a
b
Figure 13. a. Particle paths projected on the cross-section of
flow for m = 1 (full line) and m = 0 (dash line). N and N’ represent
neutral points for m = 1 and m = 0 respectively. Source:[98], b. Paths of particles in the central plane for varying
m. Source: [98].
3.2.1 Effect of DRAs concentration on pressure drop in
bends for polymer and surfactant
solutions
No comprehensive work exists in open literature that investigates the
effect of polymer and surfactant concentration on drag reduction in
bends. Munekata et al. [95] investigated the flow of surfactant
solution (CTAB) in 90o square-cross-section bend where
reduced drag reduction as well as delayed onset of drag reduction with
increase in concentration was reported. Only two concentrations were
tested and so a reasonable conclusion cannot be drawn. Their plots
showed reduced critical Reynolds number (or flow velocity) with decrease
in concentration. The effect of concentration of drag-reducing agents
for flow in bends such as 45o and
180o remain unclear. Notwithstanding the limited
literature in this area, DRA concentration is expected to impact DR
efficiency since concentration can influence flow redevelopment and the
overall ability of the DRA to supress turbulence [10].
3.2.2 Effect of curvature on effectiveness of DRAs in
bends for flow of polymer and surfactant
solutions
Bend angles and curvatures is known to influence centrifugal forces in
the bends and consequently affects fluid redistribution in and around
the bend. While in coils, increased curvature delays the unset of
turbulence and supresses turbulence, in the case of bends, increased
curvature may result in increased flow fluctuations, even under laminar
flow conditions. There are contracting reports from the limited studies
that have been carried out to investigate the effect of curvature ratio
hence, its effect remains unclear. For low Reynolds number flows, Jones
and Davies [96] reported that the effect of curvature on drag
reduction is negligible. Nonetheless, the observed drag is higher than
that which occurred in straight pipes. Yokoyama and Tomita [97]
studied the flow of polyethylene-oxide in a 360o bend
of varying curvature ratios. They recorded a decrease in drag reduction
with increase in curvature ratio. The drag reduction recorded was
predominant at high Reynolds numbers. Only three curvatures were tested
and so the effect of curvatures on the effectiveness of drag-reducing
agents in 360o bends was inconclusive. In the
knowledge of the Authors, the effect of curvature ratio for flow in
45o, 90o and 180obends had not been reported in open literature. Another area of interest
is in determining the effect of pipe diameter on drag reduction in
elbows. This is because; the effect of flow separation is expected to
reduce with increase in pipe diameter.
3.3 Effect of micro-bubble injection
on pressure drop for single phase liquid flows in coiled
pipes
Application of micro-bubbles for DR in curved pipe flows have received
little scholarly attention and the effect of micro-bubbles on pressure
losses as well as the mechanism of micro-bubble DR remain unclear. It
was highlighted earlier that, the action of micro-bubbles on turbulent
flows is similar, in a number of ways, to that of polymers and
surfactant DRAs. To this end, it is expected that micro-bubbles will
result in significant DR in curved pipe flows. The application of
micro-bubbles for drag reduction in helical coils was first carried out
by Shatat et al. [75] and Shatat et al. [98] using hydrocyclone
effect to generate micro-bubbles. Their investigation involved three
helical coils of curvature ratios 0.025, 0.05 and 0.1. They reported
that, though there was significant drag reduction in helical coils by
injection of micro-bubbles, this drag reduction is less than that in
straight pipes under similar conditions of flow. The reduced drag
reduction in helical coils is linked to centrifugal forces (resulting in
suppressed turbulence) associated with the flow. Though the theories for
micro-bubble drag reduction in helical coils are in agreement with
existing micro-bubble DR theories for other geometries, further research
is needed to establish these theories. The effect of various parameters
such as pipe geometry, micro-bubble fraction, flow rate and micro-bubble
size remains unclear due to the limited research in this area. Based on
the limited data available, only a brief outline of the effect of these
parameters is presented in this review.
