Review of mechanical degradation and de-aggregation of drag reducing polymers in turbulent flows

Review of mechanical degradation and de-aggregation of drag reducing polymers in turbulent flows

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Review of mechanical degradation and de-aggregation of drag reducing polymers in turbulent flows Edson J. Soares PII: DOI: Reference:

S0377-0257(19)30419-7 https://doi.org/10.1016/j.jnnfm.2019.104225 JNNFM 104225

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Journal of Non-Newtonian Fluid Mechanics

Received date: Revised date: Accepted date:

22 February 2019 13 December 2019 16 December 2019

Please cite this article as: Edson J. Soares, Review of mechanical degradation and de-aggregation of drag reducing polymers in turbulent flows, Journal of Non-Newtonian Fluid Mechanics (2019), doi: https://doi.org/10.1016/j.jnnfm.2019.104225

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Highlights • The main theories and advances in polymeric DR are discussed • Important aspects of mechanical polymer degradation are pointed out • The importance of some important variables to accelerate or delay the mechanical polymer degradation is discussed • The role that polymer de-aggregation plays on the fall of polymeric DR is discussed • The single available numerical work on mechanical polymer degradation is discussed

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Review of mechanical degradation and de-aggregation of drag reducing polymers in turbulent flows Edson J. Soares LABREO, Department of Mechanical Engineering, Universidade Federal do Espirito Santo, Avenida Fernando Ferrari, 514, Goiabeiras, 29075-910, ES, Brazil

Abstract The efficiency of drag reduction (DR) is known to be impaired by mechanical scission of molecules, which is a great obstacle to the practical use of flexible polymeric drag reducers. In this review, I try to give an account of the main aspects of this problem, pointing to the important variables (concentration, molecular weight, temperature, Reynolds number, quality of solvent, residence time, and relaxation time) that accelerate or delay the molecular scission of linear polymer chains in turbulent flows. The impairment of DR can also be related to de-aggregation, owing to the rupture of the intermolecular associations, instead of intra-molecular mechanical scission. Keywords: Polymer degradation, polymer de-aggregation, drag reducing flows 1. Main aspects of drag reduction by polymeric additives Drag reduction (DR) was unexpectedly observed by Toms [1] when he was trying to investigate polymeric degradation in turbulent flows. He obEmail addresses: [email protected] (Edson J. Soares), +55 27 33579500 5028 (Edson J. Soares) Preprint submitted to Elsevier

December 19, 2019

served that a small quantity of polymethyl methacrylate (a synthetic flexible polymer) diluted in monochlorbenzene was able to significantly reduce the pressure drop in turbulent flows through straight tubes1 . After this pioneering work, DR by polymers has been much analysed, mainly due to its many applications, including fire fighting [3], transport of liquids in pipelines [4, 5, 6], agriculture [7] and medicine [8, 9, 10, 11, 12, 13]. From the point of view of the friction factor, polymeric DR occurs when the skin friction of a polymer solution (fp ) is below that of a Newtonian fluid (fs ) flowing at the same Reynolds number [14], which leads to

DR = 1 −

fp . fs

(1)

Lumley [14] mentions a number of important features of DR, which have been widely analyzed. Among them are: the polymer’s concentration, molecular weight, temperature, molecular length and structure; the polymeric flexibility and relaxation; the Reynolds number; and the quality of the solvent. Virk et al. [15] checked a range of concentrations and molecular weights of many kinds of polymers and found that the DR is bounded by the so called maximum drag reduction asymptote (MDR) or, simply, Virk’s law (blue line in Fig. 1). The MDR can be reached by increasing the polymer concentration (c) or molecular weight (Mv ) at fixed Re (red triangles in Fig. 1). Alternatively, the MDR can be reached by increasing Re (brown circles), but often, it is not reached, even at high Re (black crosses). The point where the symbols detach from the Blasius law (red line) is the onset of DR. A great overview of 1

The history of Toms’s early experiments on DR can be found in [2]

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polymeric DR, including the effects of concentration, molecular weight, and quality of the solvent can be found in Virk [16]. Good recent reviews of the progress in understanding and predicting polymeric DR are [17, 18, 19, 20].

Figure 1: A scheme illustrating the onset of DR and how the MDR (blue line) can be reached. The black line and the red line are, respectively, the friction factor for the laminar flow and the for the turbulent flow in a straight smooth tube. The red triangles represent the case in which Re is fixed and DR increases by increasing c or Mv . The brown circles and black crosses represent the cases in which c or Mv are fixed, and Re is increased.

Much effort has been devoted to describing the mechanisms of DR. The first idea comes from Lumley [14], who argues that polymers are stretched by the turbulent structures and, as a result, the extensional viscosity increases, 3

which works to dissipate the small vortices. Ryskin [21] and Benzi [22] used this idea to build their model. Based on different concepts, Tabor and de Gennes [23] proposed the so called cascade theory of drag reduction, which is based on elastic effects. They do not attribute the dissipation of vortices to an increasing extensional viscosity, but consider that the polymers directly interact with the turbulent structures in a coil-stretch cycle. Under specific flow conditions, the turbulent kinetic energy is transformed by a counter-torque like mechanism into elastic energy, some vortices are destroyed, and the elastic energy is released to the mean flow. As a consequence, the Kolmogorov cascade has its smaller scales cut off. This idea is followed by many authors who enumerate important entities that interact in cyclic mechanisms: the mean shear, the turbulent structures, and the polymers [24, 25, 26, 27, 28]. The polymers are partially stretched by the mean shear and this fact is especially interesting. At the beginning of the polymeric DR, the molecules are mainly coiled. Since they take energy from the mean shear flow, the drag instantaneously increase before starting to decrease. Dimitropoulos et al. [29] numerically analysed the evolution of DR in a boundary layer flow of a FENE-P fluid, in which this drag increase can be clearly observed. Very recently, Pereira et al. [26] analysed the transient flow of FENE-P fluids between parallel plates and also numerically showed the drag increase. It seems that the drag increase is closely related to the conformational state of the molecules before the test, as supported by experimental data reported by Andrade et al. [30]. It is evident that the understanding of the mechanisms of DR has been constantly improved, but many aspects are still not well understood, such

