3.8 Flux models for non-Newtonian
fluids
Several theoretical and semi-theoretical solutions to the governing
equations of flow in curved pipes have resulted in series equations for
fluid flux. Most of the theoretical correlations proposed for fluid flux
in curved pipes are functions of material constants
(\(\alpha_{1},\ \alpha_{2},\ \alpha_{3},\ \alpha_{5},\ \beta_{1}\text{\ and\ }\beta_{2}\)).
The normal stress difference for flow in curved pipes involve\(\alpha_{2}\) and \(\alpha_{3}\) only and the term
(\(\alpha_{5}+\ \beta_{1}\ \)) represents the departure from a
constant viscosity. For straight pipe flow the flux is determined by the
viscosity constant \(\alpha_{1},\ \alpha_{5}\text{and\ }\beta_{1}\) and
is independent of normal stress terms\(\alpha_{2}\text{\ and\ }\alpha_{3}\) [63]. The material constants
are also expressed
as,\(\ \alpha_{2}^{{}^{\prime}}=\frac{\alpha_{2}}{\left(\rho a^{2}\right)}\),\(\alpha_{3}^{{}^{\prime}}=\frac{\alpha_{3}}{\left(\rho a^{2}\right)}\),\(\alpha_{5}^{{}^{\prime}}=\frac{\alpha_{5}\alpha_{1}}{\left(\rho^{2}a^{4}\right)}\)and\(\beta_{1}^{{}^{\prime}}=\frac{\beta_{1}\alpha_{1}}{\left(\rho^{2}a^{4}\right)}\).
Others presented their correlations for flow rates as functions of
Weissenberg number, We , Reynolds number and ratio of polymeric to
shear viscosity. Table 2 gives a summary of fluid flux models.
3.9 Two phase liquid-liquid flow and
the effect of drag-reducing agents on two-phase (gas-liquid and
liquid-liquid) flow in curved
pipes
Limited reports are available in open literature on the effect of
drag-reducing agents on gas-liquid flow in bends and curves. In the
knowledge of the Authors, no report is available in the public domain
that investigates the effect of drag-reducing agents on liquid-liquid
flows in bends and curves. Unlike like the flow of gas-liquids in curves
and bends, two-phase liquid-liquid flows in curves and bends have
received little attention till date. Research has shown that
liquid-liquid properties such as density, viscosity and interfacial
tension have profound effects of pressure drop and flow pattern
characteristics [7], [119], [120]. It has also been
establish that flow patterns and fluid characteristics such as
interfacial tension play important role in determining the effectiveness
of drag reducing agents [121], [122].
3.9.1 Effect of drag-reducing agents on gas-liquid flows
in curved
pipes
Though there is limited literature on the flow on gas-non-Newtonian
fluids in curved pipes, a reasonable body of knowledge exist for the
case of Gas-Newtonian fluid flows in curved pipes. A few reports on the
flow of air-CMC solution in helical coils have shown that the CMC
solution has a significant influence on both in-situ volume fraction and
pressure drop [123]. Pressure drop reduction was reported by Mujawar
and Rao [120] when the drag coefficient approach [124] was used
for analysis but not when the Lockhart and Martinelli [122] approach
was used. This highlights the limitations to the applicability of these
correlations for predicting gas-non-Newtonian liquid flows. Some more
recent reports on air-SCMC systems have shown that polymer
concentration, pipe curvature, pipe diameter and to a lesser extent
helical coil pitch does influence frictional pressure losses, phase
distribution and liquid holdup of gas-liquid flows in helical coils
[126]–[128]. Although the objective of these studies was not to
determine DR, useful information on DR can be derived from them. It is
important to state here that SCMC solutions can behave as
pseudo-plastic, dilatant or thixotropic fluids depending on the
concentration and temperature of the polymer solution [129]. In
general, however, it behaves as a shear thinning (pseudo-plastic) fluid
at low concentrations (< 1 wt%). This behaviour probably
explains the results of Biswas and Das [123] where an increase in
frictional pressure loss with increase in fluid viscosity (increase in
concentration) was reported. The implication of the result is that DR
(if any) occurs at low polymer concentration. Thandlam et al. [124]
reported that stratified flow (ST) regime (in the case of air-SCMC
solution) occupied a larger region of the flow pattern map compared to
air-water flow. The extension of the ST flow regime is an indication of
turbulence suppression and this effect has been reported for DRPs
application in straight pipes [122]. In a separate report that
focuses on mass transfer characteristics of air-SCMC solution in helical
coils, it was reported that mass transfer (kl) was
higher for Newtonian water flow than for SCMC solution and it decreases
with SCMC concentration [128]. Since mass transfer is proportional
to frictional force [130], [131], it may be inferred that the
frictional force in water was higher than that of SCMC at the test
concentration of < 3 kg/m3 used in that
study. The effective viscosity for shear thinning liquids is higher when
it flows as a single phase liquid in coils than when it flows together
with gas [132]. This is due to the higher shear rates for two-phase
gas liquid flows compared to the simple shearing flow of single-phase
liquid flow. The implication of this is higher drag reduction than
single phase flow of polymer solution due to reduced effective
viscosity. It may be deduced that for shear thickening liquids lower
drag reduction would be obtainable in two-phase flow compared to single
phase flow (though no research is available to confirm this
postulation). Clearly further research is needed to interrogate the
effect of drag-reducing agents on gas-liquids flows in curved pipes.