a
b
Figure 3. (a) Schematic representation of polymer-induced
turbulent drag reduction (DR) mechanism [32] (b) Illustration
of turbulence suppression mechanism Source: [13].
Between the Prandtl-Karman Law and the maximum drag reduction curve is a
roughly linear polymeric regime characterised by the wall shear stress\(\left(\tau_{w}\right)\) and the increment in slope\(\left(\delta\right)\) [33]. This regime is represented by Eq.
(9).
\(\frac{1}{\sqrt[f]{}}=\left(4.0+\delta\right)\text{Re}\sqrt[f]{}\ -0.4-\delta\left[\frac{\sqrt[2d]{}}{v_{s}}\left(\frac{\tau_{w}^{*}}{\rho}\right)^{\frac{1}{2}}\right]\ \)(9)
The centrifugal forces associated with flow in curved pipes results in
secondary flows which appear in the form of vortices [34].
Centrifugal force causes faster-moving fluids in the middle of the pipe
to move to the outer wall while fluids in the outer wall to move to the
centre resulting in secondary flow [10]. These vortices flow
behaviour results in flow fluctuations and higher pressure drop in
curved pipes compared with that in straight pipes of equivalent length.
The higher pressure drops observed in curved pipes prompted Shah and
Zhou [30] to propose a modified Virk’s envelop. They replaced the
Prandtl-Karman Law (Newtonian line in Fig. 1) with a Newtonian friction
factor correlation for coiled tubing given by Srinivasan [32].
Findings have revealed that maximum drag reduction asymptote (MDRA) for
curved pipes is lower than that of straight pipes and depends on the
curvature ratio of the pipe. Shah and Zhou [33] proposed an
expression for MDRA for flow of drag-reducing polymers (DRPs) in curved
pipes as a function of curvature ratio given by Eq. (10). Fig. 2 shows
the MDRAs for coils of various curvatures, as determined by Eq. (10).
\(\frac{1}{\sqrt[f]{}}=AN_{\text{Re}}\sqrt[f]{}\ +B\) (10)
where;\(A=\left[c_{1}+c_{2}\left(\frac{a}{R}\right)^{0.5}\right]^{-1},\ \ c_{1}=0.053109965,\ c_{2}=0.29465004\)and
\(B=\left[c_{3}+c_{4}\left(\frac{a}{R}\right)^{0.5}\right]^{-1},\ \ c_{3}=0.0309447,\ c_{4}=0.245746\)
when \(\left(\frac{a}{R}=0\right),\ \ A=18.83\) and \(B=32.32\),
and Eq. (10) reduces approximately to the Virk’s MDRA.
Based
on the redefined MDRA for curved pipes, Shah and Zhou [33] described
a new drag reduction envelop. This drag reduction envelop is bounded by
three lines – the laminar flow line, the MDRA for curved pipe and the
zero-drag reduction line given by the Srinivasan [32] correlation
for Newtonian turbulent flow in curved pipes. The laminar flow
correlation chosen for their work was that of Liu and Masliyah [34].
Studies have shown that phenomenological models for MDRA, developed for
polymers, are not applicable to surfactants. An interesting
characteristic of surfactants is their higher shear viscosity compared
to polymer solutions. This makes surfactant solution more shear rate
dependent and makes the definition of the Reynolds number all the more
difficult [21]. Zakin et al. [35] showed that fanning friction
factor curves of most surfactants in straight pipes lies below the Virk
MDRA. They proposed an MDRA for surfactant solutions in straight pipes
given by:
\(f=0.32N_{\text{Re}}^{-0.55}\) (11)
Their work did not account for how
viscosity depends on the shear rate in surfactants. Aguilar et al.
(2006) used surfactant with viscosity similar to that of the solvent and
recorded friction factors slightly lower than those given by the Zakin
MDRA. They proposed a new correlation for MDRA given by;
\(f=0.18N_{\text{Re}}^{-0.50}\)(12)
Surfactant solutions exhibit higher MDRA than polymers. Kamel and Shah
[36] therefore extended Zakin et al. [35] MDRA for straight
pipes to curved pipes and proposed a correlation for MDRA for surfactant
in coiled pipes given by Eq. (13).
\(f=\left[-32200.42\left(\frac{a}{R}\right)^{3}+1830.62\left(\frac{a}{R}\right)^{2}+0.32\right]N_{Re^{\prime}}^{\left[7210.95\left(\frac{a}{R}\right)^{3}-316.97\left(\frac{a}{R}\right)-0.55\right]}\)(13)
They further suggested a modified maximum drag reduction envelop for
surfactant in coiled pipes bounded by Liu and Masliyah [34]’s
equation for laminar flow, the Srinivasan et al. [32] correlation
and Eq. (13).
