Single and two-phase flows of shear-thinning fluids: analytical and
computational fluid dynamics modelling
by
Giovanni Elvio Langella
Thesis
Faculty of Energy Engineering University of Pisa
Energy Engineering MSc
Academic year 2013/2014
Supervisor: Prof. Ass. Chiara Galletti
Supervisor: PhD Aris Twerda
ii
Acknowledgements
The author wishes to acknowledge:
the fluid-dynamics department of TNO, and the supervisors PhD Aris Twerda and Mr Pejman Shoeibi Omrani for the opportunity to work with them;
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Abstract
Single and two-phase flows of shear-thinning fluids: analytical and
computational fluid dynamics modelling
The injection of small amount of high molecular weight long-chain polymers into turbulent flows could decrease the pressure drop in the pipe flow and consequently decrease the required pumping power. This work is aimed to the modeling of viscoelastic fluid, specifically polymer, in single phase and two-phase flows.
A literature review of drag-reduction by polymers in multiphase flow is firstly presented, then two different approaches are analyzed: one-dimensional analytical approach and CFD approach. Stratified gas-liquid flow and gas-liquid slug flow regimes are analyzed by means of analytical models developed by Taitel and Dukler (1976) and Taitel and
Barnea (1990), respectively. The models are adapted for polymer solutions in order to
estimate the pressure gradient and the holdup.
For single-phase flow, the modified k-ε model for power-law fluids developed by Ro and
Ryou (2012) was implemented in Fluent through a User Defined Function (UDF) and
validated for two polymer solutions Carboxy-Methyl Cellulose (CMC) and Xanthan Gum (XG) in a turbulent pipe flow. The applicability and the effect on the structure of the flow of the modified k-ε model were studied for stratified and slug flow using the ANSYS Fluent software package.
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Table of Contents
List of Figures ... vi
List of Tables ... viii
List of Symbols ... ix
1. INTRODUCTION ...12
1.1 Non-Newtonian fluids ...12
1.2 Theories of drag reduction ...16
1.3 Drag-reducing polymers in multiphase flow ...17
1.4 Outline of the thesis ...20
2. ANALYTICAL MODELLING ...21
2.1 Overview of the models ...21
2.1.1The Taitel and Dukler (1976) stratified-flow model ...21
2.1.2The Taitel and Barnea (1990) slug-flow model ...25
2.2 Implementation of the models ...29
2.2.1Validation for water and air ...29
2.2.2Application to drag-reducing fluids ...30
2.3 Results ...34
3. SINGLE-PHASE CFD MODELLING ...35
3.1 The modified k-ε model for power-law fluids ...35
3.2 Results in a turbulent pipe flow ...36
4. TWO-PHASE CFD MODELLING ...40
4.1 Literature Survey ...40
4.2 Methodology description ...40
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4.2.2 Boundary Conditions ...42
4.2.3 Numerical settings ...43
4.3 Results ...43
4.3.1Results for water and air ...43
4.3.2 Application of the modified k-ε model for power-law fluids ...48
5. SUMMARY AND CONLUSIONS ...50
6. REFERENCES ...53
7. APPENDICES ...57
7.1 Literature review of two-phase gas-liquid flow ...57
vi
List of Figures
Figure 1 Fluctuating velocity vector field in the cross section for pure water (left) and
1.24-ppm polymer solution (right). Regions with are enclosed with solid lines
(for ) and dashed lines (for . ... 17
Figure 2 Stratified flow: and are the gas and liquid perimeters, and the gas and liquid cross-sectional areas, is the perimeter of the interface. ... 22
Figure 3 Flow diagram of the analytical stratified model ... 24
Figure 4 Schematic description of a slug unit ... 25
Figure 5 Moving control volume being considered for the gas and liquid mass balances ... 26
Figure 6 Flow diagram of the slug-flow analytical model ... 28
Figure 7 Comparison of the pressure gradient (left) and of the holdup (right) between the model (line) and LedaFlow simulator data (dots). ... 29
Figure 8 Comparison of the pressure gradient (left) and of the holdup (right) between the model (line) and LedaFlow simulator data (dots). ... 29
Figure 9 Photos of a stratified flow with (left) and without polymers (right). ... 30
Figure 10 Effect of decreasing on the pressure gradient (left) and on the holdup (right) : (continuous line), 0.3 (dashed line). ... 30
Figure 11 Effect of decreasing on the pressure gradient (left) and on the holdup (right): (continuous line), 0.5 (dashed line). ... 31
Figure 12 Effect of a combined decrease of and on the pressure gradient (left) and on the holdup (right): , (continuous line), 0.5 , 0.3 (dashed line). ... 31
Figure 13 Measured effect of polymers on the interfacial friction factor (left) and on the liquid wall stress (right). ... 32
Figure 14 Measured slug frequency with (full dots) and without polymers ... 32
Figure 15 Comparison with experimental data for the pressure gradient (left) and for the holdup (right) at VSG=10 m/s. Model (line), data from Soleimani et. al (dots). ... 34
Figure 16 Single-phase turbulent pipe flow: non-dimensional velocity profile ... 38
Figure 17 Single-phase turbulent pipe flow: velocity profile ... 38
Figure 18 Plots of turbulent kinetic energy (left) and dissipation rate (right) with the model for power-law fluids ... 39
Figure 19 Plots of turbulent viscosity (left) and turbulence intensity (right) with the model for power-law fluids ... 39
Figure 20 Comparison between CFD and experimental results of the velocity (left ) and turbulent kinetic energy (right) ... 44
Figure 21 Detail of the velocity and turbulent kinetic profile of the liquid phase. ... 44
Figure 22 Contour of the water volume fraction after 38 seconds... 45
Figure 23 Monitor of the holdup at the test section ... 46
Figure 24 Three bubbles at different axial location: from the top to the bottom, from the inlet to the outlet ... 47
Figure 25 Monitor of the holdup at the test section: comparison between water and XG 0.2% solution ... 48
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Figure 26 Taylor bubble close to the outlet of the pipe. Top: air-water. Bottom:
Air-XG0.2% solution... 49
Figure 27 Comparison with experimental data for the pressure gradient (left) and for the
holdup (right) at VSG=10 m/s. Model (line), data from Soleimani et. al (dots). ... 50
Figure 28 Monitor of the holdup at the test section: comparison between water and XG
0.2% solution ... 51
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List of Tables
Table 1 Flow conditions for the validation of the model ... 37
Table 2 Comparison between analytical and CFD pressure drop ... 37
Table 3 Input parameters for the simulations of stratified flow ... 43
Table 4 Parameters for the slug-flow simulation ... 46
Table 5 Comparison between analytical and CFD pressure drop ... 51
Table 6 Summary of the studies on polymer effect in multiphase flow (Part1). ... 57
Table 7 Summary of the studies on polymer effect in multiphase flow (Part2). ... 58
ix
List of Symbols
Acronyms
CFD Computational Fluid Dynamics
CMC CarboxyMethyl Cellulose
DNS Direct Numerical Simulation
DR Drag Reduction
DRA Drag Reducing Agent
DRP Drag Reducing Polymer
FENE Finitely Extensible Non-linear Elastic