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as the effect of mechanical polymer degradation. Another important aspect is the so called de-aggregation, as pointed out in Shetty and Solomon [31]. The main point of the present review is to discuss these two points in more detail, with particular attention to linear polymers, both flexible and rigid. 2. Main aspects of mechanical polymer degradation The mechanical polymer degradation occurs when polymers cannot sustain the hydrodynamic force imposed by the flow. In order of importance, it is the first cause of decrease in the efficiency of flexible drag reducers. A sketch of molecular scission in a turbulent flow is shown in Fig. 2. Essentially, an initially coiled flexible polymer needs to be entirely stretched before breaking [32]. Such a fully extended state can be allowed by a pure elongational flow. An example is a flexible polymer flowing close to the center of a cross-slot channel. The polymer can be easily extended in this geometry and eventually breaks [33, 34]. It is worth noting that the shear intensity has secondary importance in the degradation mechanism. Even a powerful pure shear flow is not able to break a polymer chain, since in this kind of flow the polymer tumbles, preventing the complete polymer stretching [35]. However, if a polymer is previously sheared before being submitted to an elongational flow, it can reach its fully extended state faster. In this case, the shearing can favor mechanical degradation [33]. In a pure extensional flow, a macromolecule is stretched by the mean extensional flow, and the force is maximal at its center. A force balance on the chain leads to the following relation applied to its center [32]: f ∝ εηM ˙ v 2, 5

(2)

where ε˙ is the mean elongation rate, η the solvent viscosity and Mv the molecular weight. Hence, when a critical force is reached (a rupture force), a long molecule mainly breaks on its center, reducing the molecular weight to half of its initial value. In turbulent flows, the force f can also be scaled by Mv 2 [36], but different from laminar elongational flows, the critical kinematic is not a mean extensional flow. The mean shear flow plays a role, but polymers are mainly stretched by the turbulent structures (hyperbolic and elliptical structures [28]) (see Fig. 2) by means of a counter torque like mechanism [23], whose intensity produce a local flow extension that I will call from now on ε˙ap . The main difference from the laminar flow is that ε˙ap is not constant in time (Eulerian point of view) and it is highly dependent on polymer concentration and molecular weight and such a fact is crucial to understand how these variables affect the efficiency of DR. Mean shear flow Vortex

Stretching Mechanical scission

Figure 2: Sketch of molecular scission.

There is a great deal of experimental evidence for very intense molecular scission under turbulent flows, such as that reported by Merril and Horn [37, 38], who analysed a turbulent flow of solutions of polystyrene through a 6

straight tube and compared the molecular weight distribution, with the aid of chromatography, of the degraded solution (after one single pass through the system) with the original one, a sample that was not passed through the system. They not only proved that the polymers are highly degraded by the turbulent flow, but also gave good arguments that the scissions mainly occur at the midpoint of the macromolecules, identical to laminar elongational flows. The molecular weight distribution of the degraded solution is shifted to the left after one pass through their system and, in addition, its peak is centered at Mw = Mw0 /2, reduced by one-half from the original solution’s peak. Their data are consistent with the measurements of the chain’s fraction induced by the cross-slot flow provided by Odell [32], who showed that when an isolated macromolecule is subjected to a strain rate above a critical value, its molecular weight is reduced to one-half of its initial value. The chromatography analysis conducted by Vanapalli et al. [39] using a solution of polyethylene oxide (PEO) further supports the midpoint scission idea in turbulent flows of polymers. They used an experimental apparatus very similar to that used by [37, 38]. Vanapalli et al. measured the molecular weight distribution in PEO by passing the solutions through their system multiple times and one can infer that the molecular weight’s peak is also basically reduced by half after one pass through the system. It is well known that after a long enough time, the molecular weight distribution reaches a final form. Such a fact was independently demonstrated, for laminar elongational and turbulent flows, by different researchers. Using chromatography, Lee at al. [40] demonstrated that the mean molecular weight of polyisobutylene solubilized in kerosene and its distribution reaches 7

Number of molecules

Asymptotic state, 30 min

15 min 5 min Initial state

1x106

2x106

3x106

Molecular weight (g/mol) Figure 3: Sketch of the evolution of the molecular weight distribution of 100 ppm solution of polyisobutylene in kerosene. The turbulent flow is produced in a rotating disk apparatus at high angular velocity (ω = 77.8 rad/s).

(Data taken from Lee et al. [40]). a steady state in a turbulent flow very quickly, in a rotating apparatus with high speed, after around 30 min. From Fig. 3, one can see that, initially, the molecular weight spectrum is very broad, but tends to a narrow shape after 30 min. This means that the turbulent flow preferentially breaks the long molecules, and in the final state, all macromolecules have almost the same size. Vanapalli et al. [39] further proved such a fact. They performed many experiments quite similar to those reported in [41], in which a turbu8

lent flow of a polymer solution is studied without using pumps. By the aid of gel chromatography, Vanapalli et al. [39] showed that the molecular weight distribution reaches a final form very quickly, after only 10 passes through their system, composed basically of pipelines. As expected, the molecules do not keep breaking indefinitely. The breakage is accentuated at the beginning, but stops after a long enough time when a state of equilibrium between the polymers and the turbulent flow (mean shear flow and turbulent structures) is reached. 3. The transient nature of drag reduction As sketched in Fig. 4, the available results concerning flexible polymers [29, 26, 30] show that at the very start of a DR test with a well mixed solution (called homogenous drag reduction), the DR decreases from DR0 to DRmin , before achieving its top level of efficiency at DRmax . The maximum drag reduction occurs when a sufficient number of the molecules in a coil-stretch cycle are in a state of equilibrium with the turbulent structures [24]. The time to achieve DRmax is called developing time, td [42]. The increasing friction factor at the beginning of the process is related to an instantaneous increment of the local extensional viscosity after a high polymer stretching. Following td , it is observed a constant value of DR for a period of time, which is denoted tr , the resistance time [42]. Finally, after this period, DR begins to fall, reaching a minimum level after a long enough time, when the degradation or de-aggregation process has reached its steady state and DR assumes an asymptotic value, DRasy . The time to reach DRasy , ta , is relatively large compared with the stretching time of a single molecule, 9

because the molecules are stretched and degraded step-by-step [43]. Thus, I could presume that during tr the increasing number of molecules in the coilstretch cycle is balanced by the molecular degradation and the ultimate level of drag reduction is sustained for a period. Following that, with a continuous degradation, the DR starts to decrease until achieving its final steady state, DRasy . As an example, for a DR test with 100 ppm of PEO in a double gap device (Mv = 5.0 × 106 g/mol, T=25o C, Re =1360), td ≈10 s; tr ≈ 50 s; ta ≈ 3000 s [44]. The process for rigid materials is quite different. The DR does not assume negative values 2 and td is very fast [30]. The coil-stretch process, for example, probably does not exist in this kind of material, once the rigid molecules stay almost well-extended at rest. The decrease in the DR efficiency caused by the mechanical molecular scission or de-aggregation can be evaluated by the relative drag reduction, DR0 , given by DR0 =

DR . DRmax

(3)

For heterogeneous drag reduction, in a process where a polymeric solution is injected into a channel or duct, for example, td additionally depends on the time for the total polymer dilution. Even if the dilution is very fast, since a period of time is required to reach a state of equilibrium between polymers and turbulent structures, DRmax is expected to be observed far 2

In Fig. 4 DR0 ≈ 0. Hence, the values of DR in the range of DR0 ≤ DR ≤ DRmin are

negative, which means ”drag increase”.

