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The effects of turbulent drag reducing additives on hydrocyclone operation : an evaluation of the flow… MacKenzie, Jordan Alexander 2014

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The Effects of Turbulent DragReducing Additives on HydrocycloneOperationAn Evaluation of the Flow Behaviour and ParticleMechanicsbyJordan Alexander MacKenzieB.ASc., The University of British Columbia, 2009A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinThe Faculty of Graduate and Postdoctoral Studies(Chemical and Biological Engineering)THE UNIVERSITY OF BRITISH COLUMBIA(Vancouver)August 2014c© Jordan Alexander MacKenzie 2014AbstractThe effects of adding drag reducing additives to a pulp processing hydrocy-clone were experimentally investigated. This work was found to contributetowards improving particle separation efficiencies and reducing energy con-sumption. To effectively evaluate the performance improvements, the flowfield was initially measured using laser Doppler velocimetry. In the presenceof a particulate phase, the motions of variously sized particles were measuredusing a three dimensional, dual camera set-up. Quantification of the dragreducing potential of the various polymer solutions and fibre suspensionsstudied was experimentally determined using an integral analysis for a fixedcontrol volume.The addition of drag reducing polymer additives was found to fundamen-tally change the hydrocyclone flow field from what is classically observedwith water alone. For the conditions studied in this work, the effectivenessof a hydrocyclone towards removing contaminants would likely be reduced,as a particles separation zone was limited.The addition of polymer additives to a hydrocyclone was found to in-crease the size of particles susceptible to overflow removal. It was foundthat particles of density 1280 kg/m3 and diameter 500 – 600 µm displayedinwards motion in a 0.03% APAM solution, where purely outwards motionwas measured for identically sized particles suspended in water. The flowfield, however, indicated that overflow removal is limited to only a smallregion near the vortex finder.Polymer additives were found to be effective in reducing energy con-sumption in a hydrocyclone. Maximum drag reduction was found to occurat a reject ratio of 50% for polymer solutions, independent of inlet veloc-ity. The energy savings potential for polymer additives in a pulp processinghydrocyclone, however, was found to be limited to the inlet velocity. Mostin process hydrocyclones operate well above the minimum inlet velocitiesmeasured for rejects ratios of 25% and 50%, suggesting that additional en-ergy savings would likely occur. The phenomenological degradation of thepolymer agents investigated in this work suggests that the practical use ofthese additives would be difficult. This was found to be most significant withiiAbstractcellulose fibre suspensions containing cationic polyacrylamide (CPAM), aspolymer adsorption resulted in rapid polymer degradation.iiiPrefaceThis thesis is original work by Jordan MacKenzie, hereafter referred to theauthor.The research outline was proposed by Dr. James Olson and Dr. MarkMartinez. The research, including experimental design, experimental pro-cedures and data analysis were performed by the author.The tracking algorithm implemented in the MATLAB image processingand particle tracking velocimetry program was originally developed by J.Crocker, D. Grier and E. Weeks and made available in MATLAB by D.Blair and E. Dufresne. Segments of the code used to condition the inputand output data for the tracking algorithm were developed by a researchcolleague T. Mithrush. The head loss data for a pulp suspension and wa-ter in a pipe is that of A. Nikbakht, a research colleague. The results ofthe computational fluid dynamics (CFD) simulation presented in this workare those of E. Zaman, another research colleague. The image processing,quantification of turbulent drag reduction, and data analysis were the solework of the author.ivTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . vList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiiList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ixNomenclature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xivAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . xvii1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . 42.1 Hydrocyclone Flow Field . . . . . . . . . . . . . . . . . . . . 42.1.1 Newtonian fluids . . . . . . . . . . . . . . . . . . . . . 42.1.2 Hydrocyclone turbulence with Newtonian fluids . . . 62.1.3 Non-Newtonian fluids . . . . . . . . . . . . . . . . . . 72.1.4 Hydrocyclone turbulence with non-Newtonian medi-ums . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.2 Particle Motion in a Hydrocyclone . . . . . . . . . . . . . . . 92.3 Numerical and Experimental Discrepancies . . . . . . . . . . 102.4 Turbulent Drag Reduction . . . . . . . . . . . . . . . . . . . 122.5 Polarity Effects in Fibre Suspensions . . . . . . . . . . . . . 142.6 Summary of Literature . . . . . . . . . . . . . . . . . . . . . 153 Experimental Design . . . . . . . . . . . . . . . . . . . . . . . . 183.1 Apparatus . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.2 Fluid and Fibre Characteristics . . . . . . . . . . . . . . . . 20vTable of Contents4 Investigating the Flow Field of a Hydrocyclone . . . . . . . 254.1 The Effects of a Polymer Additive on the Hydrocyclone FlowField . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254.1.1 Laser Doppler velocimetry . . . . . . . . . . . . . . . 264.1.2 Laser Doppler velocimetry of shear thinning fluids . . 274.1.3 The flow field of a hydrocyclone . . . . . . . . . . . . 274.1.4 The flow field of a hydrocyclone with polymer additive 314.1.5 Inlet condition study . . . . . . . . . . . . . . . . . . 334.1.6 Interpreting the hydrocyclone flow field . . . . . . . . 384.1.7 Predicting the motion of spherical particles in a hy-drocyclone . . . . . . . . . . . . . . . . . . . . . . . . 444.2 Turbulence . . . . . . . . . . . . . . . . . . . . . . . . . . . . 545 An Investigation of Solid Particle Motion in a Hydrocyclone 565.1 Particle Tracking Set-Up . . . . . . . . . . . . . . . . . . . . 565.1.1 Particle properties . . . . . . . . . . . . . . . . . . . . 575.1.2 Image processing . . . . . . . . . . . . . . . . . . . . 585.2 LDV vs. PTV Comparison: Water Alone . . . . . . . . . . . 635.3 Effect of Particle Size on Slip Velocity: Water Alone . . . . . 665.4 LDV vs. PTV Comparison: 0.03% APAM . . . . . . . . . . 715.5 Effect of Particle Size on Slip Velocity: 0.03% APAM . . . . 736 A Quantitative Analysis of Turbulent Drag Reduction . . 776.1 Evaluating Turbulent Drag Reduction in a Hydrocyclone . . 776.2 Drag Reduction Experiment Outline . . . . . . . . . . . . . . 826.3 Polymer Degradation . . . . . . . . . . . . . . . . . . . . . . 846.4 Drag Reduction in a Pulp Processing Hydrocyclone . . . . . 886.5 Polymer Adsorption Characteristics . . . . . . . . . . . . . . 937 Conclusions and Recommendations . . . . . . . . . . . . . . 957.1 Strengths and Limitations of Research . . . . . . . . . . . . . 977.2 Recommendations for Future Work . . . . . . . . . . . . . . 97Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99AppendicesA Model Parameters . . . . . . . . . . . . . . . . . . . . . . . . . 106B Analytical Study of a Swirling Cross-flow . . . . . . . . . . 107viTable of ContentsC Numerical Error of the Radial Velocity Profiles . . . . . . 115D Pressure Drop Relationships . . . . . . . . . . . . . . . . . . . 119E A Discussion on the PTV Seeding Particle Study . . . . . 123F Examples of Linear Lost Energy Relationships . . . . . . . 126G Economic Evaluation of Energy Savings . . . . . . . . . . . 128viiList of Tables3.1 Hydrocyclone dimensions. . . . . . . . . . . . . . . . . . . . . 203.2 SPF fibre quality analyser results. . . . . . . . . . . . . . . . 214.1 Overview of the LDV experiments. . . . . . . . . . . . . . . . 254.2 Summary of experiments . . . . . . . . . . . . . . . . . . . . . 344.3 Overview of the constants used to solve Equation 4.1 to modelthe flow of water and APAM. . . . . . . . . . . . . . . . . . . 414.4 Forces caused by particle-fluid interactions. . . . . . . . . . . 455.1 Overview of the PTV experiments. . . . . . . . . . . . . . . . 565.2 Polyethylene particle properties. . . . . . . . . . . . . . . . . 586.1 Summary of drag reduction experiments. . . . . . . . . . . . . 836.2 CPAM adsorption in a 0.7%(w/w) SPF fibre suspension . . . 94A.1 Regression coefficients of Ostwald-de Waele power-law model(Equation B.5). . . . . . . . . . . . . . . . . . . . . . . . . . . 106A.2 Regression coefficients of Carreau model (see: Equation 4.4). 106A.3 Power law parameters for normal-stress variation (N1 = bτm). 106A.4 Solution constants to k(RR) polynomial fit (see: Equation6.12). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106B.1 Overview of the values used to solve Equation B.15. . . . . . 111G.1 BC hydro’s monthly demand charge: Large general serviceconservation rate. . . . . . . . . . . . . . . . . . . . . . . . . . 128G.2 Electrical energy consumption rates for water and pulp mix-tures. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129G.3 Total operation cost. . . . . . . . . . . . . . . . . . . . . . . . 129viiiList of Figures1.1 Illustration of the hydrocyclone geometry used for this thesis. 22.1 Hydrocyclone flow pattern [63]. . . . . . . . . . . . . . . . . . 52.2 Mantle and short circuit flow in a hydrocyclone [3]. . . . . . . 62.3 Comparison of LDA and CFD data for three turbulence-closure models [15]. . . . . . . . . . . . . . . . . . . . . . . . . 112.4 Head loss curves for water and a pulp suspension in pipe flow. 133.1 Diagram showing interactions between test environment anddata processing. . . . . . . . . . . . . . . . . . . . . . . . . . . 183.2 Experimental flow loop. . . . . . . . . . . . . . . . . . . . . . 193.3 Viscosity curve of 0.03% APAM. . . . . . . . . . . . . . . . . 223.4 First normal stress difference vs. shear stress for 0.03% APAM. 233.5 Apparent extensional viscosity, ηE , as a function of Henckystrain (). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244.1 Water flow field: RR = 0%, Uinlet = 2.57 m/s. . . . . . . . . 284.2 Water flow field: RR = 25%, Uinlet = 2.57 m/s. . . . . . . . . 304.3 Water flow field: RR = 50%, Uinlet = 2.57 m/s. . . . . . . . . 304.4 0.03% APAM flow field: RR = 25%, Uinlet = 2.57 m/s. . . . 324.5 0.03% APAM flow field: RR = 50%, Uinlet = 2.57 m/s. . . . 334.6 Comparison of 〈uθ〉(z = -0.1 m) for conditions: W25(♦),UM25(∗) and PM25(◦). . . . . . . . . . . . . . . . . . . . . . 354.7 Comparison of 〈uz〉(z = -0.1 m) for conditions: W25(♦),UM25(∗) and PM25(◦). . . . . . . . . . . . . . . . . . . . . . 354.8 Comparison of 〈uz〉(z = -0.3 m) for conditions: W25(♦),UM25(∗) and PM25(◦). . . . . . . . . . . . . . . . . . . . . . 364.9 Comparison of 〈uθ〉(z = -0.1 m) for conditions: W50(♦),UM50(∗) and PM50(◦). . . . . . . . . . . . . . . . . . . . . . 374.10 Comparison of 〈uz〉(z = -0.1 m) for conditions: W50(♦),UM50(∗) and PM50(◦). . . . . . . . . . . . . . . . . . . . . . 37ixList of Figures4.11 Comparison of 〈uz〉(z = -0.3 m) for conditions: W50(♦),UM50(∗) and PM50(◦). . . . . . . . . . . . . . . . . . . . . . 384.12 Validity of the axially fully developed flow assumption forwater. RR = 50%, z = -0.1 m, Uinlet = 2.57 m/s. . . . . . . . 394.13 Validity of the axially fully developed flow assumption forAPAM. Uinlet = 2.57 m/s, z = -0.1 m. © RR = 25%; • RR= 50%. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 404.14 Comparison between the tangential velocity profiles of waterin a hydrocyclone and in a swirling cross-flow.  Water: z= -0.1 m, Uinlet = 2.57 m/s, RR = 50%; © Water: Swirlingcross-flow solution. . . . . . . . . . . . . . . . . . . . . . . . . 424.15 Numerically calculated tangential velocity profiles for the 0.03%APAM model. © Model for RR = 25%; • Model for RR =50%. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 434.16 Experimentally measured tangential velocity profiles of the0.03% APAM solution for Uinlet = 2.57 m/s at z = -0.1 m.© RR = 25%; • RR = 50%. . . . . . . . . . . . . . . . . . . 434.17 Estimated particle trajectories released from z = -0.1 m. Uinlet= 2.57 m/s and Dp = 50 µm. Particle streamlines (—), waterstreamlines (—), water vector field (↗). . . . . . . . . . . . . 474.18 Estimated particle trajectories released from z = -0.27 m.Uinlet = 2.57 m/s and Dp = 50 µm. Particle streamlines(—), water streamlines (—), water vector field (↗). . . . . . 484.19 Estimated particle trajectories released from z = -0.36 m.Uinlet = 2.57 m/s and Dp = 50 µm. Particle streamlines(—), water streamlines (—), water vector field (↗). . . . . . 494.20 Estimated particle trajectories released from z = -0.14 m.Uinlet = 2.57 m/s and Dp = 200 µm. Particle streamlines (—), 0.03% APAM streamlines (—), 0.03% APAM vector field(↗). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 514.21 Estimated particle trajectories released from z = -0.24 m.Uinlet = 2.57 m/s and Dp = 200 µm. Particle streamlines (—), 0.03% APAM streamlines (—), 0.03% APAM vector field(↗). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 524.22 Estimated particle trajectories released from z = -0.34 m.Uinlet = 2.57 m/s and Dp = 200 µm. Particle streamlines (—), 0.03% APAM streamlines (—), 0.03% APAM vector field(↗). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 534.23 Local Turbulence Intensity: RR = 25%. (a) Water, (b) 0.03%APAM. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55xList of Figures4.24 Local Turbulence Intensity: RR = 50%. (a) Water, (b) 0.03%APAM. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 555.1 Studied window locations. . . . . . . . . . . . . . . . . . . . . 575.2 Visualized X-Y coordinate system. . . . . . . . . . . . . . . . 585.3 Pixel values in a 10x10 neighbourhood (left) of a greyscaleraw image (right). . . . . . . . . . . . . . . . . . . . . . . . . 595.4 Example of a filtered binary image. . . . . . . . . . . . . . . . 605.5 Border and centreline analysis. . . . . . . . . . . . . . . . . . 615.6 Eulerian cell analysis (not to scale). . . . . . . . . . . . . . . 635.7 PTV vs. LDV: Uθ(r). Reinlet = 47,000, z = -0.12 m. PTV(◦), LDV (•) CFD (). . . . . . . . . . . . . . . . . . . . . . 655.8 PTV vs. LDV: Uz(r). Reinlet = 47,000, z = -0.12 m. PTV(◦), LDV (•) CFD (). . . . . . . . . . . . . . . . . . . . . . 655.9 PTV vs. LDV: Ur(r). Reinlet = 47,000, z = -0.12 m. PTV(◦), LDV (•) CFD (). . . . . . . . . . . . . . . . . . . . . . 665.10 Φ(r, z) for the smallest (a) and biggest (b) particle studied inthe top window location; measured particle vector field (↗). . 685.11 Φ(r, z) for the smallest (a) and biggest (b) particle studiedin the middle window location; measured particle vector field(↗). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 695.12 Φ(r, z) for the smallest (a) and biggest (b) particle studied inthe bottom window location; measured particle vector field(↗). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 705.13 PTV vs. LDV: Uθ,polymer(r). Reinlet = 47,000, z = -0.22 m.PTV (◦), LDV (•). . . . . . . . . . . . . . . . . . . . . . . . . 725.14 PTV vs. LDV: Uz,polymer(r). Reinlet = 47,000, z = -0.22 m.PTV (◦), LDV (•). . . . . . . . . . . . . . . . . . . . . . . . . 725.15 PTV vs. LDV: Ur,polymer(r). Reinlet = 47,000, z = -0.22 m.PTV (◦), LDV (•). . . . . . . . . . . . . . . . . . . . . . . . . 735.16 ΦAPAM (r, z) for the smallest (a) and biggest (b) particle stud-ied in the top window location; measured particle vector field(↗). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 745.17 ΦAPAM (r, z) for the smallest (a) and biggest (b) particle stud-ied in the middle window location; measured particle vectorfield (↗). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 755.18 ΦAPAM (r, z) for the smallest (a) and biggest (b) particle stud-ied in the bottom window location; measured particle vectorfield (↗). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76xiList of Figures6.1 Hydrocyclone energy analysis. . . . . . . . . . . . . . . . . . . 816.2 APAM degradation curves: k(RR)APAM/k(RR)H2O vs. time.(a) RR = 0%, (b) RR = 25%, (c) RR = 50%, (d) RR = 75%.◦ 100 ppm APAM (11.2 kW); 4 150 ppm APAM (11.2 kW);♦ 100 ppm APAM (7.5 kW). . . . . . . . . . . . . . . . . . . 846.3 CPAM degradation curves (pumping power = 11.2 kW): k(RR)CPAM/ k(RR)H2O vs. time. (a) RR = 0%, (b) RR = 25%. ◦ 100ppm CPAM; ♦ 150 ppm CPAM. . . . . . . . . . . . . . . . . 856.4 APAM degradation curves in a 0.7% (w/w) SPF pulp suspen-sion (pumping power = 11.2 kW): k(RR)additive/k(RR)H2Ovs. time. (a) RR = 0%, (b) RR = 25%, (c) RR = 50%, (d)RR = 75%. ◦ 100 ppm APAM; ♦ 150 ppm APAM. . . . . . . 866.5 CPAM degradation curves in a 0.7% (w/w) SPF pulp suspen-sion (pumping power = 11.2 kW): k(RR)additive/k(RR)H2Ovs. time. (a) RR = 0%, (b) RR = 25%. ◦ 100 ppm CPAM;♦ 150 ppm CPAM. . . . . . . . . . . . . . . . . . . . . . . . . 876.6 Water lost energy curve: k(RR) vs. RR.  Raw data; ·−Cubic model. . . . . . . . . . . . . . . . . . . . . . . . . . . . 886.7 APAM lost energy curve: k(RR) vs. RR.  water; ◦ 100 ppmAPAM; B 300 ppm APAM; ♦ 500 ppm APAM. . . . . . . . . 896.8 CPAM lost energy curve: k(RR) vs. RR.  water; ◦ 100 ppmCPAM; B 300 ppm CPAM; ♦ 500 ppm CPAM. . . . . . . . . 906.9 Pulp fibre drag reduction curves: DR vs. U2inlet. (a) RR =0%, (b) RR = 25%, (c) RR = 50%, (d) RR = 75%. ♦ 0.5%SPF; ◦ 0.7% SPF; 4 0.9% SPF. . . . . . . . . . . . . . . . . 916.10 APAM DR evaluated in a 0.7% SPF pulp suspension: DRvs. U2inlet. (a) RR = 0%, (b) RR = 25%, (c) RR = 50%, (d)RR = 75%. • 0.7% SPF pulp suspension; ◦ 150 ppm APAM;4 300 ppm APAM; ? 500 ppm APAM. . . . . . . . . . . . . . 926.11 CPAM DR evaluated in a 0.7% SPF pulp suspension: DR vs.U2inlet. (a) RR = 0%, (b) RR = 25%, (c) RR = 50%, (d) RR= 75%. • 0.7% SPF pulp suspension; ♦ 100 ppm CPAM; ◦150 ppm CPAM. . . . . . . . . . . . . . . . . . . . . . . . . . 92B.1 Numerically predicted tangential velocity profiles assuminglaminar flow (`T = 0). © Water;  Power-law fluid (n′ = 0.5). 112B.2 Numerically predicted tangential velocity profiles assumingturbulent flow (κ = 0.05). ©Water;  Power-law fluid (n′ =0.5). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113xiiList of FiguresB.3 The effect of flow behaviour index on the tangential velocityprofiles of power-law fluids (κ = 0.05). ♦ Power-law fluid(n′ = 0.25); 4 Power-law fluid (n′ = 0.75). . . . . . . . . . . . 114C.1 Continuity error for the water case normalized to the inletflux using a second order difference approximation: RR = 0%. 116C.2 Continuity error for the water case normalized to the inletflux using a second order difference approximation: RR = 25%.116C.3 Continuity error for the water case normalized to the inletflux using a second order difference approximation: RR = 50%.117C.4 Continuity error for the 0.03% APAM solution normalized tothe inlet flux using a second order difference approximation:RR = 25%. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117C.5 Continuity error for the 0.03% APAM solution normalized tothe inlet flux using a second order difference approximation:RR = 50%. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118D.1 Pressure drop relationship for water calculated using LDVresults. Uinlet = 2.57 m/s, RR = 0%. . . . . . . . . . . . . . . 120D.2 Pressure drop relationship for water calculated using LDVresults. Uinlet = 2.57 m/s, RR = 25%. . . . . . . . . . . . . . 120D.3 Pressure drop relationship for water calculated using LDVresults. Uinlet = 2.57 m/s, RR = 50%. . . . . . . . . . . . . . 121D.4 Pressure drop relationship for APAM calculated using LDVresults. Uinlet = 2.57 m/s, RR = 25%. . . . . . . . . . . . . . 121D.5 Pressure drop relationship for APAM calculated using LDVresults. Uinlet = 2.57 m/s, RR = 50%. . . . . . . . . . . . . . 122F.1 Lost energy versus U2inlet, RR = 0%. © water;  0.03% APAM.126F.2 Lost energy versus U2inlet, RR = 25%. © water;  0.03%APAM. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127F.3 Lost energy versus U2inlet, RR = 50%. © water;  0.03%APAM. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127xiiiNomenclatureα Reduction factorαk Volume fraction of phase kδ¯k,m Random error valueL¯w Weighted fibre length(mm)L¯ww Weighted weighted fibrelength (mm)U¯ Bulk velocity (m/s)U¯r,slip Radial particle slip velocity(m/s)U¯z,slip Axial particle slip velocity(m/s)W¯ Fibre width (µm)∆p Pressure drop (Pa)∆t Time step (sec)γ˙ Carreau model shear rate(s−1)m˙ Mass flow rate (kg/s)`T Turbulent lengthscale (m)`energy Lost energy (m2/s2) Hencky strainη Experimentally determinedviscosity (Pa s)ηE Extensional viscosity (Pa s)ηo Carreau model zero shearrate viscosity (Pa s)η∞ Carreau model infinite vis-cosity (Pa s)γ Stretch ratioΓk Rate of mass generation ofphase k at the interface(kg/m3s)uˆr Relative radial velocity(m/s)κ Turbulent lengthscale con-stantλ Carreau model time con-stant (s)〈U〉 Mean streamwise velocity(m/s)〈u′z〉 Axial fluctuating compo-nent (m/s)〈u′θ〉 Tangential fluctuatingcomponent (m/s)〈ur〉 Mean radial velocity (m/s)〈uz〉 Mean axial velocity (m/s)〈uθ〉 Mean tangential velocity(m/s)〈uv〉 Reynolds stress (m2/s2)µ Fluid viscosity (Pa s)xivNomenclatureµ′ Simplified laminar consis-tency index (Pa sn)µ′′ Laminar consistency index(Pa sn)∇ Gradient operatorνT Turbulent viscosity (m2/s)ω Angular velocity (s−1)ωR Angular velocity at radiusR (s−1)∂x Eulerian cell width (mm)∂y Eulerian cell depth (mm)∂z Eulerian cell height (mm)Φ Slip factorΨ Non-dimensional angularvelocityρ Density (kg/m3)ρp Particle density (kg/m3)σ Non-dimensional radiusτ Shear stress (Pa)τT Turbulent stress (Pa)τw Wall shear stress (Pa)θ(t) Tangential position at timet (rad)ε Rate of strain tensor~U(n, i) Velocity vector of nth tra-jectory~V (k,m) Mean velocity vector of(k,m)th cella Length of a single monomerAparticle Particle area (m2)B Integration constantC Continuity constantCD Particle drag coefficientCf Skin friction coefficientD Pipe diameter (m)dp Particle diameter (m)Dmid(t) Sample diameter as afunction of time (mm)Do Initial sample diameter(mm)DR Drag reductionf Friction factorjk Volumetric flux of phase k(m/s)k Lost energy constantL Axial distance (m)m Carreau model fluid be-haviour indexmp Particle mass (kg)N Number of repeatingmonomersn Free vortex constantn′ Flow behaviour indexP Pressure (Pa)Pparticle Particle perimeter (m)Q Volumetric flow rate(m3/s)r(t) Radial position at time t(m)Rz Hydrocyclone radius (m)RR Reject ratioSp Particle circularityStk Stokes numbert Time (sec)Tz Polymer relation time (sec)xvNomenclatureU Mean velocity (m/s)u Instantaneous radial veloc-ity (m/s)u∗ Turbulent velocity scale(m/s)u+ Dimensionless velocityuτ Friction velocity (m/s)Ui Average inlet velocity(m/s)u(r,θ,z) Velocity uncertainty fromparticle location error(m/s)U∞ Free stream velocity (m/s)ur,p Particle radial velocity(m/s)v Instantaneous tangentialvelocity (m/s)w Instantaneous axial veloc-ity (m/s)y Spanwise position (m)y+ Viscous lengthsz Axial position (m)z(t) Axial position at time t (m)[A] PVSK uptakeAPAM Anionic polyacrylamideCPAM Cationic polyacrylamideE(f) Energy spectra in terms offrequency fI Turbulent intensityk Boltzmann constant(m2kg/s2K)LDA Laser Doppler anemometryLDV Laser Doppler velocimetryN1 First normal stress differ-ence (Pa)PDA Phase Doppler anemome-tryPM Pressure matched LDVtrialR Total pipe radius (m)Re Reynolds numberSpeed√〈uz〉2 + 〈uθ〉2 (m/s)SPF Spruce, pine and firT Temperature (K)TDR Turbulent drag reductionUM Velocity matched LDV trialxviAcknowledgementsI would like to express my sincere gratitude to those who have assisted mein completing this work. I would like to thank my supervisors, Dr. JamesOlson and Dr. Mark Martinez for their guidance, and support throughoutthe course of this project. I would like to also give a special thanks to Dr.Richard Branion for the many helpful discussions throughout my