GNF Generalized Newtonian Fluid
GRN Generalized Reynolds Number
ID Internal Diameter
LDV Laser Doppler Velocimetry
PDE Partial Differential Equation
PIV Particle Image Velocimetry
RANS Reynolds Averaged Navier-Stokes
VOF Volume Of Fluid
XG Xanthan Gum
Subscripts
Liquid phase in the film region of the slug
Gas phase Interface Liquid phase Mixture Newtonian fluid Non-Newtonian Polymer
Based on the superficial gas velocity
Based on the superficial liquid velocity
At the wall
Symbols
Cross sectional area of the pipe
Parameters
Polymer conformation tensor
Pipe diameter
, Deformation or strain rate tensor
Degree of drag reduction
x
Maximum drag reduction percentage
Fanning friction
Fanning friction factor at maximum drag reduction
Slug frequency
Froude number
Gravity acceleration
Height of the liquid layer
Holdup in the slug cylinder
Holdup in the film region
Total holdup in the slug unit
Flow consistency index
Turbulent kinetic energy
L Maximum extension of the polymer chain
Pipe length
Slug cylinder length
Film region length
Slug unit length
Flow index behavior
Pressure drop across the pipe
Volumetric flow rate
Pipe radius
Reynolds Number based on bulk velocity
Generalized Reynolds Number
Reynolds Number based on wall viscosity
Perimeter of the cross section of the pipe
Bulk velocity
Shear velocity
Velocity vector component
Non-dimensional velocity
Slug translational velocity
Actual liquid velocity in the film region of a slug Actual gas velocity in the film region of a slug
Actual liquid velocity in the slug cylinder
Actual gas velocity in the slug cylinder
Position vector component
Non-dimensional wall distance
Greek Symbols
Volume fraction
Volume fraction of the i-th phase
̇ Shear rate
xi
Turbulent dissipation rate
Kronecker delta
Pipe inclination angle
No-slip holdup
Polymer relaxation time
Surface tension
Density
Viscosity
Turbulent viscosity
Viscosity at zero shear rate
Viscosity at infinite shear rate
Kinematic viscosity
Apparent or effective viscosity
, Stress tensor Yield stress Definitions
1. INTRODUCTION
The addition of high molecular weight long-chain polymers into turbulent flows can decrease the pressure drop across a pipe. The phenomenon was first observed by Toms in the 40s. The finding has received much interest in the oil and gas industry because of the possibility to reduce pumping costs and increase pipelines capability, with resulting economic benefits.
Two-phase gas-liquid flow is frequently encountered in many industrial units such as distillation columns, pipelines, boiler tubes, condensers, evaporators, and chemical reactors. Two-phase flow is common in offshore production as it necessitates transportation of both gas and liquid phases over long distances before separation. This type of flow has many unique characteristics, which must be evaluated in each situation. However, one common and undesirable feature is the high axial
pressure gradient, resulting in substantial energy consumption per unit volume of liquid throughput. Many experimental studies have revealed that the injection of polymers in multiphase flow can decrease the pressure gradient along the pipe achieving a drag reduction as high as 60%, even higher than that achieved in single-phase flow. Aspect even more interesting is that the
configuration of the phases can be changed. For instance a slug flow can be turned into a stratified smooth pattern by adding a small amount of polymer into the liquid. This phenomenon could be used in many applications in order to suppress the formation of slugs which are undesired as they cause vibration and stress on the structures due to their intermittent nature.
As in many other fields of engineering, simple analytical models provide a tool which can be quickly employed to get a first insight into a problem or to get the order of magnitude of a phenomenon. Also, they are able to show the qualitative effect which the variables exert on each other. On the contrary, a computational fluid dynamics approach could provide deeper
understanding of small details at the cost of employing many hours of calculation.
Aqueous solutions of polymers are essentially non-Newtonian in the turbulent flow and Newtonian in many laminar flows. In the contest of non-Newtonian fluids, e.g. polymer solutions, in single phase and multiphase flow both the analytical and CFD approaches are currently being studied by oil and gas companies, research centers and other industries. The studying of analytical and numerical modelling of drag-reducing polymer solution in single and two-phase flow, is also the aim of the present work.
1.1 Non-Newtonian fluids
Given a velocity field, the stain rate or rate of deformation is defined as:
In a non-Newtonian fluid the relationship between the shear stress and the strain rate is not linear, and generally is time dependent. It is sometimes useful to introduce the apparent or effective viscosity, defined as:
For a non-Newtonian fluid this quantity can be therefore either a function of the shear rate ̇ or of its history. The shear rate is related to the second invariant of which is defined as:
̇ √
is the double-dot product, defined as: ∑ ∑ .
A first classification splits fluids into two categories: viscoelastic fluids which exhibit partial elastic recovery, and time-independent viscosity fluids. We can therefore distinguish between:
Thixotropic Fluids: apparent viscosity decreases with stress over time (e.g. honey, yogurt, paints)
Rheopectic Fluids: apparent viscosity increases with stress over time (e.g. some lubricants, whipped cream)
Shear Thinning or Pseudoplasic Fluids: apparent viscosity decreases with increased stress (e.g. blood, whipped cream)
Shear thickening or Dilatant Fluids: apparent viscosity increases with increased stress (e.g. Oobleck)
Fluids for which the shear rate at any point is determined only by the value of the shear stress at that point and at that instant are also known as Generalized Newtonian Fluids (GNF), and they can represent either the shear thinning or the shear thickening fluids. For these fluids, the shear stress can be expressed in the same way as for Newtonian fluids, by considering the apparent viscosity ̇ in place of the usual viscosity:
̇
Amongst the GNF models, the most common constitutive equations are: Power Law Fluids:
( ) or equivalently:
̇ ( ̇ )
where K is the flow consistency index and n the flow behavior index. For n>1 the fluid is dilatant, for n<1 it is pseudoplastic.
Cross Fluids:
( ̇ )
̇
At low shear rate, ̇ the fluid behaves as a Newtonian fluid while at high shear rate as a power-law fluid. is the natural time, and n the flow index behavior.
Carreau Fluids:
where is the viscosity at zero shear rate, the viscosity at infinite shear rate, a relaxation time and n a power index. At low shear rate, ̇ 1/ the fluid behaves as a Newtonian fluid, at high shear rates as a power-law fluid.
Herschel-Bulkley fluids:
̇ { ̇ ̇ ̇ ̇ ̇ ̇
where K and n are rheological parameters. If the stress is larger than the yield stress the fluid does not deform, otherwise it does. This captures the Bingham behavior: the material behaves as a rigid body at low stresses and as a viscous fluid at high stresses.
