DRAG REDUCTION OF A CATIONIC SURFACTANT SOLUTION AND ITS SHEAR STRESS RELAXATION*

CAI Shu-peng

Institute of High Pressure Water Jet, Hunan University of Technology, Zhuzhou 412007, China,

E-mail: nagoya_cai@sohu.com

DRAG REDUCTION OF A CATIONIC SURFACTANT SOLUTION AND ITS SHEAR STRESS RELAXATION*

CAI Shu-peng

Institute of High Pressure Water Jet, Hunan University of Technology, Zhuzhou 412007, China,

E-mail: nagoya_cai@sohu.com

In order to study the mechanisms of the turbulent frictional drag reduction by surfactant additives, the drag reduction, the shear viscosity and the shear stress relaxation were measured for solutions of a cationic surfactant cetyltrimethyl ammonium bromide (CTAB) with the same molar sodium salicylate as a counter-ion. It is found that the first step relaxation time decreases with increasing concentration and, thus, with the maximum drag-reducing rates, which indicates that the stiffness of the micellar structures with the first relaxation time, increases with the increase of the concentration of CTAB. Furthermore, for this surfactant, a viscoelastic property is necessary for reducing drag, while a stronger viscoelasticity characterized by a tail relaxation time does not necessarily mean a higher drag-reducing rate.

drag reduction, shear stress relaxation, relaxation time, viscoelasticity

Introduction

The frictional drag reduction in a fluid-transporting pipe is one of the most important problems and has been much studied experimentally and theoretically. The typical methods used for drag reduction include Tom’s effect, riblets[1], large eddy breakup devices[2], compliant walls or flexible tubes[3]and suspensions[4]. In these methods, with a small amount of certain surfactants added into the water flow, the turbulent frictional drag reduction can reach up to 80%. This promising technology to lower the pumping power of district heating-cooling systems has recently been widely accepted because no mechanical degradation is caused by the strong mechanical shear. In the existence of certain counter-ion, molecules of amphiphilic surfactant would form rod-like micelles, which have been believed to be necessary for drag- reduction by surfactants.

The mechanisms of turbulent drag reduction have been much studied[5,6], focusing on the viscoelastic behavior of drag-reducing cationic surfactant systems. It is generally believed that the viscoelastic rheological characteristics of a surfactant solution are responsible for the effect of turbulent drag reduction[7-10]. For a surfactant solution, the rod-micelles and the Shear-Induced Structure (SIS) were reported to be responsible for the behavior of the viscoelasticity, thus, for the turbulent drag reduction[11-14]. Generally speaking, the viscoelastic properties are characterized by the overshoot of the shear stress with a start-up, the first normal stress difference, and recoil after a swirling of a surfactant solution sample. In other words, if a fluid is viscoelastic, its viscoelastic properties would be evident. To confirm the viscoelastic properties of a surfactant system, the first normal stress difference would be firstly examined, but can not be accurately measured due to the very significant inertial effects for a value less than 1 000 Pa[9]. Secondly, the stress overshoot with a start-up would be observed, but the overshoot is often not present for a dilute surfactant solution used for experimental studies. Some Direct Numerical Simulation (DNS) studies[12,14]on cationic surfactant drag-reducing flows with viscoelastic characteristics have been performed in the last decade, although, it is quite difficult, so far, to perform the DNS analysis of the drag reducing flows in a surfactant solution with rheological parameters exactly of the real turbulent flows. On the other hand, there areonly a few studies for the quantitative relations between the SIS and the viscoelastic property and in particular between the drag reduction and the shear stress relaxative characteristics[5,9]. Therefore, the mechanism of surfactant drag-reducing is still not clearly understood. In the present study, the dragreducing characteristics of solutions of a cationic surfactant, cetyltrimethyl ammonium bromide (CTAB) and sodium salicylate as a counterion flowing in a circular pipe of 0.005 m in inner diameter are first measured, and the shear-rate dependent shear viscosities are measured with a MR301 rheometer, to confirm the SIS rheological behavior. Also, the relaxation of the shear stress after the constant shear rate is removed is examined with an ARES-LS G2 strain-controlled rheometer, in order to investigate the relaxation characteristics of the shear stresses and the relation between the drag reduction and the viscoelasticities characterized by shear stress relaxation times.

