been examined.  In further experiments, the entrance length of fully developed twophase flow in annular geometries needs to be investigated.

76

References

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References Steiner, D. and Kind, M. (2010), “Flow Patterns in Evaporator Tubes”, in VDI e.V. (Ed.), VDI Heat Atlas, VDI-buch, Online-Ausg., Springer-Verlag Berlin Heidelberg, Berlin, Heidelberg, pp. 796–800. Sun, L. and Mishima, K. (2009), “Evaluation analysis of prediction methods for two-phase flow pressure drop in mini-channels”, International journal of multiphase flow, Vol. 35 No. 1, pp. 47–54. Sunthankar, A.A. (2000), “Study of the Flow of Aerated Drilling Fluids in Annulus under Ambient Temperature and Pressure Conditions”, Master Thesis, The University of Tulsa, Tulsa, 2000. Sunthankar, A.A., Kuru, E., Miska, S. and Kamp, A. (200 ), “New developments in aerated mud hydraulics for drilling in inclined wells”, SPE drilling & completion, Vol. 18 No. 02, pp. 152–158. Taitel, Y. and Dukler, A.E. (1976), “A model for predicting flow regime transitions in horizontal and near horizontal gas‐liquid flow”, AIChE Journal, Vol. 22 No. 1, pp. 47–55. Taitel, Y., Bornea, D. and Dukler, A.E. (1980), “Modelling flow pattern transitions for steady upward gas‐liquid flow in vertical tubes”, AIChE Journal, Vol. 26 No. 3, pp. 345–354. Taitel, Y., Lee, N. and Dukler, A.E. (1978), “Transient gas‐liquid flow in horizontal pipes: Modeling the flow pattern transitions”, AIChE Journal, Vol. 24 No. 5, pp. 920–934. Taylor, G.I. (1921), “Diffusion by continuous movements”, Proc. London Math. Soc. No. 2, pp. 196–211. Thome, J.R. (2004), “Engineering data book III”. Tiedt, W. (1966), “Berechnung des laminaren und turbulenten Reibungswiderstandes konzentrischer und exzentrischer Ringspalte”, CHEMIKER-ZEITUNG, Vol. 90 No. 23, pp. 813-&. Tosun, I. (1984), “Axial laminar flow in an eccentric annulus: an approximate solution”, AIChE Journal, Vol. 30 No. 5, pp. 877–878. VDI e.V. (Ed.) (2010), VDI Heat Atlas, VDI-Buch, Online-Ausg., Springer-Verlag Berlin Heidelberg, Berlin, Heidelberg. Wallis, G. (1969), One-dimensional two-phase flow, McGraw-Hill. Whalley, P.B. (1990), Boiling, condensation, and gas-liquid flow, Oxford science publications, Vol. 21, Clarendon Press; Oxford University Press, Oxford [Oxfordshire], New York. 89

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91

Appendix

Appendix A

Figure A-1: Proof of repeatability of measured liquid holdups

92

Appendix

Figure A-2: Proof of repeatability of measured pressure drop

Figure A-3: Intermittent flow at 0.4 l/s water flow rate and 0.6 kg/h air flow rate


93

Appendix



Figure A-7: Bubbly flow at 3.6 l/s water flow rate and 0.3 kg/h air flow rate


94

Appendix

Appendix B Table B-1: Comparison of the evaluated diameter concepts used in the Barnea model in order to predict the flow pattern Model

Barnea (Equivalent Diameter)

Barnea (Lamb/Wien Diameter)

Barnea (Slot Diameter)

Barnea (Ozbayoglu Diameter)

Barnea (Hydraulic Diameter)

Correctly Predicted (-)

Total number (-)

Percentage (-)

80

84

0.952

79

84

0.940

79

84

0.940

79

84

0.940

76

84

0.904

76

84

0.904

73

84

0.869

73

84

0.869

Barnea (Knudsen/Prengle/Rothfus Diameter)

Barnea (Crittendon Diameter)

Barnea (Omurlu Diameter)

95

Appendix

Table B-2: Comparison of the evaluated diameter concepts used in the Beggs and Brill model in order to predict the flow pattern Model

Beggs and Brill (Crittendon Diameter)

Beggs and Brill (Omurlu Diameter)

Beggs and Brill (Hydraulic Diameter)

Correctly Predicted (-)

Total number (-)

Percentage (-)

48

84

0.869

48

84

0.869

43

84

0.904

43

84

0.904

40

84

0.940

40

84

0.940

40

84

0.940

39

84

0.952

Beggs and Brill (Knudsen/Prengle/Rothfus Diameter)

Beggs and Brill (Lamb/Wien Diameter)

Beggs and Brill (Ozbayoglu Diameter)

Beggs and Brill (Slot Diameter)

Beggs and Brill (Equivalent Diameter)

