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UBC Theses and Dissertations

Experimental investigation of a model forming fabric Gilchrist, Seth 2006

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Experimental Investigation of a Model Forming Fabric by Seth Gilchrist B.Sc., The University of Wyoming, 2003 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF Master of Applied Science in The Faculty of Graduate Studies (Mechanical Engineering)  The University Of British Columbia September, 2006 c Seth Gilchrist 2006  Abstract Paper making involves three fabrics: forming, pressing, and drying. The forming fabric is responsible for sheet forming, the initial dewatering of a low concentration pulp suspension into a wet sheet of paper. In the process of forming, topographical and hydrodynamic marks can be transferred from the drainage media (the forming fabric) to the sheet produced. An experimental investigation of a model forming fabric was performed to identify the geometric parameters having the largest influence on hydrodynamic wire mark. The data were also compared with the numerical simulations of Huang. To simplify the problem, justifiable engineering simplifications were made. The second phase (the fibres) was removed and the machine-direction filaments were neglected. This reduced the problem to investigation of flow through a bank of dissimilar cylinders. It was desired to find the most important geometrical parameter to reduce flow non-uniformity in the paper side flow field. Particle image velocimetry, pressure drop and flow visualization tests were conducted to investigate the flow through the array of cylinders. It was found that with a cylinder surface separation of 0.75× the paper side cylinder diameter the pressure drop tended toward the sum of the rows, and the paper side flow field was nearly identical to the paper side row only flow field, regardless of the backing side cylinder dimensions and configuration. It was seen that when the pressure drop through the bank of cylinders was equal to the sum of the rows’ pressure drops the paper side flow field was the same as the paper side row only flow field. As such, pressure drop can act as an indication of when the machine side row will not affect the paper side flow field.  ii  Table of Contents Abstract  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  ii  Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  iii  List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  v  Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  viii  Statement of Co-Authorship . . . . . . . . . . . . . . . . . . . . . . . . .  ix  Nomenclature  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  x  1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  1  Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  4  2 Motivation . . . . . . . . . . . . . . . . . . . . . 2.1 Fabric Basics . . . . . . . . . . . . . . . . . 2.2 Development of the Modern Forming Fabric 2.3 The Problem of Wire Mark . . . . . . . . .  . . . .  5 5 7 9  Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  11  3 Project Goals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  12  4 Literature Review . . . . . . . . . 4.1 Introduction . . . . . . . . . . . 4.2 Previous Research . . . . . . . . 4.2.1 Investigating the Fabrics 4.2.2 Investigating the Sheet .  . . . . .  13 13 13 13 16  Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  21  5 Manuscript . . . . . . . . . . . 5.1 Introduction . . . . . . . . . 5.2 Literature Review . . . . . . 5.3 Experimental Methods . . . 5.4 PIV Results and Discussion 5.5 ∆P Results and Discussion .  22 22 23 24 27 32  . . . . . .  . . . . . .  . . . . .  . . . . . . iii  . . . . .  . . . . . .  . . . . .  . . . . . .  . . . . .  . . . . . .  . . . . .  . . . . . .  . . . . .  . . . . . .  . . . . .  . . . . . .  . . . .  . . . . .  . . . . . .  . . . .  . . . . .  . . . . . .  . . . .  . . . . .  . . . . . .  . . . .  . . . . .  . . . . . .  . . . .  . . . . .  . . . . . .  . . . .  . . . . .  . . . . . .  . . . .  . . . . .  . . . . . .  . . . .  . . . . .  . . . . . .  . . . .  . . . . .  . . . . . .  . . . .  . . . . .  . . . . . .  . . . .  . . . . .  . . . . . .  . . . .  . . . . .  . . . . . .  . . . .  . . . . .  . . . . . .  . . . . . .  5.6 5.7 5.8  H2 Bubble Visualization Results and Discussion . . . . . . . . . . . . Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . .  35 37 38  Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  39  6 Discussion . . . . . . . . . . . . . 6.1 Introduction . . . . . . . . . . 6.2 Forming Fabric Application . 6.3 Validation of Huang’s Results  . . . .  41 41 41 42  Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  48  7 Recommendations for Further Research . . . . . . . . . . . . . . . .  49  Appendix A Experimental Design . . . . . . . . . . . . . . . . . . . . . . A Appendix . . . .  51 51 51 52 55  Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  56  Appendix B Experimental Setup and Methods B Appendix B.1 Introduction . . . . . . . . . . . . . . . . . . . B.2 Experimental Setup . . . . . . . . . . . . . . . B.3 Test Section Flow Characteristics . . . . . . . B.3.1 Test Section Sensor Details . . . . . . . B.4 PIV Setup . . . . . . . . . . . . . . . . . . . . B.4.1 PIV Equipment Details . . . . . . . . . B.5 Pressure Drop Setup . . . . . . . . . . . . . . B.5.1 Pressure Drop Equipment Details . . . B.6 Hydrogen Bubble Generation System Setup .  . . . . . . . . . .  57 57 57 59 60 65 66 67 67 68 68  Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  70  C Appendix Appendix C Uncertainty Analysis . . . . . . . . . . . . . . . . . . . . . C.1 Single Valued, Multi-Variable Functions . . . . . . . . . . . . . . . . C.2 PIV uncertainty . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  71 71 76  A.1 A.2 A.3 A.4  . . . .  . . . .  . . . .  . . . .  Introduction . . . . . . . . . . . . . . Phase Simplification . . . . . . . . . Geometry Simplification . . . . . . . Flow Velocity and Reynolds number .  iv  . . . .  . . . .  . . . .  . . . .  . . . .  . . . .  . . . .  . . . .  . . . .  . . . .  . . . .  . . . .  . . . . . . . . . .  . . . .  . . . .  . . . . . . . . . .  . . . .  . . . .  . . . . . . . . . .  . . . .  . . . .  . . . . . . . . . .  . . . .  . . . .  . . . . . . . . . .  . . . .  . . . .  . . . . . . . . . .  . . . .  . . . .  . . . . . . . . . .  . . . .  . . . .  . . . . . . . . . .  . . . .  . . . .  . . . . . . . . . .  . . . .  . . . .  . . . . . . . . . .  . . . .  . . . .  . . . . . . . . . .  . . . .  . . . .  . . . . . . . . . .  List of Figures 1.1 1.2 1.3  The forming section of a fourdrinier machine. . . . . . . . . . . . . . Forming section of a twin wire former. . . . . . . . . . . . . . . . . . Parts of a fine paper machine. . . . . . . . . . . . . . . . . . . . . . .  2 2 3  2.1 2.2 2.3 2.4 2.5  Left to right: Single-layer, double-layer, and triple-layer fabrics. . . Cross sections of six popular fabric geometries . . . . . . . . . . . . Forming fabric coordinate system. . . . . . . . . . . . . . . . . . . . Shed definition for single-layer, double-layer and triple-layer fabrics. Topographical wire mark. . . . . . . . . . . . . . . . . . . . . . . .  5 6 7 8 9  4.1  Helle’s results of formation with flow orientation parallel and perpendicular to the long knuckles. . . . . . . . . . . . . . . . . . . . . . . . A comparison of drainage times for the two long knuckle orientations. Forming wire with formed sheet, ground at an incline to show webpenetration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Linear FFT of a sheet of paper. . . . . . . . . . . . . . . . . . . . . . Results of Danby’s multiple sheet split tests for twin-wire SC paper. .  4.2 4.3 4.4 4.5 5.1 5.2 5.3 5.4  . . . . .  Flow loop test section. . . . . . . . . . . . . . . . . . . . . . . . . . . Schematic of test section flow loop. . . . . . . . . . . . . . . . . . . . Schematic of the simplified forming fabric geometry . . . . . . . . . . Stream wise velocity variation for changing XS /d for D/d = 1.5, G2/G1 = 1 and Y/G1 = 0.5 at Re = 65. . . . . . . . . . . . . . . . . . . . . 5.5 Cross stream velocity variation for changing XS /d for D/d = 1.5, G2/G1 = 1 and Y/G1 = 0.5 at Re = 65. . . . . . . . . . . . . . . . 5.6 Stream wise velocity variation for changing Y/G1 for D/d = 1.5, G2/G1 = 1, XS /d = 0.75 at Re = 65. . . . . . . . . . . . . . . . . . 5.7 Cross stream velocity variation for changing Y/G1 for D/d = 1.5, G2/G1 = 1, XS /d = 0.75 at Re = 65. . . . . . . . . . . . . . . . . . 5.8 Stream wise velocity component for changing staggering with D/d = 3, G2/G1 = 2, and XS /d = 0 at Re = 25. . . . . . . . . . . . . . . . 5.9 Pressure drop through a bank of cylinders with D/d = 1.5, G2/G1 = 1 and Y/G1 = 0.5 at Re = 65 with changing XS /d. . . . . . . . . . . 5.10 Pressure drop through a bank of cylinders with D/d = 1.5, G2/G1 = 1 and Y/G1 = 0 at Re = 65 with changing XS /d. . . . . . . . . . . . 5.11 Pressure drop through a bank of cylinders with D/d = 1.5, G2/G1 = 1, and XS /d = 0.75 at Re = 65 with changing Y/G1. . . . . . . . . .  v  14 14 16 17 19 25 26 26 27 28 29 30 31 32 33 34  5.12 Pressure drop through a bank of cylinders with D/d = 1.5, G2/G1 = 2, Y/G1 = 0.5 at Re = 65 and changing XS /d. . . . . . . . . . . . . 5.13 Comparison of pressure drop as a function of Re through cylinders with D/d = 1.5, G2/G1 = 1, XS /d = 0.75 and two staggerings. . . . 5.14 Hydrogen bubble visualizations for variations of XS /d with D/d = 1.5, G2/G1 = 2, Y/G1 = 0.5 at Re = 65. . . . . . . . . . . . . . . . . . . 5.15 Relative velocities for variations of XS /d with D/d = 1.5, G2/G1 = 2, Y/G1 = 0.5 at Re = 65. . . . . . . . . . . . . . . . . . . . . . . . 6.1  6.2  6.3 6.4 7.1  Huang’s computational stream wise velocity results plotted with the PIV results for D/d = 1.5, G2/G1 = 1, XS /d = 0.75, g1 = 2 and Y/G1 = 0.5 at Re = 65. . . . . . . . . . . . . . . . . . . . . . . . . . Huang’s computational cross stream velocity results plotted with the PIV results for D/d = 1.5, G2/G1 = 1, XS /d = 0.75, and Y/G1 = 0.5 at Re = 65. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Comparison of PIV and computational stream wise velocities for a case of D/d = 3, G2/G1 = 2, XS /d = 0, Y/G1 = 0.5 at Re = 25 . . . . . Comparison of PIV and computational cross stream velocities for a case of D/d = 3, G2/G1 = 2, XS /d = 0, Y/G1 = 0.5 at Re = 25 . . Method for determining the internal 3D geometry of a forming fabric.  A.1 The support of CD filaments . . . . . . . . . A.2 Cross section of a forming fabric showing the the CD face. . . . . . . . . . . . . . . . . . . A.3 The resulting 2D bank of cylinders obtained cations. . . . . . . . . . . . . . . . . . . . . B.1 B.2 B.3 B.4 B.5 B.6 B.7 B.8 B.9 B.10 B.11 B.12 B.13  . . . . . . . . . . . . . . CD filaments contacting . . . . . . . . . . . . . . from the above simplifi. . . . . . . . . . . . . .  