3.3.1 Effect of curvature ratio on micro-bubble drag
reduction in coiled
pipes
Fig. 12a gives an illustration of the effect of curvature on
effectiveness of micro-bubble drag reduction in helical coils. It can be
observed that increase in curvature resulted in decrease in drag
reduction as well as a shift of both the onset of drag reduction and
maximum drag reduction to higher values of Reynolds number. The figure
also shows higher drag reduction in straight pipes compared to helical
coils. Though there is limited data on the effect of curvature, two
important hydrodynamics properties may play important roles: first,
unlike flows of polymer and surfactant solutions in curved pipes,
gravity/centrifugal forces may result in significant phase separation
(micro-bubble and liquid phases) for the case of micro-bubble DR. If
this occurs, the concentration distribution of micro-bubbles
(particularly in the buffer region where it is most effective) becomes
inhomogeneous and this is likely to reduce DR efficiency; second, the
curvature effects in coils is expected to suppress turbulence and thus
it should be expected that the percentage DR is affected by the degree
of curvature. Further study is, however, required to fully investigate
the effect of curvature ratio on drag reduction in curves.
3.3.2 Effect of air micro-bubble fraction on drag
reduction in curved
pipes
The effect of micro-bubble fraction is illustrated in Fig. 12b. It can
be seen that the effect of air fraction on the onset of drag reduction
is insignificant. However, the air fraction has a profound effect on the
percentage drag reduction and the range of Reynolds numbers over which
drag reduction occurs. In general, the percentage drag reduction
increased with increase in air fraction. Again, additional data is
needed in order to understand the effect of micro-bubble fraction on
drag reduction since very scanty reports are available.
3.3.3 Effect of flow rate on micro-bubble drag reduction
in curved
pipes
Similar to flow of polymer and surfactant solutions in curved pipes,
where DR is reported predominantly in the turbulent flow regime, the
limited reports on the application of micro-bubbles in curved pipe DR
also report it in the turbulent flow regime. Since the degree of
turbulence increases with increase in flow rate, it is expected that
flow rate will affect the efficiency of micro-bubble DR. It can be seen
from Figs. 12a and 12b that drag reduction occurs above a critical
Reynolds number and increases with Reynolds number until a maximum drag
reduction is achieved. Further increase in Reynolds number decreases the
drag reduction. At very high Reynolds numbers, there is increased
centrifugal forces [103] resulting in lower shear stress near the
inner wall and higher shear stress in the region close to the outer
wall. The implication of this is the uneven distribution of air bubbles
and thus reduced drag reduction.
3.3.4 Effect of micro-bubble size on drag reduction in
curves and bends
In the knowledge of the Authors, no published research is available that
investigates the effect of micro-bubble size on DR in curved pipes.
There is therefore need for more research to enhance understanding of
any possible effect of micro-bubble size on DR [28]. In the
application of micro-bubbles as drag-reducing agents for straight
channel flow, conflicting reports exist on its effect on DR. It suffices
to say, however, that bubble behaviour is size dependent, thus DR is
expected to be influenced by micro-bubble size. In general, small sized
bubbles will be better retained in the liquid under the action of
centrifugal forces. Hence, it would be expected that the smaller the
size of the bubbles the more effective it’ll be as a DRA.
3.4 Effect of polymer and surfactant
DRAs on fluid flux in curved
pipes
A number of early researchers chose to
present their results in terms of flow rates rather than drag. The
limited studies in this area have focussed on the application of polymer
DRAs in curved pipe flows. There appears to be an agreement among the
limited reports that addition of DRPs results in increased flow rate
particularly at low and moderate Dean numbers [63], [100].
Barnes and Walters [60] reported that, for fully developed turbulent
flows in curved pipes, there is decrease in flow rate after adding
polymer. It was suggested that the suppression of turbulence may have an
adverse effect on the flow rate at high Reynolds numbers. Given that a
number of recent studies have reported DR in the turbulent flow regime,
it is possible that the polymer used in that study has degraded at the
turbulent flow conditions studied. They also reported an increase in
flow rate with increase in polymer concentration and a negligible
influence of pipe curvature on the effectiveness of the DRPs in the
laminar and transition flow regimes. Though further research is required
to understand the effect of fluid characteristics and pipe geometry on
the flow rate of DRAs, the limited research available suggest that flow
rates would increase in the region of DR.