Table B-3: Comparison of the evaluated diameter concepts used in the Beggs and Brill model in order to estimate the liquid holdup Model

Beggs and Brill (Crittendon Diameter)




Maximum Error (-)

Minimum Error (-)

0.117

0.465

0.00773

0.118

0.465

0.00896

0.135

0.472

0.00407

0.136

0.472

0.00338

0.141

0.474

0.00761

0.141

0.474

0.00774

0.141

0.474

0.00774

0.147

0.476

0.00039

Beggs and Brill (Knudsen/Prengle/Rothfus Diameter)





96

Appendix

Table B-4: Comparison of the evaluated diameter concepts used in the McAdams model in order to estimate the pressure drop Model

McAdmas (Hydraulic Diameter)

McAdmas (Lamb/Wien Diameter)


Maximum Error (-)

Minimum Error (-)

0.135

1.005

0.00291

0.265

1.586

0.03467

0.122

1.057

0.00472

0.632

0.771

0.25401

0.614

2.340

0.05011

0.269

1.593

0.03759

0.623

0.765

0.23481

0.269

1.593

0.03759

McAdmas (Knudsen/Prengle/Rothfus Diameter)

McAdmas (Crittendon Diameter)

McAdmas (Equivalent Diameter)

McAdmas (Slot Diameter)

McAdmas (Omurlu Diameter)

McAdmas (Ozbayoglu Diameter)

Table B-5: Comparison of the evaluated diameter concepts used in the Beattie and Whalley model in order to estimate the pressure drop Model

Beattie and Whalley (Hydraulic Diameter)

Beattie and Whalley (Lamb/Wien Diameter)


Maximum Error (-)

Minimum Error (-)

0.088

0.587

0.00019

0.271

1.037

0.05638

0.080

0.628

0.00041

0.630

0.766

0.39835

0.624

1.616

0.07794

0.274

1.042

0.05371

0.620

0.760

0.38315

0.274

1.042

0.05371

Beattie and Whalley (Knudsen/Prengle/Rothfus Diameter)

Beattie and Whalley (Crittendon Diameter)

Beattie and Whalley (Equivalent Diameter)

Beattie and Whalley (Slot Diameter)

Beattie and Whalley (Omurlu Diameter)

Beattie and Whalley (Ozbayoglu Diameter)

97

Appendix

Table B-6: Comparison of the evaluated diameter concepts used in the Sun and Mishima model in order to estimate the pressure drop Model

Sun and Mishima (Hydraulic Diameter)

Sun and Mishima (Lamb/Wien Diameter)


Maximum Error (-)

Minimum Error (-)

0.149

0.392

0.00033

0.141

0.352

0.00079

0.129

0.374

0.00449

0.696

0.817

0.61811

0.461

0.754

0.09700

0.143

0.356

0.00299

0.688

0.811

0.60813

0.143

0.356

0.00299

Sun and Mishima (Knudsen/Prengle/Rothfus Diameter)

Sun and Mishima (Crittendon Diameter)

Sun and Mishima (Equivalent Diameter)

Sun and Mishima (Slot Diameter)

Sun and Mishima (Omurlu Diameter)

Sun and Mishima (Ozbayoglu Diameter)

Table B-7: Comparison of the evaluated diameter concepts used in the MüllerSteinhagen and Heck model in order to estimate the pressure drop Model


Maximum Error (-)

Minimum Error (-)

0.135

1.002

0.01062

0.234

1.548

0.02787

(Knudsen/Prengle/Rothfus Diameter)

0.119

1.051

0.00277


0.634

0.773

0.21920

0.563

2.246

0.03780

0.237

1.555

0.02520

0.625

0.767

0.20003

0.237

1.555

0.02520

Müller-Steinhagen and Heck (Hydraulic Diameter)

Müller-Steinhagen and Heck (Lamb/Wien Diameter)



Müller-Steinhagen and Heck (Equivalent Diameter)

Müller-Steinhagen and Heck (Slot Diameter)

Müller-Steinhagen and Heck (Omurlu Diameter)

Müller-Steinhagen and Heck (Ozbayoglu Diameter)

98

Appendix

Table B-8: Comparison of the evaluated diameter concepts used in the Beggs and Brill model in order to estimate the pressure drop Model


Maximum Error (-)

Minimum Error (-)

0.138

0.386

0.00467

0.135

0.623

0.01054

(Knudsen/Prengle/Rothfus Diameter)

0.119

0.370

0.01316

Beggs and Brill

0.668

0.775

0.51036

0.437

1.075

0.03125

0.138

0.628

0.00780

0.659

0.769

0.49818

0.138

0.628

0.00780



Beggs and Brill






99

100

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Energy Systems And Thermal Processes Two Phase Flow (Air-Water) Characteristics In Annulus

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