Cylinder frame equipt with 9.53mm and 19mm diameter cylinders. . . Locations of the pressure and hydrogen bubble ports . . . . . . . . . The test section fully installed. Flow is from left to right . . . . . . . Schematic of the Test Section Flow Loop . . . . . . . . . . . . . . . . Image of test section flow at 2cm/s. Flow is from bottom to top of image. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Image of test section flow at 4cm/s. Flow is from bottom to top of image. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Data extracted from Figs. B.5 and B.6 . . . . . . . . . . . . . . . . . Normalized test section velocity with a 135.3mm field of view. . . . . Empty test section velocity vectors at location of PIV data collection. Stopwatch measured test section velocity for comparison with PIV. . The interrogation areas from the two masked PIV images with a known time separation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The two interrogation areas are then compared by cross-correlation to find the average particle displacement and hence velocity. . . . . . . . PIV arrangement. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  vi  34 35 36 36  42  43 45 46 49 53 54 54 58 59 59 60 61 62 63 63 64 65 66 67 68  C.1 Screen shot of the spread sheet used to calculate uncertainty. . . . . . C.2 Screen shot of the spread sheet used to find the venturi constant and uncertainty. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  vii  74 76  Acknowledgements I would like to thank Sheldon Green, Ali Vakil, Dan Dressler, and Barb Murry from the University of British Columbia’s department of Mechanical Engineering for their professions and personal help throughout the course of this project. I would also like to thank Dale Johnson, Roger Danby, Graham Jackson and the rest of the team at Asten Johnson for continued financial and material support.  viii  Statement of Co-Authorship Seth Gilchrist conducted all the experiments and analysis described in this thesis. My role in the thesis was primarily supervisory; most of the intellectual content of the thesis is Seth’s. Dr. Sheldon I. Green, P.Eng.  ix  Nomenclature CD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cross machine direction d . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Paper side cylinder diameter D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Machine side cylinder diameter D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Inlet pipe diameter (§C.1) D1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Venturi inlet diameter D2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Venturi throat diameter ∆P . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pressure drop G1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Paper side row centre spacing G2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Machine side row centre spacing H . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Test section height k . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Venturi constant k . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Viscosimeter constant m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mass M D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Machine direction µ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Viscosity Re . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Reynolds number ρ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Density u . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x-direction velocity U0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Upstream flow velocity UT S . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Test section velocity Upipe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Inlet pipe velocity v . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . y-direction velocity V . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Volume W . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Test section width XC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cylinder row centre separation XS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cylinder row surface separation y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Horizontal Position Y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Row staggering z . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Fabric normal  x  Chapter 1 Introduction Of the common commodities in the modern world, there are few that have as great an impact on our daily lives as paper. Paper, in its various forms, has been used for thousands of years to record and disseminate information. In modern times the uses of paper have expanded greatly into medicine, packaging, manufacturing, and industrial production. The continual development of paper and papermaking techniques has advanced to the point that it requires advanced scientific and engineering methods.  To be classified as paper a product must be produced using particular means. In its purest form paper is made from a fibrous material that has been separated into individual fibres. These fibres are then suspended in water and a screen, or other drainage medium, is used to dewater the suspension, leaving a thin sheet of material. Paper has been made this way for nearly two thousand years, since the initial development by Ts’ai Lun in A.D. 105[2]. Today the process is essentially the same. The first, and still most common, paper machines are called fourdriniers. They are named after Henry Fourdriner who filed the first patent for them in England in 1806[2]. On these machines the pulp/water mixture is distributed into a long, thin jet in the headbox and is then sprayed onto a moving fabric. The forming section of a fourdrinier machine is shown in Fig. 1.1. In this image the water/pulp solution is coming out of the headbox in the background of the image and is being filtered through the forming fabric, the white, woven structure visible in the fore- and mid-ground. As the water is drained off, a mat of pulp and additives is formed on the forming fabric. In an effort to reduce the one-sided nature of paper produced on fourdrinier machines the twin-wire former was developed. On these machines the pulp suspension is injected into a gap between two forming fabrics. The forming section of a twin wire former is shown in Fig. 1.2. Regardless of the type of forming section used, the end result is a mat of pulp fibres and additives. This mat is then transfered into a mechanical pressing section for additional dewatering and finally to a heated dryer section where water is removed by heat and air. Once the page has been 1  Chapter 1. Introduction  Figure 1.1: The forming section of a fourdrinier machine.[1].  Figure 1.2: Forming section of a twin wire former[3].  2  Chapter 1. Introduction  Figure 1.3: Parts of a fine paper machine[4]. dried it can be further processed by calendering, coating or other special treatments. A schematic of a full twin wire paper machine is shown if Fig. 1.3.  3  Bibliography [1] Papermachine service industries http://www.papermachine.com/photos/deckles/d8.jpg , March 2006. [2] Dard Hunter. Papermaking: The History and Technique of an Ancient Craft. Dover Publications, Inc., New York, New York, 1974. [3] James Olson. University of British Columbia, Vancouver, BC CHBE-401 Class Notes: Papermaking Papermachine - Forming, Mar 11, 2005. [4] Hannu Paulapuro. Papermaking Part 1, Stock Preparation and Wet End. Fapet Oy, Helsinki, Finland, 2000.  4  Chapter 2 Motivation 2.1  Fabric Basics  It can be argued that of the three fabrics used in papermaking – forming, pressing and drying – the forming fabric is the most important. Once the sheet has been formed there is little that can be done in later sections of the papermaking machine to correct errors in properties like fibre alignment, sheet strength, formation, wire mark, surface feel and optical quality. In addition to having a significant influence on these physical sheet properties, the forming fabric heavily influences the efficiency of the papermaking process. Efficiency in papermaking is a very broad term, it applies not only to the energetic performance but also to the retention of fibres, fines and chemical additives. The process of depositing pulp fibres onto a forming fabric in an industrial setting happens at very high speed, often in the rage of 100km/hr or nearly 5500ft/min, so even small changes in the energy required to move the fabric can have a large affect on operating cost. In regards to retention, the fabric’s capability of retaining fibres, fines and additives in the paper mat while allowing for high drainage rates helps to reduce machine size, and allows smaller machines to run faster. When the pulp suspension is deposited onto the forming wire, fines and additives, which are often smaller than the openings in the fabric structure, must be retained, while water quickly and evenly drains from the paper surface through the forming fabric.  Figure 2.1: Left to right: Single-layer, double-layer, and triple-layer fabrics[5].  5  Chapter 2. Motivation  Figure 2.2: Cross sections of six popular fabric geometries[9]. With all of these factors in mind, engineers and technologists have been working for many years to develop forming fabrics that have a low running resistance and and good pulp and fines retention. Many designs have been conceived over the years and a few of the more common types are show in Fig. 2.1. The cross sections for these types of fabrics are shown in Fig. 2.2.  Before entering a discussion of development of fabric geometries, one must become familiar with the common coordinate system and other defining terms. The coordinates of a forming fabric are generally defined by its alignment on the papermachine. There are two principal directions, the machine direction (MD) and the cross-machine direction (CMD or CD). As the names imply, MD is oriented with the unit vector that points in the direction of fabric movement. CD lies in the plane of the fabric, perpendicular to MD. This is shown graphically in Fig. 2.3. The definition of a forming fabric as woven structure that retains solids while allowing liquids to pass through is very broad. As can be seen in Fig. 2.2 there are many cross-sectional variations of this definition. There are also a number of different weaves that are possible in the perpendicular plane, that is, the fabric plane. The geometry in this plane is commonly 6  Chapter 2. Motivation  $            Figure 2.3: Forming fabric coordinate system[5]. referred to as the fabric shed. The shed defines the number of cross machine direction filaments before the weave pattern is repeated. In the simplest case of a forming fabric that has only one layer, the shed is given as a single number. As the complexity of the fabric increases the shed definition must account for the two sided nature of the forming fabric. For this reason, multi-layered fabrics are defined using two numbers, the first for the paper side and the second for the machine side. Shed patterns for single, double and triple-layer fabrics are shown in Fig. 2.4. Another commonly used term in forming fabric design is frame length. This term refers to the physical length of a particular opening in a fabric structure. For example, it could be said that the CD frame length on the wear side of the triple-layer fabric shown in Fig. 2.4 is longer than the CD frame length on the paper side.  2.2  Development of the Modern Forming Fabric  The technological complexity of forming fabrics has gone through a significant maturation in the last 50 years. Initially, paper was formed on a bronze wire mesh similar to a heavy duty screen. For this reason, many still refer to the woven structure as the “forming wire”, even though they have not been made of wire for many years. In 1951 a papermachine in East Germany used the first synthetic forming fabric[8]. This fabric, made 7  Chapter 2. Motivation  Figure 2.4: Shed definition for single layer, double layer and triple layer fabrics[9].  