A simpler formulation is:
{
̇
Viscosity of polymers most often exhibits a shear-thinning behavior: as the shear rate increases, the apparent viscosity decreases, resulting in a reduced stress at the wall and hence a lower drag. A GNF model is adequate to predict the properties of the solution accurately: Escudier et al. (1998) used a Cross model to represent aqueous solutions of 0.09%, 0.25%, 0.3%, 0.4%
carboxymethylcellulose (CMC), 0.09%, 0.2% xanthan gum (XG), 0.125%, 0.2% polyacrylamide (PAA); Ptasinski et al. (2001) used a Carreau model for a partially-hydrolyzed polyacrylamide solution at different concentration. However the power-law model is usually preferred since makes easier to derive a turbulence model. A low-Reynolds number turbulence model for power-law and Herschel-Bulkley fluids was developed by Malin (1997) and Malin (1998), but it has weakness of grid generation at near wall region. A more conventional turbulence model for non-Newtonian fluids, which uses standard wall function, is described in 3. Another approach of modeling with drag-reducing polymers is the split-stress tensor approach:
The stress is split into two components: is relative to the solvent (usually Newtonian) and is relative to the polymer additive, which can be modeled as a viscoelastic fluid. Between the available choices:
Finitely Extensible Nonlinear Elastic (FENE-P) Model
It derives from the kinetic theory of dilute polymer solutions, and treats the polymer as two spherical beads connected by a massless spring. Hydrodynamic drag is taken to be isotropic. The dumbbell is specified by a connector vector Q, which defines a dimensionless
conformation tensor in combination with a scale length √ with k the Boltzmann constant, T the absolute temperature, and H the spring constant (the reader is referred to
Tesauro 2007).
( )
with , polymer intrinsic viscosity, relaxation time, L maximum extension of the chain. The three parameters characterize the molecular behavior of the polymer.
A transport equation for the conformation tensor will be necessary.
The model has been used for instance by in the DNS work of Ptasinski et al. (2007).
Giesekus Model
Intermolecular forces are described by a Hookean spring.
On one hand this model is convenient because despite the fact it needs only two material parameters to describe the polymer (i.e. the viscosity and the relaxation time ), it catches the realistic behavior for shear flows, which is anisotropic. On the other hand the Giesekus constitutive law is strongly nonlinear since it involves a quadratic term in the stress tensor. The constitutive law is as follows:
̌
where G is the elastic modulus given by and ̌ is the upper convective derivative, defined as
̌
Oldroyd-B Model
It is a linear Hookean dumbbells model, whose constitutive equation is:
̌
This model is a limiting case of the FENE-P model, for . The two models give therefore close results, for high values of the dumbbell extensibility.
An application of the model in the DNS framework was done in Min et al. (2003).
The reader is referred to Favero et al.(2010) for a deeper description of viscoelastic fluids rheology. An interesting application of the split-tensor approach in the framework of the RANS equations was done by Iaccarino et al. (2010). Polymer solutions are represented using the FENE-P dumbbell model. Only one transport equation for the elongation of the polymer chain is considered. The stress anisotropy due to the storing of energy in the stretched chain and its release in the streamwise direction, is taken into account by solving two additional transport equations according to the v2-f model. The final model requires the solution of five PDEs instead of two as in the standard k-ε model, but is claimed to be able to reproduce fairly the level of drag reduction, the mean velocity
and the turbulent fluctuations. The approach is therefore attractive because of its limited increase in computational cost and good accuracy.
1.2 Theories of drag reduction
The qualitative characteristics of turbulent polymer flows in channel and pipes have been extensively reported in the literature. During the past three decades, numerous experimental and numerical studies have been performed on polymeric drag reduction in turbulent pipe flow. Comprehensive reviews of the subject were provided in Hoyt (1972), Lumley (1973) and Virk
(1971). It has been known since the late 1940s that the addition of small concentration of high
molecular weight polymer to water or other solvent can produce large reduction in frictional pressure drop for turbulent flows, phenomenon first observed by Toms (1948). One of the most important findings by Virk (1971) is the existence of a maximum drag reduction, which suggests that drag reduction is not caused purely by viscous effect of the dilute polymer solution (if the viscosity were a dominant parameter for drag reduction, the drag would decrease regardless of the amount of polymer concentration). Drag reduction appears bounded between two universal
asymptotes, namely, the Prandtl-Karman law and Virk’s asymptote. Virk (1979) also proposed that the mechanism of drag reduction must take place somewhere between the viscous sublayer and the logarithmic zone, and he pointed out that the width of this layer is an increasing function of the degree of drag reduction.
Two main theories have been developed. In the first one, proposed by Lumley (1969), the
stretching of the polymer molecules by the flow increases the effective extensional viscosity outside the viscous sublayer, leading to a thickening of the buffer layer and, as a result to drag reduction. Lumley also mentions that the influence of the polymers on the turbulence only becomes important when the time scale of the polymers (e.g. the relaxation time) becomes larger than the time scale of the flow (onset of drag reduction). The second theory was proposed by de Gennes (1990), who argued that drag reduction is caused by the elasticity of polymer molecules rather than by viscous properties of polymers. They modify the Kolmogorov scale: the turbulent kinetic energy is
absorbed at small scales and radiated away in form of shear waves energy. Due to the viscoelastic nature of polymer solutions, energy is stored at a characteristic frequency providing a natural cut-off for velocities which fluctuates at high frequency. This cut-cut-off would be responsible for the suppression of the small eddies and presumably for the drag reduction.
Recent experimental studies, Pinho & Whitelaw (1990), Harder & Tiederman (1991), Wei &
Willmarth (1992) have applied laser Doppler velocimetry (LDV) to measure the turbulence
statistics. One of the results is that turbulence is not simply suppressed by polymer addition. Indeed, turbulence structure is changed, rather than attenuated: the streamwise turbulence intensity is
increased, while the normal turbulence intensity is decreased. Wei & Willmarth (1992) found that the turbulent energy in the normal velocity component is suppressed over all frequencies, while there is a redistribution of turbulent energy from high frequencies to low frequencies for the streamwise component. This means that polymer addition in turbulent boundary layer leads to a suppression of small coherent structures, as clearly shown in the Particle-Image Velocimetry (PIV) study by Warholic et al. (2001), shown in Figure 1.
Gyr and Tsinober (1997) concluded that drag reducing fluids are essentially non-Newtonian in
the turbulent flow state, while Berge and Solvik (1996) reported that, in general, a higher degree of fluid turbulence results in a higher drag reduction.
Warholic et al. (1999) believed that the principal effect of the polymer is to reduce Reynolds
shear stresses and velocity fluctuations in a direction normal to the wall. At flows close to maximum drag reduction the Reynolds stresses were approximately zero over the whole cross section of the channel.
In the last decade, due to advances in numerical simulation techniques, an extensive data set has been collected through DNS. Ptasinski et al. (2003) modeled polymers by means of a realistic constitutive equation, the FENE-P model. They showed that the ability of polymers to stretch is an essential ingredient for high polymer drag reduction (in agreement with Lumley), and contrary to observations of Warholic, they reported that Reynolds stresses do not vanish at maximum drag reduction. They also indicated that polymer stretching effectively separate the (thickened) viscous sublayer from the outer region, resulting in a decoupling of the structures above and below this layer.
The FENE-P formulation represents the behavior of dilute polymer solutions realistically, and is currently the basis for most of the work in the characterization of turbulent polymer solution flows.
1.3 Drag-reducing polymers in multiphase flow
One of the earliest experiments on drag reduction in gas–liquid flows were reported by Scott
and Rhodes (1972) investigated Polyhall295, a polyacrylamide polymer drag reducer in co-current,
two-phase, gas-liquid slug flow. Water and air flowing in 2.5 cm ID pipelines was considered. The liquid Reynolds number was held constant at 13000 and the gas Reynolds number was varied from 1500 to 6100. It was concluded that two phase drag reduction exceeded that of single phase flows for the same superficial liquid velocities.