1.Experiments

1.1 Materials

The cationic surfactant tested is cetyltrimethyl ammonium bromide, CTAB, a commercial product donated by Japan WAKO chemical company. Sodium salicylate, NaSal is used as a counterion. CTAB with the same molars of NaSal is dissolved in distilled water at 50 ppm, 100 ppm, 300 ppm, 500 ppm, 750 ppm and 1 000 ppm.

Fig.1 Schematic diagram of pressure loss test apparatus

1.2 Drag reduction measurements

The drag reduction measurements were carried out in the experimental facility as shown in Fig.1. In order to make the flow fully developed in the test section, the entrance length is 360 times of the pipe diameter, much longer than what is required of 50 times of the pipe diameter for Newtonian fluids, since the flow development is much slower for a drag-reducing surfactant solution than that for a Newtonian fluid[5].

The drag reduction measurements follow the procedures as follows. The storage tank is pressured, then the control valve is opened to let the test solution pass through the test pipe. Once the solution flows through the test pipe, both the pressure in the storage tank and the flow rate are decreased gradually due to the limited volume of the pressured reservoir tank. The pressure drop and the flow rate are measured by a differential pressure transducer and a magnetic flow meter, respectively. Each drag reduction measurement takes about 55 s. A pseudo-steady state is assumed in the calculation of the frictional factors from the recorded pressure drop and the flow rates. Reynolds numbers are calculated based on the water viscosity.

1.3 Shear viscosity measurements

Shear viscosity is measured by using a MR301 stress-controlled rheometer with a cone-plate flow cell, with cone angle of 0.0395 radians, cone-plate diameter of 0.05 m and gap of 9.9×10–5m. The sampling frequency for shear viscosity readings is 0.04, to ensure the viscosity to reach the equilibrium at each shear rate. Viscosity for each shear rate is taken as the average reading of clockwise and counter clockwise measurements.

1.4 Shear stress relaxation measurements

Measurements of the shear stress relaxation are carried out by using an ARES-LS G2 strain-controlled rheometer with a cone-plate of 0.04 radians in cone angle, 0.05 m in diameter and 3.9810–5m in gap. A constant shear rate is imposed on the test samples until the shear stress reaches the equilibrium value. It is then removed and the transient shear stress as a function of time is recorded.

Fig.2 Friction factor versus Reynolds numbers

2. Results and discussions

2.1 Drag reduction

The friction drag factors for the solutions with various concentrations at 20oC are shown in Fig.2, in which the Blasius empirical formula and Virk’s maximum drag reduction asymptote are also plotted for comparison. The calculations from the measured pressure drop and the flow rate are based on the following formula

where f is the friction factor, D is the pipe inner diameter, Δp is the pressure drop, ρ is the solu-tion density, L is the test section length, and umis the bulk velocity.

As shown in Fig.2, the CTAB solutions can reduce drag only at concentrations larger than 50 ppm, but not below or equal 50 ppm (data for the 50 ppm solution are not included). It can also be seen from the measured data that the critical Reynolds number with a maximum drag reduction rate increases with increasing concentration of CTAB. This drag reduction behavior indicates that with an increase in the surfactant concentration, the network structures of micelles in the solution can better resist the shearing and make the flow pseudo-laminar up to a higher Reynolds number, and, thus, further inhibit the turbulent energy dissipation, resulting in a higher turbulent drag reduction. This suggests that the aggregated microstructures of rod-like micelles are strengthened following an increase in the concentration. The critical Reynolds number Recrtand the critical wall shear stress τcrtas a function of the surfactant concentration are shown in Fig.3.

Fig.3 Critical wall shear stress versus concentration

2.2 Shear viscosity

The shear viscosity measurements for the solutions with various concentrations at 20oC are shown in Fig.4.