8  Chapter 2. Motivation of the new material polyamide, provided unsatisfactory results. However, this initiative started a period of development that led to the design of the myriad of filaments and geometries that are shown in the previous figures.  Even with the development of so many complex and effective fabrics, the double-layer remains the most common type due to familiarity, wear issues, and cost. For these reasons, the double-layer fabric has gone through a number of improvements, one of the major improvements was the introduction of the extra weft added design. All of these designs are shown in Fig. 2.2.  2.3  The Problem of Wire Mark  Wire mark is the imprint or image of the forming fabric that is left on the finished paper. Wire mark is not desirable due to a number of problems it can cause in production and application. During production, wire marking can cause release problems of the pulp mat from the forming fabric; in application, wire mark is associated with printing problems and undesirable optical properties.  Figure 2.5: Topographical wire mark[6]. Wire mark can be broken down into two categories, topographical and hydrodynamic. Topographical wire mark is the physical imprint of the forming fabric on the paper. This is common on papers that are made on fabrics with a high plane difference, such as tissue papers. A closeup of topographical wire marking on a sheet of paper is shown in Fig. 2.5. The second general category, hydrodynamic wire mark (also called shadow mark ), is created by the fluid mechanics of the flow through the forming 9  Chapter 2. Motivation fabric. Under the right conditions, flow non-uniformities can be transmitted significant distances upstream and can have an affect on the paper making surface of the fabric. These affects manifest themselves as changes in paper density and can change the printing characteristics of the paper.  Hydrodynamic wire mark is generally less visible than topographical wire mark. It tends to show up only on printed surfaces and is therefore only a concern for finer paper grades. In the manufacturing of finer paper grades, a tighter weave is used on the forming surface of the fabric, therefore the length scale of the density changes is small and consequently less obvious to the casual observer. However, the results can be seen after printing with undesirable characteristics such as ink strike-through (ability to see the ink on the opposite side of the paper) and patterns in fully inked regions with characteristics that make them highly visible to the human eye[7]. The research in this paper is primarily focused on the mechanism of hydrodynamic wire marking. Specifically, the hydrodynamic wire marking of double and triple-layer fabrics. If a section of a multi-layer fabric is taken in the z-MD plane there are essentially two layers of CD filaments that will be seen, the small diameter paper side filaments and the large diameter machine side filaments. Based on the relative diameter, spacing and separation of these filaments the upstream flow will be modified, possibly redistributing the pulp fibres, causing hydrodynamic wire mark. The goal of this research is to gain an insight into the geometrical configurations with the highest probability of significant wire mark.  10  Bibliography [5] Sabit Adanur. Paper Machine Clothing. Asten, Inc, Basel, Switzerland, 1997. [6] R. Danby. The impact of multilayer fabrics on sheet formation and wire mark. Pulp & Paper Canada, 87(8):69–74, Aug 1986. [7] T. Helle. Analysis of wire mark in printing paper. Journal of Pulp and Paper Science, 14(4):J91–J94, Jul 1988. [8] T. Helle. Paper forming wires over 75 years. Pulp & Paper Canada, 91(6):107–114, Jun 1990. [9] Hannu Paulapuro. Papermaking Part 1, Stock Preparation and Wet End. Fapet Oy, Helsinki, Finland, 2000.  11  Chapter 3 Project Goals As has been discussed in the previous sections, the characteristics of the final sheet of paper are very susceptible to manufacturing conditions. The conditions at the initial stage of paper formation are determined largely by the pulp stock, headbox flow, and forming fabric geometry and speed. Characterization of the affects of the forming fabric geometry are very difficult and in the past have largely relied on experience. During sheet formation, water is drained through the matrix of the forming fabric. The geometry of the filaments in the fabric influence the path and local drainage velocity. Water will flow more readily in locations with large gaps or less obstruction. This increased flow volume directs the flow of pulp and fines, increasing the mass of material in the flow path, possibly creating a hydrodynamic wire mark. In double and triple-layered fabrics there are essentially two layers of CD filaments. The paper side filaments are small diameter in order to provide support for the pulp fibres, and the machine side filaments are large diameter to reduce the running load and increase the wear life of the fabric. Depending on the relative diameters, horizontal filament spacings, vertical layer separations and layer off-sets, the fabric may have a larger propensity to hydrodynamically mark the sheet of paper. The purpose of the current research is to use experimental means to characterize and identify how the geometry of two rows of dissimilar cylinders influence the paper side pattern. This will then be applied to forming fabric filament geometries in order to find configurations that could influence the paper making surface of the forming fabric. The end goal is to produce a body of information that can be applied during the design of future forming fabrics to produce a finer, more predictable sheet of paper.  Additionally, the results of the experiments will be used to validate the work of Zhaolin Huang who performed the computational version of this study for his masters work at the University of British Columbia in 2003.  12  Chapter 4 Literature Review 4.1  Introduction  Although forming fabrics are possibly the most important cloth on the papermachine, the design of a suitable geometry for a given paper grade and machine has been left largely to experience and intuition. There has been much research into a suitable method to design and chose a fabric using a quantitative scale like the Beran FSI or Johnson DI (discussed in more detail in §4.2.1). However, these methods are based only on the paper side of the fabric and rely on gross properties like fabric air permeability and weave density. It is only recently that the tools have advanced to a degree that engineers can make quantitative measurements of the flow field through a model forming fabric. The previous research in this field can be roughly broken down into two sections, the investigation of the fabric itself, and the investigation of the sheet produced. They happened roughly in that order. While investigations of fibre orientation in the final sheet[16] were conducted before detailed investigations of the fabric-paper relationship, it wasn’t until the development of optical FFT methods in the mid 1980’s that engineers could evaluate the post production sheet for density variations on the order of a typical forming fabric frame length.  4.2 4.2.1  Previous Research Investigating the Fabrics  The earliest work done on the effects of the micro-scale geometry of the forming fabric was published by Torbjorn Helle in 1978[17]. In this paper, Helle investigated the affect of the orientation of the long knuckles on the paper side of the fabric. There are two possible orientations, MD and CD. Helle modified a hand-sheet former to induce a flow similar to that which is found in the formation region. He then formed extremely light weight paper with the long knuckles both MD and CD. He timed the drainage and then used a scanning electron microscope to view the final sheet of paper. His results are shown in Figs. 4.1 and 4.2.  13  Chapter 4. Literature Review  Flow Direction  Flow Direction  Figure 4.1: Helle’s results of formation with flow orientation parallel and perpendicular to the long knuckles. Low weight paper is shown on the left with a higher weight paper on the right[17].  Figure 4.2: A comparison of drainage times for the two long knuckle orientations[17].  14  Chapter 4. Literature Review As is visible in both figures, the orientation of the paper making surface filaments in a forming fabric were found to be extremely important to the drainage speed and paper quality. This paper was followed a year later by Robert Beran’s milestone paper, The evaluation and selection of forming fabrics[11]. In this paper, Beran coins one of the most influential terms in the world of forming fabric design and selection, the fibre support index (FSI). FSI is a quantitative way of evaluating the performance of a particular forming fabric on a particular papermachine. It is a function of the forming fabric filament geometry, and the statistical distribution of fibre orientations and lengths. The first mention of the possibility of hydrodynamic wire marking was put forward by Helle in his 1980 paper on the influence of the forming fabric structure on the final sheet[18]. In this paper Helle states, It is frequently claimed that there also is a ‘drainage wire mark’; that is a fines and fillers distribution pattern in the surface layer of the wireside of the paper, reflecting the strand knuckle pattern...There seems, however, not as yet to be real experimental evidence for this, in spite of several attempts to prove it. The paper goes on to corroborate the conclusions of Beran’s 1979 paper on FSI. He justifies the importance of the CD filaments using a simple beambending model and discusses the results in the context of the preferred MD alignment of the pulp fibres. Using a simple beam bending analysis of individual pulp fibres, and data on the consistency of the paper web at the wet line, Helle predicts a possible basis weight increase of 25% in locations above drainage passeges. The extent of the penetration of the paper web into the forming fabric structure is qualitatively displayed using a techneque where a sheet of paper and forming fabric were potted and ground at an angle to visualize different depths into the structure. This is shown in Fig. 4.3. The next paper of consequence attempting to define a quantitative measure of the forming fabric structure’s affect on the final sheet was published by Dale Johnson in 1984[22]. This paper conveys the methods and results of experiments to determine the relationships between fibre length, MD frame length, paper mat weight and fabric drainage rate. This investigation showed that, despite their higher flow resistance when clean, double layered forming fabrics had lower flow resistance and higher drainage rates when compared to the single layer fabrics once a mat developed. It was hypothesized that this was due to the superior support and consequently less clogging due to the low MD frame length. 