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Dragenhancement of aqueous electrolyte solutions in turbulent pipe flow.

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ASIA-PACIFIC JOURNAL OF CHEMICAL ENGINEERING
Asia-Pac. J. Chem. Eng. 2007; 2: 225–229
Published online 26 July 2007 in Wiley InterScience
(www.interscience.wiley.com) DOI:10.1002/apj.056
Short Communication
Drag enhancement of aqueous electrolyte solutions in
turbulent pipe flow
E. Benard,1 J. J. J. Chen,2 A. P. Doherty3 and P. L. Spedding1 *
1
School of Mechanical and Aerospace Engineering, Queen’s University Belfast, Belfast BT9 5AH, UK
Department of Chemical Materials Engineering, University of Auckland, Auckland, New Zealand
3
School of Chemistry and Chemical Engineering, Queen’s University Belfast, Belfast BT9 5AG, UK
2
Received 1 December 2006; Revised 14 February 2007; Accepted 19 February 2007
ABSTRACT: Experimental measurements have indicated that drag enhancement occurs when aqueous electrolyte
solutions are flowing in the turbulent regime. The primary electroviscous effect due to the distortion by the shear field
of the electrical double layer surrounding the ions in solution is invoked to explain the drag enhancement. Calculations
using the Booth model for symmetrical one-to-one electrolytes enabled the increased viscosity in the turbulent regime
to be calculated.  2007 Curtin University of Technology and John Wiley & Sons, Ltd.
KEYWORDS: electroviscosity; aqueous electrolyte; turbulent flow; pipe flow
INTRODUCTION
Frictional pressure loss of single-phase fluid flow in
pipes has been the subject of extensive study over past
years (e.g. Drew et al . (1932)), eventually resulting in
the development of the now familiar Moody diagram
(Moody, 1944) linking the friction factor φ with the
Reynolds number Re and, under turbulent conditions,
the relative pipe roughness δ/d. The flow relation is
divided into three regions; streamline flow where the
Hagan–Poiseuille relation (Hagan, 1939) applies above
an Re>3.0, the transition region between Re 2300 and
3000, and the turbulent region above Re > 3000 where
the relative pipe roughness assumes importance. The
Moody diagram has been shown to be applicable to
a large variety of fluids possessing totally different
physical properties such as gases, liquid, vapours, etc.
(Pigott, 1950a,b; 1957).
One significant deviation from the Moody diagram is
the drag reduction exhibited by dilute polymer solutions
in the turbulent regime (Toms, 1948; Savin and Virk,
1971; Sylvester, 1973). There have been a number of
models advanced to explain the effect and it appears
that the polymer acts by adhering to the inner wall
of the pipe where normal frictional loss is generated
and reduces the drag by a surface interfacial effect.
Actual drag reduction occurs in other systems such as
*Correspondence to: P. L. Spedding, School of Chemistry and
Chemical Engineering, Queen’s University Belfast, Keir Building,
Stranmillis Road, Belfast BT9 5AG, UK.
E-mail: e.benard@qub.ac.uk
 2007 Curtin University of Technology and John Wiley & Sons, Ltd.
paper pulp suspensions (Sylvester, 1973) and certain
gas liquid two-phase regions (Beattie, 1977; Ferguson
and Spedding, 1996; Spedding and Benard, 2006).
It has previously been assumed that aqueous electrolyte solutions followed the Moody diagram. Indeed,
Gould and Levy (1928) and Kratz et al . (1931) reported
that within the normal experimental accuracy aqueous
CaCl2 solutions followed the Moody diagram up until
the start of the turbulent regime. However, Spedding
and Chen (1980) obtained drag enhancement in the turbulent flow region with aqueous electrolyte solutions.
No fundamental mechanism was invoked to explain the
observed drag enhancement but pressure drop prediction of NaCl solutions was achieved in terms of the
eddy viscosity. In this work the whole subject of drag
enhancement in aqueous electrolytes is revisited to seek
a fundamental mechanism for the effect observed (Spedding and Chen, 1980).
EXPERIMENTAL
The apparatus consisted of three horizontal PVC pipes
(internal diameters of 2.93, 4.22 and 5.24 cm) 6.2 m
long connected one above the other between two
vertical headers. The aqueous solution contained in the
apparatus was circulated by a centrifugal pump from
the base of one of the headers in turn through a cooling
heat exchanger, a rotameter bank, a thermometer pocket
and into the top of the second header and through
one of the connecting pipes back to the pump. The
system was valved (ball type) so as to give flexibility
226
E. BENARD ET AL.
Asia-Pacific Journal of Chemical Engineering
of use and was fitted with a bypass. Operation could be
either electrically earthed or insulated. This facility was
included because preliminary work on aqueous polymer
solutions indicated a pronounced effect on the observed
pressure loss if static charge was present in the system.