Al-Sarkhi and Hanratty (2001) studied the effect of drag reducing polymers on annular
air-water flow in a horizontal pipe with a diameter of 0.0953 m and 23 m of length. Their polymer solution was a co-polymer of polyacrylamide and sodium-acrylate (formally sold under the trade named Percol 727 but now called Magnafloc 101l) in water. The injection of polymer solution (without using a pump) produced drag reduction of 48% with concentrations of only 10–15 ppm in water. Also, they found that annular flow regime was changed to a stratified pattern at large drag reductions. In addition, they reported that the injection of the DRP to the liquid in the pipe should not involve the use of high shear pump. The effectiveness of a drag-reducing polymer is sensitive to the method used to introduce it into the flow and to the concentration of the injected master polymer solution. Degradation occurs by the destruction of aggregates of polymers, which have a more important effect on the turbulence than individual molecules. Other details about the experimental approach are outlined by Al-Sarkhi and Soleimani (2004).
Al-Sarkhi and Hanratty (2002) studied the effect of pipe diameter on the performance of
drag-reducing polymers in annular air-water flows by varying the diameter of the pipe from 0.0953 m to
Figure 1 Fluctuating velocity vector field in the cross section for pure water (left) and 1.24-ppm polymer solution (right). Regions with 𝑢𝑣 𝑢𝑣 are enclosed with solid lines (for 𝑢 ) and dashed lines (for 𝑢 .
0.0254 m. Drag reductions up to 63% were observed in the 0.0254 m pipe compared with 48 % previously achieved in the 0.0953 m pipe.
Soleimani et. Al (2002) investigated an air-water flow in a stratified configuration in a
horizontal 2.54 cm pipe. The additive was Percol727. Mixed mean concentrations of the polymer solution in the pipe of 10, 50 and 100 ppm were studied. They measured holdup, pressure gradient and they looked at the influence of polymer on the stress at the interface and on the onset of the critical condition for the initiation of the roll waves and slug. At a fixed gas velocity, roll waves appear on the interface of an air-water flow at sufficiently large liquid flow. The addition of polymers causes the frequency of roll-waves formation to decrease and the length of the waves to increase. The onset of the roll-waves regime was found to be delayed because of the damping of the small-wavelength waves, which also results in an interfacial stress decreased of a factor 1/2 - 1/4. As the liquid flow increases, the waves eventually touch the top of the pipe establishing a pattern defined as pseudo-slugs, which is characterized by high interfacial stresses. The addition of
polymers causes the onset of the pseudo-slug regime to be delayed: when DRP are added a regime identified as incipient-slugging is established over a range of USL. Increasing of USL in this region leads to higher velocity of the roll waves with the liquid height constant at its critical value. The addition of polymers does not diminish the importance of the roll waves, which actually show a larger holdup because of their increased length, but they damp small-wavelength waves, which have a dominant influence on the interfacial stress. While the reduced interfacial stress leads to a
reduction in the pressure gradient, the increase of the holdup has a counterbalancing effect as it causes the gas to flow faster.
Al-Sarkhi and Soleimani (2004) studied the effect of the addition of drag reducing polymers on
air–water flow patterns in a horizontal pipe of 0.0254 m diameter and 17 m long. The additive was Percol727. They reported that the addition of drag-reducing polymers is accompanied by changes in the flow pattern map and pressure drop reduction occurs in almost all flow pattern configurations. Their study indicated that maximum drag reduction usually occurs when a slug, pseudo-slug or annular flow changes to stratified flow.
Mowla and Naderi (2005) studied the effect of drag-reducing agents on the pressure drop in air
and crude oil slug flow in a horizontal pipe, for different wall roughness. The set-up consisted of: a smooth pipe of polycarbonate with 2.54 cm ID, a rough pipe of galvanized iron with 2.54 cm ID and a rough pipe of galvanized iron with 1.27 cm ID. They used Polyisobutylene as additive. A drag reduction of 40% was obtained for some conditions. They observed a more important drag reduction in rough pipe than in smooth pipe, and in 1.27 cm ID pipe than in the 2.54 cm ID pipe. They explained the phenomenon in terms of the relative roughness ε/D: increasing values of this parameter result in higher degree of turbulence, which shows better the effect of DRA.
Fernandes et al. (2009) investigated the drag reduction in vertical two-phase annular flow. The
experiments were conducted using air and water in a 22m high, 0.0254m diameter pipe at an
average absolute pressure of approximately 552 kPa. Superficial gas velocities between 8.6 m/s and 20.1 m/s, and superficial liquid velocities between 0.29 m/s and 0.55 m/s were used. The drag reducers had two opposite effects on the pressure gradient: on one hand they reduced the frictional component by up to 74%, on the other hand they resulted in an increasing of the liquid holdup by up to 27% (“DRA-induced flooding”) which in turn increased the hydrostatic component of the pressure gradient, since the flow was vertical. The net result in the experiments was however an overall reduction in the pressure gradient.
Al-Sarkhi (2010) worked on a literature survey on drag reduction in two-phase gas-liquid and
liquid-liquid flow. The experimental approach is described, as well as theoretical-analytic-mechanistic approaches. A suggested mechanism of drag reduction is described. The following interpretation for the effect of DRP in multiphase flow is reported:
Annular flow: the wave damping causes the discontinuance of atomization and makes it difficult for the liquid layer to climb up the walls of a horizontal pipe. A change to stratified flow occurs.
Stratified flow: polymers damp waves. This can have two effects, a decrease of interfacial drag and an increase of gas velocity because of the increase of the height of the stratified liquid layer. These have opposite influences on the pressure gradient.
Slug flow: a damping of turbulence in the liquid decreases the wall drag and changes the behavior of the gas bubble just behind the back of the slug. DRA also reduces the slug frequency.
Bubbly flow: polymers have the possibility of affecting turbulence and bubble size.
Al-Sarkhi (2011) investigated air-water flow in horizontal pipes of 25.4 and 95.3 cm ID. The
various flow pattern configurations were observed at different gas and liquid velocities, with and without DRP. Results are given for a concentration in the liquid inside the pipe of 50 ppm. The additive was a high molecular weight anionic polyacrylamide flocculant. The maximum drag reduction was noticed when the annular flow pattern changes to stratified, or slug to pseudo-slug flow pattern. The results revealed that the addition of polymers cause the transition from annular to stratified flow with a relatively smooth surface and a negligible number of entrained drops in the gas phase. They explained the fact in terms of damping of the disturbance waves, which in turn reduces the rate of atomization and the ability of liquid to spread along the wall, and it also damps the waves on the resulting stratified flow.
Al-Sarkhi (2013) studied a stratified and slug air-water flow in a horizontal 2.54 cm ID pipe.
Percol727 was used as additive. Experiments cover stratified flow, roll-wave (pseudo-slug), and slug flow in the range USL=0.2-0.8 m/s and USG=1-8 m/s. The results show the slug frequency, the frictional pressure drop and the liquid holdup for each condition, with and without DRP. The slug frequency in general increases with superficial liquid velocity, with and without polymers. The effect of the DRP is to decrease the slug frequency, especially with increasing USL and at high USG. There is an optimum range where DRP can reduce the frequency of the slugging effectively. The major influence of USL was explained as DRP affect the turbulence in the liquid, indeed the stability of the slug is regulated by the turbulence in the liquid. A larger reduction of the pressure drop was noticed for those cases with a larger ReSL, fact which can be explained as at low Reynolds number the disturbance waves and the turbulence in the liquid are small and then the effect of the DRP is less noticeable. The pressure drop is not always decreased by DRP because of the increase in the gas velocity. Such an effect is more important at higher USG. When the transition from slug to stratified flow occurs, the pressure drop results highly decreased.