15  Chapter 4. Literature Review  Figure 4.3: Forming wire with formed sheet, ground at an incline to show webpenetration. The lowest penetration is at the top-left of the figure, and highest penetration is at the bottom-right.[18]. Perhaps the most critical development from Johnson’s 1984 paper was the drainage index (DI). This quantitative number related the coefficients of the Beran FSI, the structure of the forming fabric and the air permeability to predict the drainage characteristics of a given design. The validity of the DI in predicting the drainage characteristics of multilayered fabrics was displayed in Johnson’s next paper in 1986[23]. In this document, Johnson showed that the decreasing MD frame lengths in newer double and triple-layer fabrics increased both first pass and overall retention. He also presented results that showed a correlation between DI and cumulative drainage, in which drainage increased as a function of DI, regardless of the fabric’s air permeability.  4.2.2  Investigating the Sheet  The next stage in the research involved taking a closer look at the final sheet that was formed. The earliest work published that discussed wire mark from a final product reference frame was published by Roger Danby in 1986[12]. In this paper, Danby compared the wire mark on sheets of paper that were produced on the same machine running single, double and 16  Chapter 4. Literature Review triple-layer fabrics. In doing so he was able to postulate that wire mark was not only a function of optical severity, but also relied on the “frequency and continuity”, more of which made the marking more obvious.  Figure 4.4: Linear FFT of a sheet of paper is shown in A with the corresponding forming fabric shown in B[19]. Building on this paper, Helle released a work in 1988 that used a linear FFT of the light transmission through the final sheet to investigate the frequency of the wire mark and compared that to the forming fabric used[19]. He found that there is a direct correlation between the density changes in the sheet and the forming fabric imprint. Figure 4.4 shows the results of one of his FFTs of a sheet formed on a double-layer fabric. In this figure, Helle identifies the origins of each of the peaks that show up in the FFT. One thing that is of particular importance to the work done in this thesis is the MD direction peak at λ = 2.22mm. Helle has no explanation for this peak based on the paper side of the forming fabric. It has been hypothesized that this peak is, in fact, due to the geometry of the machine side of the forming fabric. Since there was no feature on the paper side of the forming fabric that could create this density variation, and all other density variations were related to the forming fabric geometry, it is not unreasonable to believe that this peak may originate from another structure within the fabric.  17  Chapter 4. Literature Review Helle also touches on one of the more difficult hurdles for double layer fabrics – the type and direction of the wire mark. He briefly discusses the impact of the direction and continuity of the marking and its overall effect on printability and, more importantly, readability. He recognizes that the human eye has particular sensitivities that make certain wire marks more obvious and intrusive than others. The next paper to discuss the origins and impact of wire mark was published in 1994 by Danby[13]. This paper was pragmatic and discussed the impact on the previously recognized density variations on the printing of the sheet. Danby showed that the density variations produced by wire marking affect the printing of half-tone dots by absorbing the ink differently. The areas with higher density have lower ink absorption and and more dot spreading; areas with lower density have higher absorption and a greater tendency for strike through. Later that same year Sabit Adanur published a milestone paper correlating Beran’s FSI and Johnson’s DI to many sheet properties including breaking length, burst strength, tear strength, and sheet thickness[10]. Adanur also correlated the same properties to the much more general warp × weft count and related the plane difference to the sheet thickness and sheet density. The plane difference is defined as the z -direction (see Fig. 2.3 on page 7) height difference between the most prominent weft and warp knuckles. Adanur found that breaking load increased linearly with both FSI and DI in both the MD and CD directions. Tear strength was found to be a function of DI and FSI. Burst strength was linear with DI. Sheet thickness was found to be a function of FSI. There was also a correlation between breaking load, burst strength and tear strength with warp × weft count however, the correlations were much more complicated than when these properties were compared to DI and FSI. Sheet thickness was found to decrease with a tighter weave (higher warp × weft count) and increase with higher plane difference. Adanur hypothesized that the increase in strength with the increasing DI and FSI was related to the amount of interaction between fibres. FSI and DI both increased with greater fibre support and when the fibres had more support they interacted with each other more since there was less room for fibre deflection and movement. This increased interaction had a corresponding increase in the strength and number of bonds formed between fibres, leading to a stronger piece of paper.  18  Chapter 4. Literature Review In 2000 Roger Danby published a paper that used 2D FFT analysis to investigate transmitted light wire mark (density variations) and compared the results to the top-side structure of the forming fabric[15]. He used this analysis to encourage an informed, engineering style selection of forming fabrics based on the individual needs of the end user. Danby proceeded to discuss the evolution of fabrics for various uses from tissue to liner board.  One thing that was noticeable in his discussion was that there had been a tendency to higher FSI and the use of double and triple-layer fabrics even, though they have a reduced drainage area. Often, especially with triple layer fabrics, the drainage area decrease significantly, but a corresponding increase in FSI and DI lead to a better forming fabric.  Figure 4.5: Results of Danby’s multiple sheet split tests for twin-wire SC paper[14]. Danby’s next publication, in 2002, investigated the uses of Asten Johnson’s computer simulated printing software and also explored the internal structure of a sheet of super calendared (SC) paper[14]. In his investigation of the internal structure, Danby split a sheet of SC paper three times. He then observed the light transmission properties of the sheet sections and determined that the wire mark existed only in the 12.5% of the sheet that was closest to the forming wire. These results are shown in Fig. 4.5.  19  Chapter 4. Literature Review The final author that has published work that investigated the interaction of the forming fabric and its affect on the final sheet was Zhaolin Huang. In 2006, Huang published his paper on the numerical simulation of flow around two rows of cylinders of different diameters[21]. While at first glance this may seem to have little to do with forming fabrics, his masters thesis, defended in 2003, relates the work in his 2006 paper to an investigation of the affects of different CD filament geometries in a double and triple-layer forming fabric[20]. The work that is presented in this thesis is the experimental version of the work conducted by Huang in his 2003 document. Huang investigated a variety of geometries and two Reynolds numbers in his work. He limited his Re to 6.5 for steady simulations, and 65 for unsteady simulations. For the majority of his work Huang performed his calculations at Re = 6.5. He determined that the comparative results (that is, the trends observed between different configurations) upstream of the bank of cylinders were unaffected by the lower Re, and since the steady calculations were faster and less computationally intensive, he used them as his primary data source. Huang modelled his geometries off of a commercial triple-layer forming fabric. He used a 73 x 75 mesh fabric that, when sectioned in the MD-z plane, showed two distinct layers of CD filaments. After performing an investigation of the commercial fabric, Huang computed flow fields for a number of different geometries to investigate the affects of the different parameters. He presented them as normalized velocities, 1/4 of a paper side filament diameter upstream of the front row of cylinders. The results of Huang’s investigation can be summed up as: • Variations in Re from steady to unsteady values changed the magnitude of the velocity variations in the upstream (US) flow field, but did not change the shape or trends of the variations. • When cylinder surface separation between the two rows was ≥ 0.7 times the paper side filament diameter, the US flow field was identical to the single row flow field, regardless of second row configuration. • When paper side and machine side filament spacings were equal the US flow field was nearly identical to the single row flow field, regardless of second row configuration.  20  Bibliography [10] S. Adanur. Effects of forming fabric structural parameters on sheet properties. Tappi Journal, 77(10):187–195, Oct 1994. [11] R.L. Beran. The evaluation and selection of forming fabrics. Tappi, 62(4):39–44, Apr 1979. [12] R. Danby. The impact of multilayer fabrics on sheet formation and wire mark. Pulp & Paper Canada, 87(8):69–74, Aug 1986. [13] R. Danby. The impact of forming fabric structures on print quality. Pulp & Paper Canada, 95(1):48–51, Jan 1994. [14] R. Danby. Sc print quality influenced by fibre length, fabric structures, and machine drainage characteristics. Tappi Journal, 1(9):3–9, Nov 2002. [15] R. Danby and P. Plouffe. Print quality improvements through forming fabric design changes. Pulp & Paper Canada, 101(9):66–69, Sep 2000. [16] P. Glynn, H.W.H. Jones, and W. Gallay. The fundamentals of curl in paper. Pulp and Paper Magazine of Canada, 60:T316–T323, November 1959. [17] T. Helle. How forming fabric design affects drainage and release. Pulp & Paper Canada, 79(11):91–98, November 1978. [18] T. Helle. The influence of wire structure on sheet forming. Paper Technology and Industry, 21(4):123–131, May 1980. [19] T. Helle. Analysis of wire mark in printing paper. Journal of Pulp and Paper Science, 14(4):J91–J94, Jul 1988. [20] Z. Huang. Numerical simulations of flow through model paper machine forming fabrics. Master’s thesis, The University of British Columbia, Vancouver, British Columbia, Canada, 2003. [21] Z. Huang, J.A. Olson, R.J. Kerekes, and S.I. Green. Numerical simulation of the flow around rows of cylinders. Computers & Fluids, 35(5):485–491, June 2006. [22] D.B. Johnson. Retention and drainage of forming fabrics. Pulp & Paper Canada, 85(6):T167–T172, June 1984. [23] D.B. Johnson. Retention and drainage of multi-layer fabrics. Pulp & Paper Canada, 87(5):56–59, May 1986.  