Pressure loss measurements were taken over a central 4 m length of pipe. Water and aqueous solutions of
sugar, glycerol, NaCl and Na2 CO3 were examined at
various concentrations. Physical properties of the electrolyte solutions, etc. were obtained from International
Critical Tables for actual experimental conditions. The
rotameters used were checked by timed volume output
measurements.
RESULTS
The pressure loss obtained for water in the three
pipes followed the Moody diagram with a roughness
of 0.00156 cm. All solutions tested gave agreement
with the Hagan–Poiseuille equation for laminar flow.
Aqueous sugar and glycerol solutions followed the
Moody diagram in the turbulent regime. However, the
electrolyte solutions exhibited a positive deviation in
the turbulent region as shown in Figs 1 to 3 for NaCl.
There was no observable variation with the insulated
or earthed system. The data indicated that the increase
in pressure loss above that for water depended on the
tube diameter and electrolyte concentration. Evaluation
of apparent viscosity is detailed in the Appendix.
Figure 2. The friction factor – Reynolds number relation
for various NaCl solutions flowing in a 4.22 cm internal
diameter PVC pipe.
DISCUSSION
By assuming that the observed increase in pressure drop
over that of the Moody diagram occurs in the boundary
Figure 3. The friction factor – Reynolds number relation
for various NaCl solutions flowing in a 5.24 cm internal
diameter PVC pipe.
layer adjacent to the inner pipe wall, calculation by the
Born equation
−zi2 e 2 NA
1
◦
(1)
1−
G =
8π ∈o ai
∈
Figure 1. The friction factor – Reynolds number relation
for various NaCl solutions flowing m a 2.93 cm internal
diameter PVC pipe.
 2007 Curtin University of Technology and John Wiley & Sons, Ltd.
indicated that there was sufficient Gibbs free energy
of ionic solution to account for the pressure loss
enhancement observed in Figs 1 to 3. Attention was
directed to the electroviscous effect of solid particles
suspended in ionic aqueous liquids, which has been
shown to increase the viscosity of the suspension
over that of the pure liquid. Three effects have been
identified: The primary electroviscous effect occurs due
to distortion by the shear field of the electrical double
Asia-Pac. J. Chem. Eng. 2007; 2: 225–229
DOI: 10.1002/apj
Asia-Pacific Journal of Chemical Engineering
DRAG ENHANCEMENT IN TURBULENT PIPE FLOW
layer surrounding the solid suspended particles; The
(supposed larger) secondary effect results from the
overlap of the electrical double layers of neighbouring
particles; The tertiary effect arises from changes in size
and shape of the suspended solid phase, e.g. colloids,
by the shear field. The first two effects depend on:
liquid led to an increase in the apparent viscosity ηa of
the system. Thus for the primary electroviscous effect,
Eqn (2) was presented without derivation.
1
5ν
ζ ∈ 2
(2)
1+
ηa = η 1 +
2V
σ ηa 2 2π
1. The size of the Debye thickness of the electrical
double layer compared to the size of the suspended
particles;
2. The potential at the slipping plane between the
particle and the bulk fluid;
3. Variations in the Stern layer around the particle;
4. The Peclet number, that is diffusive to hydrodynamic
forces;
5. The Hartmann number, that is electrical to hydrodynamic forces.
It is possible to apply the relation to the case
of an aqueous electrolyte by assuming that the ions
themselves are the charged particles dispersed in water.
The values of the various parameters used are given in
Table 1. The Vν volumetric ratio was calculated from
the density data. The results given in Table 2 show that
the Smoluchowski Eqn (2) gave a result well below
the data of Figs 1 to 3 and about 10% below the
actual viscosity value reported in the literature for these
solutions.
Street (1958) gave a simple derivation for the apparent viscosity taking into consideration both the primary and secondary electroviscous effects. The equation was eventually confirmed by others (Whitehead,
1969; Stone-Masui and Watillon, 1970).
