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

Abstract

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

=

$=$ 18x106 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

m

′

<

$^{\prime }_{\mathrm {m}} <$ 0.4 developed akin to solvent, with Le,p/D

=

$=$ Le,n/D

<

$<$ 42.3, while those at maximum drag reduction showed entrance lengths Le,m/D

∼

$\sim $ 117, roughly 3 times the solvent Le,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

m

′

=

$^{\prime }_{\mathrm {m}} =$ 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 kdeg s− 1 and entrance severities −ln(S

′

$^{\prime }$ /S

′ ∧

$^{{\prime }\wedge }$ )0 were extracted. At constant C, kdeg increased linearly with increasing wall shear stress Tw, and at constant Tw, kdeg was independent of C, providing a B1120 degradation modulus (kdeg/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

=

$=$ 11x106, 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

m

′

<

$^{\prime }_{\mathrm {m}} <$ 0.4 developed akin to solvent, with Le,p/D

=

$=$ Le,n/D

<

$<$ 23, while those at MDR had entrance lengths Le,m/D

∼

$\sim $ 93, roughly 4 times the solvent Le,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; kdeg was found to increase linearly with increasing Tw at constant C and was independent of C at constant Tw, providing a P309 degradation modulus (kdeg/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 (kdeg/Tw) data from laboratory pipes of D

∼

$\sim $ 0.01 m to lie on a simple extension of (kdeg/Tw) data from pipelines of D

∼

$\sim $ 0.1 m and 1.0 m, along a power-law relation (kdeg/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

β =

$\beta =$ 0.053

±

$\pm $ 0.036 enabling estimation of flow enhancement S

′

=

$^{\prime } =$ C.Lc.β for all Type B drag reduction by polymers.