time atthe Pulp and Paper Centre.I would like to sincerely thank the help of the technical staff and studentsof the Pulp and Paper Centre at the University of British Columbia, foryour assistance was essential to the successful completion of this project.In particular, I would like to thank George Soong, for your devotion tothe students at the Pulp and Paper Centre. Your commitment to keepingthe Pulp and Paper Centre running efficiently, was imperative towards thecompletion of this work.A special thanks is due to Dr. Jens Heymer, Dr. Sean Delfel, FrankSaville and Troy Mithrush, good friends and colleagues, for their words ofencouragement and support during my time at the Pulp and Paper Centre.I truly appreciate your assistance in this project from start to finish.Most of all, I would like to pay a special tribute to my family for theirunconditional love and unselfish support. I owe a great deal to my fa-ther, Gordon MacKenzie, brother and sister, Trevor and Taylor MacKenzie,and grandparents, John and Lorena Woronuk, who have been behind methroughout this long journey. I can’t thank you enough for your patience,and understanding during all this time.Lastly, I would like to thank the Natural Sciences and Engineering Re-search Council of Canada (NSERC), and Aikawa Fiber Technologies (AFT)for their financial support and interest in this research.xviiChapter 1IntroductionHydrocyclones, also known as centrifugal cleaners, are extensively used in avariety of industries as solid-solid, solid-liquid and liquid-liquid separationdevices, as well as particle classifiers. Figure 1.1 illustrates the basic shapeof a hydrocyclone, where Q refers to volumetric flow rate, and L refers tolength. In the pulp and paper industry they are used to remove undesirableparticles and unsatisfactorily pulped fibres from pulp suspensions. The re-moval of these contaminants has been found to improve paper quality, andreduce downstream equipment degradation. This work aims at reducing thecosts associated with operating a pulp processing hydrocyclone without anyalterations to the geometry.At the present time, the pulp and paper sector is seeing an increase inthe costs associated with hydrocyclone operation. Very little improvementshave been made to the separation efficiencies of hydrocyclones, making thema costly and only a mildly effective contaminant removal device. If this prob-lem continues to be ignored, it is likely that hydrocyclone costs will continueto rise, where their ability to remove contaminants from concentrated, andmore complex suspensions will remain suspect. In an attempt to improvethe contaminant removal efficiencies, and reduce the energy losses in a pulpprocessing hydrocyclone, the effects of adding various drag reducing agentswere analysed using a variety of experimental methods.1Chapter 1. IntroductionFigure 1.1: Illustration of the hydrocyclone geometry used for this thesis.To effectively evaluate the changes to the operation of a hydrocyclonewith the addition of drag reducing agents, it was deemed important to char-acterize the flow field, measure the motions of variously sized particles, andquantify the energy savings for water and all of the drag reducing agentsconsidered in this thesis. Unfortunately due to opacity constraints, the flowfield and particle motions within suspensions containing cellulose fibres werenot measurable. It was known before the onset of this work that the flowin a hydrocyclone is complex in nature, as such, it was believed to be nec-essary to characterize all of the findings to quantities which can be easilycarried over to industry. An effective quantity which gives some insightinto a hydrocyclone’s internal operation is it’s reject ratio, defined as thefollowing:RR =QunderflowQinlet(1.1)21.1. ObjectivesWhere RR is reject ratio, Qunderflow is the underflow volumetric flowrate, and Qinlet is the inlet volumetric flow rate. Visually, these terms arerepresented on Figure 1.1, where RR is defined as the portion of the totalflow entering the hydrocyclone which exits through the underflow outlet.The benefit in relating the flow field, particle motions, and energy loss resultsto a hydrocyclone’s RR is the direct applicability of the findings to a moreindustrial setting. This assists in reducing the difficulties associated withapplying laboratory scale findings to a large scale operation.1.1 ObjectivesTo identify the effects of adding turbulent drag reducing additives to a pulpprocessing hydrocyclone, three key objectives were required for this work.First, characterize the flow field in a hydrocyclone with and without thepresence of drag reducing agents. Second, measure the motions of variouslysized particles when suspended in water and a drag reducing agent. Third,quantify the energy savings of water, pulp, and pulp and polymer mixturesat a variety of inlet conditions.An introduction to some of the important findings obtained by previousinvestigators are presented in Chapter 2. This review assisted in determiningsuitable measurement techniques to achieve the goals of this work. Theexperimental apparatus, and a discussion of the fluid and fibre propertieswhich were considered in this thesis are presented in Chapter 3. The resultsof this work are presented and discussed in three separation categories. First,the flow field of water and a drag reducing additive are discussed in Chapter4. An analysis of the motions of variously sized particles suspended in waterand a drag reducing additive follows, and can be found in Chapter 5. Finally,quantification of the energy losses inside the experimental hydrocyclone fora range of fibre suspension consistencies and drag reducing polymer agentsare presented in Chapter 6. A summary of the work presented in this thesiscan be found in Chapter 7.3Chapter 2Literature ReviewThe following chapter outlines some of the relevant literature needed tobetter understand the problem described in this work. A review of somekey experimental studies on the hydrocyclone flow field, particle motions,and fundamentals of turbulent drag reduction using fibre suspensions andpolymer additives is presented and discussed.2.1 Hydrocyclone Flow Field2.1.1 Newtonian fluidsThe flow field of Newtonian fluids in hydrocyclones has been studied ex-tensively (e.g. [3]). A summary of the flow pattern is shown in Figure 2.1and can be described as a combination of an outer helical flow and an innerhelical flow. In a hydrocyclone, the outer and inner helical flows have threecomponents of velocity, namely: tangential velocity (uθ(r, θ, z)), axial veloc-ity, (uz(r, θ, z)) and radial velocity (ur(r, θ, z)). In most investigations thedependence on tangential position, θ, has been found to be minor in hydro-cyclones, as such to reduce experimentation, the flow is generally assumedto be axially symmetric. The early studies of Kelsall [34] and Knowles et al.[37] experimentally measured the tangential and axial velocity componentsof droplets (i.e. Stokes number  1) using ultra-microscope illuminationand cine photography, respectively. They found that the tangential compo-nent of velocity follows a free vortex flow pattern throughout a majority ofthe hydrocyclone’s radius, where a small region near the axis follows thatof a forced vortex flow. The axial velocity measurements showed that theflow moves towards the underflow outlet near the hydrocyclone wall, and theflow moves towards the overflow outlet near the hydrocyclone’s axis. Theradial transition point from upwards to downwards flow was not found toalign with the free to forced vortex transition point. Assuming the flow isaxially symmetric, their results indicated that the radial velocity is domi-nantly positive (i.e. outwards) near the wall, and negative (i.e. inwards)near the axis. Authors such as Bradley [3], have reported secondary flow42.1. Hydrocyclone Flow Fieldpatterns inside hydrocyclones, the most common being the short circuit flowfrom the inlet directly to the vortex finder, and circulatory eddy flow (see:Figure 2.2).Figure 2.1: Hydrocyclone flow pattern [63].The studies by Dabir [14] and Monredon [53] experimentally determinedthe flow field within a hydrocyclone using LDA. Dabir [14] studied the effectof a vortex finder contraction, Reynolds number at the hydrocyclone inlet(i.e. Reinlet), back pressure and volume split fraction (i.e. 1 − RR) on theinternal flow pattern within a 7.62 cm diameter hydrocyclone. He found thatReinlet, back pressure and reject ratio have the most pronounced effect onthe mean axial velocity profile, defined as 〈uz〉. As expected, the tangentialvelocity scaled to that of the Reinlet.The work performed by Hsieh [27] compared numerical and experimen-tal results of glycerol-water mixtures in a 75 mm diameter hydrocyclone. Hefound that mixing length theories for the length and radial scales of turbu-lence in a hydrocyclone could be applied to a modified Prandtl mixing-lengthmodel to predict the flow field. When comparing the numerically predictedflow field to the experimentally obtained LDV results, it was found thatincreasing the effective viscosity (i.e. shear viscosity plus turbulent viscos-52.1. Hydrocyclone Flow FieldFigure 2.2: Mantle and short circuit flow in a hydrocyclone [3].ity) in his numerical simulation changed the distribution of 〈uz〉 to a muchgreater extent than the mean tangential velocity, 〈uθ〉.2.1.2 Hydrocyclone turbulence with Newtonian fluidsVarious studies have established that the flow field within a hydrocycloneexhibits turbulent behaviour (e.g. [8, 9, 26]). Hou et al. [26] acquired noisespectra using an acoustic sensor mounted on a hydrocyclone. They foundthat when plotting the log-log scale of the energy spectra (E(f)) versus fre-quency (f), a substantial range agreed with Kolmogorov’s third hypothesis,namely: E(f) ∝ f−5/3, where f is the frequency. Their power density spec-tra findings indicated that the air core oscillation frequency may be stronglydependent on feed pressure. Qian et al. [58] investigated the local radialand tangential turbulent intensities with and without an air core at identi-cal feed pressures. The local radial turbulence intensity was, for example,calculated as: Ir = 〈u′r〉/〈ur〉 where 〈u′r〉 is the mean turbulent fluctuation ofthe radial component of velocity. A near 2% decrease in the mean tangentialturbulence intensity and an up to 30% decrease in Ir were observed near thecyclone axis when the air core was not present.One of the early studies of Hsieh [27] presented the mean axial tur-bulent fluctuations, 〈u′z〉, and the mean tangential turbulent fluctuations,62.1. Hydrocyclone Flow Field〈u′θ〉, for glycerol-water mixtures, concluding that the axial fluctuations wereslightly greater than the tangential fluctuations for most radial positions.He observed very high turbulent fluctuations in uθ towards the core of thehydrocyclone, where the axial fluctuations stayed relatively constant. Anincrease in glycerol concentration, resulting in an increase in viscosity, in-creased both the axial, 〈u′z〉, and tangential 〈u′θ〉 fluctuations towards thecore of the hydrocyclone for the axial positions studied. A more invasivestudy by Chu et al. [8] found, using a strain gage, a similar result. Chuet al. [9] extended their previously mentioned work and studied the meanand root mean square (RMS) time averaged pressure fluctuations using astrain gauge in a modified hydrocyclone containing a so called winged core.A winged core hydrocyclone is generally a standard hydrocyclone with apiece of material suspended in a portion of the hydrocyclone which consistsof four equally spaced wings. This ultimately produces a secondary flowpattern between each wing, and was found to increase capacity, and reduceenergy losses. Although the flow pattern of a winged core hydrocyclone isdifficult to characterize due to it’s complexities, and the effects of using astrain gauge are unclear, a decrease in pressure fluctuations near the axis ofthe hydrocyclone was observed.Laser Doppler anemometry (LDA) has proven to be a desirable methodin obtaining non-periodic turbulent fluctuations within a hydrocyclone, asthese fluctuations are measurable with relatively low exposure times (i.e.≤1000 Doppler bursts). Jirun et al. [32], for example, presented the timeaveraged turbulence and relative turbulence for the radial, axial and tangen-tial components of velocity using LDA. Their results indicated dampeningof the radial relative turbulence near the wall for flows containing no forcedvortex, suggesting that hydrocyclones operating with no forced vortex maybe more suitable for treating dilute suspensions. A later study by Chine´et al. [5] compared the turbulent flow field in a conical and cylindricalhydrocyclone using LDA. They found the root mean square of the axialfluctuations to be greater than the tangential fluctuations throughout thebody of a conical and cylindrical hydrocyclone.2.1.3 Non-Newtonian fluidsThe effect of liquid medium viscosity on hydrocyclone performance has beenstudied extensively, as most hydrocyclone suspensions and various liquidmediums display shear dependent viscosities. Bradley [3] observed a de-crease in static pressure drop with an aniline/water solution when comparedto water alone at the same feed flow rate. Fontein et al. [21] demonstrated72.1. Hydrocyclone Flow Fieldthe effect of liquid viscosity on feed flow rate, and clarification number (i.e.clarification number = feed concentration−overflow concentrationfeed concentration ) operating atconstant pressure drop. It was found that an increase in the temperature ofa suspension containing potato-starch extract resulted in a decrease in feedflow rate and an increase in clarification number, where this difference wasattributed to the decrease in the liquid medium viscosity.The influence of suspension rheology on the tangential velocity compo-nent in a 65 mm diameter conical hydrocyclone was investigated by Bergstro¨met al. [1]. Tangential velocity profiles throughout the hydrocyclone wereobtained using a pitometer for fibre consistencies ranging from 0% to 0.19%.An increase in fibre consistency resulted in a reduction of the maximum tan-gential velocity near the core of the hydrocyclone at all axial measurementpositions. The radial position of the maximum tangential velocity, Uθ,max,increased with an increase in fibre consistency with constant reject ratio andinlet velocity. Similarly to Bergstro¨m et al. [1], Lilge´ [43] investigated theeffect of magnetite concentration by characterizing the resultant change inthe tangential velocity field by the following relationship:〈uθ〉rn = 〈uθ,wall〉Rnz = αUi(Rz)n (2.1)where Rz is the outer radius at an axial position z, α is a constant, andUi is the inlet velocity. His data suggested that the tangential velocity fieldof suspensions containing magnetite have similar flow constants: n and α,to water alone. This may have been due to high shear in the hydrocyclone,causing the suspension to act like the liquid medium alone.2.1.4 Hydrocyclone turbulence with non-NewtonianmediumsTurbulence effects in the bulk of a hydrocyclone undoubtedly can have aneffect on particle motion (e.g. [63]), reducing the expected separation ef-ficiencies. The early work of Dabir [14] presented minor differences in theaxial and tangential velocity components with the addition of 100 ppm byweight of Separan AP-30, an anionic polyacrylamide, at identical Reinletand reject ratio compared to water alone. Drag reduction properties of theSeparan AP-30 polymer were measured using the Von Karman coordinates,where up to 50% drag reduction was observed for all solutions studied. Thisimplies that turbulent fluctuations may have been reduced within the hydro-cyclone. A later study by Walker et al. [71] presented measurements of the82.2. Particle Motion in a Hydrocyclonemean tangential, and axial fluid velocities using laser Doppler anemometryfor various concentrations of carboxy-methylcellulose (CMC) throughout a0.05 m diameter hydrocyclone. An increase in concentration from 0.3% to0.5% CMC reduced the region of positive axial flow towards the overflowoutlet, where 〈uθ〉 decreased from the wall towards to the core throughoutthe hydrocyclone body. Turbulent effects in this study were shown in theform of Reynolds stresses for the 0.3% and 0.5% CMC solutions; an increasein CMC concentration reduced 〈u′zu′θ〉 throughout the hydrocyclone, mostsignificantly near the core.2.2 Particle Motion in a HydrocycloneThe centrifugal force developed by the swirling motion of a suspension in ahydrocyclone creates a differentiation between the liquid medium and thesuspended particles. Characteristics such as: particle and fluid density, par-ticle diameter, particle concentration, and fluid viscosity are known to effectthe relative motion between particles and the suspending fluid. Experi-mental studies on the motion of solid particles in a hydrocyclone has un-fortunately seen little work. The difficulties associated with measuring themotion of particles, and high fluid turbulence levels, leading to variabilityin particle trajectories, are some of the limitations to producing statisti-cally sound results. Most experimental work has focused on characterizingthe radial and axial motion of variously sized particles in hydrocyclones todetermine separation characteristics. This is largely due to the difficultiesassociated with measuring the motion of particles three dimensionally (e.g.particle matching between two cameras), and because overflow/underflowseparation can be directly linked to the particle’s radial and axial motion.Particles which move downwards and towards the hydrocyclone wall willlikely be removed through the underflow, whereas particles which move up-wards and towards the hydrocyclone axis will likely be removed through theoverflow. The most predominant techniques for measuring particle motionin complex geometries are high speed video imaging, and phase Doppleranemometry (PDA).Chu et al. [10] studied the 2-D motion of solid particles using phaseDoppler anemometry (PDA) with modifications to the hydrocyclone geome-try and operating condition. They observed an increase in a particle’s radialvelocity with an increase in inlet pressure or underflow diameter, as well asa decrease in feed concentration. Their data also suggested that an increasein a particle’s density or diameter increased the particle’s relative radial ve-92.3. Numerical and Experimental Discrepancieslocity against the suspending fluid, namely water. The axial velocity profilesthroughout the hydrocyclone showed an increase in the particles upwardsaxial velocity towards the overflow, and an increase in a particles downwardsaxial velocity near the apex.The study of Wang et al. [72] described particle motion in a hydro-cyclone using a stochastic, Lagrange method for various particle sizes (i.e.700 µm, 800 µm, and 900µm) and injection points. This work focused onoverflow/underflow separation where particle velocities were not considered.The results of this study indicated that smaller particles have an increasedresidence time when exiting from the overflow when compared to largerparticles of identical density. The opposite effect was observed for smallparticles (i.e. 700 µm) exiting from the underflow. Their analysis also sug-gested that particles entering a hydrocyclone on the near wall side of theinlet were more susceptible to being removed through the underflow, whichis likely a result of the strong negative axial (towards the underflow) flownear the wall.2.3 Numerical and Experimental DiscrepanciesSignificant work has been done with regards to modelling hydrocyclone flowfields in hopes to improve separation efficiencies without the need of exper-imental studies. The time necessary to experimentally study hydrocycloneflow fields, and the motion of particles, is significant when considering thelow cost associated with CFD models. Unfortunately, experimental resultsrarely coincide with CFD models, largely due to the inaccuracies associ-ated with the turbulence-closure models. Figure 2.3 illustrates the changein 〈uθ,z〉 for three different turbulence-closure models.102.3. Numerical and Experimental DiscrepanciesFigure 2.3: Comparison of LDA and CFD data for three turbulence-closuremodels [15].112.4. Turbulent Drag ReductionDelgadillo et al. [15] found that a large eddy simulation (LES) pro-duced results that agreed to the experimental data of Hsieh [27] betterthan the results obtained with a Reynolds stress model (RSM) or k- turbu-lence model (see Figure 2.3). This suggests that resolving a portion of theturbulent eddies improves the accuracy of a hydrocyclone CFD simulation.The review of Narasimha et al. [54] details the effects of model choice onthe resulting velocity profiles, and Lagrangian particle trajectories. It wasshown that numerically estimating particle motions by including particledrag forces, centrifugal and Coriolis forces, resulted in good agreement withexperimental data (e.g. [28]) at low feed solid loadings. A more rigorousmodel which includes the effects treated by Basset, Boussinesq, and Oseen,coupled with a stochastic model for the turbulence effects on particle motion,would undoubtedly improve the accuracy of the computationally predictedparticle trajectories. This detailed computation, however, is still a researchchallenge.2.4 Turbulent Drag ReductionThe term turbulent drag reduction is best defined in a pipe by the decreasein pressure drop per unit length at the same characteristic velocity when, forexample, compared to water alone. Various additives display turbulent dragreducing capabilities in a wide range of industrial flows, assisting in reducingpumping power requirements and fluid energy losses. Specific to this review,some characteristics of fibre and polymer induced turbulent drag reductionwill be discussed.The unique characteristics of pulp slurries in pipe flows have been stud-ied extensively (e.g. [19], [52]). Figure 2.4 shows the onset of drag reduction(point D) where the pressure drop per unit length (i.e. ∆P/L) of the sus-pension, is less than that of water alone for all velocities above this point.The point of maximum drag reduction occurs at point E, where above thispoint the increase in turbulence will continue to disrupt the plug. The flowbecomes fully turbulent (plug dispersed) beyond point E where there is adecrease in the degree of drag reduction in comparison to the maximum (see[19]).Vaseleski et al. [68] obtained flow data for fibre suspensions in pipes withdifferent diameters and concluded that fibres in the turbulent core region offlow are an important factor in fibre-induced drag reduction. Lee et al. [39]found that turbulent velocity profiles in suspensions containing fibres couldbe extrapolated to intersect at the point typically given as the end of the122.4. Turbulent Drag Reduction101 102 103103104log Q (L/min)log ∂ P/L (Pa/m)B C DEAPulp Curve Water Curve(Turbulent flow)Figure 2.4: Head loss curves for water and a pulp suspension in pipe flow.buffer layer in Newtonian fluids. This implies that the turbulent mechanismchanges and occurs in the turbulent core and not near the wall. Although themechanism of drag reduction in pulp suspensions is still not well understood,fibres and flocs of fibres have been shown to dampen turbulence.Fibre suspensions are known to increase the pseudo-viscosity of a sus-pension, which allows more momentum transport to occur without Reynoldsstresses, ultimately increasing drag. To obtain a net reduction of drag, fi-bres must reduce turbulent momentum transfer without increasing otherforms of momentum transfer. Many authors (e.g. [35]) believe that dragreduction in fibre suspensions in a pipe, is due to dampening of turbulencein the radial direction; however, this cannot solely decrease the Reynoldsstresses. For example, Sharma et al. [62] studied the effects of near walland centerline injection of asbestos fibres in a pipe, concluding that bound-ary layer injection is superior as a result of the decrease