Figure 7. Drag reduction in
straight and coiled pipes for flow of surfactant solution using both DR
and TRD notations. Source: [10].
DR by
surfactant DRAs in coiled pipes have generally been reported in the
turbulent flow regime. While majority of investigations reported no DR
in the laminar flow regime, a few others (e.g. Gasljevic & Matthys
2009) reported increased drag in the laminar flow regime. The effect of
surfactant DRAs on pressure drop/friction factor as reported by Aly et
al. (2006) is illustrated in Fig. 5. They investigated the effect of
oleyldihydroxyethylamineoxide (ODEAO) surfactant on single phase water
flow in straight and coiled pipes. DR was observed in the turbulent flow
regime as indicated by the reduction in friction factor with addition of
the surfactant. They linked this to turbulence suppression by
well-ordered network of rod-like micelles structure of the surfactant.
A few early reports on the effect of polymer DRA for flow in curves
indicate a reduction in friction loss in the laminar and transition flow
regime [63], [83], [84]. Their results were, however
presented in terms of fluid flux (not flow resistance). They all
reported increased flow rate and reduced friction loss in the laminar
flow regime. It should be stated that the percentage reduction in
friction loss, in the laminar flow regime, reported by some early
researchers are generally very small. The reports of limited drag
reduction may be explained by the interaction of drag-reducing polymer
with secondary flows in the laminar flow regime in curved
pipes. In more recent investigations,
using advanced instrumentation to study the effects of polymer DRA on
flow in curves, DR was mostly recorded in the turbulent flow regime
[11], [31], [85].
3.1.1 Effect of varying polymer
and surfactant DRA concentration on drag reduction in coiled
pipes
The effect of concentration of drag-reducing polymer on drag reduction
in hydrodynamically developed flows in coiled pipes remains unclear.
Some early investigations on the effect of DRP concentration on DR (e.g.
Kelkar and Mashelkar [83] – 12.5 mm internal diameter curved
pipe/polyacrylamide polymer solution and Rao [84] – 9.35 mm
internal diameter coil/Carbocol) reported decrease in friction factor
(increase in DR) with increase in concentration. More recently, it was
shown that the effect of DRP concentration on DR depends on the pipe
diameter [33], [88]. For larger diameter pipes, higher
concentrations resulted in lower drag reduction and even enhanced the
drag at lower flow rates (Fig. 6a). In addition, higher concentration
for the larger pipes delayed the onset of drag reduction. This becomes
obvious when the plots are done on the Prandtl-Karman Coordinates (Fig.
6b). For smaller pipe diameter, Shah & Zhou (2001) reported that higher
concentration of polymer resulted in higher drag reduction. The effect
of concentration on the onset of drag reduction for small pipes is not
clear. Their plots showed no consistent patterns on the effect of DRA
concentration.
Other
even more recent studies did not investigate the coupled effect of
concentration and pipe geometry, and the reports on the effect of DRP
concentration on DR are rather inconsistent. For example Zhou et al.
[31] and Shah and Zhou [33] reported higher DR when low
concentration polymer was used in coiled pipes. However, reports of Shah
et al. [82], Gallego and Shah [86] and Kamel [10] showed
that DR in coiled pipes increased with concentration of DRA until a peak
value where further increase in concentration increased drag. Shah et
al. [82] used AMPS-copolymer for their study as opposed to Xanthan
used by Shah & Zhou [30, 85] and reported a peak concentration of
0.07 % by volume polymer. This concentration was employed in subsequent
works by Gallego & Shah [86] and Kamel [10]. The optimum
concentration recorded by Gallego and Shah [86] for Nalco
ASP-820 and Nalco ASP-700 were 0.05% and 0.03% by volume
respectively. However, the drag reduction recorded for these
concentrations were very close to that of 0.07%.
Reports on the effect of concentration on DR for surfactant solution
flow in curves are scanty. It has been reported that below a certain
surfactant concentration in the turbulent regime, no drag reduction was
observed. However, beyond this concentration, the percentage drag
reduction increased with increase in concentration until a value of
concentration beyond which no further drag reduction was achieved (Fig.
5) [23], [73]. The reason given for this (where further increase
in concentration results in no further drag reduction) is the saturation
of the network structure of the rod-like micelles. Therefore, further
increase in concentration was ineffective in producing additional drag
reduction. Plots of Inaba et al. [70]
(\(\frac{f_{C}}{f_{\text{SL}}}\) versus \(N_{Dn^{\prime}}\)) shows a negative
drag reduction for higher surfactant concentration at low Dean number\(N_{Dn^{\prime}}\). However, at high Dean number it appeared that higher
concentration of surfactant results in higher drag reduction. For both
polymer and surfactant solution flows at fairly high Reynolds numbers,
majority of reports indicate an increase in DR with increase in
concentration up to an optimum concentration beyond which further
increase in concentration produces no further increase in DR.
3.1.2 Effect of fluid
velocity/Reynolds number on the drag reduction in coiled pipes for
polymer and surfactant
DRA
Similar to flow in straight pipes, drag
reduction using polymer or surfactant DRAs in coiled pipes increases
with increase in Reynolds number or flow rate (Figs. 9a and 9b)
[34], [88]. In both straight and coiled pipes, the increase in
drag reduction with flow rate is limited by critical shear stress above
which polymer and surfactant DRAs degrade either permanently or
temporarily. The difference in effectiveness (as defined by Eq. 2) of
drag-reducing agent in coiled and straight pipes reduces with increase
in Reynolds number (Fig. 7) [10], [34], [85]. Beyond the
critical shear stress, drag reduction decreases with increase in flow
rate (see Figs. 9a and 9b). Similar to observations in straight pipes,
Gasljevic and Matthys [9] reported that there is no significant drag
reduction in laminar flow regime in coiled pipes.
3.1.3 Effect of coil curvature
ratio on the effectiveness of polymer and surfactant
DRA
The curvature ratio plays an
important role in determining the friction losses in coils [23]. In
general, when polymer DRA is used, an increase in curvature results in a
delay in the onset of drag reduction (Figs. 8a and 8b). This is linked
to the delay of turbulence with increase in curvature [31],
[34], [85]. Shah and Zhou [33] proposed a correlation for
determining the Reynolds number at the onset of drag reduction for
polymer drag-reducing agents given
by;
\(N_{Re^{\prime}}^{*}=c_{1}-\frac{c_{2}}{\left(\frac{a}{R}\right)^{0.5}},\ c_{1}=13172,\ c_{2}=835.33\)(14)
The effectiveness of polymer drag-reducing agents generally reduces with
increase in curvature (Fig. 9a and 9b) [19], [31], [85]. In
the case of surfactants DRAs, there is increase in friction factor with
increase in curvature ratio and this is linked to increase in the
intensity of secondary flows [23], [38]. For a special case of
very low Reynolds number \(N_{\text{Re}}<25\), Robertson and Muller
[87] reported that extremely small drag reduction occurred and it
increases with the curvature of the pipe. Their report requires further
investigation to be validated.