In 7.1 a summary of the studies and of the findings of the effects of polymer in multiphase flow is provided in Table 6 and Table 7.
1.4 Outline of the thesis
The present work is aimed to provide a wide picture of the problem of modelling viscoelastic flow. In the previous sections a review of the main theories of drag reduction and of the most relevant findings were presented. Also, the problem of rheology and mathematical modelling of non-Newtonian fluids in turbulent flow was introduced.
A double approach to the prediction of viscoelastic flows is then developed: analytical in 2 and CFD in 3 and 4. The analytical approach is based on two existing mechanistic models for the prediction of pressure gradient and holdup in stratified and slug flow. The models were validated for air-water flow and then they were modified for polymer solutions.
The problem was study numerically in the framework of RANS equations, by using Fluent. In 3 a modified k-ε model for power-law fluids was chosen as case study and implemented for Fluent through User Defined Functions (UDFs). The turbulence model was validated for two polymer solutions in a turbulent pipe flow at three different Reynolds.
In 4 two-phase flows were considered. The Volume of Fluid method (VOF) was applied for stratified and slug flow, and the effect of the novel turbulence model on the structure of the flow was investigated.
2. ANALYTICAL MODELLING
In this chapter two analytical models for the prediction of the holdup and pressure gradient for stratified and slug flow are considered. The models were developed by Taitel and Dukler (1976) and Taitel and Barnea (1990), respectively. Scope of this section is to study the applicability of these models, originally developed for Newtonian fluids, for the case of polymer solutions. The stratified flow model, on which is based part of the slug flow model, is described first because of its simplicity due to only one closure relationship needed for the interfacial friction factor. The slug flow model requires three additional closure relationships for the holdup in the slug cylinder, for the slug frequency and for the bubble translational velocity.
The models were implemented in Matlab, validated against data from LedaFlow simulator for the case of water and air, and finally tested for drag-reducing fluids: the interfacial and the liquid friction factor were reduced according to the values reported in Soleimani et al. (2002) and the effect of their change on the pressure gradient and on the holdup was investigated and compared with the one observed during experiments.
In order to avoid unrealistic results for the case of slug flow, the slug frequency was reduced according to the value reported in the literature Al-Sarkhi (2013), and the correlation for the bubble translational velocity which is based on physical considerations was modified for drag-reducing fluids.
The slug-flow modified for drag-reducing fluids showed a good qualitative response when compared with measured data.
2.1 Overview of the models
The models here described are one-dimensional two-fluid models for the simple case of steady, isothermal flow without mass transfer between the two phases. The two momentum equations will then lead to two equations for the two unknowns: the phase holdup and the two-phase pressure gradient.
2.1.1 THE TAITEL AND DUKLER (1976) STRATIFIED-FLOW MODEL
Consider a stratified flow. The momentum equations for the two fluids, under the assumptions of no acceleration, no axial variation of the holdup (equilibrium holdup) and same pressure gradient for the two phases are:
Eliminating the pressure gradient by combining the two equations, one gets the following equation:
( )
From now on, the equation will be simply referred to , with:
The equation is an implicit function of the liquid height, which can therefore be determined iteratively. For the calculation of the liquid and gas friction factors, we use the liquid and gas Reynolds numbers:
The calculation of the hydraulic diameters, perimeters and holdup as a function of the liquid height is described further. For a rough interface with small waves, the suggested correlation for the interfacial friction factor is (Oliemans 1998).
Once the equation is solved iteratively and the height of the liquid layer is obtained, the pressure gradient can be calculated as:
The reader is suggested to refer to the calculation procedure, drawn in Figure 3.
Calculation of hydraulic diameters, perimeters and holdup as a function of the liquid height
The analytical models for stratified and slug flow described here require an iterative resolution of the implicit equation for the unknown liquid height .
( )
All these quantities are functions of the liquid height, and they can be derived analytically with reference to Figure 2.
Figure 2 Stratified flow: 𝑆𝐺 and 𝑆𝐿 are the gas and liquid perimeters, 𝐴𝐺 and 𝐴𝐿 the gas and liquid cross-sectional areas, 𝑆𝐼 is the perimeter of the interface.
At each iteration the equation is calculated according to the following set of equations: [ √ ] √
With and actual velocities in the stratified flow, or in the film region for the case of slug flow, (named and ). and are the Fanning friction factors, which can be calculated for instance through the Colebrook equation1:
, for laminar flow
√ ( √ ) , for turbulent flow
1 The fanning friction factor should not be confused for the Darcy friction factor, defined by:
which is , for example:
, for laminar flow √ (
Calculation procedure for stratified flow
OUTPUT PARAMETERS:
Holdup, Velocities, Pressure Gradient
GUESS A VALUE FOR THE LIQUID HEIGHT IN THE FILM REGION 𝐿 𝐶𝑎𝑙𝑐𝑢𝑙𝑎𝑡𝑖𝑜𝑛 𝑜𝑓 𝐻𝐿 𝑆𝐹 𝑆𝐺 𝑆𝐼 𝐴𝐹 𝐴𝐺 𝐷𝐹 𝐷𝐺 𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛 𝐿 𝑉𝐿 𝑉𝑆𝐿 𝐻𝐿 𝑉𝐺 𝑉𝑆𝐺 𝐻𝐿 𝐶𝑎𝑙𝑐𝑢𝑙𝑎𝑡𝑖𝑜𝑛 𝑜𝑓 𝜏𝐹 𝜏𝐺 𝜏𝐼 𝐹 𝜏𝐹𝑆𝐹 𝐴𝐹 𝜏𝐺𝑆𝐺 𝐴𝐺 𝜏𝐼𝑆𝐼(𝐴𝐹 𝐴𝐺) 𝜌𝐿 𝜌𝐿 𝑔 𝜃 CALCULATION OF 𝑭 𝒉𝑳 𝟎 𝑭 𝒉𝑳 𝟎? N O YES Update 𝐿 according to regula-falsi method INPUT PARAMETERS: Flow Parameters: 𝑉𝑆𝐿, 𝑉𝑆𝐺, D Fluid Properties: 𝜌𝐿 𝜌𝐺 𝜇𝐿 𝜇𝐺, σ Closure relationship for: 𝑓𝐼
2.1.2 THE TAITEL AND BARNEA (1990) SLUG-FLOW MODEL
The model being considered is the one presented by Taitel and Barnea (1990). Refer to Figure
4 for the schematic description of the model.
A simplification of the calculation procedure can be introduced by assuming a constant film
thickness along the entire film zone. In such a situation the gas velocity and the holdup in the film region are constant.
The model requires as input parameters the flow properties , , D, the fluid properties , σ, as well as the translational velocity of the bubble , the slug frequency , the liquid holdup and the gas velocity in the slug cylinder, for which four closure
relationships are required.