21  Chapter 5 Manuscript By: Seth Gilchrist and Dr. Sheldon Green  5.1  Introduction  When paper is made, a dilute suspension of pulp and water (generally about 0.7% pulp by mass) is passed through a forming fabric. This process removes the water and leaves a mat of pulp. This pulp mat is drained using inertia and suction and is then passed to a mechanical press, thermal dryer, and, in some cases, special processing before being wound onto a role for transportation. In order to create a high quality, even density sheet of paper the initial drainage of the pulp suspension through the forming fabric must occur uniformly. For this reason much effort has gone into the development of forming fabrics that allow for even and fast drainage, low running resistance, and good fibre retention. However, specific engineering research into flow patterns and governing dynamics of flows through the forming fabrics has only recently been initiated. Another unique aspect of forming fabrics is their two-sidedness. This is generated by the need to have a very fine structure to support the paper on the top side of the fabric (filaments of ∼0.15mm diameter) and a courser mesh on the bottom of the fabric (filaments of ∼0.3mm diameter). These different sides of the fabric serve to provide substantial support for the pulp fibres while increasing wear life and lowering the running resistance of the forming fabric. Due to their relevance, flows through banks of cylinders have been well studied. Flows that concern single cylinders and groupings of similar cylinders are the most well understood[25, 35, 36, 38, 39, 40, 41]. There are also a number of studies that focus on flows through arrangements of cylinders at higher Reynolds numbers[27, 29, 33, 37]. Even with all the work that has been done on flows through banks of cylinders there are aspects of these flows that are not well characterized. These include flows through banks of cylinders at low Re, banks of cylinders of nonuniform sizes and 0  A version of this chapter will be submitted for publication.  22  Chapter 5. Manuscript spacings, and examination of the flows upstream of the bank of cylinders.  In order to investigate paper formation some simplifications were required to reduce the complexity of the problem. In the real case, flows are multiphase and highly 3-dimensional. However, a review of the literature allows for reasonable engineering simplifications to make the problem manageable, yet still applicable. Due to the low concentration of pulp fibres in the flow it is possible to model the fluid as pure water. Additionally, due to the flow dynamics encountered just before the forming fabric (in the headbox region) it becomes possible to neglect one direction of the filaments in the forming fabric and model the fabric as a 2D bank of nonuniform cylinders. Previous authors have used both theory and experiments to provide evidence that the machine-direction (MD) orientation of the fibres as they exit the headbox make it possible to reasonably neglect the MD filaments of the forming fabric[24, 26, 30, 34]. However, it is worth mentioning that the current study is concerned with purely incident flows, that is there is only a x -component in the approach flow, but in many of the studies, particularly those performed by Helle, the flows impinge the forming fabric at an angle, so there is a x and y -component of velocity. In order to effectively model the forming fabric, a notion of the flow’s Re must be obtained. The drainage velocity in the region of jet impingement can be quite high, however studies have shown that the bulk velocity in this region is on the order of 0.05m/s to 0.50m/s[28]. Based on this velocity, the top-side filament diameter, and modelling the fluid as pure water, it is found that Red ∈ [6.5, 65]. While flows at low Re are encountered in a number of applications, such as those utilizing drainage screens and flow conditioners, it is not as common to experience flows through banks of nonuniform cylinders. Additionally, most research into cylinder flow is concerned with the nature of vortex shedding due to its impact on the vibrational characteristics of a bank of cylinders. For this reason, flows upstream of a bank of cylinders have not been well investigated.  5.2  Literature Review  A number of previous studies have examined the flow structures associated with flows through banks of cylinders. Some of these papers can be found in the references, but includes many more authors and works.  23  Chapter 5. Manuscript However, with all of the work that has been published regarding the interactions of two or more cylinders in cross flow, only one author was found to investigate low Re flow through two rows of non-uniform cylinders. Zhaolin Huang’s paper and thesis[31, 32] are the numerical equivalent to the experimental work presented in this paper. In these documents, Huang examined the flow through a single row of cylinders, as well as symmetric and asymmetric banks of cylinders at different separations and staggerings. In his paper, Huang found that for cylinder flows in a confined channel at Re = 100, the vortex shedding is dependent on the cylinder centre separation. He also showed that the second row of cylinders had little affect upstream for a row surface separation of 0.7 times the upstream cylinder diameter (0.7d ). In his thesis, Huang did a more in depth investigation of the affect upstream of the cylinder bank due to changing the configuration of the two rows of cylinders. He gave extensive results comparing the upstream flow field of one row of cylinders to that of multiple rows of cylinders. He also investigated the effect of adding small diameter filler cylinder in the upstream row as well as that of an asymmetric alignment in the second row. These last two were to find the impact of a misplaced filament in the weaving of a forming fabric. The distillation of his thesis was that if the cylinder surface spacing between the two rows was greater than 0.7d, the upstream flow field was largely undisturbed from that of a single row of cylinders.  5.3  Experimental Methods  The experiments were carried out in a flow loop circulating a glycerol solution for Reynolds number matching. The target Re for the experiments was 10 ≤ Re ≤ 65. For the velocities that the flow loop was capable of, and the diameter of the upstream cylinders, a viscosity of 12 ≤ µ ≤ 18 cP was required, or an ∼65(m/m)% glycerin/water mixture. The test section of the flow loop is shown in Fig. 5.1. The test section measured 30x30 cm, and contained a frame for holding the bank of cylinders. It was of a closed loop configuration driven by a 20hp electric pump. Velocity measurement was via a venturi flow meter located downstream of the pump and upstream of the test section. The flow then passed through a diffuser, conditioning screens and honeycomb. The test section was located immediately downstream of the flow conditioner and was constructed of acrylic to facilitate flow field observation. Flow was  24  Chapter 5. Manuscript  $            Figure 5.1: Flow loop test section. then returned to the tank through a 4” return line. A schematic of the flow loop is shown in Fig. 5.2. The primary means of data collection were through particle image velocimetry (PIV), pressure drop measurement, and hydrogen bubble generation. The PIV provided instantaneous flow field observations and was arranged to give the best resolution of the flow field just upstream of the bank of cylinders. The pressure drop measurements were made with a Validyne Engineering DP103 calibrated for a differential pressure of 0– 0.1” of water (0–25Pa). The pressure data were collected through pressure ports located on the bottom of the test section upstream and downstream of the bank of cylinders. The test section was also fitted with an hydrogen bubble generation wire for qualitative flow visualization. The pressure ports and hydrogen bubble wire can be seen in Fig. 5.1. Flow within the test section was observed with the bubble generation wire to be even to within 5% over 83% of the test section, with the boundary layers being somewhat less than 2cm thick at the location of the cylinder bank. These measurements were made using the H2 bubble wire and a high-definition video camera at multiple velocities. The simplification of the forming fabric structure made it possible to model the fabric as a 2-dimensional bank of cylinders of various sizes and staggerings. Even in this simplified format there were a large number of possible configurations. Huang developed a nomenclature that has been used to define the cylinder arrangement and coordinate system. This system is shown in Fig. 5.3. 25  Chapter 5. Manuscript  Figure 5.2: Schematic of test section flow loop.  XS  XC  MD-z face $           Figure 5.3: Schematic of the simplified forming fabric geometry.  26  Chapter 5. Manuscript  5.4  PIV Results and Discussion  A number of different configuration were evaluated using PIV. In order to keep the plots uncluttered only one uncertainty bar has been included on each plot. The given uncertainty bars represents the maximum uncertainty for 80% confidence. The values of these bars are fairly typical across all plots. The PIV data was also smoothed using a moving average and in some cases a polynomial fit with a 10% span.  XS/d XS/d XS/d  $           Figure 5.4: Stream wise velocity variation for changing XS /d for D/d = 1.5, G2/G1 = 1, Y/G1 = 0.5 at Re = 65. The results of each curve are statistically the same. Fig. 5.4 shows the effects on the stream wise component of the velocity d /4 upstream of the bank of cylinders when changing the cylinder surface separation for an arrangement of D/d = 1.5, G2/G1 = 1, and Y/G1 = 0.5 (fully staggered), at Re = 65. The cross stream component for the same arrangement is shown in Fig. 5.5. Figs. 5.6 and 5.7 show the effects d /4 upstream of the cylinders on the stream wise and cross stream velocity components of staggering for an arrangement of D/d = 1.5, G2/G1 = 1, XS /d = 0.75 at Re = 65. 