Fig.4 Shear viscosity versus shear rate

As seen from the measured results, for all the CTAB solutions with the drag-reduction effect, the shear viscosity is close to the Newtonian behavior at very low rates, and it suddenly increases at a critical shear rate, then, followed by a shear thinning. The structures of micelles with the behavior that its shear viscosity increases with the shear rate are called the SIS. The SIS rheological behavior referred to by some researchers is a large scale agglomerated structure of micelles induced by shear flow under sufficiently high shear rates, and is considered to be the vital condition for the surfactant drag reduction effect. A more likely explanation is that the SIS promotes viscoelastic properties of surfactant solutions to suppress small scale turbulent eddies, thus reducing turbulent energy losses. The peak value of shear viscosities in the SIS state increases with the increase of the concentration, perhaps due to the fact that the length of rod-like micelles increases with the increase of the concentration. Therefore, it is important to clarify the relation between rheological characteristics and viscoelasticities in SIS conditions.

2.3 Shear stress relaxation

Shear stress relaxationmeasurements are very important for examining the aggregated structures of fluid molecules induced by a shear flow and for making sure whether the fluid is viscoelastic or not. The relaxation time of shear stress is an important time scale parameter for a viscoelastic fluid. It is generally believed that a long shear stress relaxation time is associated with a strong viscoelastic property. The shear stress relaxation measurements for the solutions with various concentrations at 20oC are shown in Fig.5. The constant shear rate is 100 s–1at which the solution is in the SIS state.

Fig.5 Relaxa tion of shear stress at 20oC

As widely known, the shear stress for water without v iscoelastic property will be instantaneously relaxed. The shear stress relaxations of surfactant solutions with drag-reducing behavior used in this study are obviously different from water as seen from Fig.5, which indicates that a shearing flow in the SIS condition brings about a shearing strain elastic energy in the agglomerated structures of micelles. The shear stress relaxation is a discharging process of the shearing strain elastic energy accumulated by shearingflow.

The shear stress relaxation of the surfactant solution often involves multiple-step relaxation times[9]. To fit the relaxation data tested as well as possible, the following Maxwell model of four orders is used.

where τand τ0are the shear stress at time t and the inertial shear stress at inertial time t0, λ1, λ2, λ3and λ4are the relaxation times in respective different steps,A, B, C and D are the coefficients indicating the contributions from the respective surfactant micelle structures with respective relaxation times related with the initial shear stress.

It is found that for the solutions of 300 ppm, 500ppm, 750 ppm, and 1 000 ppm, a four-step Maxwell model can fit very well the relaxed processes of shear stresses, for the solution of 100 ppm, a three-step Maxwell model is adequate, while for the solution of 50 ppm without drag reducing effect, a two-step Maxwell model can well fit its shear stress relaxation data measured.

Fig.6 The Maxwell model and shear stress relaxation for the solution of 750 ppm

Table 1 The relaxation times at various concentrations

A comparison between the Maxwell model and the measurements of the shear stress relaxation for the solution of 750 ppm is shown in Fig.6, and the relaxation times for the respective step for the solutions with various concentrations, are listed in Table 1.