in radial momentumtransfer. They noted that the presence of fibres in the core enhances thedrag-reducing capability of polymer additives. However, as discussed, fora decrease of the Reynolds stresses to occur in a pipe, the radial and lon-gitudinal velocity fluctuations must decouple. How fibres or flocs of fibresinduce this decoupling of the longitudinal and radial velocity fluctuations isstill up for discussion.In 1948 Toms [66] reported that the addition of polymethylmethacrylateto high Reynolds number, turbulent pipe flow of monochlorobenzene reducedthe pressure drop per unit length below that of the solvent alone at a similar132.5. Polarity Effects in Fibre Suspensionsflow rate. Since then, much work has been done on studying the effects ofpurely viscous, shear-thinning liquids in practical applications such as pipeflow, channel flow (Dimitropoulos et al. [17] and Min et al. [51]) and marinevehicles (Canham et al. [4]). Unlike fibre-induced drag reduction, dragreduction through polymer additives is generally divided into two classes.The most predominant class focuses on the viscous effect (e.g. [44] and [61])where in wall bounded shear flows, the polymers are believed to be stretchedin the buffer layer. The strain rate and vorticity fields associated with thebuffer layer are believed to cause full extension of the polymers, leading toan increase in the elongational viscosity (e.g. [49]), suppression of turbulentfluctuations, and a reduction in wall friction. The second class associatesthe effects of polymer additives with drag reduction as an elastic effect.Tabor et al. [64] argues that the elastic energy stored by partially stretchedpolymers becomes comparable to the kinetic energy in the buffer layer atsome turbulent length scale above the Kolmogorov scale. The Kolmogorovenergy cascade from the energy-containing range to the dissipation range isthus terminated, where elastic behaviour is seen at scales below this limit.This elastic behaviour is believed to increase the buffer layer thickness andreduce the Reynolds stresses.Previous work (e.g. [29, 41, 65]) has shown that the effects of polymermolecular weight and molecular weight distribution are considerable towardsoptimization of drag reduction in shear flow. Flow-assisted degradation be-haviour of polymer additives has been correlated to solvent, polymer con-centration, molecular weight, and molecular weight distribution. It has beenobserved that scission of molecular structures does occur; however, the sig-nificant contribution to drag reduction losses is related to the decrease instructure formation. Liberatore et al. [41] noted that the large drag re-duction with solutions that did not form such structures would suggest thatirreversible aggregate deformation contributes to DR. However, this couldoccur due to incomplete mixing of the solution within the study volume.2.5 Polarity Effects in Fibre SuspensionsThe rheology of fibre suspensions has been well reviewed (e.g. [36]), con-cluding that fibre contact, network strength, medium viscosity and chemicaladditives all play an important role in characterizing fibre suspensions inshear flow. Retention aids, generally cationic polymers, are commonly usedin a range of fibre suspensions in which the fibres are negatively charged.Various authors have studied the effects of cationic polymer addition in fibre142.6. Summary of Literatureflows, noting that the flow mechanisms change significantly. Cationic ad-sorption in dilute fibre suspensions promotes fibre-fibre bridging (e.g. [40]),whereas, in high consistency fibre suspensions, mechanical entanglement be-tween fibres dominate. Cationic polymer addition increases the number offibre-fibre contact points and the shear strength of the fibre flocs in dilutefibre suspensions. It has been proposed previously that both interparticlebridging and charge reduction are important flocculation mechanisms forfibre suspensions. A distinct rheological difference is seen at low bulk flowrates, where in dilute suspensions without polymer, a loose and continuousnetwork entraps all the solvent and moves as a plug. With the addition ofcationic polymer, denser flocs form with a fluid phase containing few fibres,indicating an increase in suspensions yield stress. The increase in suspen-sions yield stress is a result of the increase in fibre-fibre contact points andstrength of the mechanical entanglements.Anionic polymer addition in dilute fibre suspensions decreases the bridg-ing effect between fibres due to the increase in negative charge density. Noadsorption between the fibres and polymer occurs, increasing the effects offibres, not fibre networks, on drag reduction under turbulent conditions. Inpipe flow, for example, a strong fibre network, or plug, is present at low bulkflow rates where an increase in flow rate or Reynolds number reduces thesize of the plug due to the increase in shear. The addition of anionic polymereffectively alters the deformation of the plug as a function of Reynolds num-ber. This de-flocculation effect stimulated by the elevated negative chargedensity results in a decrease of the initial plug core size, possibly decreasingthe maximum drag reduction potential of the fibres (see: Figure 2.4). Theeffects of fibre network strength on drag reduction, however, is still not wellunderstood.2.6 Summary of LiteratureThe literature survey for this thesis revealed many important features con-cerning the flow field of Newtonian and non-Newtonian fluids in a hydro-cyclone. There is evidence to suggest that the flow field of water inside ahydrocyclone is different than that of a non-Newtonian fluid. For example,the works of Bergstro¨m et al. [1] and Walker et al. [71] revealed the flowfields of cellulose fibre suspensions and CMC solutions, respectively, are dif-ferent to that of water alone (e.g. [14]). Turbulence results in the studyby Walker et al. [71] showed an increase in CMC concentration resultedin a decrease in the Reynolds stresses throughout a 50 mm diameter hy-152.6. Summary of Literaturedrocyclone. How non-Newtonian fluids change the degree of the axial andtangential turbulent fluctuations is still up for discussion, as historically ithas been found that for water, the axial turbulent fluctuations are greaterthan the tangential turbulent fluctuations throughout the body of a hydro-cyclone (e.g. [5, 27]). This type of characterization of the turbulence insidehydrocyclones is still a research challenge, and has shown to be a majorinfluence in CFD simulations.The study by Delgadillo et al. [15] showed the effects of turbulence-closure model choice varied the predicted axial and tangential velocities ofwater in a 75 mm diameter hydrocyclone. This is a challenge, as CFDsimulations are used to predict the motions of variously sized particles forindustrial applications. This is largely due to the experimental measurementdifficulties associated with hydrocyclones, as the flow field is three dimen-sional and turbulent. To this author’s knowledge, the three dimensionalmotion of particles has yet to be instantaneously measured experimentally,which is crucial to statistically show the effects of fluid and particle size onthe resulting motions. This type of analysis is, however, limited to verydilute suspensions, which in many cases is difficult to use towards charac-terizing more concentrated suspensions.Cellulose fibre suspensions display complex properties in bounded shearflows where, depending on consistency, they will initially move as a plugunder small amounts of shear. An increase in shear, however, results inthe fibre plug breaking down and an increase in turbulence at the plugboundary. In a pipe, it has been found that these fibre suspensions displayturbulent drag reducing properties, as the pressure drop per unit length issmaller than that of water alone at the same characteristic velocity. Thishas been similarly observed for polymer solutions in pipe flow; however, themechanisms behind drag reduction of these additives are not closely related.Nevertheless, how fibres, flocs of fibres, or polymer additives alter turbulencein pipe flow, is still not well understood.Cellulose fibre suspensions are commonly flocculated or dispersed by us-ing either a cationic or anionic polymer additive, which stems from cellulosefibres being naturally negatively charged. The synergism between cellulosefibres and polymer additive is difficult to characterize, as cationic polymeradsorbs on to the surface of cellulose fibres. The effective drag reductionof fibre suspensions containing cationic or anionic polymer is likely heavilydependent on polymer concentration, as fibre contact and percentage of ad-sorption are likely to change. How polymer charge, fibre consistency, andpolymer concentration influence the drag reducing properties of these sus-pensions in industrial flows is still not well understood and continues to be162.6. Summary of Literaturea research challenge.17Chapter 3Experimental DesignExperiments were carried out using an acrylic hydrocyclone, which was de-signed and fabricated to dimensions commonly seen in the pulp and papersector. Following the installation of the hydrocyclone to the experimentalflow loop, experiments were carried out to better understand the effects ofturbulent drag reducing additives on hydrocyclone operation. Firstly, theinternal flow field was measured for both water and a polyacrylamide solu-tion using laser Doppler velocimetry. Under identical operating conditions,the three dimensional motion of variously sized particles was measured us-ing high speed video imaging, accentuating the effects of surface area andpolymer addition. Lastly, turbulent drag reduction in a pulp processinghydrocyclone was quantified under a variety of operating conditions, em-phasizing the potential for industrial application. This chapter details thelaboratory apparatus, and fluid and fibre properties which were investigatedin this study. A schematic of the interactions between the test environmentand data processing is shown in Figure 3.1.Figure 3.1: Diagram showing interactions between test environment anddata processing.183.1. ApparatusFigure 3.2: Experimental flow loop.3.1 ApparatusThe flow loop used for the experiments is shown schematically in Figure3.2. Flow was provided by a centrifugal pump (Flowserve 0.088 m) drivenby a 15 kW, variable speed, Teco motor fed directly from a 1 m3 tank via a0.10 m diameter PVC schedule 80 pipe through a Rosemount 8712E mag-netic flow-meter. The line size constricts over 1.5 m to a 0.025 m diameterschedule 80 PVC pipe where a Rosemount 2088 pressure transducer is con-nected to a threaded tee fitting. The loop constricts again to 0.019 m over0.4 m until entering the hydrocyclone. The flow rate and pressure in theoverflow and underflow lines are recorded using 0.025 m Rosemount 8712Emagnetic flow-meters and Rosemount 2088 pressure transmitters, respec-tively. Both outlets have schedule 80 ball valves to control the flow-rateand pressure in each exit stream. Data logging for the pressure transducersand flow-meters is provided using a NI-6210 USB-DAQ connected to a datalogging computer running LABVIEW. The transducers were calibrated atperiodic intervals using air against a Druck DPI 601 high-precision differen-tial pressure transducer with an accuracy over the calibration range of 0.1%of reading. The accuracy of the Rosemount transducers are estimated tobe better than ±0.5% of fsd (full scale deflection). An alcohol thermometerwas used to record the tank temperature to an accuracy of ±1◦.The optically clear experimental hydrocyclone used in this study was193.2. Fluid and Fibre CharacteristicsHydrocyclone Diameter 102 mmInlet Diameter 19.2 mmVortex Finder 12.7 mmDepth of Vortex Finder 63.5 mmWall Thickness of Vortex Finder 3.1 mmSpigot Diameter 6.0 mmLength of Cylindrical Section 102.0 mmLength of Conical Section 405.2 mmCone Angle 6.76 ◦Table 3.1: Hydrocyclone dimensions.made from a 0.15 m ×0.18 m × 0.50 m block of acrylic glass, where theblock was cut in half and milled to the desired dimensions. The block isheld together with four dowel pins for alignment and six 0.21 m threadedrods for stability. The hydrocyclone has two 0.28 m × 0.20 m × 0.0084 mstainless steel rectangular blocks on the top and bottom for elevation sup-port. The top piece consists of a threaded fitting for the 0.013 m vortexfinder, allowing flexibility in the length of the vortex finder for future ex-periments. The bottom piece is aligned with the 6 mm underflow diameterwhere a connection to the schedule 80 PVC piping was attained withoutdisrupting the acrylic glass. Rubber o-rings throughout the block providedsufficient sealing, however, limited the optical range near the cyclone axis(see Chapter 5). The dimensions of the hydrocyclone are listed in Table 3.1.3.2 Fluid and Fibre CharacteristicsThe fluids used in the present work were aqueous solutions and suspensionsof the following polymers and pulp fibres, respectively:• Anionic Polyacrylamide (APAM), Superfloc A-110 supplied by Kemira.• Cationic Polyacrylamide (CPAM), Superfloc C492PWG supplied byKemira.• 100% SPF (spruce, pine and fir) Kraft pulp supplied by Canfor. SeeTable 3.2 for fibre properties.The rheological properties of the SPF pulp suspensions used in this studywere not measured, as characterizing the flow behaviour was outside the203.2. Fluid and Fibre CharacteristicsL¯w (mm) L¯ww (mm) W¯ (µm) %fines2.1± 0.035 2.6± 0.037 28.1 ± 0.39 35± 0.5Table 3.2: SPF fibre quality analyser results.scope of this work. As discussed in Chapter 2, the flow of pulp suspensionsare complex, where the transition of a fibre plug to fully turbulent flow ina pipe occurs over various stages. Ja¨sberg [30] summarized the transition,indicating that a plug flow regime with incipient fluid phase turbulence ex-ists, where with an increase in Reynolds number, transforms to a mixedflow regime. Both stages were found to contain a fibre plug, indicating thatfibre suspensions contain a yield stress at relatively high Reynolds numbers.Nikbakht et al. [56] found that SPF fibre suspensions had yield stressesbelow 6 Pa when the consistency was below 1%. An increase in fibre consis-tency beyond 1% resulted in a significant increase in yield stress for Reynoldsnumbers greater than 2 x 105. Although pulp suspensions are known to beshear thinning, the yield stress at the edge of the fibre plug was found toincrease with increasing Reynolds number until a critical value. Beyond thisvalue, a decrease in yield stress occurred with increasing Reynolds number.The polymers used in this study have been extensively investigated indrag reduction and rheology literature. The pseudoplasticity of aqueouspolyacrylamide has been found to be largely dependent on concentration,thus, some of the rheological properties of the polymer used in this studywere measured. Directly measuring the yield stress and the viscoelastic andviscous response, however, was outside the scope of this work. Previousinvestigations (e.g. [23]) have reported yield stress values (i.e. up ramp)near 0.36 Pa for 0.25% polyacrylamide solutions, where the yield stress wasfound to increase close to linearly with polymer concentration. It is likelythat the yield stress value for the polymer used in this study is below 5 Pa.The rheology study of aqueous anionic polyacrylamide was carried outusing a Bohlin Gemini 2 viscometer and Thermo HAAKE CaBer 1 exten-sional viscometer. A 4◦, 40 mm cone and plate configuration was used witha linear controlled shear rate (CR) between 5 - 1000 1/s. An initial samplediameter and height of 6 mm and 1.5 mm, respectively, was found to beoptimal for the extensional viscometry study using the HAAKE CaBer 1unit, where the temperature was held constant at 24◦C for all experiments.Figure 3.3 shows the flow behaviour of a 0.03% APAM solution in termsof viscosity. The solution experiences shear thinning behaviour up to a crit-ical shear rate. At this critical shear rate (γ˙c), the viscosity of the tested213.2. Fluid and Fibre Characteristicssolution experiences a slight shear thickening behaviour. The shear thicken-ing behaviour that appears at γ˙c is generally observed for polymer solutions,and has been reported by several authors (e.g. [23]). It is believed that theshear thickening phenomena is a result of the formation of a flow-inducedreversible structure. The viscous behaviour in Figure 3.3 can be describedbest by a non-linear regression of the Carreau viscosity model shown inEquation 4.4. The parameters of this regression cover a shear rate range upto 1000 s−1 and are listed in Appendix A.10−1 100 101 102 10310−310−210−1100γ (1/s)η (Pas)Figure 3.3: Viscosity curve of 0.03% APAM.The measured variation of the first normal stress difference, N1, whichis a good indicator of the level of elasticity of the fluid, versus shear stressτ is shown in Figure 3.4. A power-law fit to the N1(τ) data can be found inAppendix A. In the measured range for the 0.03% APAM, the recoverableshear N1/2τ is  0.5 indicating a highly elastic fluid.223.2. Fluid and Fibre Characteristics100 101100101102Shear Stress (Pa)First normal stress difference N1 (Pa)Figure 3.4: First normal stress difference vs. shear stress for 0.03% APAM.A further investigation on the elasticity of the anionic polyacrylamideis shown in Figure 3.5, where the measured extensional viscosity curve wasfound using the HAAKE extensional viscometer. The Hencky strain forthis study is defined as  = ln(γ) where γ = Dmid(t)/Do is the stretch ra-tio, Do is the initial diameter and Dmid(t) is the midpoint diameter underthe current strain. The extensional viscosity data agrees well with previouspolyacrylamide investigations of concentrations near 0.03% (e.g. [22]). Itcan be seen from Figure 3.5 that the extensional viscosity, ηE , rapidly in-creased at a Hencky strain of 16, followed by a steady increase in ηE to ≈ 45Pa s where sample scission occurs. The extensional viscosity measurementsindicate that the 0.03% APAM solution is highly visco-elastic within thestudied strain range.233.2. Fluid and Fibre Characteristics101 102100101102εη E (Pas)Figure 3.5: Apparent extensional viscosity, ηE , as a function of Henckystrain ().24Chapter 4Investigating the Flow Fieldof a HydrocycloneThe results presented in this chapter of the work are based on two ap-proaches: experimentally measuring the flow field at various locations usinglaser Doppler velocimetry (LDV), and predicting a portion of the flow fieldusing analytical formulations. Emphasis on reject ratio, was taken to quali-tatively show the change in a particle’s separation zone size throughout thehydrocyclone body with and without polymer addition. An overview of thelaboratory experiments conducted for this portion of work is listed in Table4.1.Sample Measured Velocities (m/s) RR (%)Water 〈uz,θ〉 and 〈u′z,θ〉 0, 25, 500.03% APAM 〈uz,θ〉 and 〈u′z,θ〉 25, 50Table 4.1: Overview of the LDV experiments.4.1 The Effects of a Polymer Additive on theHydrocyclone Flow FieldThis section provides quantitative evidence which characterizes the resultsof adding a drag reducing polymer additive, namely anionic polyacrylamide(APAM), to a hydrocyclone. Laser Doppler velocimetry was used to mea-sure the mean tangential (〈uθ〉) and axial (〈uz〉) velocities of polyamid (PSP)seeding particles of Stokes number (Stk)  1. The polyamid seeding parti-cles were between 5 – 10 µm in diameter. The mean radial velocity (〈ur〉)was numerically solved for using the continuity equation, where errors wereobserved up to O(1) (refer to Appendix C for full error analysis). Turbulenceresults are displayed in the form of local turbulence intensities for only 〈u′z〉and 〈u′θ〉 throughout the measurement domain for two reject ratios, specif-ically 25% and 50%. The results indicate that a moderately concentrated254.1. The Effects of a Polymer Additive on the Hydrocyclone Flow Fieldpolymer solution alters all dimensions of the hydrocyclone flow field whencompared to that of water alone. Table 4.2 outlines the inlet conditionsassociated with the polymer, and water studies presented in this section.4.1.1 Laser Doppler velocimetryMeasurements of the mean tangential (Uθ = 〈uθ〉+ 〈u′θ〉 = f(r, z)) and axial(Uz = 〈uz〉+ 〈u′z〉 = g(r, z)) velocity components were obtained throughoutthe hydrocyclone using a MiniLDV frequency shifting system comprisinga G5-240 (658 nm) laser diode with a sensor drive and VioBP-LDV burstprocessor. The LDV optical parameters were as follows: fringe spacing 9.6µm, probe volume distance 260 mm, probe volume size at FWHM 300 ×150 µm, probe volume length at FWHM 4 mm. The probe head, housingthe transmitting and receiving optics, was mounted on a two axis traversecontrolled by an external computer with a spacial resolution is 0.5 mm.Alignment for data collection was achieved by centring the beams with twocentred wires located at the front and back of the hydrocyclone block. Noflat-faced optical box was required to minimize beam refraction at the curvedsurface.Data analysis and validation was carried out using software providedfrom MSE. A minimum sample size of 100 with an average minimum signal-to-noise ratio (SNR) of 6.5 was achieved for each validated measurement.The change in the mean velocity components were found to be negligiblewith the inlet condition held within a measurable error of approximately3%. It must be noted that little variance in the full cross section velocityprofile was observed for H2O (i.e. ≤ 5%) for 〈uz,θ〉, therefore, all full planevelocity profiles presented are under the assumption of axisymmetry.Uncertainties associated with LDV measurements were confined to twofeatures: location error and statistical error. Axially, the measurement vol-ume was controlled to within ± 5 mm of the desired position, whereas,radially the measurement volume was confidently within ± 2 mm. Randomerror, or statistical error, was calculated using a t-distribution uncertaintyanalysis for finite data sets. All measurements retained a 95% confidenceunder the application of Equation 5.6, detailed in Chapter 5.1.2. A maxi-mum error of ± 0.82 m/s in 〈uθ〉 and ± 0.16 m/s in 〈uz〉 were measured nearthe core (i.e. region of highest turbulence) for water alone. The mean tan-gential and axial velocities at these positions were found to be 7.79 m/s and1.03 m/s, respectively. The errors observed for LDV results with polymeradditives were all ≤ O(10−1) for 〈uz,θ〉.264.1. The Effects of a Polymer Additive on the Hydrocyclone Flow Field4.1.2 Laser Doppler velocimetry of shear thinning fluidsThe shear thinning effect associated with the anionic polyacrylamide usedin this work limited the maximum LDV exposure time. To minimize theinfluence of polymer degradation on the tangential and axial velocity com-ponents, the studied axial positions and LDV exposure time were reducedwhen compared to water alone. The cylindrical section for the polymer trialswas ignored, as the bulk of the flow field, the conical section, is of more inter-est in this study. A maximum exposure time of 30 sec was specified, yielding≥50 validated measurements per studied position. The coarser measurementconditions reduced the number of solutions needed for one studied condition,minimizing the effect of polymer rheology alterations, from pre-trial mixing,etc., on LDV results. Similar to water, the single inlet had little effect onthe symmetry of the internal velocity profiles, thus, all profiles presented areunder the assumption of axisymmetry.4.1.3 The flow field of a hydrocycloneThe effect of RR on the flow field within a hydrocyclone operating withwater is evaluated in Figures 4.1, 4.2 and 4.3. The speed shown is definedas√〈ur〉2 + 〈uz〉2; Ut is 〈uθ〉.274.1. The Effects of a Polymer Additive on the Hydrocyclone Flow FieldFigure 4.1: Water flow field: RR = 0%, Uinlet = 2.57 m/s.The speed contour and vector fields in Figures 4.1 - 4.3 demonstrate theseparation zones within the conical section of the hydrocyclone, where theseparation zone is