# Glimpses of Flow Development and Degradation During Type B Drag Reduction by Aqueous Solutions of Polyacrylamide B1120

## Abstract

Flow development and degradation during Type B turbulent drag reduction by 0.10 to 10 wppm solutions of a partially-hydrolysed polyacrylamide B1120 of MW \(=\) 18x10^{6} was studied in a smooth pipe of ID \(=\) 4.60 mm and L/D \(=\) 210 at Reynolds numbers from 10000 to 80000 and wall shear stresses Tw from 8 to 600 Pa. B1120 solutions exhibited facets of a Type B ladder, including segments roughly parallel to, but displaced upward from, the P-K line; those that attained asymptotic maximum drag reduction at low Re*√* f but departed downwards into the polymeric regime at a higher retro-onset Re*√* f; and segments at MDR for all Re*√* f. Axial flow enhancement profiles of S\(^{\prime }\) vs L/D reflected a superposition of flow development and polymer degradation effects, the former increasing and the latter diminishing S\(^{\prime }\) with increasing distance downstream. Solutions that induced normalized flow enhancements S\(^{\prime }\)/S\(^{\prime }_{\mathrm {m}} <\) 0.4 developed akin to solvent, with L_{e,p}/D \(=\) L_{e,n}/D \(<\) 42.3, while those at maximum drag reduction showed entrance lengths L_{e,m}/D \(\sim \) 117, roughly 3 times the solvent L_{e,n}/D. Degradation kinetics were inferred by first detecting a falloff point (Re*√**f*^{∧}, S\(^{{\prime }\wedge }\)), of maximum observed flow enhancement, for each polymer solution. A plot of S\(^{{\prime }\wedge }\)vs C revealed S\(^{{\prime }\wedge }\)linear in C at low C, with lower bound [S\(^{\prime }\)] \(=\) 5.0 wppm^{− 1}, and S\(^{{\prime }\wedge }\) independent of C at high C, with upper bound S\(^{\prime }_{\mathrm {m}} =\) 15.9. The ratio S\(^{\prime }\)/S\(^{{\prime }\wedge }\) in any pipe section was interpreted to be the undegraded fraction of original polymer therein. Semi-log plots of (S\(^{\prime }\)/S\(^{{\prime }\wedge }\)) at a section vs transit time from pipe entrance thereto revealed first order kinetics, from which apparent degradation rate constants k_{deg} s^{− 1} and entrance severities −ln(S\(^{\prime }\)/S\(^{{\prime }\wedge }\))_{0} were extracted. At constant C, k_{deg} increased linearly with increasing wall shear stress Tw, and at constant Tw, k_{deg} was independent of C, providing a B1120 degradation modulus (k_{deg}/Tw) \(=\) (0.012 \(\pm \) 0.001) (Pa s)^{− 1} for 8 \(<\) Tw Pa \(<\) 600, 0.30 \(<\) C wppm \(<\) 10. Entrance severities were negligible below a threshold Twe \(\sim \) 30 Pa and increased linearly with increasing Tw for Tw \(>\) Twe. The foregoing methods were applied to Type A drag reduction by 0.10 to 10 wppm solutions of a polyethyleneoxide PEO P309, MW \(=\) 11x10^{6}, in a smooth pipe of ID \(=\) 7.77 mm and L/D \(=\) 220 at Re from 4000 to 115000. P309 solutions that induced S\(^{\prime }\)/S\(^{\prime }_{\mathrm {m}} <\) 0.4 developed akin to solvent, with L_{e,p}/D \(=\) L_{e,n}/D \(<\) 23, while those at MDR had entrance lengths L_{e,m}/D \(\sim \) 93, roughly 4 times the solvent L_{e,n}/D. P309 solutions described a Type A fan distorted by polymer degradation. A typical trajectory departed the P-K line at an onset point Re*√* f* followed by ascending and descending polymeric regime segments separated by a falloff point Re*√**f*^{∧}, of maximum flow enhancement; for all P309 solutions, onset Re*√* f* = 550 \(\pm \) 100 and falloff Re*√**f*^{∧} = 2550 \(\pm \) 250, the interval between them delineating Type A drag reduction unaffected by degradation. A plot of falloff S\(^{{\prime }\wedge }\) vs C for PEO P309 solutions bore a striking resemblance to the analogous S\(^{{\prime }\wedge }\) vs C plot for solutions of PAMH B1120, indicating that the initial Type A drag reduction by P309 after onset at Re*√* f* had evolved to Type B drag reduction by falloff at Re*√**f*^{∧}. Presuming that Type B behaviour persisted past falloff permitted inference of P309 degradation kinetics; k_{deg} was found to increase linearly with increasing Tw at constant C and was independent of C at constant Tw, providing a P309 degradation modulus (k_{deg}/Tw) \(=\) (0.011 \(\pm \) 0.002) (Pa s)^{− 1} for 4 \(<\) Tw Pa \(<\) 400, 0.10 \(<\) C wppm < 5.0. Comparisons between the present degradation kinetics and previous literature showed (k_{deg}/Tw) data from laboratory pipes of D \(\sim \) 0.01 m to lie on a simple extension of (k_{deg}/Tw) data from pipelines of D \(\sim \) 0.1 m and 1.0 m, along a power-law relation (k_{deg}/Tw) \(=\) 10^{− 5.4}.D^{− 1.6}. Intrinsic slips derived from PAMH B1120 and PEO P309-at-falloff experiments were compared with previous examples from Type B drag reduction by polymers with vinylic and glycosidic backbones, showing: (i) For a given polymer, [S\(^{\prime }\)] was independent of Re*√* f and pipe ID, implying insensitivity to both micro- and macro-scales of turbulence; and (ii) [S\(^{\prime }\)] increased linearly with increasing polymer chain contour length Lc, the proportionality constant \(\beta =\) 0.053 \(\pm \) 0.036 enabling estimation of flow enhancement S\(^{\prime } =\) C.Lc.*β* for all Type B drag reduction by polymers.

## Keywords

Drag reduction Turbulent flow Polyelectrolyte solutions Polymer degradation## Notes

### Acknowledgements

A sample of GE/Betz AP1120 flocculant was kindly provided by Chris Deeley, GE Power & Water, Foxborough, MA.

### Compliance with Ethical Standards

### **Conflict of interests**

The author declares he has no conflict of interest.

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