27  Chapter 5. Manuscript  XS/d XS/d XS/d  $         Figure 5.5: Cross stream velocity variation for changing XS /d for D/d = 1.5, G2/G1 = 1, Y/G1 = 0.5 at Re = 65. The results of each curve are statistically the same.  28  Chapter 5. Manuscript  Figure 5.6: Stream wise velocity variation for changing Y/G1 for D/d = 1.5, G2/G1 = 1, XS /d = 0.75 at Re = 65. The results of each curve are statistically the same.  29  Chapter 5. Manuscript  Figure 5.7: Cross stream velocity variation for changing Y/G1 for D/d = 1.5, G2/G1 = 1, XS /d = 0.75 at Re = 65. The results of each curve are statistically the same.  30  Chapter 5. Manuscript  Figure 5.8: Stream wise velocity component for changing staggering with D/d = 3, G2/G1 = 2, and XS /d = 0 at Re = 25. Fig. 5.8 shows the stream wise velocity component d /4 upstream of the cylinders for variations in staggering with D/d = 3, G2/G1 = 2, XS /d = 0 at Re = 25. Examination of the PIV data shows a number of interesting characteristics. Figs. 5.4 and 5.5 show that for and equal cylinder spacing in each row, changing the row separation had negligible effect. This was not true for uneven cylinder spacings, as can be seen in Fig. 5.8. Fig. 5.8 shows that for a surface separation of XS /d = 0 the second row alignment with the first row was more critical than diameter ratio or relative spacing. Figs. 5.6 and 5.7 show that for a cylinder surface separation of XS /d = 0.75 there were no effects seen in the upstream flow field for any staggerings. This is a similar result obtained by Huang. He found that for a separation of XS /d = 0.7 at an Re = 65 changing the row staggering had little affect on the upstream flow field.  31  Chapter 5. Manuscript  5.5  ∆P Results and Discussion  Pressure drop data was collected for all of the presented PIV cases. In addition to that data there were many cases for which only ∆P data was. The PIV data was more laborious to take, so only extremes had been evaluated. The ∆P data was much more straight forward and was used to take measurements on intermediate configurations. The uncertainty bars on the ∆P plots were determined using: ⎡  ∂f δx1 δf (x1 , x2 , . . . , xn ) = ⎣ ∂x1  2  ∂f + δx2 ∂x2  2  ∂f + ... + δxn ∂xn  ⎤ 2 1/2 ⎦  (5.1)  In this equation, the value of δn was given by manufacturer specifications, or by two standard deviations in the case of sampled data.  XS/d  , XS/d $           Figure 5.9: Pressure drop through a bank of cylinders with D/d = 1.5, G2/G1 = 1 and Y/G1 = 0.5 at Re = 65 with changing XS /d. Fig. 5.9 shows the pressure drop as a function of changing XS /d for a configuration of D/d = 1.5, G2/G1 = 1, and Y/G1 = 0.5 (fully staggered) at Re = 65. The negative value of XS /d indicates that the leading edge of the downstream cylinder was upstream of the trailing edge of the upstream cylinder. This was only possible for staggered configurations. Fig. 5.10 shows the pressure drop through the same bank of cylinders as presented in Fig. 5.9, but with Y/G1 = 0 (tandem). 32  Chapter 5. Manuscript  XS/d  , XS/d $           Figure 5.10: Pressure drop through a bank of cylinders with D/d = 1.5, G2/G1 = 1 and Y/G1 = 0 at Re = 65 with changing XS /d. It can be seen in Fig. 5.9 that the pressure drop through the bank of cylinders converged to the sum of the rows’ pressure drops with a surface separation of XS /d = 0.75. Fig. 5.10 shows the same result. This indicates that for XS /d ≥ 0.75 the flows through each row of cylinders was essentially independent. Fig. 5.9 corresponds to the PIV data in Figs. 5.4 and 5.5. Fig. 5.11 shows the pressure drop for changing staggerings for a configuration of D/d = 1.5, G2/G1 = 1, and XS /d = 0.75 at Re = 65. The results shown in Fig. 5.11 show that for XS /d = 0.75 and G2 = G1 the pressure drop coefficient is independent of the row staggering. When this data is considered in conjunction with the PIV data shown in Figs. 5.6 and 5.7 it is seen that for this configuration the upstream flow fields were also the same, regardless of staggering. Fig. 5.12 shows the pressure drop through a bank of cylinders with unequal G1 and G2, and changing XS /d. The plot shows that for unequal cylinder spacings there is no affect on pressure drop coefficient once the row separation is ≥ 0.75d.  33  Chapter 5. Manuscript  Figure 5.11: Pressure drop through a bank of cylinders with D/d = 1.5, G2/G1 = 1, and XS /d = 0.75 at Re = 65 with changing Y/G1.  XS/d  , XS/d $           Figure 5.12: Pressure drop through a bank of cylinders with D/d = 1.5, G2/G1 = 2, Y/G1 = 0.5 at Re = 65 and changing XS /d.  34  Chapter 5. Manuscript  Figure 5.13: Comparison of pressure drop as a function of Re through cylinders with D/d = 1.5, G2/G1 = 1, XS /d = 0.75 and two staggerings. Fig. 5.13 shows the pressure drop coefficient as a function of Reynolds number for staggered and tandem configurations of D/d = 1.5, G1/G2 = 1, XS /d = 0.75. The Reynolds number data shows that k increases at low Re, and asymptotically approaches a lower value as Re increases. These figures also show the pressure drop is the same for the staggered and tandem configurations when XS /d = 0.75, regardless of Reynolds number. Comparison of th PIV data with the corresponding ∆P data suggests a link between the pressure drop and the upstream flow field. Examination of Figs. 5.9 throuhg 5.12 shows that for XS /d ≥ 0.75 there is no change in pressure drop, regardless of the second row configuration. The corresponding PIV data show that the flow field upstream of these configurations looks like that of just 1 row of cylinders. This indicates that, ∆Prows , the upstream flow field will be like that of flow when ∆P → through only the first row of cylinders.  5.6  H2 Bubble Visualization Results and Discussion  35  Chapter 5. Manuscript  $  XS/d = -0.25  XS/d = 0.75  XS/d = 2.75  Empty          Figure 5.14: Hydrogen bubble visualizations for variations of XS /d with D/d = 1.5, G2/G1 = 2, Y/G1 = 0.5 at Re = 65.  XS/d XS/d XS/d  $          Figure 5.15: Relative velocities for variations of XS /d with D/d = 1.5, G2/G1 = 2, Y/G1 = 0.5 at Re = 65.  36  Chapter 5. Manuscript The images in Fig. 5.14 show the flow visualizations obtained via hydrogen bubble visualization for variations in XS /d for a configuration of D/d = 1.5, G2/G1 = 2, and Y/G1 = 0.5 at Re = 65. The qualitative comparison of the velocities obtained from the the visualizations are contained in Fig. 5.15. These flow fields correspond to the pressure drop data shown in Fig. 5.12. These data show that, for a cylinder surface separation of XS /d ≥ 0.75, the flow field remains the same and the pressure drop coefficient converges to the sum of the rows’ pressure drops. this result corroborates the earlier result that when the pressure drop is equal to the sum of the row’s pressure drops the upstream flow field will be the same as the upstream flow field for just the first row of cylinders.  5.7  Conclusions  From the data presented it has been shown that: • There is little or no effect on the upstream flow for row surface separations of 0.75d or more, regardless of downstream row configuration. • The pressure drop through the bank of cylinders converges to the sum of the rows’ pressure drops for surface separations of 0.75d and greater, regardless of downstream row configuration. • When the pressure drop through a bank of cylinders is equal to the sum of the rows’ pressure drops, the upstream flow field will be the same as that for a single row. • For equal values of G1 and G2 the flow upstream will be identical to the one row flow field, regardless of staggering and separation. In these cases the pressure drop may be higher than the sum of the rows’ pressure drops if cylinder surface separation is less than 0.75d. • The value of the pressure drop coefficient asymptotically approaches a value with increasing Reynolds number. Further, for a cylinder surface separation of XS /d = 0.75 and G1 = G2 the pressure drop coefficient is the same for staggered and tandem configurations, regardless of Reynolds number. In the context of a multi-layer papermaking forming fabric, this indicates that the paper side flow field contributed by the cross-machine direction filaments will be the same as a single-layer fabric provided the backing side filaments are at least 0.75d away from the paper side filaments. Additionally, it can be expected that the upstream flow will be unaffected by 37  Chapter 5. Manuscript the backing side filaments if the pressure drop of the complete fabric is approximately equal to the sum of the fabric layers.  5.8  Acknowledgements  The authors would like to thank Asten Johnson and NSERC for financial and material support.  38  Bibliography [24] S. Adanur. Effects of forming fabric structural parameters on sheet properties. Tappi Journal, 77(10):187–195, Oct 1994. [25] P.W. Bearman and A.J. Wadcock. The interaction between a pair of circular cylinders normal to a stream. Journal of Fluid Mechanics, 61(3):499–511, 1973. [26] R.L. Beran. The evaluation and selection of forming fabrics. Tappi, 62(4):39–44, Apr 1979. [27] M.J. Braun. Fluid flow structures in staggered banks of cylinders located in a channel. Transactions of the ASME, 117:36–44, March 1995. [28] B. Dalpke, R.J. Kerekes, and S.I. Green. Modelling jet impingement and the initial drainage zone in roll forming. Journal of Pulp and Paper Science, 30(3):65– 70, March 2004. [29] P. Le Gal, M.P. Chauvre, and Y. Takeda. Collective behaviour of wakes downstream a row of cylinders. Physics of Fluids, 8(8):2097–2106, 1996. [30] T. Helle. How forming fabric design affects drainage and release. Pulp & Paper Canada, 79(11):91–98, November 1978. [31] Z. Huang. Numerical simulations of flow through model paper machine forming fabrics. Master’s thesis, The University of British Columbia, Vancouver, British Columbia, Canada, 2003. [32] Z. Huang, J.A. Olson, R.J. Kerekes, and S.I. Green. Numerical simulation of the flow around rows of cylinders. Computers & Fluids, 35(5):485–491, June 2006. [33] C. Iwaki, K.H. Cheong, H. Monji, and G. Matsui. Piv measurement of the vertical cross-flow structure over tube bundles. Experiments in Fluids, 37:350–363, 2004. [34] D.B. Johnson. Retention and drainage of multi-layer fabrics. Pulp & Paper Canada, 87(5):56–59, May 1986. [35] H.J. Kin and P.A. Durbin. Investigation of the flow between a pair of circular cylinders in the flopping regime. Journal of Fluid Mechanics, 196:431–448, 1988. [36] D.R. Polak and D.S. Weaver. Vortex shedding in normal triangular tube arrays. Journal of Fluids and Structures, 9:1–17, 1995.  39  Bibliography [37] D. Ghosh Roychowdhury, Sarit Kumar Das, and T. Sundararajan2. Numerical simulation of laminar flow and heat transfer over banks of staggered cylinders. International Journal for Numerical Methods in Fluids, 39:23–40, 2002. [38] D. Sumner, S.J. Price, and M.P. Pa¨ıdoussis. Flow-pattern identification for two staggered circular cylinders in cross-flow. Journal of Fluid Mechanics, 411:263– 303, 2000. ¨ [39] D. Sumner, S.S.T. Wong, S.J. Price, and M.P. Padoussis. Fluid behaviour of sideby-side circular cylinders in steady cross-flow. Journal of Fluids and Structures, 13:309–338, 1999. [40] C.H.K. Williamson. Evolution of a single wake behind a pair of bluff bodies. Journal of Fluid Mechanics, 159:1–18, 1985. [41] S. Ziada. Flow periodicity and acoustic resonance in parallel triangle tube bundles. Journal of Fluids and Structures, 14:197–219, 2000.  