The total volumetric flow rate ̇ is a constant through any cross section of the pipe, provided that the flow is incompressible.
Consider a cross section at a point along the slug cylinder, the conservation of the volumetric flow rate states:
The only unknown in the former equation is the liquid velocity in the slug cylinder , which can be therefore determined.
The total mass flow rate is not constant at any cross section because of the intermittent nature of the flow. However, the gas and liquid mass flow rates considered separately are constant at any cross section of the pipe if one considers a coordinate system moving at the translational velocity . It is convenient to consider such a coordinate system and a control volume bounded by the slug front and a plane in the slug body, as in Figure 5.
A mass balance of the gas phase applied on such a control volume provides:
The same procedure applied to the liquid phase results in:
The two equations can be solved for the (actual) liquid and gas velocity in the film region, once a value for the liquid holdup in the film region is provided.
The liquid mass balance over the whole slug unit provides another equation through which the film region length can be determined:
The combination of the momentum equations for the gas and liquid phase which hold in the film region, leads to:
( )
All the parameters in the former equation can be determined once a value for , or equivalently a value of the height of the liquid layer in the film region is given. It therefore provides an implicit relationship of the form , which can be solved iteratively, as for the stratified flow.
Once the solution is converged, the film length , the holdup , and the actual liquid and gas velocity in the film region , can be obtained.
Finally, the pressure drop across the slug unit can be estimated as the sum of the pressure drop in the slug cylinder and in the film region.
The reader is suggested to refer to the calculation procedure, drawn in Figure 6.
Figure 5 Moving control volume being considered for the gas and liquid mass balances
Closure Relationships
Bendiksen (1984) translational velocity correlation
where
√ is the Froude number based on the mixture velocity . For √ For √
Gregory et al. (1978) liquid holdup in the slug cylinder correlation
( )
Gregory and Scott (1969) slug frequency correlation (
)
( )
Gas-bubble velocity in the slug
[
]
Calculation procedure for slug flow
𝑓𝑖
INPUT PARAMETERS:
Flow Parameters: 𝑉𝑆𝐿, 𝑉𝑆𝐺, D Fluid Properties: 𝜌𝐿 𝜌𝐺 𝜇𝐿 𝜇𝐺, σ
Closure relationships for: 𝑉𝑇𝐵 , 𝑓𝑠 , 𝐻𝐿𝐿𝑆 , 𝑉𝐺𝐿𝑆
GUESS A VALUE FOR THE LIQUID HEIGHT IN THE FILM REGION 𝐿 𝐶𝑎𝑙𝑐𝑢𝑙𝑎𝑡𝑖𝑜𝑛 𝑜𝑓 𝐻𝐿𝑇𝐵 𝑆𝐹 𝑆𝐺 𝑆𝐼 𝐴𝐹 𝐴𝐺 𝐷𝐹 𝐷𝐺 𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛 𝐿 𝑉𝐿𝑇𝐵 𝑉𝑇𝐵 𝑉𝑇𝐵 𝑉𝐿𝐿𝑆 𝐻𝐿𝐿𝑆 𝐻𝐿𝑇𝐵 𝑉𝐺𝑇𝐵 𝑉𝑇𝐵 𝑉𝐺𝐿𝑆 𝑉𝑇𝐵 𝐻𝐿𝐿𝑆 𝐻𝐿𝑇𝐵 𝐿𝐹 𝐿𝑈 𝑉𝑇𝐵 𝑉𝐿𝐿𝑆𝐻𝐿𝐿𝑆 𝑉𝑆𝐿 𝐻𝐿𝐿𝑆 𝐻𝐿𝑇𝐵 𝐶𝑎𝑙𝑐𝑢𝑙𝑎𝑡𝑖𝑜𝑛 𝑜𝑓 𝜏𝐹 𝜏𝐺 𝜏𝐼 𝐹 𝜏𝐹𝑆𝐹 𝐴𝐹 𝜏𝐺𝑆𝐺 𝐴𝐺 𝜏𝐼𝑆𝐼( 𝐴𝐹 𝐴𝐺 ) 𝜌𝐿 𝜌𝐿 𝑔 𝜃 CALCULATION OF 𝑭 𝒉𝑳 𝟎 𝑉𝐿𝐿𝑆 𝑉𝑚 𝑉𝐺𝐿𝑆 𝐻𝐿𝐿𝑆 𝐻𝐿𝐿𝑆 𝐿𝑈 𝑉𝑇𝐵 𝑓𝑆 CALCULATE: 𝑭 𝒉𝑳 𝟎? OUTPUT PARAMETERS:
Slug dimensions, Holdups, Velocities, Pressure Gradient N
O
2.2 Implementation of the models
The models were developed Newtonian fluids. The correct implementation of the models in Matlab was validated for a range of the superficial velocities against data from LedaFlow simulator. Afterwards, the presence of the polymers was taken into account and implemented into the models, with considerations based on experimental findings.
2.2.1 VALIDATION FOR WATER AND AIR
Stratified flow model
In Figure 7 good agreement is shown between the model implemented in Matlab and the one implemented in LedaFlow.
Slug Flow model
The model was validated against a set of points obtained with the LedaFlow simulator for the range of superficial gas and liquid velocities of and . The comparison in terms of pressure gradient and total holdup in the slug unit is fair enough and
Figure 7 Comparison of the pressure gradient (left) and of the holdup (right) between the model (line) and LedaFlow simulator data (dots).
Figure 8 Comparison of the pressure gradient (left) and of the holdup (right) between the model (line) and LedaFlow simulator data (dots).
is shown in Figure 8.
2.2.2 APPLICATION TO DRAG-REDUCING FLUIDS
First is considered the case of stratified flow, which is easier to deal with. The effect of the interfacial and liquid friction factors on the holdup and pressure gradient is studied. Then focus is given to the slug flow model, which needs further treatments because of the presence of more closure relationships.
Stratified flow
In Figure 9 is shown an air-water stratified flow with and without polymers. One can see that the shape of the interface is rather changed: it appears no longer rough and wavy. The phenomenon is explained as the polymers damp turbulence in the liquid leading to a smoothening of the interface. It is experimentally observed that another consequence of polymer addition is the increase of the liquid height. This is due to the reduced interface drag which let the gas to flow faster and the holdup to increase as a consequence of the conservation of the mass flow rate of the gaseous phase.
The stratified-flow model uses a correlation for the interfacial friction factor. Scope of this section is to study the effect of a variation of this coefficient on the pressure gradient and the holdup. However, when the turbulence in the liquid is damped, also the liquid friction factor is affected. Therefore, also this parameter was taken into account.
Effect of decreasing fI
As shown in Figure 10, the effect is in agreement with the experimental observations.
Effect of decreasing fL
This leads to a decrease of the pressure gradient because the stress at the wall is reduced, and to a decrease of the holdup. This can be explained because the liquid accelerate and it needs a smaller section to flow.
Combined effect of fI , fL
The combined effect shows a higher decrease in the pressure gradient, and the holdup almost unchanged. However, if different values of fI and fL are used, the holdup can either increase or decrease.
Figure 11 Effect of decreasing 𝑓𝐿 on the pressure gradient (left) and on the holdup (right): 𝑓𝐿 (continuous line), 0.5𝑓𝐿 (dashed line).