denoted as the region where the speed ≈ 0. This wasdefined to quantify the region where the fluid’s axial and radial velocitieswill only have a minor influence towards the radial motion of a particle.This is further detailed in Chapter 4.1.7. Figure 4.1 shows that particleswould not be significantly effected by 〈ur,H2O〉 or 〈uz,H2O〉 within a largeportion of the conical section for RR = 0%. The vector field indicates thata region of strong upwards flow exists near the core of the hydrocyclonewhere weak downwards flow can be seen near the wall. Full flow reversalupwards was found to occur near the underflow outlet. Figures 4.2 and4.3 were experimentally obtained with an identical inlet velocity to thatof RR = 0% and show distinct differences in the speed contour and vectorfields. The contraction of the hydrocyclone and the high volumetric flow ratetowards the underflow resulted in a reduction in the size of the separationzone towards the apex for reject ratios equal to 25% and 50%. The speedcontour near the apex, however, for RR = 50% was found to be greater thanthat presented for RR = 25% due to the increase in the underflow volumetric284.1. The Effects of a Polymer Additive on the Hydrocyclone Flow Fieldflow rate. The axial velocity distribution for RR = 50% indicated that atransition to fully downwards flow occurred 0.12 m above the hydrocycloneapex, 0.02 m greater than what was measured for RR = 25%. The vectorfield for all cases agrees well with previous studies (see [2]), where strongoutward fluid velocities were observed close to the wall and strong inwardflow was observed close to the core. The speed and vector fields for waterindicate that an increase in RR reduces the particle separation zone withinthe conical section.294.1. The Effects of a Polymer Additive on the Hydrocyclone Flow FieldFigure 4.2: Water flow field: RR = 25%, Uinlet = 2.57 m/s.Figure 4.3: Water flow field: RR = 50%, Uinlet = 2.57 m/s.304.1. The Effects of a Polymer Additive on the Hydrocyclone Flow Field4.1.4 The flow field of a hydrocyclone with polymeradditiveThe effects of adding drag reducing agents to a hydrocyclone were inves-tigated in a similar manner to that described in 4.1.3. Due to the shearthinning characteristics of the APAM used in this study, a concentration of0.03% was chosen to provide adequate time to obtain LDV results for uθor uz throughout the hydrocyclone body. The radial component of velocitywas again calculated using continuity in cylindrical coordinates, under theassumption of axisymmetry. The solution errors using an O(∆x2) differenceapproximation can be found in Appendix C. Figures 4.4 and 4.5 show thatthe visco-elastic effects of APAM have changed the flow field from what ismore classically observed with, for example, water alone. The highlightedsection (a) in Figure 4.4 shows the locus of zero vertical velocity (LZVV)to be much closer to the core than that observed with water for RR = 25%(see Figure 4.2), where, the region near the onset of the conical section hasonly a small portion of downward flow near the wall. The vector field of (a)in Figure 4.4 shows a steep change from negative (downwards) to positive(upwards) fluid flow, decreasing the separation zone in the upper half of theconical section. As can be seen from Figure 4.4 (b), the upwards portion offlow diminishes to zero at an axial location 0.20 m above the apex, wherethe total flow beyond this point, towards the underflow, accelerates down-wards. Although the speed contour in (b) displays a small section of flowwhere speed ≈ 0, flow reversal along the radius in these regions of the hy-drocyclone does not occur; consequentially leading to no overflow/underflowseparation. These observations of 〈uz〉 and the decay of 〈uθ〉 from the walltowards the hydrocyclone axis, agree well with that of Walker et al. [71].A similar analysis is applied to Figure 4.5, where differences, dependenton RR, in the flow field were observed. The vector field of (a) in Figure 4.5shows a small upwards portion of flow centred between the wall and core,where very little flow splitting exists in the bulk of the hydrocyclone. Thepositive flux in (a) is a maximum of 17% of the total flow at 0.24 m above theunderflow outlet, and has a maximum speed of 0.054 m/s. Interestingly, thepositive flux in Figure 4.5 reveals two distinct vortices in the upper sectionof the hydrocyclone within 0.14 m and 0.34 m above the apex. Consideringthe swirling motion of the fluid, the radial pressure drop to leading order (i.e.∂P∂r = ρ〈u2θ〉r ) displayed no adverse axial pressure gradients (see Appendix C),which is generally associated with vortex formation. Ignoring the effects ofthe swirling motion, however, the axial pressure gradient can be calculatedthrough the following:314.1. The Effects of a Polymer Additive on the Hydrocyclone Flow Field∂P∂z= −ρ(〈ur〉∂〈uz〉∂r+ 〈uz〉∂〈uz〉∂z)where the viscous terms in the z-component of the Navier Stokes equa-tions, shown above, were ignored. In this particular case, an axial pressuregradient exists, however, this method is hard to justify due to the high swirlcomponent measured inside the hydrocyclone. Nevertheless, this does sug-gest it is possible that the low shear rate in the r, θ plane may have increasedthe effective viscosity enough in the vortex regions to balance out the iner-tial forces governed by the tangential velocity field. The visco-elastic andturbulence effects associated with the polymer additive is, however, still upfor discussion.The speed contour of Figure 4.5 (a) and (b) shows the effect of 〈ur,z〉 issmall, whereas 〈uθ〉 increased near the wall when compared to water alone.Although the increase in 〈uθ,wall〉 could increase the particle size susceptibleto overflow removal, the dominant downwards velocity in the cyclone unitappears to restrict overflow/underflow separation.Figure 4.4: 0.03% APAM flow field: RR = 25%, Uinlet = 2.57 m/s.324.1. The Effects of a Polymer Additive on the Hydrocyclone Flow FieldFigure 4.5: 0.03% APAM flow field: RR = 50%, Uinlet = 2.57 m/s.4.1.5 Inlet condition studyThe drag reducing behaviour of the polyacrylamide used in this study variedthe inlet conditions when compared to water alone (i.e. PH2O,inlet|U1 >PAPAM,inlet|U1). As such, an analysis of the velocity field for RR = 25% and50% was performed for APAM with an inlet pressure equal to that of wateralone. The increase in pumping shear required to meet the inlet conditionsfor both RR resulted in an increase in the polymer degradation time scale;hence, select axial locations were chosen throughout the hydrocyclone foruθ and uz acquisition. For ease, Table 4.2 specifies a condition name for theinlet velocity and pressure matched trials for RR = 25% and 50%, where UMmeans velocity matched for a 0.03% APAM solution, PM means pressurematched for a 0.03% APAM solution, and W means water alone.334.1. The Effects of a Polymer Additive on the Hydrocyclone Flow FieldCondition name Uinlet (m/s) Pinlet (kPa) RR (%)W25 2.57 170.9 25UM25 2.57 66.1 25PM25 4.44 173.9 25W50 2.57 228.1 50UM50 2.57 138.5 50PM50 3.41 226.0 50Table 4.2: Summary of experimentsA comparison of 〈uz〉 and 〈uθ〉 for conditions: W25, UM25 and PM25 isshown in Figures 4.6 - 4.8. Figure 4.6 demonstrates the decay in 〈uθ〉 towardsthe hydrocyclone core is arguably due to solution rheology, as both UM25and PM25 display similar trends. As one would expect, 〈uθ,wall〉 increaseswith an increase in Uinlet, similarly to UM25. Figure 4.7 and 4.8 showsthat the increase in inlet pressure for RR = 25% results in the formationof two vortices, as a positive flux is present between the wall and core. Ofthe studied axial locations, Figures 4.7 and 4.8 were found to be the upperand lower positions that displayed an upwards flux between the wall andcore. Similarly to Figure 4.5, it is difficult to estimate what is causing theformation of these vortices, beyond possibly, an increase in effective viscosity.The polymer rheological characteristics that may be influencing the velocityfield is still unknown.344.1. The Effects of a Polymer Additive on the Hydrocyclone Flow Field0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05012345678<u θ> (m/s)Radial Position (m)Figure 4.6: Comparison of 〈uθ〉(z = -0.1 m) for conditions: W25(♦),UM25(∗) and PM25(◦).0 0.01 0.02 0.03 0.04 0.05 0.06−0.500.511.522.533.5<u z>(m/s)Radial position (m)Figure 4.7: Comparison of 〈uz〉(z = -0.1 m) for conditions: W25(♦),UM25(∗) and PM25(◦).354.1. The Effects of a Polymer Additive on the Hydrocyclone Flow Field0 0.005 0.01 0.015 0.02 0.025 0.03−0.4−0.200.20.40.60.811.2<u z>(m/s)Radial position (m)Figure 4.8: Comparison of 〈uz〉(z = -0.3 m) for conditions: W25(♦),UM25(∗) and PM25(◦).Analogous to RR = 25%, an analysis of the conditions UM50 and PM50on 〈uz〉 and 〈uθ〉 is shown in Figures 4.9 - 4.11. Figure 4.9 demonstrates theconditions of PM50 proportionally increased 〈uθ(r)〉 for 0.01 ≤ r(m) ≤ 0.05in comparison to that of UM50. The flow was as well found to transitionfrom free vortex flow to solid body rotation at a radial position (rt) 0.003m greater than that observed for the UM50 case.Figure 4.10 shows that an increase in inlet pressure resulted in the for-mation of two regions of downward flow and one region of upward flow atthe selected axial location. This implies that the position and strength ofthe vortices are largely effected by the capacity of the overflow and under-flow outlet, as the UM50 case did not possess vortices at z = -0.1 m. Thedata presented in Figure 4.11 agrees with this hypothesis, as the negative(downwards) axial velocities measured for the PM50 case were found to bemuch stronger than those measured with the UM50 case.364.1. The Effects of a Polymer Additive on the Hydrocyclone Flow Field0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05012345678<u θ> (m/s)Radial Position (m)Figure 4.9: Comparison of 〈uθ〉(z = -0.1 m) for conditions: W50(♦),UM50(∗) and PM50(◦).0 0.01 0.02 0.03 0.04 0.05 0.06−1−0.500.511.522.5<u z> (m/s)Radial position (m)Figure 4.10: Comparison of 〈uz〉(z = -0.1 m) for conditions: W50(♦),UM50(∗) and PM50(◦).374.1. The Effects of a Polymer Additive on the Hydrocyclone Flow Field0 0.005 0.01 0.015 0.02 0.025 0.03−1.5−1−0.500.51<u z> (m/s)Radial position (m)Figure 4.11: Comparison of 〈uz〉(z = -0.3 m) for conditions: W50(♦),UM50(∗) and PM50(◦).4.1.6 Interpreting the hydrocyclone flow fieldTo better understand the behaviour of water and the polymer solution inves-tigated in this study, the experimentally measured flow fields were comparedto the analytical solutions of a swirling cross-flow. The derivation of the gov-erning equations for a swirling cross-flow can be found in Appendix B. Forthis study, the tangential velocity profiles of water and the polymer solu-tion measured at the bottom of the cylindrical section (i.e. z = -0.1 m) werecompared to those numerically predicted using Equation B.15. This methodwas found to be useful in exposing some of the dominant forces associatedwith the tangential motion of water and the polymer solution investigated inthis work. All experimental measurements of the APAM solution presentedin this section are those of the inlet velocity matched trials discussed in theprevious sections.The basis of this analytical formulation originates from assuming that theflow field in a hydrocyclone is axially fully developed, and axially symmetric.This leads to characterizing the tangential velocity field in a hydrocycloneas a swirling cross-flow, where all terms dependent on axial and tangentialposition are ignored (i.e. ∂/∂z and ∂/∂θ terms). The non-dimensionalgoverning equation for the tangential velocity field of a power-law fluid anda Newtonian fluid (i.e. for n′ = 1) in a swirling cross-flow was developed inAppendix B, and is shown again below.384.1. The Effects of a Polymer Additive on the Hydrocyclone Flow Field∂2Ψ∂σ2(µ′n′σn′+ ρκ2R2σ2ω(R)2−n′(−∂Ψ∂σ)2−n′)=ρCω(R)1−n′(∂Ψ∂σ+2Ψσ)(−∂Ψ∂σ)1−n′+µ′σn′−1(−∂Ψ∂σ)(2 + n′) + 2ρσκ2R2ω(R)2−n′(−∂Ψ∂σ)3−n′(4.1)If the axially fully developed, and axially symmetric assumptions arevalid for the hydrocyclone used in this work, the constants: C, n′, µ′, andω(R) in Equation 4.1, also shown above, can be matched to those mea-sured experimentally for APAM and water. To note, the viscosity of water,namely µ′water, was not measured experimentally and was assumed to be1002 mPa s. The viscosity relationship in terms of shear rate for the 0.03%APAM solution (see Figure 3.3) was found to be best represented by theCarreau model, which is a bounded viscosity model. As such, the use of theunbounded power-law viscosity model for the APAM solution must be takenlightly, as this will only capture a portion of the true viscosity as a functionof shear rate. The flow behaviour index, n′, and the laminar consistencyindex µ′′, which model a portion of the 0.03% APAM viscosity curve can befound in Appendix A.0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04−1−0.500.51 x 10−3r (m)ru r (m2/s)Figure 4.12: Validity of the axially fully developed flow assumption for water.RR = 50%, z = -0.1 m, Uinlet = 2.57 m/s.394.1. The Effects of a Polymer Additive on the Hydrocyclone Flow FieldThe validity of the swirling cross-flow comparison, ignoring the molecularviscosity model, stems from the assumption that the flow inside the hydro-cyclone is axially fully developed. Characterizing the axial development ofthe hydrocyclone flow field was accomplished by analysing the continuityequation in a similar fashion to the method discussed in Appendix B. Onecan see from Figure 4.12 that the flow field of water inside a hydrocycloneat a RR = 50% is not independent of axial location, as rur 6= constant. Thisis not a surprise considering the axial velocities measured near the exteriorwall of the hydrocyclone decreased with decreasing axial position, and theaxial velocities measured near the axis (i.e. r = 0 m) increased with in-creasing axial position. Since the swirling cross-flow investigation is onlybeing used as a simple discussion towards interpreting the dominant forcesin hydrocyclone flow, the constant C in Equation 4.1 will be taken as theaverage value of rur between 0 ≤ r (m) ≤ 0.025. This was found to be -200mm2/s, and will be the only water case presented in this section.0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05−14−12−10−8−6−4−2024 x 10−4r (m)ru r (m2/s)Figure 4.13: Validity of the axially fully developed flow assumption forAPAM. Uinlet = 2.57 m/s, z = -0.1 m. © RR = 25%; • RR = 50%.Similarly to water, Figure 4.13 shows that the flow field of APAM ina hydrocyclone is not axially fully developed for reject ratios equal to 25%and 50%. As such, the constant C in Equation 4.1 was arbitrarily chosento be -300 mm2/s and -1000 mm2/s for a reject ratio of 25% and 50%,respectively. This was defined in an attempt to illustrate the difference inthe tangential velocity field for a swirling cross-flow of a power-law fluidwith variations in rur. The remaining constants used to model the swirling404.1. The Effects of a Polymer Additive on the Hydrocyclone Flow FieldModel RR n′ κ µ′′ (mPa sn′) ω(R) (s−1) C (mm2/s)Water 50% 1.0 0.05 1.0 40 -200APAM25% 0.5 0.05 97 50 -30050% 0.5 0.05 97 50 -1000Table 4.3: Overview of the constants used to solve Equation 4.1 to modelthe flow of water and APAM.cross-flow of APAM for this discussion can be found in Table 4.3.A comparison between the numerically estimated tangential velocity pro-file of water in a turbulent swirling cross-flow, with parameters describedin Table 4.3, and the experimentally measured tangential velocity field forRR = 50% is shown Figure 4.14. The numerical results of the turbulentswirling cross-flow were found to be closely related to a slow developingboundary layer, as the viscous forces are dominant within 0.02 ≤ r (m)≤ 0.05. Although the swirling cross-flow results do not follow the trendmeasured experimentally, some important conclusions can be draw from thedata. Firstly, the experimental data shows very little change in uθ in theregion where the total viscous forces (i.e. molecular and turbulent viscosity)are dominant. This indicates that the inertial forces are much higher in thisregion in a hydrocyclone, which is not unlikely considering rur increasednear a radius of 0.03 m. Secondly, the inertial and total viscous forces wereclosely balanced between 0 ≤ r (m) ≤ 0.015 for the swirling cross-flow case,whereas the experimental results were found to follow a trend similar to whatwould be expected of a swirling cross-flow case with high inertial forces (i.e.C  0). This indicates that the inertial effects governed by the flow notbeing axially fully developed are likely significant as r → 0. Furthermore,the turbulent viscosity could as well be decreasing with decreasing radius,as Figure 4.12 does not show any significant differences in rur as r → 0.The size of the turbulent lengthscales in a hydrocyclone, however, are stillup for discussion.414.1. The Effects of a Polymer Additive on the Hydrocyclone Flow Field0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05012345678r (m)u θ (m/s)Figure 4.14: Comparison between the tangential velocity profiles of waterin a hydrocyclone and in a swirling cross-flow.  Water: z = -0.1 m, Uinlet= 2.57 m/s, RR = 50%; © Water: Swirling cross-flow solution.The swirling cross-flow solutions for the APAM case are shown in Figure4.15, whereas the experimentally measured velocity profiles of APAM in ahydrocyclone are shown in Figure 4.16. The swirling cross-flow solution forthe 25% reject ratio model shows a slow developing boundary layer, similarto what was presented for the water case in Figure 4.14. An increase inthe constant C to -1000 mm2/s resulted in an increase in the magnitudeof the inertial forces (i.e. strongly negative) within 0.025 ≤ r (m) ≤ 0.05,whereas, the total viscous forces (i.e. molecular and turbulent) dominatedfor r ≤ 0.025 m. The experimental data for RR = 50% in Figure 4.16follows a similar trend to that of the swirling cross-flow with C equal to -1000mm2/s, however, indicates that a larger region exists where the magnitudeof the inertial forces are greater than the total viscous forces, namely within0.01 ≤ r (m) ≤ 0.05. A region of inertial dominance was as well observedfor RR = 25%, however, the magnitude of the inertial forces were onlyfound to be significant within 0.043 ≤ r (m) ≤ 0.05. Based on the swirlingcross-flow results, it is likely that this phenomena is a result of the flow ina hydrocyclone not being axially fully developed, however, it is difficult tocharacterize since the turbulent lengthscales inside a hydrocyclone are notknown.424.1. The Effects of a Polymer Additive on the Hydrocyclone Flow Field0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05−0.500.511.522.533.5r (m)u θ (m/s)Figure 4.15: Numerically calculated tangential velocity profiles for the 0.03%APAM model. © Model for RR = 25%; • Model for RR = 50%.0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.0511.522.533.5r (m)u θ (m/s)Figure 4.16: Experimentally measured tangential velocity profiles of the0.03% APAM solution for Uinlet = 2.57 m/s at z = -0.1 m. © RR = 25%;• RR = 50%.434.1. The Effects of a Polymer Additive on the Hydrocyclone Flow Field4.1.7 Predicting the motion of spherical particles in ahydrocycloneThis section investigates the flow fields measured experimentally with waterand APAM by estimating the motion of spherical particles throughout thehydrocyclone body. This was found to reveal important characteristics ofthe APAM flow field that were not initially recognized, specifically withregards to the axial development of the flow.The equation of motion for spherical particles in a centrifugal field is sim-ilar to that for rectilinear motion of particles in a gravitational field, wherethe gravitational acceleration, g, is replaced by the centrifugal accelerationrw2. Considering the particle-fluid interactions of a particle moving swirlingflow, the radial velocity of a spherical particle, neglecting particle-particleinteractions, may be determined through the following:mp∂uˆr∂t= mp(1−ρρp)rw2 + FD + FAdd + FBass+FSLF + FMLF + FPG (4.2)where mp is the mass of the particle, uˆr is the relative radial, or slipvelocity between the fluid and particle (i.e. uˆr = ur,particle − ur,fluid), ρ isthe fluid density and ρp is the particle density. Referring to the descriptionsin Table 4.4, it is reasonable to ignore the history term referred to as theBasset force, FBass, and the added mass force, FAdd, for small particlesin hydrocyclone flow. These assumptions are justified for this case as itwill be assumed that a particle radially accelerates or decelerates to that ofthe fluid instantaneously throughout the hydrocyclone, namely: ∂uˆr/∂t ≈ 0.This simplification to Equation 4.2 was considered largely because the radialvelocities calculated from the LDV data showed little change with positionthroughout the bulk of the hydrocyclone. As such, ignoring the Basset andadded mass force was deemed suitable for this discussion.The Magnus lift force, FMLF , and the pressure gradient force, FPG, willas well be neglected for this discussion as these forces are governed by thepressure gradients along a particle. Considering the spherical particle diam-eters investigated here range from 50 µm to 200 µm, and the radial pressuredistribution in a hydrocyclone is largely governed by the tangential velocityfield, these forces were considered minor and hence neglected. Lastly, theSaffman lift force, FSLF , was not considered in this investigation becausethe velocity gradient would have to be large over the particle diameter toinduce a lift force worth noting. The velocity gradients measured inside the444.1. The Effects of a Polymer Additive on the Hydrocyclone Flow Fieldhydrocyclone experimentally do not indicate high shear rates on the scaleof particle diameters considered for this investigation, as such, the Saffmanlift force was not included for this work. Considering the above assump-tions to be valid, the equilibrium radial velocity of a spherical particle canbe calculated by simplifying the force balance shown in Equation 4.2 to thefollowing:u2θ,fluidr= −34ρCD|uˆr|(uˆr)dp (ρ− ρp)(4.3)Where dp is the particle diameter, uθ,fluid = ωr, and CD is the dragcoefficient, defined as: CD = 24/Rep, where Rep = ρp|ur,f − ur,p|dp/µ ifRep  1. To note, Equation 4.3 is only valid for r > 0, as a singularityoccurs at r = 0 m.Force Source EquationDrag force,FDThe force acting on a par-ticle to move it through afluid under a uniform pres-sure and velocity field whenthere is no acceleration be-tween the particle and con-veying fluid.FD = 12ρCDpid24 |ur,f − ur,p| (ur,f − ur,p)Added massforce, FAddThe force required to accel-erate the fluid surroundingan accelerating particle.FAdd =ρVp2(∂ur,f∂t −∂ur,p∂t)Basset force,FBassThe force from the temporaldelay in boundary layer de-velopment as the relative ve-locity changes with time.FBass = 32d2√piρµto∫t(∂ur,f∂t −∂ur,p∂t)√t−sdsSaffman liftforce, FSLFThe force produced by thepressure distribution on aparticle due to rotation in-duced by fluid velocity gra-dients.FSLF = 1.615d2√piρµ(ur,f − ur,p)Magnus liftforce, FMLFThe force caused by a pres-sure difference along a rotat-ing particle.FMLF = 12ρCLpid24 (ur,f − ur,p)2Pressure gra-dient force,FPGThe force caused by the pres-sure gradient in the fluid sur-rounding a particle.FPG = −Vp∇PTable 4.4: Forces caused by particle-fluid interactions.454.1. The Effects of a Polymer Additive on the Hydrocyclone Flow FieldThe approach used in this study to predict particle trajectories in ahydrocyclone assumes particle inertia is small, and the axial and tangentialfluid velocities are equal to that of the particle, expressed as: uz/θ,p =〈uz/θ,f 〉. Although this methodology is strictly a qualitative approach topredicting particle separation characteristics for a specified hydrocycloneflow field, it is a valuable discussion when comparing the results obtainedwith water and the anionic polyacrylamide.The pseudo-plasticity of the 0.03% APAM solution investigated in thiswork was shown to be highly dependent on shear rate (see Figure 3.3), assuch a point-wise interpolation of the drag-coefficient, CD, was adopted forthis investigation. The model that was found to fit the APAM viscosity curvemost accurately was the bounded Carreau viscosity model. As discussed inChapter 4.1.6 the inertial forces were found be comparable to, or greaterthan the effective viscous forces in portions of the flow. As such, a boundedmodel was chosen to reduce the prediction error in the shear viscosity outsidethe critical shear rate region. The Carreau model is shown in Equation 4.4where the magnitude of the rate of strain tensor, to leading order, wasdefined as: γ˙ = r2∂∂r (uθr ) (see Appendix B for justification).