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enough from the injection point. Further from such a point, because of polymer degradation or de-aggregation, the DR reaches the asymptotic value DRasy < DRmax . Since the degradation does not proceed indefinitely, the ultimate value of drag reduction is usually not zero, DR(t → ∞) = DRasy > 0. This aspect of mechanical degradation is a consensus and reported by many authors [45, 46, 47, 44, 48]. The total relative loss of efficiency is 1 − DR0 (t∞ ) = 1 − DRasy /DRmax . We can find in the literature many different ways of dealing with the loss of DR by degradation [46, 44, 49]. Particularly interesting is the model proposed by Vonlanthen and Monkewitz [49], who experimentally investigated the turbulent flow of PEO solutions with the aid of the particle image velocimetry (PIV). They proposed a very interesting cascade approach to estimate the mechanical polymer degradation, which is summarized by two equations. The average molecular weight is written in terms of the number of passes3 Np through their system as follows:

Mv (Np ) =

Mv0

∞ X mi i=0

2i

.

(4)

Here, Mv0 is the average molecular weight of the non-degraded solution and mi is the fraction of the polymer, given by eq. 6.1 of [49]: 3

It is worth noting that the effect of Np will depend on the length of the pipe. Hence,

a more appropriate variable should be the residence time.

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mi =

 

exp(−0.45Np ) h i P  1 − i−1 mj exp(−0.45 × 0.18i Np ) j=0

for i=0 for i > 0

(5)

Assuming i = 2 (which is an approximation) for one pass through the system (Np = 1), Eq. 4 leads to Mv (Np = 1) = 0.637Mv0 + 0.335Mv0 /2 + 0.028Mv0 /4 + ...., which means that the mean molecular weight undergoes a cascade: Mv0 , Mv0 /2, Mv0 /4, ...., Mv0 /2i . Thus, for one pass, around 63.7% of the molecules do not break, around 33.5% break once and around 2.8% break twice, and so on. The goal of this model is the spectral representation of the molecular weight, which changes step by step until an asymptotic level is reached, when Np → ∞. Such a model is a physically good description of the polymer degradation. The parameter mi , which balances the components of the spectrum of Mv , is dependent on the kind of polymer and on the kind of system. Eq. (5) is for PEO in the specific system analysed by [49]. If one assumes that the decrease in efficiency of a flexible polymer is exclusively related to molecular scission, the relative drag reduction can be approximated as a function of Mv (Np )/Mv0 . It would be quite interesting to conduct experiments in an attempt to find accurate functions of type DR0 = f (Mv (t)/Mv0 ). Such a kind of study is missing and deserves our attention. It is worth noting that many variables play an important role in the reduction of DR and, consequently, it is very difficult to find a universal law to take into account all of them. I will discuss those variables in the next sections.

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4. Important variables for mechanical polymer degradation The mechanical scission of a flexible polymer in a specific flow depends on a number of variables: 1. the concentration of the polymer in the fluid; 2. the molecular weight of the polymer; 3. the chemical structure of the polymer 4. the linear polymer configuration; 5. the temperature; 6. the turbulence intensity (usually measured by the Reynolds number); 7. the polymer-solvent interaction (the quality of the solvent); 8. the residence time (the period of time that the molecules undergo turbulent flow); 9. the polydispersity; 10. the geometry and local occurence: contractions, expansions, etc. In this section I discuss some of the critical literature mainly for turbulent drag reducing flows in which these variables have been treated. Since the rupture force for a specific kind of polymer is a constant, in a laminar elongational flow, the critical extension ε˙c (minimal necessary to observe molecule scission) is a decreasing function of viscosity and molecular weight (Eq. (2)) [50]. In other words, if the main extension is fixed, a more concentrated solution or a higher molecular weight polymer solution are more prone to break [33]. The polymer degradation in a turbulent flow is quite different, mainly because of the interactions between the hydrodynamic of the flow and the polymers. A pioneering work with the objective of determining 13

the effects of mechanical polymer degradation in a turbulent flow is Paterson and Abernathy [41]. In fact, mechanical degradation in turbulent flows had been previously noted by others [51, 52], but in Paterson and Abernathy [41] the degradation was investigated in more detail and over time, which is quite important for understanding its mechanism. They analysed many of the variables listed above, using PEO solutions, more specifically, the role played by its concentration, molecular weight, temperature, and shape of the pipe inlet. By measuring the intrinsic viscosity, which can be directly related to the molecular weight, they noted that the PEO degradation was greatly intensified by high temperatures. Additionally, the authors observed that the rate of degradation increases with the polymer’s molecular weight (Mv ) when submitted to the same flow intensity, as would be expected. Hence, the relative molecular weight Mv (t)/Mv0 , the current molecular weigh divided by its initial value, falls with Mv0 . In other words, the number of scissions per unit of time increases with the length of the molecule. The data from [41], displayed in Fig. 5, illustrates what was just said. They measured the intrinsic viscosity, which is directly related to the molecular weight, over time, for solutions of PEO with different initial intrinsic viscosity η0 . Note that the rate of degradation is higher for the solution with η0 = 25, but the final measured value of intrinsic viscosity ηf is also larger than for those with η0 ≤ 25. Such a fact is essential to understand the effect of molecular weight on the DR efficiency. In a laminar elongation flow, if the mean extension ε˙ is fixed, the asymptotic mean molecular weight is independent of Mv0 . In a DR flow, this is different. The molecular scission is strongly dependent on interactions be14

tween polymers and flow. If the Reynolds number is fixed, the turbulent vortices are continuously reduced by the polymers until a first asymptotic steady stated is reached (DRmax is reached). After this point, mechanical polymer degradation plays a role and turbulence increases again. As a consequence of that, DR is reduced until DRasy is reached. The rate of vortexes destruction at the beginning of a DR flow is an increasing function of Mv0 , which attenuates the polymer degradation. Thus, in a turbulent flow with a specific initial flow intensity, a solution with higher initial molecular weight reaches larger asymptotic molecular weight than that with smaller Mv0 , which implies a larger DRasy . Moreover, it seems that the solutions with larger Mv0 are also more efficient (smaller loss of efficiency). Pereira and Soares [44] conducted DR tests in a rotating apparatus using PEO solutions with different Mv0 at fixed concentration, temperature and Reynolds 0 number (shown in Fig. 6). The asymptotic relative drag reduction DRasy

for Mv0 = 5.0 × 106 (hollow black circles) is significantly larger than that for

Mv0 = 6.0 × 105 (blue triangles).