40  Chapter 6 Discussion 6.1  Introduction  The results presented in §5 provide new knowledge regarding the flow characteristics through a bank of cylinders consisting of two rows of different geometries at low Reynolds number. This information has interpretations that impact the design of papermaking forming fabrics. This section discusses the applications of the new knowledge and places it into the context of multi-layer forming fabric design. In addition to being applicable to the design of forming fabrics, the results can be used to validate the work of Zhaolin Huang, who performed the computational version of this study during his masters degree at the University of British Columbia in 2003[42]. While the exact configurations that he simulated could not be tested, similar configurations show results that agree with his findings. Where there were large discrepancies, another graduate student, Ali Vakil, performed simulations using Huang’s methods on geometries that were identical to the corresponding experiments.  6.2  Forming Fabric Application  The data found in the experiments presented in §5 have direct applications to papermaking and forming fabric design. 1. The data presented in Figs. 5.4 and 5.5 show that for an identical cylinder centre separation on both the paper and machine sides of the forming fabric, the paper side flow field is undisturbed from the single row solution. This is true even with a larger cylinder in the second row, full staggering and small to negative row separations. In terms of forming fabric design, this indicates that if the paper side and machine side have the same weave density (weft × warp count), the separation, staggering, and filament diameter ratio would not affect the paper side flow field. 2. If the paper side of the fabric and the machine side have different weave density, the separation between the filaments must be ≥ 0.75d, otherwise there is a possibility that the machine side weave will alter the paper side flow field. The 41  Chapter 6. Discussion results in Figs. 5.8,5.12, 5.14 and 5.15 show that if G1 = G2 then for row separations of less than 0.75d there will be an effect on the paper side flow field. 3. Figs. 5.9 – 5.12 can be compared with the corresponding flow fields shown in Figs. 5.4 – 5.7 and 5.14 – 5.15. These figures show that when the pressure drop through the two rows of filaments is equal to the sum of the rows’ individual pressure drops that the paper side flow field will be undisturbed from the single row case. This can be used in the design of triple-layer fabrics. Traditional triple-layer fabrics are essentially made by binding two single layer fabrics together. If, once bound, the pressure drop of the final fabric is approximately equal to the sum of the pressure drop of the two individual fabrics, then in can be assumed that the paper side flow field will be largely undisturbed from the single, paper side fabric flow field.  6.3  Validation of Huang’s Results  Figure 6.1: Huang’s computational stream wise velocity results plotted with the PIV results for D/d = 1.5, G2/G1 = 1, XS /d = 0.75, g1 = 2 and Y/G1 = 0.5 at Re = 65.  42  Chapter 6. Discussion  Figure 6.2: Huang’s computational cross stream velocity results plotted with the PIV results for D/d = 1.5, G2/G1 = 1, XS /d = 0.75, and Y/G1 = 0.5 at Re = 65.  43  Chapter 6. Discussion Most of the results found by Huang in his computational investigation of a model forming fabric show similar trends and numbers to the data found by the current study. Huang did not present any pressure drop results in his thesis or paper, so that data is unavailable for aiding validation. Figs. 6.1 and 6.2 show the PIV data for D/d = 1.5, G2/G1 = 1, g1 = 2, XS /d = 0.75 and Y/G1 = 0.5 and Huang’s data for D/d = 1.615, G2/G1 = 1, g1 = 2.692, XS /d = 1.923 and Y/G1 = 0.5 plotted on the same axes. It is seen that for these similar configurations the plots are nearly coincident. The uncertainty bars shown on the plots are for the PIV and represent the uncertainty for 80% confidence. Huang did not report any uncertainties. Examination of these figures shows that Huang’s data corresponds to the PIV data. In the stream wise velocity field, Fig. 6.1 shows that the maximum velocity seen by Huang was approximately 1.26U0 , the PIV data shows similar values of between 1.23U0 and 1.27U0 . The minimum velocity seen by Huang for these test was in the vicinity of just over 0.5U0 , while the minimum velocity seen in the PIV was between 0.5U0 and 0.6U0 . In the cross stream direction, the plots show good agreement in terms of magnitude and shape. Huang shows the velocity ranging from -0.37U0 to 0.37U0 , with a linear change between. The PIV data ranges from approximately -0.40U0 to 0.35U0 . This is shown in Fig. 6.2. For the G1 = G2 comparisons, a new set of simulations were performed. The computations discussed here were exicuted by Ali Vakil of the University of British Columbia’s department of Mechanical Engineering using Huang’s methods. Vakil simulated the exact configuration that was tested by PIV and reported velocity fields for the viscous and inviscid flows and pressure drop for the viscous flow. Figs. 6.3 and 6.4 show the d/4 upstream velocity fields for a configuration of D/d = 3, G2/G1 = 2, XS /d = 0, Y/G1 = 0.5 at Re = 25. The uncertainty bars represent a typical 80% confidence uncertainty. It is seen from these images that the PIV results don’t show as good of an agreement for these simulations as the results for the G1 = G2 cases. The results are closer to the inviscid solution, but in general fall between the viscous and inviscid solutions. Vakil also solved for the pressure drop of the viscous solution and obtained a value of k = 22.9. The measured value for the experiments was k = 21.24 ± 2.04.  44  Chapter 6. Discussion  Figure 6.3: Comparison of PIV and computational stream wise velocities for a case of D/d = 3, G2/G1 = 2, XS /d = 0, Y/G1 = 0.5 at Re = 25  45  Chapter 6. Discussion  Figure 6.4: Comparison of PIV and computational cross stream velocities for a case of D/d = 3, G2/G1 = 2, XS /d = 0, Y/G1 = 0.5 at Re = 25  46  Chapter 6. Discussion The source of the discrepancies between the PIV and computations is unknown. It has been suggested that the problem lies is the computation’s ability to model the onset of separation. However, even with these differences, the computations remain a powerful tool in examination of the affect of the second row filament geometries. In order to determine the effect of the velocity discrepancy, it must be determined how the fibres and fines are influenced by the flow conditions. If the u-component of the velocity dominates the behaviour of a fibre, the higher sheer rates in the viscous solution would tend to align the fibres with the x -direction and they would pass though the forming fabric. In this case the computations would under estimate the fibre retention. If, on the other hand, the v -component of the velocity dominates the flow, the higher shear rates seen in the viscous simulation in the machine direction (the computational y -direction) would help the fibre retain its initial MD orientation, bridging more fabric filaments, increasing retention. In this case the computations would over estimate the fibre retention. The PIV served to validate Huang’s results for the G1 = G2 cases. It was not possible to reproduce the G1 = G2 computational flow fields in the experiments, however, the pressure drop data from the computations matched the experimental value, giving credence to the results. In the case of the G1 = G2 simulations, the viscous solution consistently showed higher shear in the x and y -directions than the experiments. To determine if this would lead to an under estimate, or an over estimate of the retention more must be known about the influence of the flow on the behaviour of the pulp fibres.  47  Bibliography [42] Z. Huang. Numerical simulations of flow through model paper machine forming fabrics. Master’s thesis, The University of British Columbia, Vancouver, British Columbia, Canada, 2003.  48  Chapter 7 Recommendations for Further Research There are two avenues that are of interest and should be investigated, either coincidently or consecutively. The next step in the modelling of the flow is to move from 2D to 3D and include the effects of the machine direction filaments on the upstream flow field. The other direction of interest is determining how the flow field influences the deposition of the pulp fibres and fines.  Figure 7.1: Method for determining the internal 3D geometry of a forming fabric. In the modelling of the 3D forming fabric one of the more significant challenges is likely to be the determination of an appropriate geometry. Because little is known about the internal structure of forming fabrics, the exact geometries of the filament knuckles and path of the woven strands will have to be obtained. One method of doing this is shown in Fig. 7.1. In this figure, the forming fabric in question is potted in a hard material. An end mill is then used to make consecutive, thin cuts. Each cut will be on the order of 0.001” inch and by imaging the filament geometries 49  Chapter 7. Recommendations for Further Research after each cut it will be possible to construct a fully 3D, accurate CAD model of the complete forming fabric. This model can then be used as a computational domain, or could be physically replicated using rapid prototyping for experimental analysis. The other direction of interest is the modelling of the effects of the flow field on the pulp fibres that will constitute the final sheet of paper. As an initial investigation, this is best done computationally. The flexibility of the computational domain and flow conditions make it relatively easy to investigate affects of the different components of the flow field. It also allows for the modelling of a standardized pulp fibre, something that would be very difficult to obtain experimentally. Initially, the fibres could be considered on a 2D basis, with the 3D evaluation performed once a satisfactory 3D computational domain has been determined using the methods discussed previously.  50  Appendix A Experimental Design A.1  Introduction  Investigation of flow through a forming fabric poses a number of experimental and engineering challenges. In application, the flow through a forming fabric is multi-phase, highly 3-dimensional, and happens at a very small scale and hence low, but not Stokes flow, Reynolds numbers. In order to totally evaluate the flow structures these complications have to be overcome, however, before doing a fully 3D, multi-phase, unsteady simulation of the sheet forming process there are some reasonable simplifications in the geometry and flow conditions that can made to make the problem manageable, yet still applicable. This appendix details the simplifications, and their justifications, that were used in the design of the experiments.  