2.4 Viscoelastisity and drag-reduction

Combining the measurements of drag reduction and those of shear stress relaxation, it can be seen that From themeasurements of the drag-reduction and the relaxation times of shear stresses, the following conceptual model is proposed for the drag reduction in a surfactant solution flow with viscoelastic property. In the first step, corresponding to micelle structures with the first relaxation time, the nodes on the solution of 50 ppm does not have the drag-reducing capacity due to its shorter tail relaxation time indicating a weaker viscoelastic property. This dragreducing behavior is interpreted as resulting from the fact that the structures of micelles with a very weak viscoelastisity would not interact with small scale eddies associated directly with the dissipation of the turbulent kinetic energy in a turbulent flow, and, thus, would not reduce the turbulent frictional drag. On the other hand, the first relaxation time decreases with an increase in concentration, which suggests that an increase in concentration can reinforce the stiffness of the agglomerated structures of micelles with the first relaxation time and, therefore, the micelle structures against shear would become stronger. This observation agrees with the above analysis about the reason why the drag-reducing capacity increases with increasing concentration. Also, the tail relaxation time (the longest among all relaxation times) increases with increasing concentration of CTAB up to 750 ppm, while a concentration of 1 000 ppm would not result in the longest relaxation time. Since a very long tail relaxation time is often considered to be accompanied by a very strong viscoelasticity, it is clearly confirmed from the tail relaxation times listed in Table 1 that for the CTAB solutions, there is not an obvious positive correlation between the viscoelasticity characterized by the tail relaxation time and the drag reducing capacity. Thus, the CTAB solution with a stronger viscoelastic property does not necessarily bring about a larger reduction in friction drag. For a surfactant dragreducing flow, both in a channel and in a pipe, some studies show that the sum of the Reynolds stress and the viscous stress may not be equal to that calculated from measurements of the pressure gradient. This stress deficit is due to the generation of additional shear stresses caused by interaction of micellar structures with the turbulent flow field. A micellar structure with a strong viscoelasticity would cause a strong viscoelastic shear stress to obtain turbulent kinetic energy from a mean flow field, which could explain why a surfactant solution with a stronger viscoelasticity does not necessarily cause a larger dragreducing rate.

From themeasurements of the drag-reduction and the relaxation times of shear stresses, the following conceptual model is proposed for the drag reduction in a surfactant solution flow with viscoelastic property. In the first step, corresponding to micelle structures with the first relaxation time, the nodes onthe network structure entangled by rod-like micelles would not be dislocated in the short time scale such as 0.1 s, but become loose so that some drag-reducing effect will appear but in a way of slow decrease. In the second step, the nodes on the network structures of micelles are partly dissociated, therefore, the dragreducing effects are considerably reduced. In the third step, the network structures partly connected by rodlike micelles are completely destructed, replaced by a lot of individual rod-like micelles, thus, the drag-reducing effect is very small and close to zero. In the fourth step, the rod-like micelles are changed to spherical shapes and the drag reducing effect is completely lost. Micellar structures with the first relaxation time are dominant in the first step, while micellar structures with other relaxation times play respective different roles in the other steps.

Monotonously increasing shear rate which means an increase in the Reynolds numbers in an actual flow can realize the adverse transition of micellar network structures and, thus, raise monotonously the dragreducing rates.

3. Conclusions

In the present work, using a cationic surfactant CATB, the drag reduction and the shear stress relaxation were measured. The following conclusions are drawn.

(1)For the CTAB solutions, the shear stress shows complicated relaxation characteristics with multiple-step relaxation times. This experimental evidence can explain why some direct numerical simulation studies using the constitutive equation modelsviscoelastic drag-reducing flow.

(2) For the CTAB solutions, to have frictional drageffects, a rheological viscoelastic property is necessary, but viscoelasticity characterized by one or two shear stress relaxation times is not enough to reduce the frictional drag.

(3) The critical Reynolds number with a maximumdrag-reducing rate increases with the increase of the concentration of the CTAB solutions, since the stiffness of micellar structures with the first relaxation time increases with the increase of the concentration of the CTAB solutions.

(4) In the concentration range for an effective dragreduction, there is not an obvious positive correlation between the maximum drag-reducing rate and the viscoealsticity characterized by the tail relaxation time of the shear stress relaxation. This experimental observation indicates that a viscoelastic property is necessary for the surfactant drag reduction, but a stronger viscoelasticity does not necessarily means a larger drag-reducing rate.

Acknowledgement

The present author gratefully acknowledges the support provided by Professor Suzuki Hiroshi and Professor Takahashi Tsutomu.

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June 29, 2011, Revised December 22, 2011)

* Project supported by the Natural Science Foundation of Hunan Province (Grand No. 09JJ6068).

Biography: CAI Shu-peng (1963-), Male, Ph. D., Professor