2.1 Drag-reducing Agents
(DRAs)
Drag-reducing agents include additives
such as polymers, surfactant, fibres and micro-bubbles. The use of
polymer as a drag-reducing agent is most common because only small
concentrations is needed to produce drag reduction [13], [39],
[40]. Drag-reducing agents can either be soluble or insoluble
resulting in homogeneous and heterogeneous fluids mixtures respectively
[41]. The benefits of DRAs include reduced operation cost and ease
in application [42]. Its
application in oil and gas ranges from petroleum product transport to
enhance oil recovery [43].
2.1.1 Polymer
DRAs
Synthetic and natural polymers are
classes of polymer DRAs. Examples of synthetic polymers include;
polyethylene oxide (PEO), polyisobutylene (PIB), polyacrylamide (PAM),
partially hydrolysed polyacrylamide (HPAM) etc. Synthetic polymers
generally produce high percentage drag reduction. They are, however,
mostly non-biodegradable thereby posing environmental challenges.
Natural polymers include; carboxymethylcellulose (CMC), guar gum (GG),
xanthan gum (XG), tragacanth, karaya, locust bean, chitosan and okra
[14]. Natural polymers are biodegradable thus making them
environmentally friendly [41]. However, this biodegradability
reduces their shelf life thus reduces their effectiveness for
long-distance transport. Grafting the artificial polymers into the rigid
structures of natural polymers have been suggested as a means of
controlling biodegradation [14], [44]. Recent advances in
polymer technology have seen the rise in high performance biodegradable
polymers. Some of the recent synthesis have been centred around improved
cross-linking of polymer chains [45], [46]. A common
characteristic of DRAs is the increase in efficiency with increase in
molecular weight of polymer. A drawback of polymers DRAs is their
susceptibility to both chemical and mechanical degradation. High
molecular weight (Mwt >
106) polymers are the most commonly employed DRAs
possibly because of their unique rheological properties which makes them
effective and economical [14]. Various theories exist seeking to
explain the mechanism of polymer drag reduction. These theories includes
those based on shear thinning, viscoelasticity, vortex stretching,
molecular stretching, flow anisotropy and turbulence suppression
[16], [32].
A number of researchers have tried to explain the mechanism of polymer
DR by molecular stretching of polymer molecules. In this model, the
shear-hardening characteristic of drag-reducing polymers (DRPs) is
assumed to increase resistance to extensional flow, thereby inhibiting
turbulent burst at the near wall region. The Lumley [44] model,
which is based polymeric chain extension, suggest that DR involves
increased elongational viscosity. This results in increased thickness of
the viscous sub-layer which dampens and suppresses small eddies and
turbulent fluctuations. The overall effect is higher turbulence
dissipation, reduction of both velocity gradient and shear stress near
the wall and consequently reduction of drag. It has also been suggested
that stretching of polymer molecules results in the storage of elastic
energy (see Fig. 3a) emanating from flow very close to the wall
[48]. Thus if there is sufficient relaxation time, the elastic
energy is transported to the buffer layer and dissipated there by the
vortex motion resulting in DR ([49].
A number of proposed DRP drag reduction mechanisms are based on
polymer’s spring like behaviour. A bead-spring model was used by
Armstrong and Jhon [47] to describe the mechanism of DR. The polymer
molecule is assumed to be a chain of identical beads linked by an
arbitrary spring potential. Here the effect of the stochastic velocity
field on the polymer molecule is associated with arenormalisation of the connector potential and the dumb-bell
probability density is derived for the arbitrary connector potential. At
certain degree of turbulence, the second moment of the probability
density becomes infinite. The renormalisation of the connection
potential between the beads reduces the connection force, thus making
the beads extend (or polymer molecules expand). A mechanism analogues to
the dumb-bell model wherein stretched polymer molecule are simplified as
springs with masses at their ends was also proposed by [49]. The
theory assumes that there is a balance between centrifugal stretching
force and centripetal restoring force acting on rotating polymer chains.
The rotational flow kinetic energy is converted to polymer elastic
energy and subsequently becomes damped by the surrounding viscous fluids
when the polymer relaxes.
A common view is that interaction of polymer with turbulence (resulting
in flow laminarization ) is the main reason for its efficiency as
a drag-reducing agent [42]. The complex rheological properties of
DRPs such as viscosity and elasticity play important role in the process
[14]. The non-axial component of turbulent flows results in wasteful
turbulent eddy dissipation and the implication of this is increased drag
[16]. The ability of DRP to induce flow laminarizationtranslates to reduction of wasteful energy dissipation and consequently
DR. In effect, the action of DRPs in flow laminarization is to
reduce radial velocity fluctuations and Reynolds stresses [32],
[41], [42].