Figure 12 Effect of a combined decrease of 𝑓𝐿 and 𝑓𝐼on the pressure gradient (left) and on the holdup (right): 𝑓𝐿, 𝑓𝐼 (continuous line), 0.5𝑓𝐿 , 0.3 𝑓𝐼 (dashed line).
Slug flow
If the liquid is non-Newtonian, the two momentum equations and the three balance equations remains formally the same. The introduction of polymers affects the stress at the liquid wall and at the interface, modifying the force balance on the due phases.
Soleimani et al. (2002) measured a reduction in the interfacial friction factor of about 1/2-1/4, and a
reduction in the liquid wall stress of a factor about 1/2 respect the Newtonian flow. The reader is referred to Figure 13.
Figure 13 Measured effect of polymers on the interfacial friction factor (left) and on the liquid wall stress (right).
A first investigation on the pressure gradient and holdup was done only playing around with these two friction factors. However, it was concluded that also the effect of the slug frequency and of the slug translational velocity should be taken into account to get consistent results. Indeed the holdup is calculated through a balance equation for the mass flow rate of the liquid, which involves the slug unit and the slug cylinder length. These parameters are unaffected unless the translational velocity and the frequency are modified.
Al-Sarkhi (2013) measured a reduction in the slug frequency when polymers are added, as shown in Figure 14.
The slug-flow model also uses correlations for the holdup in the slug cylinder, for which however there are no available measurements or correlations, and for the slug translational velocity. For this it is possible to follow a physical model summarized below. This two parameters should also be changes to avoid unrealistic results.
Figure 14 Measured slug frequency with (full dots) and without polymers
Physical Model for the slug translational velocity
Consider a horizontal slug flow of a Newtonian liquid. It is possible to obtain as described in
Shoham (2006) the slug translational velocity as a function of the slug velocity :
Where is defined as:
and is the shedding rate (the liquid mass flow rate shed backwardly into the liquid film).
By assuming that the turbulent velocity profile in the slug cylinder is fully developed, it is possible to calculate the shedding rate. Indeed, close to the wall, the fluids moves at a local velocity less than
, thus the fluid in this region is shed behind.
∫
or, equivalently,
∫
where is the radius where ( ) .
By using the dimensionless “wall coordinate” parameters, it is possible to get the equation in a non-dimensional form, which shows that the parameter is a unique function of the Reynolds number in the slug cylinder, . For as in most cases of slug flow, results to be 1.2, in agreement with the correlation proposed by Bendiksen (1984) for a
Consider now a horizontal slug flow of a polymer solution. The same procedure can be replicated once one knows the turbulent velocity profile for the given polymer solution. At the maximum drag reduction such a distribution is an analytical result and it is given by the Virk’s asymptote:
with and instead of and for the Newtonian turbulent flow. For other values of the drag reduction less than the maximum, and could be found by fitting experimental data.
Following the described procedure it is possible to find new input values for the translational velocity when polymer solutions are being used. For instance, at the maximum drag reduction, for a turbulent regime in the slug cylinder, is found to be 1.35.
Summarizing, the following changes were applied for the slug flow of a polymer solution, respect the case without polymers:
Liquid wall stress reduced by a factor of 0.5 (both in the slug cylinder and in the film liquid film):
Slug frequency reduced by a factor of 0.5: Holdup in the liquid slug unchanged:
Slug translational velocity calculated as:
2.3 Results
In this chapter, two analytical models were considered for stratified and slug flow. The models, implemented in Matlab, are able to predict fairly the pressure gradient and the holdup for the case of air and water. The possibility to adapt this model for drag-reducing fluids was considered, by
applying some experimental findings.
The effect of the polymer on the slug flow was compared to the data measured by Soleimani et al.
(2002), as shown in Figure 15.
The following changes were used:
Good agreement between the model and the experiments was achieved.
Figure 15 Comparison with experimental data for the pressure gradient (left) and for the holdup (right) at VSG=10 m/s. Model (line), data from Soleimani et. al (dots).
3. SINGLE-PHASE CFD MODELLING
In the introduction the need for a turbulence model for non-Newtonian fluid was underlined and possible approaches were explained. The DNS approach in combination with the viscoelastic FENE-P model for the polymer stress is one of the most complete, however unattractive for industrial application due to its high required computational cost.
In the framework of the RANS equations a low-Reynolds number turbulence model for power-law and Herschel-Bulkley fluids was developed by Malin (1997)-(1998), but it has weakness of grid generation at near wall region. An interesting model was developed also by Iaccarino et al. (2010) starting from the v2-f, a four-equation eddy viscosity closure for Newtonian fluids. The polymeric stresses are modelled using a Boussinesq-type relationship in combination with an additional transport equation for trace of the conformation tensor (FENE-P model). The model is attractive because it shows good agreement with DNS results, at the cost of three additional transport equations to be solved with respect to the Newtonian counterpart.
In this chapter an attractive modified k-ε model for power-law fluids, based on the standard k-ε model with wall function and damping function including the drag reduction phenomenon is selected as case study. The model was implemented in Fluent and validated for single-phase turbulent pipe flow, by considering two different solutions flowing at three turbulent Reynolds numbers. The model is then applied to two-phase flows through the VOF method in Fluent, as described in 4.
3.1 The modified k-ε model for power-law fluids
The model being considered is a high Reynolds number model for power-law fluids, developed by Ro & Ryou (2012) and is based on the standard k-ε model with wall function, and damping function including drag reduction phenomenon. According to Escudier et al. (1998),
Ptasinski et al. (2001), non-Newtonian fluids have specific value of drag reduction in accordance
with the Reynolds numbers and viscosity, which is the main idea of the model. Although drag reduction has an effect on the buffer region and on logarithmic layer, at large distance from the wall the slope of non-Newtonian mean velocity profile approaches the slope of the wall function of a Newtonian turbulent flow. Besides, because the slope of the mean velocity profile is varied by drag reduction, turbulent viscosity changes according to the variation of drag reduction (according to Reynolds number and viscosity of the non-Newtonian fluid).
According to the model, the turbulent viscosity is therefore given by:
where is the damping function which varies according to drag reduction as follows:
with A and B correlation factors:
is the degree of drag reduction:
represents the quantity of drag reduction, and it is defined as:
and are the (fanning) friction factors for the Newtonian and Non-Newtonian fluid,
respectively, and they can be both calculated through the correlation method developed by Dodge
and Metzner (1959) called “generalized Reynolds number”(GRN) as follows:
( ) √ ( )
is calculated by the maximum drag reduction theory of Virk (1975) as:
is the friction factor at the maximum drag reduction, for which an empirical relation exists:
√
( )
is the wall Reynolds number and is the viscosity at the wall for non-Newtonian fluid. The model is based on the power-law equation, therefore the non-Newtonian fluid viscosity has to be expressed in the form:
3.2 Results in a turbulent pipe flow
The model was implemented in a User-Defined Function UDF in Fluent through the macro DEFINE_TURBULENT_VISCOSITY. A single-phase 3D pipe flow in a 2.54 cm internal diameter pipe with a length of 20 D was considered. For each case two simulations were run to get first a fully developed velocity profile in the pipe, as to exclude entrance phenomena.
