1.0
Introduction
Locations of petroleum wells are
sometimes several kilometres from processing plants. There is therefore
need to transport single and multi-phase fluids through pipes, including
pipe fittings such as bends, to processing plants and for separation
[1]–[3]. A large percentage of the energy cost in petroleum
production and transport results from pressure losses. Most of the
pressure losses in pipeline flows are associated with the production of
turbulence eddies resulting in non-axial components of flow. Unlike
laminar flows where pumping power is directed at providing axial
unidirectional fluid flow, turbulent flows are characterised by both
axial and radial flows. The implication of this is loss of pumping power
or increased drag. A common view is that any process mechanism that
results in flow laminarization would also result in drag reduction
[4].
Drag reduction (DR) is a process of reducing pressure losses associated
with flows [5]. Additives, called drag-reducing agents (DRAs), are
often used for drag reduction. After the pioneering work credited to Tom
[5], several studies have examined the effect of DRAs on liquid
flows through straight pipes and channels of various orientations
[7], [8]. A few others investigated this effect in curved pipes
[9]–[11]. Other methods of drag reduction involving pipe
modifications such as riblets, dimples, wavering walls and amenable
surfaces are also common [12]–[14].
Drag reduction (DR) as originally defined by [15] is given by Eq.
(1).
\(\text{DR}\left(\%\right)=\frac{\left(\frac{\text{dp}}{\text{dl}}\right)_{s}-\left(\frac{\text{dp}}{\text{dl}}\right)_{\text{DRA}}}{\left(\frac{\text{dp}}{\text{dl}}\right)_{s}}\times 100\%\)(1)
where \(\left(\frac{\text{dp}}{\text{dl}}\right)_{s}\) and\(\left(\frac{\text{dp}}{\text{dl}}\right)_{\text{DRA}}\) are
frictional pressure gradients for solvents and DRA solution
respectively, under the same flow conditions. Where the viscosity and
density of solvent and polymer solution are almost the same, Eq. (2)
gives an equivalent measure of drag reduction.
\(\text{DR}\left(\%\right)=\frac{f_{s}-f_{\text{DRA}}}{f_{s}}\times 100\%\)(2)
where \(f_{s}\) is the fanning friction factor before the addition of
DRA. fDRA is the fanning friction factor after
the addition of DRA.
\(f=\frac{\tau_{w}}{\frac{1}{2}\rho U^{2}}\) (3)
The wall shear stress \(\tau_{w}\) is given by
\(\tau_{w}=\frac{dp}{4l}\) (4)
where d is the internal diameter of pipe and \(P\) is the
frictional pressure drop over the pipe length l .
Eqs. (1) and (2) are referred to as pressure drop drag reduction and
friction factor drag reduction [16].
A measure of drag reduction, in curved
and straight pipes, called turbulence reduction drag (TRD) given by Eq.
(5) is sometimes used [17].
\(\text{TRD}\left(\%\right)=\frac{f_{T}-f_{T\_DRA}}{f_{T}-f_{L}}\times 100\%\)(5)
where T and L denote turbulent and laminar flow of the
solvent respectively.
The definition given by Eq. (5) enables comparison of only the degree of
turbulence suppression in curved and straight pipes. In general, the
difference between Eq. (2) and (5), for straight pipes is small.
However, the respective difference is large in the case of flow in
curved pipes due to suppressed turbulence and secondary flow
effects [18]. It should be stated that at the same Reynolds number
of flow, the degree of turbulence in straight pipes is higher than that
in curved pipes [19].
Drag-reducing agents also influence turbulent heat
transfer [20]–[22]. In certain applications, the effect of DRAs
on heat transfer reduction (HTR) outweighs its effect on drag reduction
[23]. Besides heat transfer and drag reduction, DRAs affects flow
structure, phase-distribution and flow regime
transitions [24]–[27].
Till date, most of the drag reduction studies have focussed on flows
through vertical, horizontal, inclined and undulated pipes. Application
of DRAs for flows in curved pipes has received little
attention. Moreover, the flow of single and multiphase fluids through
curved pipes is a common occurrence in the petroleum and chemical
industries. Such a flow is associated with large pressure drop and
pressure fluctuations among other effects [10]. It is important to
gain insight into drag reduction in curved pipes to improve the
economics of pipeline design and operation. Fsadni [27] provided a
brief review of pressure drop reduction studies for flow in helical
coils. Besides this review, the Authors are not aware of any other
reviews pertaining to drag reduction in curved pipes. Hence this work is
devoted to the review of existing research on single and two-phase drag
reduction for flows through curved pipes.