Despite many observations of the evolution of the molecular weight distribution in drag reducing flows, specific tests to accurately show how the distribution of Mv affects DR and DR0 are missing. A straightforward way to do that is by producing polymers with the same mean molecular weight, but with the different molecular distribution. I expected a significant effect of molecular weight distribution on DR, mainly at the beginning of the flow. A sample with specific molecular weight distribution with a more massive amount of long molecules would affect more the turbulent structures at the beginning of a DR flow. In this case, the DR would reach high asymptotic

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values. Such kind of investigation is missing and deserves attention. The role played by polymer concentration on the decrease of DR efficiency is very similar to that played by the molecular weight. That is, different from laminar elongational flows, in turbulent flows, a more concentrated solution is more resistant to mechanical degradation due to the hydrodynamic interactions. For a fixed Re, the turbulent intensity decreases with concentration. Soares et al. [48] recently demonstrated such a fact by measuring the molecular weight of PEO solutions (by means of its intrinsic viscosity) with different concentrations under the same initial flow intensity in a Taylor-Couette apparatus. They showed that Mv (t) is strongly dependent on concentration, c. In more detail, they demonstrated that the asymptotic molecular weight increases with c, as it also increases with the initial molecular weight Mvo (shown by [41]). Hence, a more concentrated polymer solution is, in fact, more resistant, as observed in many other works in which many different flexible polymers or geometries were used [53, 44, 42]. One remarkable paper, in which many of the variables listed previously were treated all together, is Moussa and Tiu [54], who conducted a number of tests using capillary tubes with different lengths and diameters, through which were passed solutions of polyacrylamide in water and polyisobutylene in cyclohexane and in toluene. Their main tests are displayed in terms of f × Re, as sketched in Fig. 7. Most of their values of the friction factor initially fit Virk’s MDR (maximum drag reduction) asymptote but depart from it when the degradation starts to play an important role. This value is called the critical Reynolds number, Re∗ . It is an increasing function of 16

c and Mv . That is a further proof that an increase of these two parameters makes the solution more efficient. The same is observed when the quality of the solvent is improved. As pointed by Moussa and Tiu, it is important to note that degradation also occurs below Re∗ , but during a first pass through the system at low Reynolds numbers, it can not be perceived because the amount of non-degraded polymer is still sufficient to maintain the MDR state. They also tried to investigate the effect of contractions and expansions but their data are not conclusive. In fact, it seems that analyses of the effects of local occurrences, such as contractions and expansions, among others, on polymer degradation are scarce, and that this question deserves more attention. An interesting study is Vanapalli et al. [39], where it is argued that the degradation is dominated by the entrance contraction. A different conclusion was reached by Soares et al. [48], who compared the values of DR obtained with a smooth entrance with those obtained with a straight one. It is worth noting that [39] used tubes with approximately 1.0 m of extension, while the apparatus in [48] had pipelines of 13.0 m. Hence, the residence time (the period in which the polymers undergo the turbulent flow) is totally different in these two studies. Thus, the role played by local occurrences (channel accessories, abrupt contractions, and other ) must be considered in terms of the pipeline’s extension. In other words, in short closed systems the role played by contractions or expansions can be of great significance, but this is not necessarily true in open or long closed systems. As mentioned before, the residence time is obviously quite important for degradation. As showed by Kalashnikov [47], studying a turbulent flow of PEO solutions in a rotating apparatus, the relative drag reduction starts to

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decrease only after a certain period of time, when the number of degraded molecules is sufficiently large to affect DR. As argued by Elbing et al. [55], who analysed experimentally a drag reducing flow over a flat plate, only a small percentage of the molecules start to stretch at each instant of time and, consequently, a period of time is required to have a significant number of molecules degraded before starting to see a fall in DR. This period of time, called by Pereira and Soares [44] the resistance time, is significantly larger than the stretching time for an individual polymer molecule. For the same reason, the asymptotic DR is not reached instantaneously. The resistance time tr and the total time to reach the asymptotic DR decrease with the Reynolds number, as expected, since the turbulence intensities increase with Re. Vanapali et al. [36] give a measure of the maximum drag force on a polymer chain as a function of the Reynolds number (Fmax ∼ Re2/3 ). Pereira et al. [44] carried out many tests to investigate how the resistance time depends on the concentration c, temperature T , and molecular weight Mv , in addition to any dependence on the Reynolds number Re. They show that tr is an increasing function of c and Mv and decreases with Re, as was seen in [47]. Another recurrent question is what kind of linear polymer offers more resistance to mechanical degradation [39, 56, 44, 42]. Clearly, those called rigid polymers are the most stable and, normally, with them, the DR does not decrease as much as it decreases when flexible polymers are used. Among the rigid polymers can be mentioned xanthan gum (XG), guar gum (GG), and many other bio-polymers. The water soluble flexible polymers most used are PEO and PAM. Pereira et al. [42] carried out a number of tests to

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compare PEO, PAM and XG. They found that PAM is less susceptible to degradation than PEO. Fig. 6 shows that DR0 for PAM (full black circles) is significantly larger than that for PEO (hollow black circles) for the same Mv . These data were confirmed later in [48]. Both are in agreement with Vanapalli et al. [39]. In terms of resistance to degradation, the linear biopolymers (rigid materials as okra and aloe vera, for example) are, in fact, very stable, as reported by many researchers [57, 58, 59]. Critical applications for stables materials include multi-pumping flows or systems with multiple local occurrences. 4.1. Conformational configuration of linear polymer chains Some of the variables listed at the beginning of Section 4 play a direct role in the relatively conformational configuration of the polymers (coiled, partially-expanded or highly-expanded); this is the reason I will now discuss it as a special topic. Polymers having linear structures are the most effective drag reducers and their conformational configuration has a special importance for their mechanical degradation [14]. A linear polymer in a solution at rest can be coiled, partially expanded, or highly expanded, depending on specific conditions, such as the quality of the solvent [60, 53]. Breakage is possibly favored by a state in which polymers are relatively less extended (more coiled) during the DR process [61, 62, 53, 63]. An arrangement in spiral form or in a sequence of rings makes the rupture of the bonds easier, possibly because a concentration of stress can occur at many points. Harrington and Zimm [61] showed that the critical stress could be significantly smaller in poor solvents in comparison with good ones. As the quality of the solvent is improved, the initial polymer conforma19