3.5 Secondary flow in bends and
curves
The secondary flow observed for the flow of Newtonian fluids in bends
and curves results from centrifugal forces associated with such flow.
The secondary flow of spiral form superimposes on the axial primary flow
and there is also reduction in flow rate as a result of higher
dissipation resulting from secondary flow compared to primary flow. The
maximum axial velocity in curves and bends is shifted to the outer side
of the curve. As Dean number increases the secondary flow become more
confined to a thin area near the pipe wall [19]. At higher Reynolds
number additional pairs of vortices appear and multiple solutions exist
[104].
3.5.1 Effect of DRAs on secondary flow for single phase
flows in
curves
It has been suggested that drag-reducing agents would have an effect on
secondary flows [10]. At high flow rates, the secondary flow field
can be categorised into two regions. These are the shear free mid-region
and the offside boundary layer region [105]. The non-Newtonian
characteristic of fluid changes the thickness of the shedding layer. For
pseudo-plastic fluids the shedding layer becomes thicker, whereas for
dilatant fluid flow it is thinner than that of Newtonian fluids. This
thickening or thinning effect may, to a small or large extent, alter the
secondary flow. Fig. 13a shows the paths of fluid particles projected on
the cross section of the pipe. The extremes of m=1 and m=0represents viscoelastic and Newtonian viscous liquids respectively. It
is seen from the figure that, the effect of elasticity (measured roughly
by m ) on the projected streamlines is small. However, the neutral
point for the viscoelastic liquid is slightly nearer to the outer edge
of the pipe compared to that for the Newtonian liquid.
The elasticity of the liquid has a
profound effect on the pitch of the spirals in which the liquid
particles move along the central plane (Fig. 13b). Fig. 13b shows that a
decrease in m leads to a major increase in the curvature of the
streamlines in the central plane. The main effect of elasticity on the
flow of viscoelastic liquids through a curved pipe is to decrease the
curvature of the streamlines in the central plane and to increase the
fluid flux through the pipe [63], [98].
For a third-order fluid (see Coleman
and Noll [103]), Jones [80] presented correlation (Eq. 15) for
the streamline function, which describes the secondary flow in the
cross-section of curved pipes.
\(\Psi=\frac{2L\alpha_{1}}{\text{ρa}}\left[\left(\frac{1}{144}+\frac{\alpha_{3}^{{}^{\prime}}}{48}\right)r_{1}-\left(\frac{1}{64}+\frac{\alpha_{3}^{{}^{\prime}}}{24}\right)r_{1}^{3}+\left(\frac{1}{96}+\frac{\alpha_{3}^{{}^{\prime}}}{48}\right)r_{1}^{5}-\frac{r_{1}^{7}}{576}\right]\cos\cos\ \alpha\ \)(15)
Eq. 15 indicates that, for third-order fluids, the non-Newtonian effect
on secondary flow streamline could be associated mainly to the elastic
behaviour (\(\alpha_{3}\)) of the fluid.
It has also been suggested that an
analogy existed between the counter-rotating secondary flow vortex
superimposed on the primary flow in curved pipes and the vortex pair at
the near wall region of turbulent shear flow in straight pipes [65].
Since drag reduction is a phenomenon of the near wall region where the
flow is primarily a shear flow, it is suggested that any mechanism that
results in this phenomenon would also affect the secondary flow in
curved pipes, at least in the laminar flow regime. It should be stated
here that this assumption relies on the notion that secondary flow, like
turbulent flow, is dissipative. Though a few other studies [87],
[107], [108] made brief mention of the effects of DRAs on
secondary flows, there is insufficient data from which concrete
conclusions can be drawn.
3.5.2 Effect of DRAs on secondary flow in
bends
There are very few studies on the effect of DRAs on secondary flows in
bends and though the limited reports agree that such effects exist,
there is no clarity on whether DRAs suppresses or enhances secondary
flows. In the study carried out by Jones and Davies [96] using very
dilute polyacrylamide and Kezan solution, the onset of non-Newtonian
effects was around Dean number of 300. This is the region where
secondary flow with a Newtonian fluid is sufficiently strong enough to
cause appreciable deviation from Poiseuille flow . Munekata et al.