η − η∞ηo − η∞=[1 + (λγ˙)2](m−1)/2(4.4)Particle streamlines of the simulations are finally computed using thefollowing relationship:ur,p(r, z) = −1r∂φp∂z(4.5)uz,p(r, z) = uz,f (r, z) =1r∂φp∂r(4.6)where φp is the particle stream function.Figures 4.17 - 4.19 display particle rakes in water at various axial lo-cations for RR = 25% and 50%. The particle diameter chosen for thissimulation is quite small, as the large LDV probe volume and high turbu-lence made it difficult to measure the region of solid body rotation near thehydrocyclone core. As such, high terminal radial velocities were calculatednear the centre, where generally much smaller velocities were expected. Nev-ertheless, it was found that particles achieve an equilibrium orbit where thecentrifugal and drag forces are balanced in areas with minimal axial veloc-ity. An increase in the particle density reduces the effect of the local radial464.1. The Effects of a Polymer Additive on the Hydrocyclone Flow Fieldand axial fluid velocities on particle motion and shifts the equilibrium orbitoutwards.0 0.01 0.02 0.03 0.04 0.05−0.5−0.45−0.4−0.35−0.3−0.25−0.2−0.15−0.1−0.050ρp = 1020 kg/m3Radial Position (m)Axial Position (m)0 0.01 0.02 0.03 0.04 0.05−0.5−0.45−0.4−0.35−0.3−0.25−0.2−0.15−0.1−0.050ρp = 1100 kg/m3Radial Position (m)Axial Position (m)(a) RR = 25%0 0.01 0.02 0.03 0.04 0.05−0.5−0.45−0.4−0.35−0.3−0.25−0.2−0.15−0.1−0.050ρp = 1020 kg/m3Radial Position (m)Axial Position (m)0 0.01 0.02 0.03 0.04 0.05−0.5−0.45−0.4−0.35−0.3−0.25−0.2−0.15−0.1−0.050ρp = 1100 kg/m3Radial Position (m)Axial Position (m)(b) RR = 50%Figure 4.17: Estimated particle trajectories released from z = -0.1 m. Uinlet= 2.57 m/s and Dp = 50 µm. Particle streamlines (—), water streamlines(—), water vector field (↗).474.1. The Effects of a Polymer Additive on the Hydrocyclone Flow Field0 0.01 0.02 0.03 0.04 0.05−0.5−0.45−0.4−0.35−0.3−0.25−0.2−0.15−0.1−0.050ρp = 1020 kg/m3Radial Position (m)Axial Position (m)0 0.01 0.02 0.03 0.04 0.05−0.5−0.45−0.4−0.35−0.3−0.25−0.2−0.15−0.1−0.050ρp = 1100 kg/m3Radial Position (m)Axial Position (m)(a) RR = 25%0 0.01 0.02 0.03 0.04 0.05−0.5−0.45−0.4−0.35−0.3−0.25−0.2−0.15−0.1−0.050ρp = 1020 kg/m3Radial Position (m)Axial Position (m)0 0.01 0.02 0.03 0.04 0.05−0.5−0.45−0.4−0.35−0.3−0.25−0.2−0.15−0.1−0.050ρp = 1100 kg/m3Radial Position (m)Axial Position (m)(b) RR = 50%Figure 4.18: Estimated particle trajectories released from z = -0.27 m. Uinlet= 2.57 m/s and Dp = 50 µm. Particle streamlines (—), water streamlines(—), water vector field (↗).484.1. The Effects of a Polymer Additive on the Hydrocyclone Flow Field0 0.01 0.02 0.03 0.04 0.05−0.5−0.45−0.4−0.35−0.3−0.25−0.2−0.15−0.1−0.050ρp = 1020 kg/m3Radial Position (m)Axial Position (m)0 0.01 0.02 0.03 0.04 0.05−0.5−0.45−0.4−0.35−0.3−0.25−0.2−0.15−0.1−0.050ρp = 1100 kg/m3Radial Position (m)Axial Position (m)(a) RR = 25%0 0.01 0.02 0.03 0.04 0.05−0.5−0.45−0.4−0.35−0.3−0.25−0.2−0.15−0.1−0.050ρp = 1020 kg/m3Radial Position (m)Axial Position (m)0 0.01 0.02 0.03 0.04 0.05−0.5−0.45−0.4−0.35−0.3−0.25−0.2−0.15−0.1−0.050ρp = 1100 kg/m3Radial Position (m)Axial Position (m)(b) RR = 50%Figure 4.19: Estimated particle trajectories released from z = -0.36 m. Uinlet= 2.57 m/s and Dp = 50 µm. Particle streamlines (—), water streamlines(—), water vector field (↗).Figures 4.20 - 4.22 present calculated particle trajectories within themeasured flow field shown in Chapter 4.1.4. Unfortunately, during velocityprofile acquisition the polymer fluid temperature fluctuated between 20◦Cand 28◦C. The rise in fluid temperature would inevitably reduce the mea-sured viscosities in Chapter 3.2 within the range of studied shear rates,increasing the error associated with this analysis. The effective viscosity of494.1. The Effects of a Polymer Additive on the Hydrocyclone Flow Fieldthe polymer additive was determined to be between 0.011 ≤ η(Pa s) ≤ 0.16for both reject ratios studied.The simulated particle diameter was chosen to be 200 µm due to thedrastic differences in the tangential velocity field and fluid viscosity of wa-ter and APAM. The inward fluid velocity is dominant in the upper partof the hydrocyclone for RR = 25% and 50%, where RR = 50% displaysoverflow/underflow separation. The strong axial velocities are shown to im-pact the trajectories of these particles to a much larger extent than thatobserved with water; however, overflow/underflow separation is not possiblein the bottom half of the hydrocyclone. Again, the vortex regions discussedin Chapter 4.1.4 are clearly visible in Figures 4.21b and 4.22b, where itwas found that the fluid vorticity is strong enough to entrap both particlesnumerically investigated.504.1. The Effects of a Polymer Additive on the Hydrocyclone Flow Field0 0.01 0.02 0.03 0.04 0.05−0.5−0.45−0.4−0.35−0.3−0.25−0.2−0.15−0.1−0.050ρp = 1020 kg/m3Radial Position (m)Axial Position (m)0 0.01 0.02 0.03 0.04 0.05−0.5−0.45−0.4−0.35−0.3−0.25−0.2−0.15−0.1−0.050ρp = 1100 kg/m3Radial Position (m)Axial Position (m)(a) RR = 25%0 0.01 0.02 0.03 0.04 0.05−0.5−0.45−0.4−0.35−0.3−0.25−0.2−0.15−0.1−0.050ρp = 1020 kg/m3Radial Position (m)Axial Position (m)0 0.01 0.02 0.03 0.04 0.05−0.5−0.45−0.4−0.35−0.3−0.25−0.2−0.15−0.1−0.050ρp = 1100 kg/m3Radial Position (m)Axial Position (m)(b) RR = 50%Figure 4.20: Estimated particle trajectories released from z = -0.14 m. Uinlet= 2.57 m/s and Dp = 200 µm. Particle streamlines (—), 0.03% APAMstreamlines (—), 0.03% APAM vector field (↗).514.1. The Effects of a Polymer Additive on the Hydrocyclone Flow Field0 0.01 0.02 0.03 0.04 0.05−0.5−0.45−0.4−0.35−0.3−0.25−0.2−0.15−0.1−0.050ρp = 1020 kg/m3Radial Position (m)Axial Position (m)0 0.01 0.02 0.03 0.04 0.05−0.5−0.45−0.4−0.35−0.3−0.25−0.2−0.15−0.1−0.050ρp = 1100 kg/m3Radial Position (m)Axial Position (m)(a) RR = 25%0 0.01 0.02 0.03 0.04 0.05−0.5−0.45−0.4−0.35−0.3−0.25−0.2−0.15−0.1−0.050ρp = 1020 kg/m3Radial Position (m)Axial Position (m)0 0.01 0.02 0.03 0.04 0.05−0.5−0.45−0.4−0.35−0.3−0.25−0.2−0.15−0.1−0.050ρp = 1100 kg/m3Radial Position (m)Axial Position (m)(b) RR = 50%Figure 4.21: Estimated particle trajectories released from z = -0.24 m. Uinlet= 2.57 m/s and Dp = 200 µm. Particle streamlines (—), 0.03% APAMstreamlines (—), 0.03% APAM vector field (↗).524.1. The Effects of a Polymer Additive on the Hydrocyclone Flow Field0 0.01 0.02 0.03 0.04 0.05−0.5−0.45−0.4−0.35−0.3−0.25−0.2−0.15−0.1−0.050ρp = 1020 kg/m3Radial Position (m)Axial Position (m)0 0.01 0.02 0.03 0.04 0.05−0.5−0.45−0.4−0.35−0.3−0.25−0.2−0.15−0.1−0.050ρp = 1100 kg/m3Radial Position (m)Axial Position (m)(a) RR = 25%0 0.01 0.02 0.03 0.04 0.05−0.5−0.45−0.4−0.35−0.3−0.25−0.2−0.15−0.1−0.050ρp = 1020 kg/m3Radial Position (m)Axial Position (m)0 0.01 0.02 0.03 0.04 0.05−0.5−0.45−0.4−0.35−0.3−0.25−0.2−0.15−0.1−0.050ρp = 1100 kg/m3Radial Position (m)Axial Position (m)(b) RR = 50%Figure 4.22: Estimated particle trajectories released from z = -0.34 m. Uinlet= 2.57 m/s and Dp = 200 µm. Particle streamlines (—), 0.03% APAMstreamlines (—), 0.03% APAM vector field (↗).534.2. Turbulence4.2 TurbulenceThis section presents the turbulence characteristics measured for water andthe 0.03% APAM solution. This section will only discuss the turbulence in-tensities measured for the velocity matched trials conducted for the polymercases, namely: UM25 and UM50 in Chapter 4.1.5. For this work, the turbu-lence characteristics are described using local turbulence intensities withinthe measurement volume, namely:I(r, z) =√12(〈u′z(r, z)〉2 + 〈u′θ(r, z)〉2)√〈uz(r, z)〉2 + 〈uθ(r, z)〉2(4.7)Due to the uncertainties of 〈ur(r, z)〉 (see Appendix C), 〈u′r〉 and 〈ur〉were not considered in I(r,z). Figure 4.23 (a) shows that I(r,z) increasestowards the underflow where the highest turbulence intensity is observednear the wall and core of the hydrocyclone’s conical section. The regionof minimal turbulence intensity is within the separation zone illustrated inFigure 4.2; indicating turbulence effects on particle trajectories are smallwhere 〈ur,z〉  〈uθ〉. Figure 4.23 (b) shows a similar trend to that observedin Figure 4.23 (a) within the conical section of the hydrocyclone, where theregion near the underflow displayed a near 30% increase in local turbulenceintensity to that of water.Figure 4.24 presents a comparison between the local turbulence intensi-ties observed for water and the 0.03% APAM solution at RR = 50% (i.e.UM50 case in Chapter 4.1.5). The 0.03% APAM was found to slightly reducethe width of the high intensity region near the core, where both exhibit anincrease in I(r,z) towards the hydrocyclone apex. Interestingly, it seems thechanges in 〈uz,θ〉 observed with the 0.03% APAM solution have little effecton the local intensity within the study volume, as the trends and magni-tudes observed for water are analogues to the polymer cases for RR = 25%and 50%. The influence of APAM on turbulence in the radial direction wasbeyond the scope of this work, however some conclusions can be drawn fromthe local turbulence intensities presented. The fluctuations of I(r,z) for thepolymer cases are less than that observed for water within a large portionof the conical section. From this observation, it can be speculated that thetotal Reynolds stresses may be less than that for water, but is difficult tojustify given the radial velocity fluctuations were not measured. A moredilute polymer solution could lead to a more traditional hydrocyclone flowfield while still reducing the turbulent fluctuations, however, the degradation544.2. Turbulencescale of this visco-elastic polymer additive would be substantial.Figure 4.23: Local Turbulence Intensity: RR = 25%. (a) Water, (b) 0.03%APAM.Figure 4.24: Local Turbulence Intensity: RR = 50%. (a) Water, (b) 0.03%APAM.55Chapter 5An Investigation of SolidParticle Motion in aHydrocycloneThe results presented in this chapter of work were used to characterize themotions of variously sized particles in a hydrocyclone. Particle tracking ve-locimetry (PTV) was used to measure the mean tangential, axial and radialvelocities instantaneously using the dual camera set up described in Chapter5.1. Both water alone and a semi-dilute polymer solution were studied tofurther investigate the effects of liquid viscosity and fluid dynamics on par-ticle motion in a hydrocyclone. An overview of the PTV experiments can befound in Table 5.1, where the properties of the three particles investigatedcan be found in Table 5.2.Sample Measured Velocities (m/s) Particle Types RR (%)Water 〈uz,r,θ〉 3 500.03% APAM 〈uz,r,θ〉 3 50Table 5.1: Overview of the PTV experiments.5.1 Particle Tracking Set-UpTwo high speed Phantom cameras were used to track solid particle trajec-tories inside the hydrocyclone. A Phantom V611 camera with a Nikon 50mm 11.4D lens, and a Phantom V121 camera with a Navitar 7000 zoomlens were set-up perpendicular (see Figure 5.2) to the hydrocyclone axis ata frame rate of 1200 frames-per-second, yielding approximately 3000 framesper video. The raw .cine files were transferred to a PC running the phan-tom camera control (PCC) software for each camera, and were converted to12 bit digital images. The resolution necessary to statistically analyse themotion of particles within the hydrocyclone limited the visualization win-565.1. Particle Tracking Set-UpFigure 5.1: Studied window locations.dow size to a 58 cm2 square. Three window positions were chosen along thehydrocyclone axis, where each window location was matched for both cam-eras. Figure 5.1 illustrates the chosen window heights for this work, wherethe location, in meters, is measured from the top of the hydrocyclone body.Window centring was achieved by aligning the window centre to the centreof the face of the acrylic block housing the hydrocyclone. Any slight offsetsobserved in the image were dealt with during image processing.The region of interest for each camera was illuminated by a pair of 500W lights, passing through a light diffuser sheet on the opposing side of thehydrocyclone block. The focal plane for both cameras were aligned withthe hydrocyclone core (using the air core as a reference image), providingsufficient clarity for image processing. Any subtle effects of focal adjustmentson image scaling, and centring, were taken care of during image processing.5.1.1 Particle propertiesThe solid spherical particles used in this study were made of polyethylenewith properties shown in Table 5.2. The pathways of the seeding particles(P1) (i.e. particles which follow fluid streamlines closely) were compared to575.1. Particle Tracking Set-UpFigure 5.2: Visualized X-Y coordinate system.the Eulerian velocity measurements obtained using LDV, as the Stk  1 inboth water alone and polymer solution (see Chapter 5.2). The remainingparticle diameters were chosen to emphasize the effect of particle drag ontheir resulting motion within the hydrocyclone. To minimize particle alias-ing during image recording, 1.5-1.9g of particles per 150 litres of water wasfound to be suitable.Particle name ρp (kg/m3) dp (µm)P1 1000 710–850P2 1280 500–600P3 1280 710–850Table 5.2: Polyethylene particle properties.5.1.2 Image processingThe task of particle identification and image processing was carried out usingin house Matlab R© software. The greyscale pixel values of the recorded 12-bit digital images ranged from 0 (black) to 212 (white). As the grey levelsdistinguishable to the human eye are approximately 64, 212 was found to bemore than adequate for image quality. A raw greyscale image output from585.1. Particle Tracking Set-UpFigure 5.3: Pixel values in a 10x10 neighbourhood (left) of a greyscale rawimage (right).the PCC software and the normalized pixel values (i.e. p¯(x, y) = p(x, y)/212)are shown in Figure 5.3.A greyscale mathematical morphology (MM) technique utilizing a diskshaped morphological structuring element, of radius 2 pixels, was used tolocally detect particles within the images bounds. A spatial edge detectionfilter was applied to the raw image, pre morphology, to highlight intensitytransitions within the image via emphasizing horizontal and vertical edges.The axial derivative correlation mask used for this work is as follows:1 2 10 0 0−1 −2 −1 (5.1)Applying a neighbourhood regional maxima filter to the post morpho-logically dilated output image, transforms the greyscale image to binaryvia:pˆ(x, y) ={1 if p¨(x, y) ≥ PT (x, y)0 otherwiseThe threshold pixel value, PT (x, y), is locally determined in regionswhere a minimum 4x4 neighbourhood of value p¨(x, y) is at least greaterthan a defined limit of the surrounding. Upon image binarization, a pixelarea bandpass filter was applied to each particle signature. A user definedlower and upper area limit, namely Alow and Ahigh, were chosen based onthe window location, and camera used. Any particle signatures outside thelimits were discarded.595.1. Particle Tracking Set-UpFinally, a circularity filter was applied to each particle signature to elim-inate any lingering noise. The circularity of a particle signature was definedas:Sp =4pi(Aparticle)(Pparticle)2(5.2)where Sp = 1 is a perfect circle and Sp = 0 is a line. The circularitythreshold applied to particle signatures was determined based on windowposition and camera calibration. The final processed binary image of Figure5.3 is shown in Figure 5.4.Figure 5.4: Example of a filtered binary image.Particle trackingThe principle behind particle tracking in a hydrocyclone starts from defin-ing the hydrocyclone 3-dimensionally, as shown in Figure 5.2. Scaling thepixel image to the internal dimensions of the hydrocyclone was carried outby relating the visualization window (7.6 cm x 7.6 cm) to the number ofpixels contained within. A similar method is applied to the exterior wallsof the hydrocyclone within the raw image to define the X and Y centrelinefor the coordinate system. This is shown explicitly in Figure 5.5. Upondimensionalizing the visualization window to the appropriate Cartesian sys-tem for the experimental hydrocyclone, the particle signature centroids werefound by applying MATLAB’s regionalprops ′′centroid′′ measurement tothe processed image. It was assumed the centroid location was accurate upto 1/16 pixels.605.1. Particle Tracking Set-UpFigure 5.5: Border and centreline analysis.Tracking of the particle centroids was carried out using an open source(Crocker et al. [13]) Brownian diffusion based method for each camerasrecorded data set. The resulting trajectories on the X-Z and Y-Z plane werepaired by minimizing |〈uz(x, z, t)〉−〈uz(y, z, t)〉| within a specified z(t) range,thus accommodating any slight axial offsets between the cameras, possiblyfrom camera tilt. The following expressions were applied to calculate thevelocity field of the nth trajectory from i to i+ 1.uj1(n, i)vj2(n, i)wj3(n, i) =1∆t∆rrp∆θ∆z (5.3)where,r(n, i)θ(n, i)z(n, i) =√x(n, i)2 + y(n, i)2arctan(y(n,i)x(n,i))z(n, i)For this work, u is the instantaneous radial velocity, v is the instan-taneous tangential velocity, w is the instantaneous axial velocity, and themean position vector, j, is defined as:rp(n, i)θp(n, i)zp(n, i) =12r(t) + r(t+ ∆t)θ(t) + θ(t+ ∆t)z(t) + z(t+ ∆t) (5.4)615.1. Particle Tracking Set-UpEulerian analysisThe resolution limitations of the PTV analysis required the visualizationwindow to be reduced to a size much smaller than that of the hydrocyclone.As such, a Lagrangian type analysis was identified as an unsuitable referenceframe due to the impossibility of tracking solid particles throughout theirresidence time for this particular study. Laboratory configuration was theleading cause for the limited visualization area. Considering a key featureof this work is to emphasize the change in particle velocities with increasingparticle weight, a fixed position analysis (Eulerian) was chosen to be moreappropriate for this work. One of the key benefits of transforming froma Lagrangian to Eulerian frame is the minimization of pathline error, asparticle inertia in even neutrally buoyant particles can play a role in pathlinemisalignment.The Lagrangian transformation was achieved by meshing the 3D geom-etry (in Cartesian coordinates) in to subdomains within the visualizationwindow. The cell (1 mm (∂x) x 1 mm (∂y) x 15 mm (∂z)) is shown in Fig-ure 5.6. The Eulerian mean vector field, assuming axially symmetric flow,was calculated as follows:~Vk,m =nmax∑n=1imax∑i=1~U(n, i)N(5.5)where ~U = (u, v, w), ∂r = ∂x = ∂y, and N is the number of measuredparticle velocities that fall within the (k,m)th cell. All profiles and correla-tions presented in this work were found using this subdomain method.The uncertainty associated with the methodologies used during the PTVanalysis is represented by three key features: calibration error, location errorand statistical error. Considering the experimental set-up, calibration errorwas attributed to camera tilt, centreline approximation and image scaling.For simplicity, the effects of camera tilt was negligible as the camera positionwas held constant for each window location. The centreline and scaling,however, can only be confidently approximated to within 6 pixels. Applyinga maximum deflection to the imaging resolution of 1024 x 512 pixels and800 x 600 pixels for the V611 and V121 camera, respectively, found the totalaverage error in particle position from calibration error to be:(V 611)x,z ≈ 5mm(V 121)y,z ≈ 4.4mm625.2. LDV vs. PTV Comparison: Water AloneFigure 5.6: Eulerian cell analysis (not to scale).The propagation of uncertainty on the resultant velocity components isthen defined as:u(r,θ,z) = ±[N∑i=1(∂y∂xi∆xi)2]1/2where y is a function dependent on the measured variables xi → xN .Random error for reported Eulerian velocities was calculated using a t-distribution statistical uncertainty analysis for finite data sets. The randomerror per (k,m)th cell was calculated as follows:δ¯k,m ± tα,vσ√n(5.6)where δ¯ denotes the upper and lower bounds for an applied tα,v confi-dence on a finite data set of size n and standard deviation σ. Retaining asample size of nearly 1600 per mesh element led to a maximum 95% con-fidence interval on 〈uθ〉 to δ¯k,m ≈ ± 0.18 m/s, which is small consideringmost measurements were greater than 1 m/s. For simplicity, summaries ofthe error range shown in Chapter 5.2 and Chapter 5.4 are combinations ofboth statistical and experimental uncertainty.5.2 LDV vs. PTV Comparison: Water AloneThe mean particle velocities of type P1, described in Table 5.2, are presentedin Figures 5.7 - 5.9. To note, axially symmetric flow was assumed for the635.2. LDV vs. PTV Comparison: Water Alonefollowing investigations as the key contribution of this work is to illustratethe mean motions of variously sized particles throughout the entire flowfield. A comparison between the velocity measurements of the neutrallybuoyant particles used for the PTV and LDV analysis was found to revealsome unique characteristics of the PTV analysis. This helped verify whetheror not the 710 – 850 µm particles were able to accurately follow all thefluid motions faithfully within the various measurement windows. All CFDresults presented in this section of work were calculated using a Reynolds-averaged Navier-Stokes (RANS) model, where the transport equations weresolved for the individual Reynolds stresses. As discussed in Chapter 2, theRANS turbulence closure model tends to poorly approximate the true flowfield in a hydrocyclone. This was similarly observed for the hydrocyclonegeometry, and operating conditions discussed in this work.It was found that the mean tangential velocities of particle P1 agreewell with LDV measurements between 0.02 ≤ r (m) ≤ 0.037 for the studiedreject ratio of 50% and an Reinlet = 47 000. The discrepancy between thetwo methods is likely a result of the large seeding particle diameter (i.e.dp,average ≈ 780 µm) used for the PTV measurements, as the diameter ofthe seeding particles used for the LDV measurements were between 5 – 10µm. Particle accumulation could explain the variability in seeding particlevelocities, however, remains suspect. A discussion on particle accumulationand drag can be found in Appendix E.Figure 5.8 and 5.9 present the radial and axial velocities measured usingthe LDV and PTV method, and those computed using CFD. The resultsshow reasonable agreement between experimental techniques, however, theCFD data shows significant differences in both velocity components. Itcan be seen that the RANS model used for the CFD simulation lead to anoverestimation of the axial and radial velocities near the wall and axis ofthe hydrocyclone. To note, the radial velocity profiles presented as LDVmeasurements are those calculated using continuity, and assuming axiallysymmetric flow.645.2. LDV vs. PTV Comparison: Water Alone0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.0500.511.522.533.54Radial Position (m)U θ (m/s)Figure 5.7: PTV vs. LDV: Uθ(r). Reinlet = 47,000, z = -0.12 m. PTV (◦),LDV (•) CFD ().0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05−0.4−0.3−0.2−0.100.10.20.30.4Radial Position (m)U z (m/s)Figure 5.8: PTV vs. LDV: Uz(r). Reinlet = 47,000, z = -0.12 m. PTV (◦),LDV (•) CFD ().655.3. Effect of Particle Size on Slip Velocity: Water Alone0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05−0.4−0.3−0.2−0.100.10.20.30.4Radial Position (m)U r (m/s)Figure 5.9: PTV vs. LDV: Ur(r). Reinlet = 47,000, z = -0.12 m. PTV (◦),LDV (•) CFD ().5.3 Effect of Particle Size on Slip Velocity:Water AloneTo effectively evaluate the change in particle motions due solely to an in-crease in particle weight, the relative velocity between a particle and thecontinuous phase was required. The relative radial and axial velocity, orslip velocity, of a particle in a hydrocyclone is defined as follows:[U¯r,slipU¯z,slip]=[Ur,particle − Ur,fluidUz,particle − Uz,fluid](5.7)where Ur,fluid and Uz,fluid are the local fluid velocities measured usingthe LDV method described in Chapter 4.1. Upon assessing particle slip, itwas found to be advantageous to introduce a scaling factor which simplifiesthe relative velocity measurements to a single term. The scaling factor thatwas found to be the most effective for this work is shown below.