tion (in a solution at rest) becomes less coiled. In fact, in a better solvent, during all the coil-stretch process, the molecule’s mean length is relatively larger, and polymers start to perform better and become less susceptible to mechanical degradation. Zakin and Hunston [62] measured the drag reduction of polystyrene solutions in turbulent flows in capillary tubes in different solvents and clearly showed that the rate of degradation is more pronounced in the worst solvent. Kim et al. [53] studied the degradation of polystyrene in three different solvents (benzene, chloroform, and toluene). The decrease in the relative DR is clearly more pronounced in toluene (the worst solvent studied by [53]). Mechanical scission is accelerated in this solvent, in comparison with the other two. Choi et al. [64], studying the DR induced by DNA, show how the mechanical degradation dramatically depends on the polymer’s linear configuration. DNA degrades very quickly in distilled water, but it turned quite stable when a better solvent was used (a complex solvent including different components). The temperature T plays a very important and complex role on DR and, specifically, polymer degradation. The quality of the solvent can be improved by an increasing T . This means that the minimal polymer length in the solvent increases with T , which can improve DR and reduce the degradation. Nakken et al. [65] show an improvement in DR for turbulent flows of polystyrene solutions in a rotating geometry. In contrast, the relaxation time falls with temperature, which works to reduce DR and increases the degradation, since the number of coil-stretch cycles over a certain period of time is larger when the relaxation time is smaller. Hadri et al. [66] studied the effect of temperature on turbulent flows in pipes and found that, below

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a certain critical T , the degradation is delayed when the temperature is increased. Otherwise, the degradation is accelerated when the temperature is increased. A complex role of temperature was also reported in the work of Pereira and Soares [44] for turbulent flows of PEO and PAM in a double gap rotating device. They showed that the resistance time decreases with T . This means that the degradation starts more quickly at higher temperatures, but the asymptotic DR increases with T . Another way to modify the quality of solvents is the addition of ionic compounds in polymer solutions, for example. Salt is a very common ionic compound and very much studied. Andrade et al. [63] carried out many tests to study the effect of salt in solutions of PEO, XG, and PAM. The two first materials are very much affected by the salt, mainly the PEO solutions. That study compared two solutions with 50 ppm of PEO. In one of them, there was synthetic seawater with a salt concentration of 3.5% by weight. Besides the loss of the DR, the mechanical degradation was also intensified. Andrade et al. [63] showed that the maximum drag reduction is much smaller in the synthetic sea water and, in addition, the loss of efficiency is also intensified. The salt imposes a state of configuration on the molecules of PEO that is more coiled than in a solvent without salt. Elbing et al. [55] also reported effects of salt on PEO. The DR is markedly reduced by the presence of salt, though, concerning the degradation specifically, the authors say that it was not significantly affected. It is well known that complex solutions including polymers can improve DR, possibly changing the conformation of the flexible polymer present in the solution [67, 68, 69, 70, 71, 72]. Such complex mixtures can be polymer–

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polymer solutions, polymer–surfactant solutions, or of many other kinds. Two pioneering works which consider multi-polymer solutions are Dingilian and Ruckenstein [67] and Reddy and Singh [68]. They show that DR can be substantially improved in solutions that include rigid and flexible polymers together. In fact, the DR of a mixture of polymers in a solution can be larger than the sum of the DR of each single solution at the same concentration. This is called additive synergy. The reason for such an increase in the DR efficiency is the change of the configuration of the flexible polymer. Supposedly, the flexible polymer in the presence of the rigid one becomes more extended [73, 74, 16]. Evidence of such a hypothesis is the observation that the viscosity of mixtures of polymers can be higher than the viscosity of individual components alone at the same concentration level [68]. Very recently, Sandoval and Soares [71] concluded that mechanical degradation is also reduced in complex mixtures. They tested mixtures of (PEO) and (XG) and (PAM) and XG, showing that the degradation of PEO (a flexible polymer) is attenuated by the presence of XG (a rigid polymer). The attenuation of PEO degradation was also clearly verified by Novelli et al. [72] in co-presence of guar gum (another very common rigid polymer). In fact, it seems that mechanical polymer degradation can be reduced in complex solutions of many different kinds of materials, such as in mixtures of polymers and surfactants, as argued by Kim et al. [69], who conducted many tests with a synthetic polymer solution with surfactant additives. It is not obvious, at a first glance, that the co-presence of a second material (a rigid polymer, surfactant, or a fiber) in a polymer solution can reduce the mechanical degradation. A recent work of Steele et al. [70] provides good

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insights for a way to prove such a fact. They analysed the DR of mixtures of PEO and carbon nanotubes (CNT). This work is very illustrative because the carbon nanotube is not a drag reducer, but it can improve the efficiency of PEO. More precisely, they tested two solutions of 330 ppm of PEO. In one of them, they added 100 ppm of CNT. The friction factor was significantly smaller in the solution with CNT. Hence, the CNT-polymer interaction allows the combined mixtures to reach a higher drag reduction performance, possibly because the presence of CNT changes the conformational structure of the PEO: from a coiled state to a more extended form, which also increases the polymer’s resistance to mechanical degradation. They showed that the friction factor of the pure PEO solution reaches the Newtonian one after some time under a turbulent flow, but the friction factor of the PEO/CNT solution is smaller for the whole time. Since only PEO can degrade in such a solution (CNT is rigid), its rate of degradation is reduced in the presence of CNT. Indeed, the mechanism of the synergy of complex mixtures containing drag reducers deserves more investigation. A direct measure of mean polymer length (by light scattering, for example) in co-presence of different materials should definitely prove that flexible polymers are more extended in this situation or another phenomenon, as aggregations, also play a role. Many different complex mixtures are well worthy of investigation. Polymerclay blends, for example, are very common and not sufficiently studied. Relevant studies come from [75, 76, 77] who show that mixtures of polymers and clays clearly exhibit drag reduction in turbulent flows. It seems that clay can aggregate, which is vital for DR. The blend appears to be not very stable [77]. Most of the authors used carboxymethyl cellulose (CMC) in their works, but

23

different kinds of polymers (flexible and rigid) should also be studied in an attempt to find more efficient and stable materials. Analyses of polymeric degradation in this kind of blends are not yet adequately investigated, and such a point deserves more attention. This review is focused on linear polymer chains, but it is worth noting that an alternative to them is the branched polymers that show higher stability against mechanical scission [78]. Branched polymers, like star polymers [79, 80, 81], do not undergo midpoint scission, loosing only their branches during elongational or turbulent flows, leading only to a small decrease in molecular weight [79, 82]. It seems that, specifically for DR applications, there are not many available works in the literature, two exceptions are [78, 82]. In most of the works with branched polymers, the degradation is studied under laminar elongation flows or by acoustic fields. Hence, experimental DR tests directly comparing branched polymers and linear polymer chains with the same molecular weight deserves to be analysed. 5. Polymer de-aggregation It is worth noting that the loss of efficiency of a drag reducing flow is not only attributed to mechanical polymer scission4 . In fact, researchers must also pay attention to another mechanism, which will here be referred to as polymer de-aggregation. Shetty and Solomon [31], using light scattering, provide evidence of the 4

