Estimating the skin-friction coefficient in drag-reduced pipe flows of a flexible polymer

In turbulent pipe flows, drag-reducing polymers are commonly used to reduce skin-friction drag; however, predicting this reduction in industry applications, such as crude oil pipelines, remains challenging. The skin-friction coefficient ( $C_f$ ) of polymer drag-reduced turbulent pipe flows can be related to three dimensionless parameters: the solvent Reynolds number ( $Re_s$ ), the Weissenberg number ( $Wi$ ) and the ratio of solvent viscosity ( $\eta _s$ ) to zero-shear-rate viscosity ( $\eta _0$ ), denoted as $\beta$ . The function that relates these four dimensionless numbers was determined using experiments of various pipe diameters ( $D$ ), flow velocities ( $U$ ) and drag-reducing polyacrylamide solutions. The experiments included measurements of streamwise pressure drop ( $\Delta P$ ) for determining $C_f$ , and measurements of shear viscosity ( $\eta$ ) and elastic relaxation time ( $\lambda$ ). This experimental campaign involved 156 flow conditions, each characterised by distinct values for $C_f$ , $Re_s$ , $Wi$ and $\beta$ . Experimental results demonstrated good agreement with the relationship: $C_f^{-1/2} = \widehat {A}\log _{10}(Re_sC_f^{1/2})+\widehat {B}$ , where $\widehat {A} = 27.6(Wi \beta )^{0.346}$ and $\widehat {B} = 122/15-58.9(Wi \beta )^{0.346}$ . Based on this relationship, onset and maximum drag reduction are predicted to occur when $Wi \beta$ equals $3.76 \times 10^{-3}$ and $3.40 \times 10^{-1}$ , respectively. This function can predict $C_f$ of dilute polyacrylamide solutions based on predefined parameters (bulk velocity, pipe diameter, density, solvent viscosity) and two measurable rheological properties of the solution (shear viscosity and elastic relaxation time) with an accuracy of $\pm 9.36$ %.