Φ =1picos−1U¯r,slip√U¯2r,slip + U¯2z,slip (5.8)665.3. Effect of Particle Size on Slip Velocity: Water AloneThe slip factor, Φ, and particle vector field shown in Figure 5.10(a)and (b) reveals a distinct line where particle motion changes from radiallyoutwards to inwards, namely a separation line, for both particles studied.The point of transition to dominantly inwards slip, Φ ≈ 1 (i.e. red contour),for the smallest particle was found to occur at a radial position near 0.021m, whereas, the radial separation line increased to 0.025 m for the largestparticle studied. Minor radial and axial particle velocities were measuredinside the high inwards slip region shown in Figure 5.10(b), indicating thereduction in centrifugal force closely balances the particle drag forces.As shown in Equation 4.3, characterizing radial slip is well defined how-ever axial slip is not as intuitive. The high swirl inside the hydrocycloneused for this study implies gravitational forces should have no impact onparticle settling, therefore the axial slip measured in regions of high shear(i.e. near the wall and axis of the hydrocyclone) is likely due to particledrag. The Stokes number for both particles, where: Stk = τp/τf and τ isthe relaxation time of the particle or fluid, was found to be greater than 1for particles P2 and P3, indicating that flow detachment is likely occurringin regions of high shear. Quantifying the drag forces acting axially on theparticle is outside the scope of this work, however, these results indicate thataxial slip is undoubtedly an important characteristic for optimizing particleseparation in a hydrocyclone.675.3. Effect of Particle Size on Slip Velocity: Water AloneRadial Position (m)Axial Position (m)  0.01 0.015 0.02 0.025 0.03 0.035−0.085−0.08−0.075−0.07−0.06500.10.20.30.40.50.60.70.80.91(a) dp = 500–600 µm (P2)Radial Position (m)Axial Position (m)  0.01 0.015 0.02 0.025 0.03 0.035−0.09−0.085−0.08−0.075−0.07−0.06500.10.20.30.40.50.60.70.80.91(b) dp = 710–850 µm (P3)Figure 5.10: Φ(r, z) for the smallest (a) and biggest (b) particle studied inthe top window location; measured particle vector field (↗).The slip contour of Figure 5.11(a) shows the particle’s equilibrium orbitposition, namely where the centrifugal (FC) and drag forces (FD) balance,is located at a radius of approximately 0.026 m at an axial location of -0.205 m. Similarly to Figure 5.11(a), Figure 5.11(b) indicates little overflowseparation potential for the 710 – 850 µm particles, as the particle vector fieldshows a dominant outwards motion. The slip contour, however, indicatesa stronger axial slip near the wall of the hydrocyclone when compared tothe 500 – 600 µm particle. This indicates that particle drag increased inthe region where: |∂uz/∂r|  0, as a result of an increase in the average685.3. Effect of Particle Size on Slip Velocity: Water Aloneparticle weight (i.e. from an increase in particle diameter).Radial Position (m)Axial Position (m)  0.01 0.015 0.02 0.025 0.03−0.225−0.22−0.215−0.21−0.20500.10.20.30.40.50.60.70.80.91(a) dp = 500–600 µm (P2)Radial Position (m)Axial Position (m)  0.01 0.015 0.02 0.025 0.03−0.225−0.22−0.215−0.21−0.20500.10.20.30.40.50.60.70.80.91(b) dp = 710–850 µm (P3)Figure 5.11: Φ(r, z) for the smallest (a) and biggest (b) particle studied inthe middle window location; measured particle vector field (↗).Figure 5.12(a) indicates that overflow removal of the smallest particleis likely occurring near a radius of 0.006 m at an axial location of -0.365m. Outside this region, the particle vector field shows outwards motiontowards the hydrocyclone wall, ultimately leading to underflow removal.The slip contour shows an increase in axial slip towards the underflow outletwhen compared to Figure 5.11. This is not surprising considering |∂uz/∂r|increases towards the underflow outlet where a transition to fully downwards695.3. Effect of Particle Size on Slip Velocity: Water Aloneflow occurs (see Chapter 4.1). In Figure 5.12(b), it can be seen that anincrease in particle diameter resulted in a larger region within the bottomwindow location where the axial slip dominates over the radial slip (i.e.Φ ≈ 0.5). It is likely that no overflow separation occurs for the 710 – 850µm particle, as the vector field shows purely outward motion within thestudy volume.Radial Position (m)Axial Position (m)  0.006 0.007 0.008 0.009 0.01 0.011 0.012 0.013 0.014−0.385−0.38−0.375−0.37−0.36500.10.20.30.40.50.60.70.80.91(a) dp = 500–600 µm (P2)Radial Position (m)Axial Position (m)  6 7 8 9 10 11 12 13 14x 10−3−0.385−0.38−0.375−0.37−0.36500.10.20.30.40.50.60.70.80.91(b) dp = 710–850 µm (P3)Figure 5.12: Φ(r, z) for the smallest (a) and biggest (b) particle studied inthe bottom window location; measured particle vector field (↗).705.4. LDV vs. PTV Comparison: 0.03% APAM5.4 LDV vs. PTV Comparison: 0.03% APAMSeeding particle velocities measured using the PTV, and LDV method arepresented in Figures 5.13 – 5.15 for a 0.03% APAM solution. The Reinlet,assuming the viscosity of the solvent (i.e. water) alone, and reject ratio werematched to that described in Chapter 5.2 and 5.3. A comparison betweenthe tangential velocity profiles measured using LDV and PTV show goodagreement at the reviewed axial location of z = -0.22 m between 0.015 < r(m) < 0.025, where similarly to water alone, the transition to a forced vortextype flow occurred further away from the core than that measured usinglaser Doppler velocimetry. As discussed in Appendix E, the discrepanciesbetween the LDV and PTV methods near the hydrocyclone axis could bea result of particle accumulation, as the volume fraction of particles in thehydrocyclone may be varying in time and space. Further analysis is requiredto quantify the slip observed for the neutrally buoyant particles used in thisinvestigation.The LDV results presented in Figures 5.14 and 5.15 show that a vortexis present at z = -0.22 m, where the portion of upwards flow at a radial po-sition of 0.02 m has a radially negative velocity towards the core. The PTVmeasurements do not reveal a similar case, as ∂uz,PTV∂r does not show multi-ple zeros along the hydrocyclone’s radius, indicating no vortex is present. Asdiscussed, it is inconclusive what is causing the neutrally buoyant particlesused for the PTV analysis to slip, however particle accumulation in time andspace is likely a factor. Other investigations (e.g. [74]) have reported thatparticles larger than the Kolmogorov scale tend to filter out high frequencyoscillations, however is difficult to justify as most works consider particleswith a specific gravity greater than unity (see [11]).715.4. LDV vs. PTV Comparison: 0.03% APAM0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.0400.511.522.533.54Radial Position (m)U θ (m/s)Figure 5.13: PTV vs. LDV: Uθ,polymer(r). Reinlet = 47,000, z = -0.22 m.PTV (◦), LDV (•).0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04−0.2−0.100.10.20.3Radial Position (m)U z (m/s)Figure 5.14: PTV vs. LDV: Uz,polymer(r). Reinlet = 47,000, z = -0.22 m.PTV (◦), LDV (•).725.5. Effect of Particle Size on Slip Velocity: 0.03% APAM0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04−0.2−0.15−0.1−0.0500.050.10.150.2Radial Position (m)U r (m/s)Figure 5.15: PTV vs. LDV: Ur,polymer(r). Reinlet = 47,000, z = -0.22 m.PTV (◦), LDV (•).5.5 Effect of Particle Size on Slip Velocity: 0.03%APAMThe slip velocities of particles P2 and P3 were calculated using Equation5.7, where Ur,fluid and Uz,fluid are the local fluid velocities measured fromthe LDV. The slip contour and vector field of the smallest particle, shownin Figure 5.16(a), reveals that the radial position of the separation lineincreases with a decrease in axial position. The high swirl near a radius of0.01 m was found to increase the outwards radial velocity of the particles,indicating little potential for overflow removal.Figure 5.16(b) shows that an increase in particle size resulted in a sepa-ration line near a radius of 0.02 m for both axial positions shown. The highcentrifugal force near a radius of 0.01 m increased the outwards radial veloc-ity, again leading to a reduction in potential overflow removal. Analogous tothe water case, the slip contour for both particle sizes show that axial slipis considerable within the high shear region close to the hydrocyclone wall.735.5. Effect of Particle Size on Slip Velocity: 0.03% APAMRadial Position (m)Axial Position (m)  0.01 0.015 0.02 0.025 0.03 0.035−0.135−0.13−0.125−0.12−0.11500.10.20.30.40.50.60.70.80.91(a) dp = 500–600 µm (P2)Radial Position (m)Axial Position (m)  0.01 0.015 0.02 0.025 0.03 0.035−0.135−0.13−0.125−0.12−0.11500.10.20.30.40.50.60.70.80.91(b) dp = 710–850 µm (P3)Figure 5.16: ΦAPAM (r, z) for the smallest (a) and biggest (b) particle studiedin the top window location; measured particle vector field (↗).The slip contour and vector field within the middle window location,shown in Figure 5.17, explicitly shows radial fractionation between the twoparticles studied. It can be seen in Figure 5.17(a) that the smallest particlestudied exhibits strong inwards slip within 0.01 ≤ r (m) ≤ 0.019. Althoughthe inwards slip region shown in Figure 5.17(a) is much stronger than thatmeasured in water (see Figure 5.11(a)), the adverse axial velocity field andhigh outwards slip region near a radius of 0.01 m indicates little overflow sep-aration potential. The 710 – 850 µm particles, shown in Figure 5.17(b), werefound to move primarily towards the hydrocyclone wall displaying strong745.5. Effect of Particle Size on Slip Velocity: 0.03% APAMoutwards slip (i.e. Φ < 0.5) throughout most of the measurement volume.This was found to lead to purely underflow removal for the larger particles,as the dominantly outward motions were again found within the bottomwindow.Radial Position (m)Axial Position (m)  0.01 0.012 0.014 0.016 0.018 0.02 0.022 0.024 0.026−0.275−0.27−0.265−0.26−0.255−0.2500.10.20.30.40.50.60.70.80.91(a) dp = 500–600 µm (P2)Radial Position (m)Axial Position (m)  0.01 0.012 0.014 0.016 0.018 0.02 0.022 0.024 0.026−0.275−0.27−0.265−0.26−0.255−0.2500.10.20.30.40.50.60.70.80.91(b) dp = 710–850 µm (P3)Figure 5.17: ΦAPAM (r, z) for the smallest (a) and biggest (b) particle studiedin the middle window location; measured particle vector field (↗).The slip contour and vector field presented in Figure 5.18(a) shows thata region of strong inwards slip exists towards the hydrocyclone axis at anaxial location of -0.365 m. Unfortunately, the LDV measurements withinthis region, presented in Chapter 4.1, indicate that overflow removal is notpossible as the flow is fully downwards.755.5. Effect of Particle Size on Slip Velocity: 0.03% APAMThe slip contour and vector field in Figure 5.18(b) revealed an equilib-rium orbit position (i.e. where FC ≈ FD) near a radius of 0.015 m, and anaxial position of -0.365 m. It was found that the radius where this equilib-rium occurs decreased with decreasing axial position.The slip contour and vector field of the 710 – 850 µm particles were foundto follow a similar trend to the measurements presented in Figure 5.12(b).Based on these results, it is clearly evident that axial slip is considerable asparticles approach the underflow outlet. This is likely a result of the highstrain rates (see Chapter 4.1) leading to notable particle drag.Radial Position (m)Axial Position (m)  0.008 0.009 0.01 0.011 0.012 0.013 0.014 0.015 0.016 0.017−0.385−0.38−0.375−0.37−0.36500.10.20.30.40.50.60.70.80.91(a) dp = 500–600 µm (P2)Radial Position (m)Axial Position (m)  0.006 0.008 0.01 0.012 0.014 0.016−0.385−0.38−0.375−0.37−0.36500.10.20.30.40.50.60.70.80.91(b) dp = 710–850 µm (P3)Figure 5.18: ΦAPAM (r, z) for the smallest (a) and biggest (b) particle studiedin the bottom window location; measured particle vector field (↗).76Chapter 6A Quantitative Analysis ofTurbulent Drag ReductionThis section provides quantitative results characterizing turbulent drag re-duction (TDR) in a pulp hydrocyclone using polymer additives. The degra-dation of the polymer additives is quantified as the decline in TDR, withtime, at constant pumping power for various reject ratios ranging from 0–100%. Upon characterization of the maximum time allowed for quantifyingpolymer induced TDR in a hydrocylcone, the influence of fibre concentra-tion, polymer concentration, and fibre+polymer on fluid energy losses in thedevice was evaluated. The results indicate the effectiveness of polymer addi-tives with and without cellulose fibres is largely dependent on inlet velocityand reject ratio.6.1 Evaluating Turbulent Drag Reduction in aHydrocycloneNumerous investigations into polymer induced turbulent drag reduction(TDR) have been investigated since Toms [66] discovery in 1948. Theworks have helped characterize the mechanisms of TDR, most notably intwo dimensional flows. One of the characteristics of polymer induced TDRin a pipe, for example, is that laminar flow of polymer solutions displayno change in skin friction compared to laminar flow of Newtonian fluids,expressed as:Cf =τw12ρU2∞(6.1)where Cf is the skin friction coefficient, τw is the wall shear stress, ρis the fluid density, and U∞ is the free stream velocity. Secondly, the min-imum Reynolds number for drag reduction to occur, in a fixed geometry,776.1. Evaluating Turbulent Drag Reduction in a Hydrocyclonedepends on the number of monomers in the macromolecule. This impliesthat turbulence dynamics largely effect polymer stretching.Experiments and scaling arguments have led to a time criterion ([44])for the onset of DR, dependent on polymer characteristics. The principalcentres around the scaling of polymer relaxation time to the characteristictime scale of the near wall turbulence. This stems from the fact that thetotal shear stress τy is the sum of the viscous stress µ∂〈U〉/∂y and theReynolds stress -ρ〈uv〉. The boundary condition U(x, t) = 0 dictates thatall the Reynolds stresses are zero, thus, the wall shear stress is due entirelyfrom the viscous effects. The resulting time criterion is as follows:Tz >µρuτ(6.2)where Tz is the polymer relaxation time, µ is the solution viscosity,and uτ is the friction velocity defined as√τwρ . An approximation to therelaxation time for flexible polymers in dilute solutions ([75]) is shown as:Tz =µ(N3/5a)3kT(6.3)where, N is the number of repeating monomers, a is the length of asingle monomer, k is the Boltzman constant, and T is the solution temper-ature. Obvious difficulties arise when characterizing drag reduction in moreindustrial flows, largely from the reacting type flow behaviour of polymersolutions. The monomer length and solution temperature change with time,thus leading to the need of a macroscopic definition of drag reduction forengineering applications.The Prandtl-Ka´rma´n coordinate system is a benchmark analysis for DRin a pipe. It illustrates the deviations from the Prandtl-Ka´rma´n law basedon Re and friction factor, where the maximum DR line (see [69]) in thiscoordinate system is thought to be when the polymer effects are felt overall flow scales, thus causing the buffer layer thickness to extend over theentire boundary. The buffer layer is defined as the region between the vis-cous sublayer and log-law region, were the turbulent energy production anddissipation is at a maximum. It is described for water in a wall boundedflow, in viscous lengths, as follows.786.1. Evaluating Turbulent Drag Reduction in a Hydrocycloney+ =uτyρµ5 < y+ < 30 (6.4)As mentioned, the buffer layer is situated before the log-law region, wherethe production to dissipation ratio, P/, and the normalized mean shearrate, Sk/ (where S = ∂〈U〉/∂y), are essentially uniform. The mean velocity,normalized by the friction velocity, of the log-law region in viscous lengthsis as follows:u+ =1kln y+ +B y+ > 30 (6.5)where, k is the von Ka´rma´n constant (≈ 0.41) and B is an integrationconstant, generally between 5.0 – 6.2. An important characteristic of thelog-law is that it predicts the velocity profile over the whole flow within apipe with only small adjustments to the constants. Slight differences aregenerally observed when compared to experimental data near the pipe’scentreline and for y/R < 0.1, where y = R− r.Traditionally, energy losses within a pipe are expressed as in terms ofthe friction factor, defined as:f =∆pD12ρU¯2L(6.6)where, ∆p is the pressure drop over an axial distance L, D is the pipediameter, and U¯ is the bulk velocity, shown below.U¯ =1piR2R∫0〈U〉2pir∂r (6.7)The friction factor f plotted against Reynolds number for a fully devel-oped flow in smooth pipes, distinctly shows the transition from laminar toturbulent flow, as the friction factor rapidly increases from the 64/Re lawnear a Re of 2000. Neglecting the transition region, Prandtl’s friction lawfor smooth pipes follows experimental data extraordinarily well for Re >3500. The principal behind this law originates from the assumption that796.1. Evaluating Turbulent Drag Reduction in a Hydrocyclonethe mean velocity profile in a pipe can be approximated by the logarithmiclaw (Equation 6.5), thus, substitution into Equation 6.6 yields Prandtl’sfriction law for smooth pipes:1√f≈ 1.99 log10(√fRe)− 0.95 (6.8)with k = 0.41, B = 0.95, and Re = 2U¯R/ν. This friction law is thecurrent standard for quantitatively expressing polymer induced turbulentdrag reduction in a pipe. The Prandtl – Ka´rma´n law presents the effectsof Re on friction factor within a pipe by graphically displaying Equation6.8 linearly. The effects of introducing turbulent drag reducing additives bythe deviation from the Prandtl – Ka´rma´n law are easily observed, boundedbetween the Hagen – Poiseuille friction law (i.e. f = 64/Re) and maximumdrag reduction asymptote.Analytically defining the hydrocyclone problem is challenging, as the flowis three dimensional, highly turbulent, and is not truly axially symmetric(i.e. for single inlet hydrocyclones). The first critical views on quantifyingfluid energy losses within a hydrocyclone were based on the approach ofBradley [3], in terms of a cyclone loss coefficient (ξCH). The energy con-sumption in Bradley’s approach, widely simplified to static pressure drop, isoften used as a benchmark for characterizing the fluid energy losses duringhydrocyclone operation. The desire, however, would be to fundamentallydefine DR in a similar manner to that of pipe flow or channel flow. Unfor-tunately, measurement difficulties and flow complexity presently make thattype of analysis a near impossibility. Nevertheless, macroscopically definingthe energy losses in a hydrocyclone essentially follows the principal behindthe derivation of Equation 6.6.806.1. Evaluating Turbulent Drag Reduction in a HydrocycloneFigure 6.1: Hydrocyclone energy analysis.Considering the hydrocyclone problem as a fixed control volume withone inlet, and two outlets, as shown in Figure 6.1, a steady flow energybalance can be written as follows:˙mi,1(Pi,1ρ+U2i,12+ gzi,1)=∑outm˙o(Poρ+U2o2+ gzo)+ `energy (6.9)where m˙ is mass flow rate, U is the mean velocity in the inlet or outletpipes, P is pressure, z is the height of the inlet or outlet pipe from theground, and `energy is the energy losses due to friction. The subscript irefers to the inlet pipe, whereas the subscript o refers to one of the outletpipes shown in Figure 6.1. The secondary subscripts used for the outletpipes follow the definitions shown in Figure 6.1. A mass balance on thesystem in terms of a hydrocyclone specific quantity, namely reject ratio,reduces Equation 6.9 down to the following:816.2. Drag Reduction Experiment Outline`energy =1ρ(Pi,1 − (1−RR)Po,1 − (RR)Po,2)+12(U2i,1 − (1−RR)U2o,1 − (RR)U2o,2)(6.10)+g (zi,1 − (1−RR)zo,1 − (RR)zo,2)where reject ratio or RR = Qunderflow/Qinlet = Uo,2/Uo,1 due to the crosssectional area of the inlet and outlet pipes being identical (i.e. Ai,1 = Ao,1 =Ao,2). Please note: the flow rate measurements for the drag reduction analy-sis were taken at the 25 mm diameter pipe, where the pressure transducer isattached. As discussed in Chapter 3.1, the 25 mm diameter pipe constrictsto a 19.2 mm diameter pipe before entering the hydrocyclone. As such, allinlet velocity measurements presented in this section of work are 1.7 timesless than the true inlet velocity entering the hydrocyclone.Modelling the `energy term as a combination of the inlet kinetic energyU2i,1, and a scalar, non-dimensional constant k, results in a similar form toEquation 6.6; the ratio of static pressure drop to dynamic pressure in a pipe.Drag reduction in a hydrocyclone is thus defined as:%DR(U2i,1, RR) = 100% ·kH2O(U2i,1, RR)− kadditive(U2i,1, RR)kH2O(U2i,1, RR)(6.11)under the assumption that kH2O,additive is non-linear with U2i,1. Theinfluence of inlet velocity, and reject ratio within a hydrocyclone is nowquantitatively represented within a scalar variable k, where DR is definedas the reduction in k to that of water alone. For this work, characterizingdrag reduction in a hydrocyclone was carried out by applying Equation 6.11for various polymer solutions, fibre suspensions, and combinations of thetwo.6.2 Drag Reduction Experiment OutlineThe performed experiments are summarized in Table 6.1. Here, the out-line illustrates the various combinations of solvent, fibre and polymer in 0.4m3 of water (see: Chapter 6.4). No NaCl was added to the combinationsof solvent, fibre and polymer described below; minimizing the influence ofpolymer chain flexibility on the results. Each sample described in Table 6.1826.2. Drag Reduction Experiment Outlinewas assessed over a range of inlet velocities, where the maximum was lim-ited to the inlet and overflow pressure dependent on reject ratio. The trialsconducted were repeated for each RR a minimum of three times; where themeasurement devices described in Chapter 3.1 were calibrated to take themean of 10 sec of data at a data-rate of 1 kHz for each inlet velocity. Thevariance between pressure and flow-rate observed with water and suspen-sions containing only cellulose fibres was quite small (i.e < 3.3%); however,the dependency on experimental run time, when introducing polymer ad-ditives was significant (see: Chapter 6.3). All evaluations of kadditive whenAPAM or CPAM was present in the sample were conducted within the first10 min of experimental run time to minimize the effect of polymer degrada-tion on the accuracy of the results. A slightly longer degradation scale wasachieved than shown in Figure 6.2 due to the range of inlet velocities foreach polymer study were within pumping powers of 2.5 kW and 11.5 kW(i.e. less shear exerted on the structures over the run time). The variancein all of the experiments run with polymer additives were found to rangebetween 1.5 – 5% at identical inlet velocities (±1.5%) with a reject ratioerror within 2%.Sample Fibre Consistency (%w/w) Polymer (ppm) RR # RRAPAM CPAM min maxH1 - - - 0 1 9A1 - 100 - 0 0.9 8A2 - 150 - 0 0.9 8A3 - 300 - 0 0.9 8A4 - 500 - 0 0.9 8C1 - - 100 0 0.9 8C2 - - 150 0 0.9 8C3 - - 300 0 0.9 8C4 - - 500 0 0.9 8P1 0.5 - - 0 0.75 4P2 0.7 - - 0 0.75 4P3 0.9 - - 0 0.75 4PA1 0.7 100 - 0 0.75 4PA2 0.7 150 - 0 0.75 4PA3 0.7 300 - 0 0.75 4PA4 0.7 500 - 0 0.75 4PC1 0.7 - 100 0 0.75 4PC2 0.7 - 150 0 0.75 4Table 6.1: Summary of drag reduction experiments.836.3. Polymer Degradation6.3 Polymer DegradationEvaluating the effectiveness of polymer induced drag reduction as a functionof experimental run time has been investigated. The polymer degradationcaused by chain scission with a constant pumping power of either 11.2 kW or7.5 kW was studied for various APAM and CPAM concentrations at rejectratios between 0 and 100%. Figure 6.2 and Figure 6.3 illustrates the declineof drag reducing capabilities through the increase in the non-dimensionallost energy constant, k(RR), for APAM and CPAM, respectively.0 10 20 30 40 50 600.20.40.60.811.21.4Time (min)k(RR) APAM/k(RR) H 2O(a)0 10 20 30 40 50 600.20.40.60.811.21.4Time (min)k(RR) APAM/k(RR) H 2O(b)0 10 20 30 40 50 600.20.40.60.811.21.4Time (min)k(RR) APAM/k(RR) H 2O(c)0 10 20 30 40 50 600.20.40.60.811.21.4Time (min)k(RR) APAM/k(RR) H 2O(d)Figure 6.2: APAM degradation curves: k(RR)APAM/k(RR)H2O vs. time.