Obviously, the bio-degradation also plays an important role, but it will not be discussed

here (for some insights, see [58, 59])

24

formation of aggregates in aqueous solutions of PEO, even in ultra-low concentration, much below its overlap values. As sketched in Fig. 8, what is here called an aggregate is a polymer structure comprised of about two molecular chains. Supposedly, aggregates increase DR. This makes sense, since two aggregated molecules seem like a single one with a larger length. Thus, an increasing degree of aggregation plays the same role in DR as that played by an increasing molecular weight. Hence, as argued by [31], part of the loss of efficiency observed in PEO, in fact in any polymer solution, can be related to the de-aggregations of the molecules, instead of simply molecular scission. I firmly believe that this kind of aggregations also exists in solutions of other kinds of molecules, such as XG, for example. Wyatt et al. [83] conducted some experiments in turbulent pipe flows with two kinds of XG solutions, but only one of them was a homogenous dilute solution. They used two stock solutions of XG, one with 1000 ppm and other with 10,000 ppm. A sample of each stock solution was diluted to 60 ppm and immediately tested in their apparatus. The values of the DR were compared with those obtained with a homogeneous solution of 60 ppm and, surprisingly, the DR was significantly larger with the non-homogeneous solution. Supposedly, some of the molecules of XG in the non-homogeneous solution were aggregated, providing an increase in the solution’s apparent molecular weight, which was consequently the cause of an increasing efficiency of DR. When the sample from the stock solution of 10,000 ppm was used, the values of DR at the beginning of the test were 5 times larger than that with the homogeneous solution. Pereira et al. [42] compared the DR induced by PEO, PAM (flexible poly25

mers) and XG (rigid polymer) in a rotating geometry (double gap device). In particular, the analysis of how the polymer concentration affects DR over time is quite interesting. Fig. 9 displays, from top to bottom, the DR as a function of time for PEO, PAM and XG for a range of concentrations (2 ppm ≤ c ≤ 100 ppm) at fixed Reynolds number and temperature. Clearly, for the flexible materials (PEO and PAM), the fall of DR decreases with c. For 2 ppm of PEO (hollow grey circles in Fig. 9 A), the DR starts from 14 % and reaches a final value of 4 %, a relative drag reduction DR0 = 0.29, while for 100 ppm (full grey circles in Fig. 9 A), the DR starts from 22 % and reaches 12 %, DR0 = 0.52. For solutions of PAM, DR0 is 0.28 for 2 ppm and 1 for 100 ppm. There is no loss of efficiency for the most concentrated solution of PAM (compare the hollow and full circles in Fig. 9 B). The behavior of XG is very different. The effect of concentration has the opposite trend. For very small concentrations (2 ppm and 5 ppm), the DR is constant over the whole of the test time. As c is increased, a clear reduction in DR is observed. Such different roles played by the concentration is attributed to polymer de-aggregation. In [42] it is supposed that aggregates do not exist in solutions with very small concentrations (2 ppm and 5 ppm, for example) and, consequently, there is no de-aggregation during the test. Since XG is a rigid polymer, it does not break and the DR is constant. For higher concentrations, the polymers can aggregate and the smaller DR observed is caused by the de-aggregation. A similar effect of concentration was reported in [48, 58, 71, 59]. Aggregation is sustained by intermolecular forces, much weaker than intramolecular forces. Thus, the necessary hydrodynamic force to induce de26

aggregation is much smaller than that needed to cause mechanical molecular scission. Fig. 10 displays the DR in terms of time for XG solutions presheared in laminar flows. It is worth noting that the fall of DR over time was reduced for the entire range of concentrations. This behavior suggests that the number of aggregated molecules of XG in the pre-sheared solutions was significantly reduced by the shearing. Further evidence that de-aggregation is the main cause of the fall of DR when XG is the drag reducer was recently reported by Santos et al. [84]. They showed that the specific viscosity and the overlap concentration of XG does not change, even after a very long time under turbulent flow with high Reynolds number. They carried out a DR test with an XG solution of 800 ppm concentration with Re fixed at 44,000. In this test, the DR fell from 68%, at the beginning, to 57%, at the end, a loss of 16%. As shown in Fig. 11, the overlap concentration obtained with the solution used in the DR test (hollow blue squares) was almost the same as that obtained with the fresh solution (full blue squares), around 180 ppm. The difference was smaller than 2.5 %. If mechanical degradation were important, the overlap concentration (c∗ ) would have changed. In fact, c∗ must increase if the molecular weight decreases. This means that a larger number of molecules is needed to have an entangled polymeric solution, if the molecular weight is smaller. Hence, if it is not the mechanical degradation that caused the decrease of efficiency in the DR test it seems that the only option is the de-aggregation. Since aggregations are related to intermolecular links, if those are broken by the flow, after some time at rest, the aggregates could be recovered. Up to now, there is no conclusive study to prove such a hypothesis. Some insights 27

comes from [42]. The authors conducted tests with pre-sheared solutions of 50 ppm of XG. The first solution was immediately tested after a pre-shearing, while the second one rested for 100 h. The DRmax (DR at the very beginning of the test) for the second solution was slightly larger, supposedly because of the regeneration. More tests must be conducted in order to verify such a point. The biodegradation is an obstacle, since the time for total or partial regeneration can be too long, but could be overcome by using biocides. 6. Modeling polymer degradation Most of the studies to further understand polymer degradation have been carried out in experimental approaches. It is a very challenging task to simulate polymer scissions in laminar or turbulent flows. The reason is obvious. The group of equations to model such flows are very non-linear, which makes the convergence difficult and limits the study to very simple geometries. However, polymer scissions do exist and must be faced up. A promising technique for numerical simulation of polymer scission is the Brownian dynamics (BD). The pioneering work is that of Reese and Zimm [85] who used BD to predict the scission of DNA in a laminar converging flow. After that, this technique has been successfully used to predict degradation of different flexible polymers in laminar extensional flows [33, 86, 87, 88, 89]. In these referred papers, the polymers were mimicked by the well-known bead-spring model (Fig. 12) in which a force balance on a single molecule leads to the position rti of each bead in a specific time step ∆t as ri,t+∆t = ri,t +

