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.