1
5ν
∈ ζ 2
a 2
1+
1+
ηa = η 1 +
2V
κ
5σ ηa 2 2π
ν 2
2.5
∈ ζ 2
a 2
(3)
+
1
+
V
κ
2σ ηa 2 2π
Theories of the electroviscosity of solid suspensions
in electrolytes have been developed, each being able
to handle some but not all of the above five variables
(Smoluchowski, 1916; Henry, 1931; Sumner and Henry,
1931; Bull, 1932; Krasny-Ergen, 1936; Overbeck, 1943;
Booth, 1950a,b; Street, 1958; Chan et al ., 1966; StoneMasui and Watillon, 1968; Whitehead, 1969; StoneMasui and Watillon, 1970; Russel, 1976; Seville, 1977;
Russel, 1978a,b; O’Brien and White, 1978; Lever, 1979;
Sherwood, 1980; Watterson and White, 1981; O’Brien,
1983; Hinch and Sherwood, 1983; Honig et al ., 1990;
Mangelsdorf and White, 1990; Dukhin and van de Ven,
1993; Sherwood et al ., 2000; Rubio-Hernandez et al .,
2001; Wada, 2005).
Smoluchowski (1916) first noted that the electrical
double layer surrounding a charged solid particle in a
Table 1. Values of parameter used in the calculations.
NaCl2 concentrations % w/w
a
ν
V
σ
∈
ξ
η
10.37
13.65
1.415 × 10−10
0.03686
15.020
695.24 × 10−12
0.088
0.001002
1.415 × 10−10
0.05076
17.834
695.24 × 10−12
0.088
0.001002
m
–
−1 m−1
F m−1
V
kg m−1 s−1
Where κ is the Debye length at 298.12◦ K
κ=
∈ kT
1
2
(4)
8π e ni
Table 2 shows again a result well below the data but
a slight improvement over the Smoluchowski equation.
The secondary electroviscous effect within Equation (3)
was only 5% of the total.
The Booth Eqn (5)
(Booth, 1950a) is accepted as being a definitive
modelling of the electroviscous effect
1
ν
∈ ζ 2 a 2
1+
.π
ηa = η 1 + 2.5
V
κ
σ ηa 2 2π
a a 2 ∈ a 1+
ζ
(5)
× 1+
κ
e
κ
2
Table 2. Calculated and experimental apparent viscosity.
NaCl2 concentrations % w/w
Smoluchowski (Eqn 2)
Street (Eqn 3)
Booth (Eqn 5)
Measured from pressure loss
10.37
13.65
0.001123
0.001162
0.002991
0.001972 ± 16%
0.001163
0.001230
0.003052
0.003131 ± 12%
 2007 Curtin University of Technology and John Wiley & Sons, Ltd.
kg
kg
kg
kg
m−1 s−1
m−1 s−1
m−1 s−1
m−1 s−1
Asia-Pac. J. Chem. Eng. 2007; 2: 225–229
DOI: 10.1002/apj
227
228
E. BENARD ET AL.
Asia-Pacific Journal of Chemical Engineering
Indeed Table 2 shows that prediction of the apparent
viscosity caused by the electroviscous effect could be
achieved within a few percent. It was not possible to
accommodate any tube diameter effect with the Booth
model.
CONCLUSIONS
Drag enhancement reported for aqueous electrolyte
solutions has been shown to be the result of the electroviscous effect due to the distortion of the electrical
double layer surrounding the ions in solution by the
shear field. Using the model of Booth, it was possible to predict the electroviscous effect and the apparent
viscosity of the electrolyte solutions in turbulent flow,
assuming the ions as the charged particles in solution.
Reynolds number
Velocity, m s−1
Particle volume, m3
Total volume, m3
Ion i charge number
Surface roughness, m
Dielectric constant, F m−1
Dielectric vacuum constant, F m−1
Potential solid/liquid interface, V
Viscosity of pure solvent liquid, kg m−1 s−1
Debye length, m
Density, kg m−3
Specific conductivity, AV−1 m−1
Friction factor
SUBSCRIPT
a Apparent
APPENDIX
Calculation of apparent viscosity ηa from pressure
loss data. The pressure loss and velocity measurement
obtained experimentally allowed for the calculation of
the friction factor and Reynolds number.
P d
l 4ρu 2
Re = duρ/η
φ=
These gave the experimentally measured dashed lines
in Figs 1 to 3. Only the Reynolds number value
depended on the value of the viscosity employed unlike
the friction factor, which was independent of viscosity.
If the friction factor derived coincided with the Blasius
relation (that is the full line in Figs 1 to 3), then
φ = 0.0396Rea−0.25
A value of Rea is obtained which included the
apparent viscosity ηa
Rea = duρ/ηa
NOMENCLATURE
a
d
e
E
G
l
ni
NA
Re
u
v
V
zi
δ
ε
εo
ζ
η
κ
ρ
σ
φ
Particle radius, m
Pipe diameter, m
Electronic charge, C
Streaming potential gradient, V m−1
Gibbs free energy, J (g mol)−1
Length, m
Number of ions i , m−3
Avagordro’s number, g mol−1
 2007 Curtin University of Technology and John Wiley & Sons, Ltd.
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229
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