X ∆t X t t Di,j Fj + v(rti )∆t + 2∆t Dti,j . kT j j 28

(6)

Here, k is the Boltzmanns constant, T the temperature and v(rti ) is the velocity of the solvent at position rti . The term Dti,j (i, j = 1, 2, 3, N ) is the component ij of a diffusion tensor, which is simply kT /ξ if hydrodynamic interactions is neglected [88], and ξ is the friction coefficient of a bead. The term Ftj is the spring force in a bead j. There are many approximations for this force, as the FENE-model (finite elastic non-linear extensibility) [90]. Fj =

HL . 1 − (L/L0 )2

(7)

Where H is the spring constant, L0 the maximal extension length and L the current length of the spring. The main goal of BD is the possibility to track an assemble of individual molecules initially placed at an inlet flow which are allowed to pass through, for example, an elongation flow (cross-slot [33]; convergent channel [88, 89]), suffering eventually successive breakages every time a scission criterion (normally based in a critical force) is reached. In laminar elongational flows, the distribution of scission fragments is qualitatively well predicted and normally indicates the predominance of the midpoint scission. A remarkable work is that of Hsieh et al. [33], who simulated the mechanical scission of around 16,000 molecules initially placed at the inlet channel of a planar cross-slot flow, in an attempt to mimic the experiments of Odell et al. [32] in which aqueous solutions of monodisperse poly(ethylene oxide) (PEO) were pumped into a cross-slot device. It is worth noting that the experiments conducted by [32] were contaminated by elastic instabilities, even though the Reynolds number was small, as shown by [36] (for more details of elastic instabilities in cross-slot flows, see the recent works [91, 92, 93]). For simplicity, the simulations performed by [33] were also for small Re but considering the flow 29

purely laminar and neglecting hydrodynamic interactions (HI), but the chain scission mechanism was well captured. The HI reduces the rate of breakage [88, 89], but it does not seem to be very important in the case of elongational flows in the absence of turbulence. Further studies on mechanical polymer scission are strongly necessary, mainly considering hydrodynamic interactions and turbulent flows, in an attempt to model drag reducing flows with molecular scission. As mentioned before, this is not so simple, and it seems practically prohibitive by Brownian dynamics because of current limits on computational power. Hence, an alternative is to use continuous viscoelastic models. An important non-Newtonian parameter which appears in most of the viscoelastic models is the polymer’s relaxation time, λ. For flexible linear polymers, λ can be approximated by λ = µs (N 3/5 c)3 /κT (see Flory [94]), where N is the number of repeating monomers in the molecule, κ is Boltzmann’s constant, µs is the solution’s viscosity, and T is the solution’s temperature. Thus, if the polymers break, N decreases and, consequently, the relaxation time falls. The Weissenberg number W e = λρu2c /µs (ρ and uc are, respectively, the solution’s density and a characteristic flow velocity), proportional to the relaxation time, also decreases. Since DR is an increasing function of W e, it must decrease. Hence, a way to incorporate polymer degradation into viscoelastic models is to consider λ as a variable dependent on the flow. Evidently, all viscoelastic properties change when the polymer breaks, e.g. the maximum polymer extension length L, a parameter used in FENE-P (finite elastic non-linear extensibility-Peterkin) model. That was the strategy 30

used by Pereira et al. [95]. As far as I know, that paper is the only attempt to simulate a drag reducing flow including polymer degradation. They considered a spatio-temporal field of the maximum polymer extension length L(x, y, z, t), instead of a constant value as typically used in the standard FENE-P model. The idea was very simple. They imposed a criterion of degradation based on the relative polymer stretching (equivalent to a critical force criterion used by [33], since a stretch corresponds to a certain value of the spring force and vice versa), which was quantified by tr(C(x, y, z, t))/Li 2 . Here, trC(x, y, z, t) is the trace of the conformation tensor C, a measure of how the polymer is stretched in a specific point, and Li is the initial maximum polymer extension. In order to spread the degraded polymers, the authors used a material transport equation. One inconvenient problem of this model is the numerical instabilities caused by the variation of the potential L over time. Fig. 13 shows the DR (grey circles) and the temporal evolution of the relative spatial average of the maximum polymer extension length < L/Li >xyz (blue triangles) for a turbulent flow between parallel plates of a modified FENE-P fluid. The time is made dimensionless by the plate velocity Uh and the distances between them, h. The parameter < L/Li >xyz starts from 1, when the DR is around 30%. To minimize the numerical instabilities, a minimal value for < L/Li >xyz was fixed at 0.7. At this stage, DR reaches its statistical steady state around 18%. The model proposed in [95] is rather simple and present a significant numerical problem. The oscillations of DR, mainly between 400 ≤ tUh /h ≤ 2000 is not physical (a stability technique could have avoided it). However,

31

the model was able to predict the fall of DR over time and the changes of the three-dimensional turbulent structures represented by vortical (or elliptical) and extensional (or hyperbolic) regions, respectively defined by the second invariant of velocity gradient tensor. If the numerical instability is minimized, which seems attainable, this simple approach proposed by [95] can be upgraded to incorporate a spatial-temporal field of λ in addition to L. 7. Summary A short review of the mechanical degradation of linear polymers in turbulent flows has been presented. At first, the main aspects of polymeric DR, as its state of art, definition, and important applications were presented in Section 1. The main theories and recent advances in the attempt to further understand DR were also discussed in Section 1. Some important aspects of mechanical degradation were discussed in Section 2. The transient nature of DR was shown in Section 3. In Section 4, the importance of some variables was pointed out (concentration, molecular weight, quality of solvent, temperature, Reynolds number, residence time, and relaxation time) to accelerate or delay the mechanical degradation. It is stressed that the polymeric conformational configuration is crucial. Next, in Section 5, the de-aggregation was discussed as well as the role that it plays in the fall of DR. Supposedly, rigid polymers do not experience mechanical scission, but de-aggregate, which is the cause of the decrease of DR in flows of solutions with such materials. At last, Section 6 reported numerical simulations to predict polymer scission in laminar flows (using Brownian dynamics) and the results of the single available work that numerically simulates a drag reducing flow, including, at least 32

in some sense, mechanical polymer degradation. 8. Acknowledgements I am grateful to Professor Renato N. Siqueira who read an early version of the paper and contributed with valuable comments and suggestions. I also acknowledge the referees for a great number of suggestions that improved a lot the quality of the final text. This research was partially funded by grants from CNPq (Brazilian Research Council). References References [1] B. A. Toms. Some observations on the flow of linear polymer solutions through straight tubes at large Reynolds numbers. Proceedings of the International Congress of Rheology, Holland, North-Holland, Amsterdam, Section II, pages 135–141, 1948. [2] B. A. Toms. On the early experiments on drag reduction by polymers. Physics of Fluids, 20:S3, 1977. [3] A. G. Fabula. Fire-fighting benefits of polymeric friction reduction. Trans ASME J Basic Engng, pages 93–453, 1971. [4] E. D. Burger and L. G. Chorn. Studies of drag reduction conducted over a broad range of pipeline conditions when flowing prudhoe bay crude oil. J. Rheology, 24:603, 1980.