(a) RR = 0%, (b) RR = 25%, (c) RR = 50%, (d) RR = 75%. ◦ 100 ppmAPAM (11.2 kW); 4 150 ppm APAM (11.2 kW); ♦ 100 ppm APAM (7.5kW).846.3. Polymer Degradation0 10 20 30 40 50 600.20.40.60.811.21.4Time (min)k(RR) CPAM/k(RR) H 2O(a)0 10 20 30 40 50 600.20.40.60.811.21.4Time (min)k(RR) CPAM/k(RR) H 2O(b)Figure 6.3: CPAM degradation curves (pumping power = 11.2 kW):k(RR)CPAM / k(RR)H2O vs. time. (a) RR = 0%, (b) RR = 25%. ◦100 ppm CPAM; ♦ 150 ppm CPAM.Figure 6.2 and 6.3 shows an increase in polymer concentration resultsin a decrease of the lost energy coefficient with constant pump power (11.2kW). The time for polymer to show negligible change in lost energy, or de-cay time, increased with an increase in initial polymer concentration for allreject ratios studied. This agrees well with previous studies which describecontributions to the decrease of DR as a reduction in the solutes averagemolecular weight and a negative shift in the molecular weight distribution(e.g. [41] and [29]). Over the range of reject ratios and additive concen-trations studied, the minimum attainable lost energy coefficient was foundto be 50% less than that of water alone with 150 ppm APAM at a rejectratio of 25% and 50%. The fluid energy losses increased to 1.18 times thatof water alone, after a run time of approximately 60 min. Interestingly,Figure 6.2(c) displays no adverse DR effects within the experimental runtime, indicating an equal volume split fraction, or reject ratio equal to 50%,may be optimal for hydrocyclone operations as DR is still seen with severelydegraded polymer.Full degradation of the polymer solutions/suspensions was interpretedto occur when the change in the lost energy constant (k) became negligiblewith time. In single pass units the degradation of polymer as a function ofpumping time is small, whereas in recycle flow systems (as in this study) theshear exerted by the pump, pipe valves, and pipe bends is significant. As856.3. Polymer Degradationmentioned, the degradation of polymer solutions is largely affected by theextent of shear exerted on the formed structures. This undoubtedly posesconsideration regarding the use of polymer additives for energy savings asthe polymer cost may exceed the reduction in energy costs for some highshear flows. Thus, a sensitivity analysis of the polymer structure rigidity wasevaluated at a reject ratio of 0% for a 100 ppm APAM solution, where thepumping power was reduced to 7.5 kW. A reduction in the shear exerted onthe formed structures (∝ pumping power) resulted in a near 20% decreasein degradation with time, when compared to the 100 ppm APAM solutionat a pumping power of 11.2 kW.In two-phase flows, a synergistic effect with respect to DR occurs betweenfibres and polymer. The degree of synergism depends on the consistency ofthe fibres, charge density of the polymer additive and concentration of thepolymer additive within the suspension. The extent of shear acting on thepolymer molecules and fibres also contributes to the effective synergism ofthe fibre-polymer mixture (e.g. [40, 50]). Figure 6.4 and 6.5 shows thedecline of %DR for various fibre-polymer mixtures similarly to Figures 6.2and 6.3.0 10 20 30 40 50 600.20.40.60.811.21.4Time (min)k(RR) Additive/k(RR)H 2O(a)0 10 20 30 40 50 600.20.40.60.811.21.4Time (min)k(RR) Additive/k(RR)H 2O(b)0 10 20 30 40 50 600.20.40.60.811.21.4Time (min)k(RR) Additive/k(RR)H 2O(c)0 10 20 30 40 50 600.20.40.60.811.21.4Time (min)k(RR) Additive/k(RR)H 2O(d)Figure 6.4: APAM degradation curves in a 0.7% (w/w) SPF pulp suspension(pumping power = 11.2 kW): k(RR)additive/k(RR)H2O vs. time. (a) RR =0%, (b) RR = 25%, (c) RR = 50%, (d) RR = 75%. ◦ 100 ppm APAM; ♦150 ppm APAM.866.3. Polymer Degradation0 10 20 30 40 50 600.20.40.60.811.21.4Time (min)k(RR) Additive/k(RR)H 2O(a)0 10 20 30 40 50 600.20.40.60.811.21.4Time (min)k(RR) Additive/k(RR)H 2O(b)Figure 6.5: CPAM degradation curves in a 0.7% (w/w) SPF pulp suspension(pumping power = 11.2 kW): k(RR)additive/k(RR)H2O vs. time. (a) RR =0%, (b) RR = 25%. ◦ 100 ppm CPAM; ♦ 150 ppm CPAM.Figure 6.4 shows a slight decrease in maximum drag reduction at a rejectratio of 25% when comparing 100 ppm and 150 ppm APAM solutions, whereall other kadditive/kH2O(t = 0) values are more or less the same. This may bea result of the interaction between the fibres and polymer additive, effectivelyhindering the formation of stable structures. The major difference betweenthe aqueous APAM/CPAM solution and polymer + fibre suspension lieswithin the steady state lost energy coefficient, where up to a 25% decreaseis achieved in comparison to polymer only solutions. It is likely that thevorticity field is not strong enough to elongate the degraded polymer strands,as the fibres in the suspension may be reducing some of the momentumtransfer.At a reject ratio of 25% the 100 ppm APAM suspension containing 0.7%(w/w) SPF fibres displays minimal change in %DR over the run time of theexperiment. This emphasizes that the synergistic effect between pulp fibresand polymer additives is not simply a weighted contribution when evaluat-ing DR in hydrocyclones. When assessing the form of the degradation scalefor the fibre+polymer suspensions as well as the polymer only solutions, itis seen that the decay is not strictly exponential. Figure 6.2 - 6.5 show anincrease in drag with an increase in time where, dependent on the suspen-sion characteristics and reject ratio, a transition from an increase in lostenergy to a decrease in lost energy with time occurs. This non-monotonic876.4. Drag Reduction in a Pulp Processing Hydrocyclonebehaviour is distinctly seen with a reject ratio of 0%, and may be attributedto the increase in the fluid residence time, as the only exit stream under thiscondition is through the vortex finder.6.4 Drag Reduction in a Pulp ProcessingHydrocycloneThe following section reviews drag reducing characteristics of various addi-tives introduced into a pulp processing hydrocyclone. The notation kH2Oand kadditive are the lost energy constants for water alone and the variousadditives including polymer, fibre and fibre and polymer suspensions solvedfrom Equation 6.9. Upon the evaluation of kH2O,additive, it was found thatwater and aqueous polymer solutions were linear with respect to `energy vs.U2inlet (see Appendix F). Figure 6.6 shows the increase in kH2O with an in-crease in reject ratio follows a higher order polynomial. The continuousform of k(RR)H2O with model error is included in Figure 6.6, which tookthe form of Equation 6.12 with n = 3 for water and polymer solutions.k(RR) = p1RRn + p2RR(n−1)...+ pn+1 (6.12)0 10 20 30 40 50 60 70 80 90 100050100150200250300RR(%)k(RR)Figure 6.6: Water lost energy curve: k(RR) vs. RR.  Raw data; ·− Cubicmodel.886.4. Drag Reduction in a Pulp Processing HydrocycloneFigures 6.7 and 6.8 show the reduction in lost energy as a function ofreject ratio for various polymer concentrations in comparison to water alone.It was found that a slight increase in DR occured with an increase in polymerconcentration from 100ppm to 500ppm APAM/CPAM for all reject ratiosstudied; however, the extent of additive concentration on drag reductionis less than that observed in other geometries (e.g. [20] and [41]). Theobserved DR increases from a reject ratio of 0% to a maximum of 55% ata reject ratio of 50%, where the %DR decreases with an increase in rejectratio beyond 50%.0 10 20 30 40 50 60 70 80 90 100050100150200250300RR(%)k(RR)Figure 6.7: APAM lost energy curve: k(RR) vs. RR.  water; ◦ 100 ppmAPAM; B 300 ppm APAM; ♦ 500 ppm APAM.896.4. Drag Reduction in a Pulp Processing Hydrocyclone0 10 20 30 40 50 60 70 80 90 100050100150200250300RR(%)k(RR)Figure 6.8: CPAM lost energy curve: k(RR) vs. RR.  water; ◦ 100 ppmCPAM; B 300 ppm CPAM; ♦ 500 ppm CPAM.The trend of `energy vs. U2inlet for suspensions containing pulp fibres wasfound to be non-linear; therefore, DR is shown as a function of U2inlet andreject ratio in Figure 6.9. Figure 6.9 shows that the level of drag reductiondecreases with an increase in inlet velocity for a reject ratio of 0% and25%; whereas, an increase to some maxima, then decrease in DR with anincrease in inlet velocity was found for the other reject ratio’s studied. Forreject ratios of 0% and 25%, an obtainable operation, due to the variablefrequency drive limiter, was achieved with U2inlet = 2.5 m2/s2, measured inthe 2.54 cm section of inlet pipe. All DR inlet velocity relationships weremeasured at this inlet position. The maximum condition correlates to amaximum %DR of 48% and 41%, respectively for a 0.9% fibre suspension.Similar to the DR characteristics of polymer additives in hydrocyclones, amaximum %DR for a 0.9% fibre suspension of 58% was achieved at a rejectratio of 50% with U2inlet = 1.5 m2/s2. The increase in DR with an increase infibre consistency agrees well with results obtained in other geometries (e.g.[62]).906.4. Drag Reduction in a Pulp Processing Hydrocyclone0 0.5 1 1.5 2 2.5 3 3.500.20.40.60.81U2inlet (m2/s2)DR(b)0 0.5 1 1.5 2 2.5 3 3.5 4 4.500.20.40.60.81U2inlet (m2/s2)DR(a)0 0.5 1 1.5 2 2.5 300.20.40.60.81U2inlet (m2/s2)DR(c)0 0.5 1 1.5 200.20.40.60.81U2inlet (m2/s2)DR(d)Figure 6.9: Pulp fibre drag reduction curves: DR vs. U2inlet. (a) RR = 0%,(b) RR = 25%, (c) RR = 50%, (d) RR = 75%. ♦ 0.5% SPF; ◦ 0.7% SPF;4 0.9% SPF.The synergism between pulp fibres and polymer additives has been stud-ied in various geometries (e.g. pipe and channel flow), where changes tothe rheological suspension properties and drag reducing characteristics havebeen observed. Figures 6.10 and 6.11 show the level of DR achieved withfibre suspensions containing polymer (APAM/CPAM) additives for rejectratios ranging from 0 - 75%.916.4. Drag Reduction in a Pulp Processing Hydrocyclone0 1 2 3 4 5 6 7−0.500.51U2inlet (m2/s2)DR(a)0 1 2 3 4 5 6 7−0.500.51U2inlet (m2/s2)DR(b)0 0.5 1 1.5 2 2.5 3 3.5 4−0.500.51U2inlet (m2/s2)DR(c)0 0.5 1 1.5 2−0.500.51U2inlet (m2/s2)DR(d)Figure 6.10: APAM DR evaluated in a 0.7% SPF pulp suspension: DR vs.U2inlet. (a) RR = 0%, (b) RR = 25%, (c) RR = 50%, (d) RR = 75%. • 0.7%SPF pulp suspension; ◦ 150 ppm APAM; 4 300 ppm APAM; ? 500 ppmAPAM.0 1 2 3 4 5 6 7−0.500.51U2inlet (m2/s2)DR(a)0 1 2 3 4 5 6 7−0.500.51U2inlet (m2/s2)DR(b)0 0.5 1 1.5 2 2.5 3 3.5 4−0.500.51U2inlet (m2/s2)DR(c)0 0.5 1 1.5 2−0.500.51U2inlet (m2/s2)DR(d)Figure 6.11: CPAM DR evaluated in a 0.7% SPF pulp suspension: DR vs.U2inlet. (a) RR = 0%, (b) RR = 25%, (c) RR = 50%, (d) RR = 75%. • 0.7%SPF pulp suspension; ♦ 100 ppm CPAM; ◦ 150 ppm CPAM.Behaviour patterns of the fibre suspensions containing APAM/CPAM926.5. Polymer Adsorption Characteristicsdiffer significantly in comparison to 0.7% (w/w) fibre suspension. For areject ratio of 0 and 25%, there is a decrease in DR with an increase inU2inlet to a minimum, where beyond this point the %DR increases with anincrease in U2inlet. Drag reducing was found to increase with an increase inU2inlet for reject ratios of 50 and 75%; opposite to that of the fibre suspensionalone.The effect of APAM concentration was found to be less influential on%DR in a fibre suspension than that observed in Figure 6.7. The nega-tive effect of polymer concentration on DR in a fibre suspension is largelypronounced for reject ratios of 0 and 75%. With typical operation, an inprocess hydrocyclone would have a reject ratio near 25%, where Figure 6.10shows an increase in DR in comparison to the fibre suspension alone whenoperated above 2.5 m2/s2, regardless of polymer concentration. Pulp pro-cessing hydrocyclones of similar size to the one used in this study operatewell above this condition, suggesting that polymer additives will improvethe energy savings. The observed drag reduction sensitivity in fibre suspen-sions containing APAM may be due to the mechanism behind the reductionin Reynolds stresses or the repulsion characteristics between the anionicpolymer and fibres.The change in DR with an increase in U2inlet for fibre suspensions contain-ing CPAM were observed to be very similar to that of the fibre suspensionalone (see Chapter 6.5). For reject ratio’s of 0% and 25% an increase inDR with the addition of 150ppm CPAM was seen beyond 4.5 m2/s2 and2.5 m2/s2, respectively. Similar %DR results for APAM and CPAM wereobserved for reject ratio’s of 25% and 50%.6.5 Polymer Adsorption CharacteristicsPolymer adsorption was measured using a Mu¨tek particle charge detectorin conjunction with a Mu¨tek automated titrator. Three cationic poly-acrylamide samples were titrated with an anionic polyvinylalcohol sulfate(PVSK) until the sample became neutral in charge. This procedure resultsin a standardized uptake per mL of sample value where no CPAM adsorp-tion occurs on a fibre surface. The described method was repeated withfibre suspensions containing various concentrations of CPAM to develop arelationship between polymer concentration in a fibre suspension and PVSKuptake. Each fibre suspension sample (250 mL) was mixed overnight in roomtemperature (i.e. 23◦C) with the CPAM polymer to ensure full adsorptionunder these conditions. The following expression defines the total CPAM936.5. Polymer Adsorption Characteristicsadsorption for the fibre suspensions studied, where [A] is the PVSK uptake(mLPV SK/mLsample).%Adsorbed = 100% ·[A]CPAMonly − [A]fibre+CPAM[A]CPAMonly(6.13)Table 6.2 presents the loss of CPAM due to surface adsorption in a0.7% (w/w) SPF fibre suspension. An increase in CPAM concentrationof 50 ppm resulted in a decrease of polymer surface adsorption by 17.7%.Although the interactive effects of CPAM and fibres in hydrocyclones is stillnot well understood, this analysis sheds light on the slight differences indrag reduction for fibre suspensions containing APAM or CPAM. Furtherstudy is required to fully characterize the effects of polymer charge density,polarity and concentration on DR in multiphase flows.Polymer Concentration (ppm) % Adsorbed100 42.8150 25.1Table 6.2: CPAM adsorption in a 0.7%(w/w) SPF fibre suspension94Chapter 7Conclusions andRecommendationsHydrocyclones are a complex problem of great interest to research institutesand industries around the world. Many experiments have been conducted topredict the performance of these units in hopes to save operation costs andincrease separation efficiencies. Operating conditions, particle-particle andparticle-fluid interactions, and hydrodynamics are all important character-istics of these units. At the present time, previous studies have lacked bothimplicit and explicit experimental results into the effects of reject ratio, andpolymer addition on the separation characteristics of variously sized parti-cles in hydrocyclones. Implicitly, the separation characteristics of particlesin a hydrocyclone flow field can be predicted by expressing the relative radialmotion of an isolated particle in terms of the driving force (i.e. centrifugal)and particle drag forces. Explicitly, the separation characteristics of par-ticles in a hydrocyclone must be directly measured using an appropriatevisualization technique, for example 3D particle tracking.This work focused on filling in the experimental gaps which are presentlyseen in literature by measuring the flow field using laser Doppler velocime-try and directly measuring the motion of variously sized particles by usinga high speed dual camera set-up. Specifically, we wanted to see whetheror not the addition of a polymer additive is a viable energy savings optionfor the pulp and paper industry. Fundamental investigation of the hydro-dynamics, and particle motions in two mediums set the groundwork of thisstudy, exposing the principles of hydrocyclone operation with a Newtonianand non-Newtonian fluid. Quantifying the energy savings as the reductionof fluid energy losses in a low consistency pulp processing hydrocyclone cor-related the more fundamental findings to an optimal industrial operatingcondition. The key findings of this work are summarized below.95Chapter 7. Conclusions and RecommendationsFlow FieldIt was found that polymer additives have the capability to fundamentallychange the internal flow pattern of a hydrocyclone. A hydrocyclone’s ef-fectiveness in isolating contaminants would likely be reduced, as a particlesseparation zone was found to be limited (see Chapters 4.1.3 and 4.1.4).Experimental TechniquesIt was found that the neutrally buoyant particles used for the PTV analysiswere unable to follow the flow field faithfully. Particle accumulation in timeand space could explain the particle slip, however remains suspect. Furtherinvestigations are required to detail the effects of particle size and concen-tration on particle slip velocities, as it was found that slip is considerable invery dilute suspensions.Particle SeparationIt was found that polymer additives increase the size of particles susceptibleto overflow removal in a hydrocyclone. Unfortunately, as shown in Chap-ter 4.1.4, the flow field of the polymer additives was found to reduce theoverflow separation potential for all particles. This effectively negates thefractionation benefits of the polymer additive.Drag Reduction (Energy Savings)Polymer additives were found to be effective in reducing the energy con-sumption in hydrocyclones, where the maximum energy savings was foundto occur at a reject ratio of 50%. Polymer additives in a multiphase cellulosefibre suspension, however, were only found to be effective for reject ratiosequal to 25% and 50%. It was found that the polymer agents increased thedrag when compared to the cellulose suspension alone if the fluid velocity en-tering the hydrocyclone was below 2.4 m/s (see Chapter 6.1 for explanation)for reject ratios of 25% and 50%. Typical pulp processing hydrocycylonesoperate well above this condition, suggesting that energy savings will likelyoccur. An economic analysis of polymer induced drag reduction can befound in Appendix G.Phenomenological Polymer DegradationPhenomenological polymer degradation was found to be excessive in the re-circulating system studied in this work. The practical use of the polymers967.1. Strengths and Limitations of Researchwas found to be largely dependent on pump shear and transport time, aspumps operating at high capacity would likely lead to faster degradation.The presence of cellulose fibres was as well found to degrade the polymersfaster than what was observed without cellulose fibres being present. Thiswas found to be most significant with the application of cationic polyacry-lamide (CPAM) in a cellulose fibre suspension, as polymer adsorption wasfound to be a contributing factor in polymer degradation time.7.1 Strengths and Limitations of ResearchThis work has unveiled several important findings that will further the un-derstanding of hydrocyclone operation in a variety of industrial settings.Laser Doppler velocimetry provided a more fundamental flow field character-ization, including turbulence intensities, setting the groundwork for particletracking velocimetry. The three dimensional particle tracking technique andquantification of drag reduction assisted in predicting true industrial opera-tion. An important limitation of the methods and results presented in thiswork is the applicability of the results to a low consistency fibre suspension.The polymers used throughout this study were highly visco-elastic, however,fibre suspensions are known to have an increased apparent viscosity whencompared to water alone. The effect of fibre flocculation, and synergism offibre + polymer suspensions on the resulting flow field is up for discussion.Secondly, the effect of fibre aspect ratio (i.e. L/W ), and rigidity on theirresulting motion within hydrocyclone’s is undoubtedly considerable. Theflow field and particle motions presented in this work should relate closelyto low consistency fibre flow, however, remains suspect.7.2 Recommendations for Future WorkThe work presented in this thesis represents a small portion of the potentialstudies that may be performed with the acrylic hydrocyclone in the UBCPulp and Paper Centre. The following recommendations provide potentialareas of study:• Study the effects of polymer concentration on the resultant flow fieldfor various reject ratios. This could possibly improve separation char-acteristics while still achieving drag reduction within the unit.• Manufacture an acrylic top plate to study the radial velocity pro-files throughout the hydrocyclone with and without polymer addition.977.2. Recommendations for Future WorkTrue measurements would further enhance the accuracy of predictedparticle trajectories.• Statistically analyse the relevant scales of turbulent motion for bothpolymer and water alone. Consider all three components velocity nearthe wall, and hydrocyclone axis to observe which regions are highlyturbulent and most influential on particle motion.• Study separation characteristics of dyed nylon fibres throughout theentire hydrocyclone body. It is possible to qualitatively, possibly quan-titatively, present the differences in particle trajectories of fibres andspherical particles at identical operating conditions.• Study the orientation of nylon fibres in hydrocyclone flow to establisha relationship between shear rate and orientation angle throughout thehydrocyclone. This shows obvious importance for predicting separa-tion efficiencies using fibre drag coefficients.98Bibliography[1] Bergstroo¨m, J. & Vomhoff, H. & So¨derberg, D. 2007Tangential velocity measurements in a conical hydrocyclone op-erated with a fibre suspension Miner. Eng. 20, 407–413.[2] Bergstroo¨m, J. & Vomhoff, H. 2007 Experimental hydrocy-clone flow field studies. Sep. Purif. Technol. 53, 8–20.[3] Bradley 1965 ’The Hydrocyclone’ Pergamon Press.[4] Canham, H.J.S., Catchpole, J.P., Long, R.F. 1971 Bound-ary layer additives to reduce ship resistance. The Naval Architect.J. 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Phys. 24(2), 269–278.105Appendix AModel ParametersSolution µ′′ (Pa sn) n0.03% APAM 0.097 0.49Table A.1: Regression coefficients of Ostwald-de Waele power-law model(Equation B.5).Solution ηo (Pa s) η∞ (Pa s) m λ (s)0.03% APAM 0.16 0.0096 0.16 1.25Table A.2: Regression coefficients of Carreau model (see: Equation 4.4).Solution Range of τ (Pa) b (Pa1−m) m0.03% APAM 0.090 - 9.41 5.29 1.14Table A.3: Power law parameters for normal-stress variation (N1 = bτm).Solution P1 P2 P3 P4Water -0.0024 0.13 -1.42 45.33100ppm APAM 0.008 -0.044 0.77 32.41300ppm APAM 0.0001 -0.004 0.98 28.92500ppm APAM 0.001 -0.039 0.14 32.46100ppm CPAM 0.002 -0.11 2.08 35.35300ppm CPAM 0.0007 -0.035 0.47 31.98500ppm CPAM -0.004 0.31 -0.88 34.53Table A.4: Solution constants to k(RR) polynomial fit (see: Equation 6.12).106Appendix BAnalytical Study of aSwirling Cross-flowThis section focuses on using fundamental equations to predict the tangentialvelocity distribution for a swirling cross-flow. The method presented herewas first introduced by Upadrashta et al. [67], and was found to be auseful analytical technique to estimate the tangential velocity profiles forNewtonian and pseudo-plastic power law fluids.The analysis starts with considering the θ-component of the Reynoldsaveraged Navier Stokes (RANS) equations for a steady, axially symmetric,turbulent flow of an incompressible fluid. Assuming gravity effects are neg-ligible, the θ-component of the RANS equations can be explicitly shownas:ρ(ur∂uθ∂r+uruθr+ uz∂uθ∂z)= −1r2∂∂r(r2(τrθ + τT ))+∂τz,θ∂z(B.1)where τT is the turbulent stress, τr,θ is the viscous stress in the r, θ plane,τz,θ is the viscous stress in the z, θ plane, ur is the radial velocity compo-nent, uz is the axial velocity component, and uθ is the tangential velocitycomponent. Equation B.1 can be slightly simplified by assuming the flow isaxially fully developed, namely, the velocity components ur and uθ and theviscous stress τz,θ are not a function of axial location z. Some experimentalinvestigations (e.g. [34]) have shown that this is not an unreasonable ap-proximation for a small portion of the flow, however, does not accuratelymodel the flow of a hydrocyclone near the wall or near the core.With the assumption that the flow is axially fully developed and sym-metric, a model equation for the radial velocity component, ur, can be foundby simplifying the continuity equation in cylindrical coordinates. The finalform of the continuity equation was found to be as follows.