[95] in their study of viscoelastic fluid flow in square-section
elbow bends suggested that centrifugal effects are suppressed by
viscoelastic effect of the fluid flow. They reported that secondary flow
for Newtonian fluids increases gradually downstream while for
viscoelastic fluids it decreases slightly resulting in DR. Their result
was not corroborated by any other research findings, and further study
is therefore required.
3.6 Flow transition and critical
Reynolds number in bends and curves of circular
cross-section
Studies show that, flow transition in
curved pipes occurs at much higher Reynolds number than in straight
pipes. There is also delayed onset of turbulence with increase in
curvature. Taylor [106] in one of the early researches in this area
showed that streamline motion persisted to Reynolds number of about 6000
in curved pipe
of\(\frac{\text{\ a}}{R}=\frac{1}{18}\).
The mechanism by which turbulence is produced in curved flow varies with
the location in the curves [110]. Turbulence near the inner wall
results from gradual superposition of higher order frequencies on the
fundamental frequency. On the other hand, turbulence, near the outer
wall, results from high frequency bursts near the outer wall. The
sinusoidal oscillations near the inner wall always precedes the
turbulent bursts [19]. The
transition region for flow of Newtonian fluids in straight pipes is
associated with violent flashes which is not the case in curved pipes.
Also, the pressure fluctuation for fully developed turbulent flow in
curved pipes is relatively damped.
3.6.1 Effect of DRAs on flow transition and critical
Reynolds number
The transition from laminar to
turbulent flow regime in curved pipes is gradual and sometimes difficult
to identify. This transition is even more gradual in the case of
non-Newtonian drag-reducing fluid flow in curved pipes [33],
[88], [111]. A delayed and gradual transition from laminar to
turbulent regime occurs for flow of DRAs through curved pipes [23].
Two factors could be responsible for this: turbulence suppression in
curved flow geometry, and effect of drag-reducing agent on flow
transition. Effect of DRAs on flow transition in curves and bends
depends on the curvature of the bend and concentration of drag-reducing
agent. Fig. 5 shows that the critical Reynolds number decreases with
curvature and increases with concentration of surfactant. Transition to
turbulent flow occurred when the wall shear of the DRA exceeded the
critical wall shear stress under strong mechanical load at high Reynolds
numbers.
The critical Reynolds numbers also
depend on the temperature especially in the turbulent regime. In
separate experiments conducted by Inaba et al. [70] and Aly et al.
[22] using surfactants in the temperature range of 5 – 20oC, it was observed that critical modified Reynolds
number \(N_{\text{Re}_{\text{crit}}}^{{}^{\prime}}\) increases with increase in
temperature. This is associated with the critical wall shear stress at
the wall which increases with temperature.
3.7 Friction factor correlations for
single phase flow in curved
pipes
Several theoretical and empirical models are available for predicting
friction factor of non-Newtonian fluids through curved pipes. In
majority of the correlations, friction factors are simple functions of
the Dean number and curvature ratio, \(\frac{a}{R}\), of the pipe. In
general, at low Dean number the friction factor can be defined as a sole
function of Dean number, \(N_{\text{Dn}}\). At higher Dean numbers, the
frictional characteristic of flow not only depend on \(N_{\text{Dn}}\),
but also on \(\frac{a}{R}\). Most of these correlations appear in the
form of ratios of friction factors in curved pipes to that in straight
pipes at the same conditions. Some
researchers [86], [112], [113] presented friction factor
correlation for drag-reducing fluids in both straight and curved pipes
in terms of the Deborah number \(N_{\text{De}}\) defined as:
\(N_{\text{De}}=\frac{\text{characteristic\ fluid\ time}}{\text{characteristic\ flow\ time}}\)(16)
Table 1 presents a summary of friction factor correlation for
non-Newtonian fluids in curved pipes.
Table
1. Friction factor correlations for non-Newtonian fluids in curved
pipes