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[71] G. A. B. Sandoval and E . J. Soares. Effect of combined polymers on the loss of efficiency caused by mechanical degradation in drag reducing flows through straight tubes. Rheologica Acta, 55:559–569, 2016. [72] G. L. Novelli, L. A. Ferrari, G. G. Vargas, and B. V. Loureiro. A synergistic analysis of drag reduction on binary polymer mixtures containing guar gum. International Journal of Biological Macromolecules, 137:1121–1129, 2019. [73] R. Kuhn and H. J. Cantow. Incompatibility of polymer mixtures: light scatering of high molecular weight polystyrenes dissoved in polymethylmethacrylate/benzene mixtures. Macromol Chemie, 122:65, 1969. [74] R. L. Patterson and R. C. Little. Drag reduction of poly(ethylene oxide) carboxylate soap mixtures. J. Colloid and Interface Sci., 53:110–114, 1975. [75] J. E. Herod and W. G. Tiederman. Drag reduction in dredge-spoil pipe flows. J. Hydraul. Div. Proc, pages 1863–1866, 1974. [76] A. S. Pereira and F. T. Pinho. Turbulent pipe flow of thixotropic fluids. Int. J. Heat Fluid Flow, 23:36–51, 2002. [77] A. Benlimane, K. Bekkour, and P. Fran¸cois. Effect of addition of carboxymethy cellulose (CMC) on the rheology and flow properties of bentonite suspensions. Appl. Rheol., 23:13475, 2012. [78] O. K. Kim, R. C. Little, R. L. Patterson, and R. Y. Ting. Polymer structures and turbulent shear stability of drag reducing solutions. Nature, 250:408–410, 1974. 43

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Figure 4: Sketch of DR over time from the very beginning of the process to its asymptotic value, after a long enough time. The period from the test beginning to DRmax is the developing time td . The interval from td to the point at which DR starts to decrease is the resistant time tr . Finally, from tr to the point when DRasy is reached is ta . (Reprinted with permission from Pereira et al. [42]

.)

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1.0

0.9

ηasy ≈2.45

η0 =2.8

[η(t)] [η0 ]

η0 =4.8

ηasy ≈4.08

0.8

η0 =25

0.7

ηasy ≈16.88

0.5

1.0

t (sec)

1.5

2.0

Figure 5: Evolution of the relative viscosity for different solutions of PEO flowing in a pipe at Re = 70, 000: (red crosses and black squares) 75 ppm; (blue balls) 50 ppm. (Data taken from Paterson and Abernathy [41].)

48

1.1 c = 50 ppm Re = 1360 T = 25ºC

Relative drag reduction, DR'

1 0.9 0.8 0.7 0.6 0.5 PEO - 6.0 x 105 g/mol PEO - 4.0 x 106 g/mol PEO - 4.6 x 106 g/mol PEO - 5.0 x 106 g/mol PAM - 5.0 x 106 g/mol

0.4 0.3 0.2

1

10

100

1000

10000

Time, t [s] Figure 6: Relative drag reduction as a function of time for fixed Reynolds number (Re = 1360), temperature (T =25o C) and polymer concentration (50 ppm). The full black circles are for the PAM solution with Mv = 5.0 × 106 g/mol and the other symbols are for a range

of PEO molecular weights (6.0 × 105 g/mol ≤ Mv ≤ 5.0 × 106 g/mol) (Reprinted with permission from Pereira and Soares [44].)

49

10−2

Friction factor, f

Blasius law

Concentration MDR

Molecular weight Quality of solvent

Critycal Reynolds number

10−3

103

104

105

Reynolds number, Re Figure 7: Sketch of the the critical Re. This kind of data is obtained when a fixed amount of solution passes through the system only once. (Data taken from Moussa and Tiu [54].)

Mean shear flow Vortex

intermolecular force

dynamic forces

de-aggregation

Figure 8: Sketch of polymer de-aggregation.

50

0.3 Poly(ethylene oxide) Mv = 5.0 x 106 g/mol Re = 1360 T = 25ºC

2 ppm 5 ppm 10 ppm 17.5 ppm 25 ppm 37.5 ppm 50 ppm 100 ppm

A

0.2

0.1

0

1

10

100

1000

10000

0.3 2 ppm 5 ppm 10 ppm 17.5 ppm

B

Drag Reduction, DR

Polyacrylamide Mv = 5.0 x 106 g/mol Re = 1360 T = 25ºC

25 ppm 37.5 ppm 50 ppm 100 ppm

0.2

0.1

0.0

1

10

100

1000

10000

0.3 2 ppm 5 ppm 10 ppm 17.5 ppm 25 ppm

C

0.2

37.5 ppm 50 ppm 75 ppm 100 ppm Xanthan Gum Mv = 2.0 x 106 g/mol Re = 1360 T = 25ºC

0.1

0.0

1

10

100

1000

10000

Time, t [s]

Figure 9: Effect of concentration on DR: comparison between XG, PEO and PAM (Reprinted with permission from Pereira et al. [42].)

51

0.3

Drag Reduction, DR

Xanthan Gum Mv = 2.0 X 106 g/mol Re = 1360 T = 25ºC

5 ppm 10 ppm 25 ppm 50 ppm 100 ppm

Samples previously submitted to a shear rate of 3995 1/s for 900 s.

0.2

0.1

0

1

10

100

1000

10000

Time, t [s]

Figure 10: Effect of concentration on DR for a XG solution previously sheared (Reprinted with permission from Pereira et al. [42].)

52

Figure 11: Overlap concentration of XG.

(Data taken from Santos et al. [84])

53

j+1

N

Fj

y

j

rj

x Figure 12: Sketch of a bead-spring model with N beads. Fj is the spring force and rj is the position in the bead j.

54

Figure 13: Drag reduction, DR (grey circles) as a function of the dimensionless time, tU h/h, together with the evolution of the spatial average of the maximum polymer extension length made dimensionless by its initial value, < L/Li > xyz (blue triangles). (Reprinted with permission from Pereira et al. [95].)

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Conflict of Interest The authors declare that they have no conflict of interest.

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