The anisotropic behaviour of DRP solutions, where shear rate, structure
and viscosity of the solution are directionally dependent, have been
used to explain polymer DR. Here the effect of DRPs is to alter the
turbulence structure and reduce drag [51]. Models based on the
finite elastic non-linear extensibility-Peterlin (FENE-P) have also been
used to explain the mechanism of polymer DR. Here pre-averaging
approximation is applied to a suspension of non-interacting finitely
extensive non-linear elastic dumb-bells, thus accounting for the finite
extensibility of the molecule [52]. The FENE-P model has been used
by Li et al. [50] as viscoelastic polymer conformation tensor
equation.
A few numerical simulation studies have been carried out to shed more
light on DR mechanism. In the Brownian dynamic simulation studies of
Terrapon et al. [51] it was demonstrated that polymers experience
significant straining around the vortices resulting in molecular
stretching. As polymer molecules stretches around the vortices, by
upward and downward fluid motion, there is extraction of energy from the
near-wall vortices. Numerical studies has also been carried out to
describe the systematic storage and release of energy to the flow by
polymer [55], [56]. Energy storage occurs at the near-wall
vortices, while the release of energy occurs at the very-near-wall
region. Numerical studies was also used to show that polymer mixing acts
as a relaxation mechanism for DR [57]. Direct numerical simulation
was used to investigate the roles of shear stress/shear rate anisotropy
and elasticity on DR [58]. The hypothesis is that, when polymer
stretches, the viscous anisotropic effect produces change in turbulent
structures and change in entropy which in turn results in DR. To shed
more light on the mechanism of DR and explain certain observed
behaviours, various studies have been carried out using laser Doppler
velocimetry (LDV) and particle image velocimetry (PIV) [41],
[59]–[62].
Overall, it appears that more than one of the suggested mechanisms is
involved in DR. Notwithstanding the mechanism(s), polymers do stretch in
the flow thereby absorbing the energy in the streak. This inhibits
turbulent burst formation (Fig. 3b) in the buffer region and results in
turbulence suppression.
The
above reports details efforts to explain the DR mechanism via
investigations of flows in straight pipes. Similar to straight pipes, DR
by polymer solutions in curved pipes and channels have been linked with
the dampening of turbulent intensities [63].
A few suggested mechanisms for polymer
drag reduction in the laminar flow regime of curved flow exist. The
general understanding is that for DRAs to be effective in the laminar
flow regime of curved pipe flows, there must be an interaction between
the DRAs and secondary flow stream lines.
A few early studies investigated the
effect of DRAs on secondary flows but the conclusions are inconsistent
and mostly speculative [64]–[66]. Frictional losses as well as
secondary flow losses contributes to pressure losses in hydrodynamically
developed flows in coils. In the case of undeveloped flows in and after
bends, additional form-drag exist due to flow redistribution. The effect
of DRAs on each of these competing forces is a subject of investigation
by the authors using a dedicated flow loop at the University of
California Berkeley.
2.1.2 Surfactant
DRAs
Surfactants are surface-active chemical
agents of relatively low molecular weight which alters the surface
tension of the liquid in which it dissolves [67]. They assume
various structures in solution such as spherical micelles, rod-like
micelles, crystals, emulsions and vesicles depending on the
concentration, temperature, salinity etc. [38]. The classes of
surfactants are ionic (examples; anionic, cationic and zwitterionic) and
non-ionic surfactants. When compared to polymer they have higher
resistance to mechanical degradation [68] and are thermodynamically
stable [23]. This is due the their ability to self-repair after
degradation [21], [69]. The efficiency of surfactants in
reducing drag depends on its concentration, temperature, geometry of
flow channel, size of micelles and bond strength. Some early investors
[70], [71] linked the mechanism of drag reduction in surfactants
to the viscoelastic rheology of the solution. However, drag reduction
has since been observed in non-viscoelastic surfactants [72]. The
ability of surfactants to act as drag reducers is associated with the
formation of thread-like micelles. These micelles changes the structure
of turbulent flow at the near wall region [34], [68]. It has
been suggested that surfactants drag reduction is achieved when
micelles, under shear stress, line up in the direction of flow and build
a huge network structure (the so-called shear-induced state) [42],
[73]. This leads to a damping of radial turbulence and subsequently
reducing pressure loss. Fig. 4a shows surfactant molecules and micelles
structures while Fig. 4b show the transmission electron microscope (TEM)
image of surfactant micelles. Different surfactants show different
response or characteristics under the influence of shear. For example,
the viscosity of Habon G decreased under prolonged shearing or mixing
while that of the mixture Ethoquad T 13/ sodium salicylate (NaSal)
increased after prolonged shearing in a rotational viscometer. The
effective velocity range for which various surfactants produce drag
reduction depends on the concentration and age of the surfactants
[74]. The effectiveness of surfactants as drag-reducing agents is
negatively influenced by disturbances in the flow, though sensitivity of
surfactants to disturbances differs. This is important in bend-flow
applications where there are high disturbances resulting from the bend.