The model was first tested with no damping function in order to compare the results with the standard k-ε model for water, then was tested with a damping function at the maximum drag
reduction in order to see if the results could match the Virk asymptote, described by:
For a drag-reducing fluid, the velocity profile is expected to be bounded between the Virk asymptote and the Prandtl-Karman law:
Two polymer solutions were considered: 0.2% XG (xanthangum) and 0.3% CMC (carboxymethyl cellulose) aqueous solutions. The rheological parameters according to the power-law fluid equation , are given below (plotted against experimental measurement from Escudier et al.
(1998) as shown in 7.2
[ ] [ ]
Simulations were run at three different Reynolds numbers for each solution. The flow conditions and the friction factors required as input for the model are reported in Table 1.
Table 1 Flow conditions for the validation of the model
XG 0.2% CMC0.3%
Re fN fV fNN Dr fNN Dr
15000 0.007 0.0021 0.0037 0.68 0.006 0.2
30000 0.0059 0.0015 0.0030 0.66 0.005 0.2 60000 0.005 0.0011 0.0025 0.65 0.0042 0.2
In order to validate the model, the pressure drop across the pipe was calculated at the end of each simulation and was compared with the one foreseen using analytical correlation. The comparison in shown in Table 2 in terms of the Fanning friction factor:
Table 2 Comparison between analytical and CFD pressure drop
XG 0.2% CMC 0.3%
Re fNNforeseen fNNcalculated error % fNNforeseen fNNcalculated error %
15000 0.0037 0.0038 3 0.006 0.0058 -3
30000 0.0030 0.0032 7 0.005 0.0049 -2
60000 0.0025 0.0025 0 0.0042 0.0041 -2
Good agreement is shown between the foreseen and calculated values of the pressure drop. Furthermore, the non-dimensional velocity profile was plotted for the case of Re=30000.
The non-dimensional coordinated are defined by using the shear velocity, √ :
The stress comparing in the definition of the shear velocity is to be calculated for each case as:
As shown in the plot in Figure 16, higher values of the degree of drag reductions correspond to a
displacement of the velocity profile toward the Virk’s asymptote.
The velocity profile is also reported in Figure 17 as to show the effects of the turbulence model on
the main flow.
The profile results in a lower centerline velocity for higher values of the degree of drag reduction, and correspondingly the gradient at the wall results lower.
The effect of the model on the characteristics of turbulence is shown in Figure 18 and Figure 19.
Figure 18 Plots of turbulent kinetic energy (left) and dissipation rate (right) with the model for power-law fluids
The turbulent kinetic energy shows a peak which increases with increasing drag reductions. The same happens for the turbulent dissipation rate, being the production and dissipation of k
proportional to those of ε (according to the transport equations being used in the k-ε model).
Figure 19 Plots of turbulent viscosity (left) and turbulence intensity (right) with the model for power-law fluids
The turbulent viscosity results an increasing function of the drag reduction. The turbulence intensity shows a peak which increases for increasing drag reductions. Warholic et al. (1999) reported a damping for the normal component of the turbulent intensity for increasing drag reductions due to the thickening of the viscous sublayer. They also reported an increasing of the streamwise
component over low frequencies. This behavior cannot be observed since the k-ε model assumes that the turbulent kinetic energy is equally divided by the three normal fluctuating components of the Reynolds stress tensor. Consequently, the overall effect is that total turbulence intensity results increased with increasing drag reductions as in Figure 19.
4. TWO-PHASE CFD MODELLING
This chapter deals with the numerical simulation of two-phase flows in Fluent. Scope of this part is to study the effect of the turbulence model developed in 3 on the structure of two-phase flows. However, the methodology for this kind of problems is not been widely approved yet, and is currently object of research. A brief literature survey of the numerical studies of stratified and two-phase flows is firstly presented. Since the turbulence model being considered is based on the standard k-ε model already available in Fluent, an attempt to model Newtonian fluids in stratified and slug flow was firstly done by using the standard k-ε model. For those cases whose results were acceptable, the modified k-ε model was then considered.
A 2D time-dependent approach was used because of the considerable numerical effort required for the correspondent 3D simulation. In spite of the rough approximation it implies especially for slug flow, some conclusions can however be drawn.
4.1 Literature Survey
Jia et al. (2010) studied non-Newtonian liquid drag reduction by gas injection with ANSYS
Fluent. Two regimes were taken in consideration: fully stratified gas shear-thinning liquid flow and gas shear-thinning liquid slug flow regimes. Three dimensional CFD simulations were performed by using the VOF method and a low-Reynolds number k-ε model. They obtained satisfactory agreement between CFD and data for the pressure gradient. They argued that discrepancy between data and CFD may be due to the fact that the VOF method could not account for a separated turbulence model for each phase.
Terzuoli et al. (2008) analysed stratified water and air flow in a rectangular duct with
NEPTUNE_CFD, ANSYS CFX and ANSYS Fluent. They used “inhomogeneous” two-phase flow model, which is a two-fluid approach, because the single-fluid approach has some limitations when interface instabilities are observed. They highlighted the need for an interphase drag model when stratified wavy flow is considered, since the turbulence in the gas is not satisfactory predicted.
Asadolahi et al. (2011) simulated with ANSYS Fluent a nitrogen-water slug flow in a
microchannel. The liquid Reynolds number was 713, so there was no need for using a turbulence model. The VOF method was used. They described and benchmarked two approaches. The first is to generate bubbles and slugs in a long tube using a time-dependent boundary condition. In the second, the flow in a single unit cell consisting of a bubble surrounded by liquid slugs, is solved in a frame of reference moving with the bubble velocity. They found very similar results by using the two methods, also in agreement with earlier verification and validation studies.
4.2 Methodology description
In this section the methodology used for stratified and slug flow is described. Since the gas-liquid interface is well-defined, interface capturing methods such as the VOF can be used. Here the k-ε is selected as turbulence model, because on this model is based the one developed in 3 and object of study of the present work.
The slug flow is modelled by using the stationary-domain approach described by Asadolahi et al.
4.2.1 THE VOF METHOD
The volume of fluid method is an Eulerian method which can model two or more immiscible fluids and track the interfaces by solving a single momentum equation shared by all the phases, which is dependent on the volume fractions of the phases through the phase-averaged density and viscosity. For each phase a new variable is therefore introduced: the volume fraction of the i-th phase in the computational cell, . Those cells which have some between 0 and 1 will be engaged by one or more interfaces.
The tracking of the interface is accomplished by the solution of a transport equation for the volume fraction of the phases, which is coupled with the momentum equation through the shared velocity field.
Assuming a flow of N phases the VOF method introduces N new variable and N-1 additional continuity equations of the form:
[
] ∑( ̇ ̇ )
The N unknowns are linked by the following closure relationship:
∑
The term in the continuity equation is a source term for the i-th phase which can be set in Fluent, while ̇ is the mass transfer from phase i to phase j.
Once the phase volume fractions are known, the averaged properties can be computed. For instance, the volume-fraction averaged density will be:
∑ Similar equations hold for the viscosity.
In interface-capturing methods, a “single-fluid” formulation is applied throughout the
computational domain in which common velocity and pressure fields are shared amongst the phases. A single momentum equation is then solved at each time step, depending on the volume fraction of all phases through the volume-fraction averaged properties.