∂∂r(rur) = 0 (B.2)107Appendix B. Analytical Study of a Swirling Cross-flowEquation B.2 indicates that rur = constant, as the partial derivative ofrur is zero. Defining rur = C, where C is a constant, and applying the fullydeveloped flow assumption stated above, Equation B.1 can be simplified tothe following:ρCr(∂uθ∂r+uθr)= −1r2∂∂r(r2(τrθ + τT ))(B.3)To fully define Equation B.3, a model equation for the viscous stressand the turbulent stress are required. First, the focus will be on definingthe viscous stress τrθ in Equation B.3. To expand the applicability of thisanalysis, a power law viscosity model was incorporated into the viscous stressτrθ to include the shear thinning behaviour of pseudo-plastic fluids (see [67]).The expression for the viscous stress τrθ and the power law viscosity modelare shown in Equations B.4 and B.5, respectively.τrθ = −µr∂∂r(uθr)(B.4)µ = µ′′∣∣∣∣∣[12(ε : ε)]1/2∣∣∣∣∣n′−1(B.5)In Equation B.5, µ′′ is the laminar consistency index, n′ is the flowbehaviour index (i.e. equal to 1 for Newtonian fluids), ε is the rate of straintensor, and µ is the effective fluid viscosity. The magnitude of the rate ofstrain tensor (i.e.√12 (ε : ε)), assuming the flow is axially fully developedand axially symmetric, can be expressed as:√12(ε : ε) =((∂ur∂r)2+(urr)2+14[r∂∂r(uθr)]2)1/2(B.6)Previous investigations (e.g. [14], [34], [43], [71]) have found that theflow of both water and fluids that exhibit shear thinning behaviour, namelymagnetite, Separan AP-30, and carboxy-methylcellulose (CMC), in hydro-cyclones show the change in ur(r) is small when compared to the changein uθ(r) throughout a majority of the flow. Similarly, the magnitude of uθis generally at least an order of magnitude greater than that of the radial108Appendix B. Analytical Study of a Swirling Cross-flowvelocity component, ur. In Chapter 4.1, this was as well found to be truefor the fluids investigated in this study. As such, the following asymptoticanalysis on the mean rate of strain tensor was found to be suitable.∂ur∂r∂uθ∂rurruθrThe mean rate of strain tensor can therefore be defined as:√12(ε : ε) =(14[r∂∂r(uθr)]2)1/2(B.7)The final form of the viscous stress, τrθ, can then be found by substitutingEquation B.5 and B.7 into Equation B.4. The final form of the viscous stressτrθ was found to be as follows:τrθ = −µ′′∣∣∣∣r2∂∂r(uθr)∣∣∣∣n′−1r∂∂r(uθr)(B.8)The second stress in Equation B.3, namely the turbulent stress, wasmodelled in a similar fashion to the viscous stress term τrθ shown above.For this analysis the Boussinesq approximation was adopted, where the tur-bulent stress is modelled as the product of the eddy viscosity and mean rateof strain tensor, shown as:τT = ρνT(12(ε : ε))1/2(B.9)where νT is the eddy kinematic viscosity, ρ is the fluid density, andsimilarly to above, ε is the rate of strain tensor. Following Prandtl’s mixinglength hypothesis and modelling the turbulent viscosity as the product ofa lengthscale (`T ) and a turbulent velocity scale (u∗), implicitly defined as:u∗ = |τT /ρ|1/2, the turbulent viscosity for an axially fully developed andsymmetric swirling cross-flow was found to be as follows.τT = ρ`2T∣∣∣∣r∂∂r(uθr)∣∣∣∣r2∂∂r(uθr)(B.10)109Appendix B. Analytical Study of a Swirling Cross-flowDefining a turbulent lengthscale model (`T ) for this problem is difficult,as the variations in the size of the turbulent eddies in a swirling cross-flow arehard to characterize to, for example, radial position. As such, for simplicity,the turbulent lengthscale will be defined as a percentage of the total radius(R) of the axially fully developed swirling cross-flow, namely:`T = κR (B.11)where κ is a constant. The final form of the turbulent stress model usedfor this investigation is shown explicitly in Equation B.12.τT = ρκ2R2∣∣∣∣r∂∂r(uθr)∣∣∣∣r2∂∂r(uθr)(B.12)Introducing Equation B.8 and B.12 into Equation B.3, yields the gov-erning equation for the swirling cross-flow considered in this investigation,shown as:ρCr(∂uθ∂r+uθr)= −1r2[∂∂r(r2µ′(−r∂∂r(uθr))n′)]−ρκ2R22r2[∂∂r(r2(−r∂∂r(uθr))2)](B.13)where µ′ = µ′′/2n′−1. Assuming that the vorticity field is non-zero, theangular velocity relationship: ω(r) = uθ(r)/r, can be introduced into Equa-tion B.13, where ω is a function of radial position. Performing the requireddifferentiations in Equation B.13 and re-arranging the terms resulted in thefollowing:∂2ω∂r2(µ′nrn′+ ρκ2R2r2(−∂ω∂r)2−n′)=ρCr(r∂ω∂r+ 2ω)(−∂ω∂r)1−n′+µ′rn′−1(−∂ω∂r)(2 + n′) + 2ρrκ2R2(−∂ω∂r)3−n′(B.14)110Appendix B. Analytical Study of a Swirling Cross-flowModel Case n′ κ µ′′ (mPa sn′) ω(R) (s−1) C (mm2/s)WaterLam. 1.0 0 1.0 50 -1.5Turb. 1.0 0.05 1.0 50 -1000PowerLam. 0.25 - 0.75 0 97 50 -1.5Turb. 0.25 - 0.75 0.05 97 50 -1000Table B.1: Overview of the values used to solve Equation B.15.Non-dimensionalizing Equation B.14 required a few assumptions, par-ticularly defining the boundary conditions. For the case considered here,the outer boundary, namely at r = R, is a free surface boundary due tour(R) 6= 0. This lead to defining the outer boundary condition for the angu-lar velocity, namely ω(R), as a positive, and non-zero constant. The innerboundary condition for the angular velocity, namely ω(r = 0), was definedto be zero, as the tangential velocity component (uθ) for this investigationis zero at r = 0. Introducing the non-dimensional quantities: Ψ = ω/ω(R)and σ = r/R into Equation B.14 yielded the following:∂2Ψ∂σ2(µ′n′σn′+ ρκ2R2σ2ω(R)2−n′(−∂Ψ∂σ)2−n′)=ρCω(R)1−n′(∂Ψ∂σ+2Ψσ)(−∂Ψ∂σ)1−n′+µ′σn′−1(−∂Ψ∂σ)(2 + n′) + 2ρσκ2R2ω(R)2−n′(−∂Ψ∂σ)3−n′(B.15)The non-dimensional boundary conditions for Equation B.15 were de-fined as:Ψ(σa) = 1Ψ(σb) = 0where σa = 1 and σb = 0. The outer radius (R) was chosen to be 0.05 mfor all of the swirling cross-flow investigations, similar to that of the outerradius of the experimental hydrycyclone described in Chapter 3.1. TableB.1 outlines the range of the remaining constants used to solve EquationB.15.A good starting point in characterizing the swirling cross-flow of waterand a power-law fluid is to assume laminar flow, namely by defining κ = 0.111Appendix B. Analytical Study of a Swirling Cross-flowThis assisted in quantifying the differences in tangential velocity profilesas a purely pseudo-plasticity (i.e. shear thinning) effect. The dimensionalsolution to Equation B.15 for two of the laminar cases described in TableB.1 are shown in Figure B.1.0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.0501234567r (m)u θ (m/s)Figure B.1: Numerically predicted tangential velocity profiles assuming lam-inar flow (`T = 0). © Water;  Power-law fluid (n′ = 0.5).The results presented in Figure B.1 indicate that the inertial effectsare dominant over the entire radius for water, whereas the viscous effectsare dominant over the entire radius for the power-law fluid model. Thetangential profile for water was found to increase with decreasing radiustowards r = 0.0025 m. As r → 0, the tangential velocity rapidly dropped tozero in near linear fashion. The power-law fluid model was found to followthat of a boundary layer, which is not surprising considering the viscouseffects are dominant in boundary layer flows. Increasing the constant Cto -1000 mm2/s and defining κ to be 0.05 (i.e. `T = 0.05R) resulted instaggering differences for the water case and the power-law fluid case withn′ = 0.5.112Appendix B. Analytical Study of a Swirling Cross-flow0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05−0.500.511.522.533.5r (m)u θ (m/s)Figure B.2: Numerically predicted tangential velocity profiles assuming tur-bulent flow (κ = 0.05). © Water;  Power-law fluid (n′ = 0.5).Figure B.2 shows that the boundary layer for the water case developedmuch further away from the axis than what was predicted for the power-law fluid. It was found the total viscous forces and the inertial forces wereclosely balanced as r → 0, whereas, the molecular and turbulent viscousforces increased within 0.03 ≤ r (m) ≤ 0.05. The power-law fluid case withn′ = 0.5 shows that the total viscous forces (i.e. molecular and turbulentviscous forces) decreased with increasing radius between 0.02 ≤ r (m) ≤ 0.05,whereas, the total viscous forces increased with increasing radius between0.005 ≤ r (m) ≤ 0.02. The results also show a region where uθ(r) < 0, dueto the inertial forces are dominant over the total viscous forces as r → 0.The inertial forces in this discussion are expressed as:ρCω(R)1−n′(∂Ψ∂σ+2Ψσ)(∂Ψ∂σ)1−n′ 0The effect of flow behaviour index, n′, on the resulting tangential velocityprofiles of the swirling cross-flow discussed here are shown in Figure B.3. Itis clearly visible that an increase in flow behaviour index from n′ = 0.25to n′ = 0.75 resulted in a tangential velocity profile much closer to thatnumerically predicted for water alone (see Figure B.2). As r → R, thetotal viscous forces were found to be dominant, hence the development of aboundary layer within 0.03 ≤ r (m) ≤ 0.05.113Appendix B. Analytical Study of a Swirling Cross-flow0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05−0.500.511.522.53r (m)u θ (m/s)Figure B.3: The effect of flow behaviour index on the tangential velocityprofiles of power-law fluids (κ = 0.05). ♦ Power-law fluid (n′ = 0.25); 4Power-law fluid (n′ = 0.75).A decrease in flow behaviour index from n′ = 0.5 to n′ = 0.25 resulted ina larger region where the dominant viscosity term increased with increasingradius. The dominant viscosity term described here is as follows:µ′n′σn′+ ρκ2R2σ2ω(R)2−n′(−∂Ψ∂σ)2−n′As r → R, the magnitude of the inertial forces were found to be greaterthan the viscous forces (i.e. inertial forces are negatively strong), as a de-crease in the tangential velocity was observed with increasing radius within0.035 ≤ r (m) ≤ 0.05. This was similarly observed for n′ = 0.5, however, adecrease in n′ to 0.25 resulted in a reduction in the inertial forces towardsthe outer radius.With these characterizations in mind, a comparison using fundamentalprinciples can be applied to the tangential and radial velocity fields mea-sured experimentally for water and a power-law fluid in a hydrocyclone.This methodology was found to assist in understanding the governing forcesassociated with a portion of the hydrocyclone flow field, as the flow of waterand pseudo-plastic fluids (i.e. shear thinning fluids) are difficult to charac-terize in such a complex flow. A comparison between the analytical solutionsof swirling cross-flows and a portion of the experimental data can be foundin Chapter 4.1.6.114Appendix CNumerical Error of theRadial Velocity ProfilesThe derivative error associated with solving for the radial component ofvelocity (〈ur(r, z)〉) is shown here. The continuity equation was specifi-cally used to calculate the radial component of velocity within the domainexperimentally measured using LDV. The ordinary differential equation isexpressed as:∂ (rur)∂r= −r∂uz∂z(C.1)Equation C.1 was found by assuming axially symmetric flow where r isthe radial position, and z is the axial position. The derivative error wasfound to be best expressed by resolving the continuity equation using asecond order difference approximation with the numerically predicted radialvelocities. The error in Equation C.1 normalized to the inlet flux, namely:Fluxinlet = UinletA, where Uinlet is the inlet velocity and A is the crosssectional area of the inlet pipe, was calculated for each LDV case. Theresults of this error analysis can be found in Figures C.1 – C.5.115Appendix C. Numerical Error of the Radial Velocity ProfilesRadial position (m)Axial position (m)0.50281 1.75980.25140.25140.502810.754210.25140.25142.26260.25140.25140.25140.754211.75982.01125.78232.514 2.26260.502810.25142.76551.5084 1.2573.77112.5142.5141.5084 2.5140.502810 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05−0.45−0.4−0.35−0.3−0.25−0.2−0.15−0.1−0.05Figure C.1: Continuity error for the water case normalized to the inlet fluxusing a second order difference approximation: RR = 0%.Radial position (m)Axial position (m)0.310150.310150.310151.24060.310150.310150.930462.1711 1.86091.24060.93046 3.72192.17115.27261.55081.55082.79144.0322.79141.55083.10150.310150.310151.55080 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05−0.45−0.4−0.35−0.3−0.25−0.2−0.15−0.1−0.05Figure C.2: Continuity error for the water case normalized to the inlet fluxusing a second order difference approximation: RR = 25%.116Appendix C. Numerical Error of the Radial Velocity ProfilesRadial position (m)Axial position (m)0.206580.206580.826310.826311.23952.27241.23952.47890.413160.619731.4462.27242.89212.27241.03291.23952.0658 1.4465.1645 1.03290.206580.206580 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05−0.45−0.4−0.35−0.3−0.25−0.2−0.15−0.1−0.05Figure C.3: Continuity error for the water case normalized to the inlet fluxusing a second order difference approximation: RR = 50%.Radial position (m)Axial position (m) 0.390060.130020.39006 1.43020.390060.130020.390060.65010.78012 1.69031.43020.260041.56020.65010.780122.21032.08030.39006 2.08030.130020 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05−0.45−0.4−0.35−0.3−0.25−0.2−0.15Figure C.4: Continuity error for the 0.03% APAM solution normalized tothe inlet flux using a second order difference approximation: RR = 25%.117Appendix C. Numerical Error of the Radial Velocity ProfilesRadial position (m)Axial position (m) 0.308170.102720.924510.513620.924511.02720.205450.308170.102720.616340.513620.616341.23271.02720.41091.43810.41092.36260.71907 0.616340.616340.513620 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05−0.4−0.3−0.2−0.1Figure C.5: Continuity error for the 0.03% APAM solution normalized tothe inlet flux using a second order difference approximation: RR = 50%.118Appendix DPressure Drop RelationshipsThe pressure drop relationships for water and the 0.03% APAM solutionwere numerically calculated using the velocity measurements acquired withLDV. Ignoring the effects of viscosity, the simplified radial component of theNavier-Stokes Equations explicitly shows the relationship for radial pressuredrop, shown as:ρ(ur∂ur∂r+ uz∂uz∂z−u2θr)= −∂P∂r(D.1)where P is the pressure in Pascals, uz is the axial velocity measuredusing LDV, uθ is the tangential velocity measured using LDV, and ur is theradial velocity solved for by applying the continuity equation. Figures D.1 –D.5 present the radial pressure drop contours solved by Equation D.1 withinthe domain experimentally measured with LDV.119Appendix D. Pressure Drop RelationshipsFigure D.1: Pressure drop relationship for water calculated using LDV re-sults. Uinlet = 2.57 m/s, RR = 0%.Figure D.2: Pressure drop relationship for water calculated using LDV re-sults. Uinlet = 2.57 m/s, RR = 25%.120Appendix D. Pressure Drop RelationshipsFigure D.3: Pressure drop relationship for water calculated using LDV re-sults. Uinlet = 2.57 m/s, RR = 50%.Figure D.4: Pressure drop relationship for APAM calculated using LDVresults. Uinlet = 2.57 m/s, RR = 25%.121Appendix D. Pressure Drop RelationshipsFigure D.5: Pressure drop relationship for APAM calculated using LDVresults. Uinlet = 2.57 m/s, RR = 50%.122Appendix EA Discussion on the PTVSeeding Particle StudyThis section of work focuses on establishing a set of possible explanationsfor the velocity discrepancies observed between the LDV and PTV methods.The results shown in Chapter 5 indicate that the concentration and size ofthe seeding particles used for the PTV analysis resulted in significant particleslip, as the velocity profiles did not match to those measured from LDV atall axial locations.This analysis will start by first introducing the phase continuity equationfor phases of arbitrary density (see [46]), shown as:∂ (αkρk)∂t+∇ · (αkρkuk) = Γk (E.1)where αk is the volume fraction of phase k, ρk is the density of phasek, uk is the velocity of phase k, and Γk is the rate of mass generation ofphase k at the interface. From Equation E.1, the continuity equation forthe mixture can be obtained by summing over all phases. This is shownexplicitly below.∂∂tn∑k=1(αkρk) +∇ ·n∑k=1(αkρkuk) =n∑k=1Γk (E.2)Since the total mass has to be conserved, the right hand side of EquationE.2 has to be zero. As such, the continuity equation of the mixture is shownas:∂ρm∂t+∇ · (ρmum) = 0 (E.3)The mixture density and velocity shown in Equation E.3 are defined as:123Appendix E. A Discussion on the PTV Seeding Particle Studyρm =n∑k=1αkρkum =1ρmn∑k=1αkρkukIf the phase densities are equal, it can be seen from Equation E.3 thatthe continuity equation for the mixture is defined as:∇ ·n∑k=1(αkuk) = 0 (E.4)This indicates that the volumetric flux of the mixture must be conserved.With this relationship, an alternative formulation of the phase continuityequation can be derived by starting with the assumption that the phasesare equal in density. This is of key importance for the PTV seeding particlestudy, as the particulate phase density was equal to the continuous phasedensity. The phase continuity equation for phases of equal density can bewritten as:∂αk∂t+∇ · jk = 0 (E.5)where jk is the volumetric flux of phase k, defined as: jk = αkuk. Thedrift velocity of a dispersed phase k relative to the volume centre of themixture, of equal density, is defined as:uvk = uk −n∑k=1αkuk = uk − jm (E.6)where jm is the mixture volumetric flux. From Equation E.4, it is clearthat the drift flux must be conserved. This is explicitly shown below.n∑k=1αkuvk = 0 (E.7)Substituting Equation E.6 into Equation E.5 yields the following conti-nuity equation of phase k in terms of the drift velocity.124Appendix E. A Discussion on the PTV Seeding Particle Study∂∂tαk + jm · ∇αk = −∇ · (αkuvk) (E.8)In the presence of a continuous phase, the drift velocity can be defined inrelation to the relative velocity, or slip velocity. Specific to the PTV analysis,only one particulate phase was present. As such, for this discussion the driftvelocity of the particulate phase, p, can be defined as:uvp = (1− αp)ucp (E.9)where ucp is the relative velocity between the continuous phase and theparticulate phase. The continuity equation can then be written as follows.∂∂tαp + jm · ∇αp = −αp (1− αp)∇ · ucp − (1− 2αp)ucp · ∇αp (E.10)Equation E.10 indicates that if the volume fraction of the particulatephase, αp, varies in time and space then by mass conservation, the relativevelocity ucp varies in time and space. If indeed the relative velocity betweenthe particulate phase and continuous phase varies in time, then additionalforces such as added mass force and Basset are likely non-zero. This is shownexplicitly when considering the total drag force acting on particle due to thevelocity relative to the fluid (see [11]), described as:FD = −12ApρpCD|ucp|ucp −12Vpρc∂ucp∂t− 6r2p√piρcµct∫0∂ucp∂s√t− s∂s (E.11)The first, second, and third terms in Equation E.11 are the viscous drag,added mass, and Basset force terms, respectively. To suggest that the neu-trally buoyant particles used for the PTV analysis were slipping due toparticle accumulation in time and space is difficult for this study, however,is not unlikely. Further investigations are required to assess seeding particlevolume fractions throughout a hydrocyclone, as the results shown in Chapter5 indicate that particle slip is considerable for very dilute suspensions. Theeffect of particle diameter, position, and proximity to surrounding particleswithin a hydrocyclone are likely of key importance, as the hydrocyclone flowfield was found to vary with axial and radial position.125Appendix FExamples of Linear LostEnergy RelationshipsThe figures shown here present the linearity of lost energy versus the squareof the inlet velocity for various reject ratios. It is clearly shown that thelost energy constant for both water, and APAM is not dependent on inletvelocity, as the change in lost energy is linear with U2inlet.0 1 2 3 4 5 6020406080100120140160180200Uinlet2  (m2/s2)Lost Energy (m2/s2 )Figure F.1: Lost energy versus U2inlet, RR = 0%. © water;  0.03% APAM.126Appendix F. Examples of Linear Lost Energy Relationships0 1 2 3 4 5 6 7020406080100120140160180Uinlet2  (m2/s2)Lost Energy (m2/s2 )Figure F.2: Lost energy versus U2inlet, RR = 25%. © water;  0.03% APAM.0 0.5 1 1.5 2 2.5 3 3.5020406080100120140160180Uinlet2  (m2/s2)Lost Energy (m2/s2 )Figure F.3: Lost energy versus U2inlet, RR = 50%. © water;  0.03% APAM.127Appendix GEconomic Evaluation ofEnergy SavingsThis section focuses on evaluating the applicability of using drag reducingpolymer agents as an economically viable option to reduce pumping costsfor hydrocyclone units in industry. The cost of polyacrylamide was assumedto be $ 2200/tonne, whereas the electrical costs were assumed to followBC Hydro’s large general service rates. The large general service rate wasadopted for this analysis as most pulp and paper mills consume greater than550,000 kWh. The large general service rates which were included in thiseconomic evaluation are summarized below.Cost Usage limit$ 0.00 per kW for first 35 kW$ 5.19 per kW for next 115 kW$ 9.95 per kW for remaining kW$ 0.1010 per kWh for first 14,800 kWh$ 0.0486 for remaining kWhTable G.1: BC hydro’s monthly demand charge: Large general service con-servation rate.To effectively determine the economic gain, the cost to operate a 15 kWmotor running a pump at a capacity of 2.6 m3/hr with water, a 0.7% SPFsuspension, and a 0.7% SPF suspension containing 100 ppm of APAM wascalculated using Equation G.1. The pump efficiency and motor efficiencywere assumed to be 100%.Cost/hr =CostkWh × Pump powerPump eff×Motor eff(G.1)The total pumping powers required to operate the experimental hydro-cyclone at a capacity of 2.6 m3/hr are shown in Table G.2. It was found128Appendix G. Economic Evaluation of Energy Savingsthat the pumping power reduced by nearly 20% with the addition of 100ppm APAM in a 0.7% SPF suspension.To incorporate the cost of polymer into this analysis, the price per cubicmeter was determined, shown explicitly in Equation G.2.Cost(polymer) = $2200(1tonne)1000−1(tonnekg)0.1(kgm3)= $0.22 /m3(G.2)Fluid Pump power (kW)Water 12.50.7% SPF 11.20.7% SPF + 100 ppm APAM 9Table G.2: Electrical energy consumption rates for water and pulp mixtures.The total operation costs for water, a 0.7% SPF suspension, and a 0.7%SPF suspension containing 100 ppm APAM are shown in Table G.3. It canbe seen that when operating the hydrocyclone with 100 ppm APAM at acost of $ 0.22/m3, the total cost is greater than with water or fibre alone.For the addition of polymer to be a financially viable energy savings option,the concentration of polymer in a 0.7% SPF suspension would have to bereduced to approximately 40 ppm. This assumes that the cost of polymeris fixed at $ 2200/tonne, and the hydrocyclone operates for each case at 2.6m3/hr.Fluid Cost ($/hr)Water 1.30.7% SPF 1.10.7% SPF + 100 ppm APAM 1.5Table G.3: Total operation cost.129

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