Friction factor with drag reduction

FIGURE 14.13 Turbulent drag reduction as evidenced by a drop in friction factors with the addition of polysaccharides to water [18]. Copyright 1972 by the American Chemical Society. Reprinted with permission of the copyright owner. 

The model for turbulent drag reduction developed by Darby and Chang (1984) and later modified by Darby and Pivsa-Art (1991) shows that for smooth tubes the friction factor versus Reynolds number relationship for Newtonian fluids (e.g., the Colebrook or Churchill equation) may also be used for drag-reducing flows, provided (1) the Reynolds number is defined with respect to the properties (e.g., viscosity) of the Newtonian solvent and (3) the Fanning friction factor is modified as follows ... 

DRAG REDUCTION IN TURBULENT FLOW. Dilute solutions of polymers in water or other solvents sometimes give the peculiar effect of a reduction in drag in turbulent flow. The phenomenon was first noted by Toms and has prompted many theoretical studies and some practical applications. As shown in Fig. 5.11, the friction factor can be significantly below the normal value for turbulent flow with only a few parts per million of polymer in water, and at 50 to 100 ppm, the drag reduction may be as much as 70 percent. Similar effects have been shown for some polymers in organic solvents. 

An exception to the generally observed drag reduction in turbulent channel flow of aqueous polymer solutions occurs in the case of aqueous solutions of polyacrylic acid (Carbopol, from B.F. Goodrich Co.). Rheological measurements taken on an oscillatory viscometer clearly demonstrate that such solutions are viscoelastic. This is also supported by the laminar flow behavior shown in Fig. 10.20. Nevertheless, the pressure drop and heat transfer behavior of neutralized aqueous Carbopol solutions in turbulent pipe flow reveals little reduction in either of these quantities. Rather, these solutions behave like clay slurries and they have been often identified as purely viscous nonnewtonian fluids. The measured dimensionless friction factors for the turbulent channel flow of aqueous Carbopol solutions are in agreement with the values found for clay slurries and may be correlated by Eq. 10.65 or 10.66. The turbulent flow heat transfer behavior of Carbopol solutions is also found to be in good agreement with the results found for clay slurries and may be calculated from Eq. 10.67 or 10.68. 

The value of hydrodynamic friction drag reduction by means of small additives of water-soluble high-molecular polymers is known to depend on many factors [1,2]. These factors are, first of all, the molecular mass of a polymer, the linearity of a molecular structure with a small number of lateral branchings which determine elasticity the concentration of a polymer the range of tangential stresses the degree of mechanical destruction etc. 

Although the above correlation is based upon data for polymer solutions which are considered concentrated for most drag reduction applications, it has been shown to be consistent with observed dependence of drag reduction upon both concentration and degradation in more dilute solutions, as well as observed dependence upon pipe diameter [1,10]. For example. Fig. 4 shows predicted friction factor characteristics for both fresh... 

In the regime below the maximum drag reduction asymptote, the friction factor varies with polymer properties and flow variables ... 

Consider the flow of glass beads of an average particle diameter of 26 nm flowing in a 0.0254-m-diameter pipe at a solids/gas ratio of 1.2. Determine the pressure drop per unit length for this system with and without drag reduction present. The ratio of the friction factor based on a mixture density to the friction factor based on gas density is given as... 

Most importantly for computational viscoelastic fluid mechanics, most of the charmel DNS calculations are not performed for a constant flux (which would have naturally resulted in a constant bulk Reynolds number) but for a constant pressure drop per unit length that results in a constant zero shear rate friction Reynolds number. These runs lead to substantial variations in the (instantaneous and average) bulk Reynolds number from which the drag reduction needs to be estimated. Knowing roughly the relationship between the friction and the average bulk Reynolds number for a Newtonian fluid (from the experimentally determined and DNS confirmed empirical relationships for the skin friction factor - see, for example. Ref [34]), one can extract such a relationship that also takes into account the already mentioned (in Section 1.2) shear thinning effect in association with viscoelastic results [78]. 

As with Newtonians, it s a pretty safe bet that flow is laminar for Re < 2100, but drag-reducing additives often seem to delay the laminar-turbulent transition to higher Re s. The friction factor-Reynolds Number relation remains a function of pipe roughness. Unlike the Newtonian case, however, the Re curve seems to depend on pipe diameter when drag reduction is observed. 

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