As reported by Gasljevic and Matthys [9], additional drag results
from the flow of surfactant solutions in the region of high flow
disturbance after the bend.
Cationic surfactants are by far the most commonly used drag-reducing
surfactants DRS . Cationic surfactants combined with suitable
counter-ions are effective drag reducers [75]. The applicability of
anionic surfactants in aqueous or hydrocarbon solutions depends on their
molecular weight. In general, low-molecular weight surfactants are used
as drag-reducing agents. Very low-molecular weight (< 10
carbon atoms in chain) anionic surfactants are too soluble to have
substantial surface effect and thus results in small drag reduction
[42]. The surface-active portion of zwitterionic surfactants carry
opposing charges on it as well as a subgroup derived from imidazoline.
Zwitterionic surfactants are more environmentally friendly than the
cationic ones. However at the recommended (low) concentration, they are
very sensitive to upstream disturbances (as is common in bends) in the
flow which may impede their drag-reducing capabilities [74].
Non-ionic surfactants are known to be chemically, mechanically and
thermally stable in comparison with ionic surfactants. In addition
non-ionic surfactants do not precipitate in the presence of calcium ions
[41]. Non-ionic surfactants are only applicable over a limited range
of temperature and concentrations and may be susceptible to chemical
degradation [14]. Glycolic acid ethoxylate, Arquad 16–50
Cetyltrimethylammonium chloride (CTAC), Ethoquad O12,
Soya-N(CH3)3Cl and Sodium oleate are
some examples of commonly used surfactants [14]. Van der Plas
[73] recently defined some essential characteristics required by
viscoelastic surfactants for them to be effective DRAs in petroleum
applications. It is safe to assume that insight into micelle formation
and rheological properties of surfactants are essential to understanding
the mechanism for drag reduction of surfactant solutions. Due to the
high shear stresses observed in curved pipes, surfactants are more
suitable as drag-reducing agents for flow through bends than polymers
[53].
2.1.3 Micro-bubbles DRAs
The application of air in micro-bubbles
drag reduction is environmentally friendly and cheaper compared to
polymers and surfactants [77], [78]. Micro-bubbles have
diameters less than ten-microns and exhibit behaviours different from
those of larger size bubbles. These differences are seen in their
chemical and physical characteristics such as the tendency to remain
suspended in the liquid phase over longer periods of time [78]. The
first work published on the application of micro-bubbles as
drag-reducing agents was by [79]. The mechanism of drag reduction by
micro-bubbles is not yet well understood. Similar to other drag
reduction techniques, the purpose of micro-bubble injection is to alter
the structure of the boundary layer. It had been suggested that
micro-bubbles reduce drag by altering both laminar and turbulent
boundary-layer characteristics [79]. It has been reported that,
injecting air bubbles results in an increase in kinematic viscosity and
decrease in the turbulent Reynolds number in the buffer layer [80].
This results in thickening of the viscous sub-layer and decrease in the
velocity gradient at the wall. Hassan and Ortiz-Villafuerte [74]
used particle image velocimetry (PIV) to study the effect of injecting
low void fraction micro-bubbles into the boundary layer of a channel
flow. Some of their results showed some similarities with drag reduction
behaviour by polymers or surfactants as well as reports of some earlier
investigations [80], [81]. These similarities include thickening
of the buffer layer as well as upward shift of the log-law region. They
stated that the micro-bubble layer formed at the top of the channel was
not responsible for the drag reduction recorded. This micro-bubble layer
served to reduce the slip between the micro-bubbles and the liquid. The
major contribution to drag reduction is the accumulation of
micro-bubbles in a critical zone within the buffer layer. The
interaction of micro-bubbles with turbulence in the buffer layer is
responsible for the observed DR. in general, injection of micro bubbles
reduces turbulent energies with the shear in the boundary layer
remaining unchanged [82]. There
appears to be some agreement on the mechanism of micro-bubble drag
reduction especially as it relates to thickening of the viscous
sub-layer and turbulence suppression.