A.2  Phase Simplification  The first simplification was the elimination of the second phase. In practise, flows through forming fabrics are primarily liquid. The consistency (c) in the forming section is low, generally less than 1% (c = mf ibre /mtotal ), but an examination of fibre-fibre interactions is also necessary. The first consideration in the phase simplification was to determine where wire mark exists in the paper. Roger Danby published a paper in 2002 in which he split a sheet of super calendar paper three times and examined each part of the sheet in succession[48]. He found that wire mark is only present in the 12.5% of the sheet that is closest to the forming fabric. Concequently, an investigation into wiremark should be limited to the inital stages of sheet formation. The results of Danby’s light transmission test are shown in Fig. 4.5 on page 19. Since Danby showed that wire mark exists only in the 12.5% of the sheet closest to the fabric, it can be said that wire mark is developed primarally in early stages of sheet formation. During formation the suspension of fibres is subjected to three basic processes: drainage, oriented shear, and turbulence[52, Ch. 1]. While, in reality, all of these processes are 51  Appendix A. Experimental Design taking place at the same time, they can be said to dominate in the order listed. Furthermore, as formation progresses the there comes a point where the deposited fibre mat dominates the flow field and the geometry of the forming fabric becomes less important[51]. Since the fibre mat will dominate the forming process in the later stages of formation, the drainage process is of primary concern. Drainage can be further broken down into two mechanisms depending on c and crowding number (CN). CN is defined as the number of fibres intersecting a sphere, centred on the middle of a particular fibre, with a diameter equal to the length of the fibre. Hence for higher consistencies, the CN is also generally higher. For low c and/or CN the predominant mechanism is filtration. In filtration fibres are highly mobile and fibre-fibre interactions are low. For higher c and/or CN thickening dominates. In this mechanism fibre-fibre interactions are high and the developing mat acts more like a tangle than a free suspension. As mentioned previously, formation is generally done at low consistency and low crowding number. Because the primary concern is in the initial stages of formation, before the fibre mat has formed, the dominating flow will be drainage undergoing the filtration mechanism. This indicates that the fibre-fibre interactions are low and can be neglected, justifying the elimination of the solid phase.  A.3  Geometry Simplification  Another complicating factor in the investigation of a forming fabric is the complicated geometry. These geometries, shown in Figs. 2.1 and 2.2 on page 5, have associated complex three dimensional flow fields. For the purpose of a preliminary investigation, modelling of the full 3D flow is difficult to verify and impractical to apply. For this reason a justifiable simplification is required. The affects of the geometry of a forming fabric on the formation process have been investigated in the past. In 1978 Torbjorn Helle published a paper investigating the influence of the forming fabric geometry on drainage rate[49]. In this paper he showed that if the prominent knuckles of the forming fabric are aligned in the CD then drainage rates are higher. This indicates that the cross direction filaments play an important part in the support of the fibre web, resisting fabric clogging. Then, in 1979, Robert Beran published the first paper addressing the subject of forming fabric classification that extended beyond the typical  52  Appendix A. Experimental Design  Figure A.1: The support of CD filaments[44]. percent open area, strand diameter, and air permeability[45]. In his paper he sought an off-machine characterization of forming fabrics based not only on the typical parameters, but also dependent on the surface structure of the fabric under investigation. To this end Beran developed the Fabric Support Index (FSI) FSI is a function of a number of different geometric and operational parameters. It accounts for the weave density in MD and CD, fibre length distribution, and fibre orientation. One of the results of his analysis is that the CD filaments in a forming fabric play a larger role in support of the fibre mat than the MD filaments[45]. This suggests that one possible geometrical simplification would be to isolate the effect of the CD filaments through use of a 2D geometry consisting of only cross-machine direction filaments. Hella and Beran’s work was expanded upon by D.B. Johnson in 1986 with the development of the Drainage Index (DI)[50]. The DI aims to predict the drainage rate of a fabric based upon the support of individual fibres and the associated blocking of flow passages. His work showed that the CD filaments played a larger role in the support of the fibre web, resisting blockage of the forming fabric and consequently allowing higher drainage rates. Another important paper influencing the simplification of a model forming fabric was published by Sabit Adanur in 1994. In his paper Adanur investigated the affect of the Beran FSI and the Johnson DI on a number of physical sheet properties, including tear strength, breaking load, burst strength, sheet thickness and sheet density[44]. Adanur also demonstrates 53  Appendix A. Experimental Design  CD z face  Figure A.2: Cross section of a forming fabric showing the CD filaments contacting the CD face[Adapted from 43].  XS  XC  MD-z face $           Figure A.3: The resulting 2D bank of cylinders obtained from the above simplifications. the previously seen effects of the CD filaments, justifying their additional importance in web support using qualitative means. Fig. A.1 shows how the CD filaments support fibres that are preferentially aligned in the MD due to the headbox hydrodynamics. These papers show that an effective simplification of a forming fabric would be the elimination of the machine direction filaments and consideration of only the cross-machine direction filaments. In the work presented in this thesis, the CD filaments are represented by a series of linear cylinders. Figure A.2 shows a cross-section of a forming fabric in the MD-z and CD-z planes. If the CD filaments that contact the MD-z face are considered, the 2D bank of cylinders shown in Fig. A.3 are obtained. This flow field is more easily investigated as flow through a bank of cylinders of different dimensions and is easily applied to the development of future forming fabrics. 54  Appendix A. Experimental Design  A.4  Flow Velocity and Reynolds number  The final flow condition that must be considered is the Reynolds number of the flow through a forming fabric. There is a relatively small amount of research in this area when compared to the impact of the geometry of the fabric, but a paper published by Barbara Dalpke in 2004 addressed the headbox jet impingement in twin wire formers using numerical methods [47]. Dalpke investigated the drainage flow for single wires[46] and twin wire gap formers[47]. Her numerical results showed drainage velocities can spike as high as 4.5m/s on single wire machines with high impingement angle[46], but showed that velocities are more commonly in the range of 0.05 to 0.5m/s on twin wire machines[47]. Using a forming fabric with paper-side filaments of ∼0.15 mm diameter and modelling the stock solution as pure water, the Reynolds number range is approximately 6.5 to 65. The value of Re = 6.5 is seen only in the later stages of formation, when the paper web will have a significant impact on the drainage flow. Consequently, a Re ≈ 65 will be seen in the initial drainage and initial layup of the paper. The literature above makes it possible to apply the following simplifications and flow conditions: 1. Elimination of the fibres due to the low consistency and CN. 2. Elimination of the MD filaments due to their smaller role in sheet support. 3. Test conditions using 6.5 ≤ Re ≤ 65 based on typical fabric geometries and formation flow velocities.  55  Bibliography [43] Astenjohnson forming fabrics page. http://www.astenjohnson.com/ , October 2005. [44] S. Adanur. Effects of forming fabric structural parameters on sheet properties. Tappi Journal, 77(10):187–195, Oct 1994. [45] R.L. Beran. The evaluation and selection of forming fabrics. Tappi, 62(4):39–44, Apr 1979. [46] B. Dalpke, S.I. Green, and R.J. Kerekes. Modelling of jet impingement in twinwire paper-machines: impingement on one fabric. TAPPI Engineering Conference, pages 513–527, Sep 2000. [47] B. Dalpke, R.J. Kerekes, and S.I. Green. Modelling jet impingement and the initial drainage zone in roll forming. Journal of Pulp and Paper Science, 30(3):65– 70, March 2004. [48] R. Danby. Sc print quality influenced by fibre length, fabric structures, and machine drainage characteristics. Tappi Journal, 1(9):3–9, Nov 2002. [49] T. Helle. How forming fabric design affects drainage and release. Pulp & Paper Canada, 79(11):91–98, November 1978. [50] D.B. Johnson. Retention and drainage of multi-layer fabrics. Pulp & Paper Canada, 87(5):56–59, May 1986. [51] J.H. Jong, W.D. Baines, and I.G. Currie. Experimental characteristics of forming fabrics and fibre mats. Journal of Pulp and Paper Science, 25(3):95–99, Mar 1999. [52] J.D. Parker. The Sheet-Forming Process. Tappi Press, Atlanta, GA, 1972.  56  Appendix B Experimental Setup and Methods B.1  Introduction  Three methods of data collection were used to collect the principal data for the evaluation of the forming fabric. Particle image velocimetry (PIV) and hydrogen bubble generation were used to evaluate the flow approaching the model forming fabric and pressure drop measurements were used to evaluate the flow through the model forming fabric. The velocity profile of the forming fabric approach flow is of primary importance to the evaluation of the possible wire marking. If the flow velocity is higher in one region than another more pulp and fines will be directed to that location, increasing the density of the paper over that flow path, creating a hydrodynamic wire mark. The pressure drop results provide an indication of the drainage rate of the forming fabric. The lower the pressure drop at a given Re the faster the drainage. Each of these three methods will be discussed in more detail following a description of the actual test facility.  B.2  Experimental Setup  There were three major stipulations for the test apparatus that was designed for these experiments. It had to accommodate the experimental methods discussed in §B.1, it was desirable to have it capable of continuous operation, and it had to include a configurable model forming fabric.  In order for it to be capable of continuous operation it was decided that a flow loop configuration would be used. The loop would be driven by a 20 HP variable speed pump and would be located at the University of British Columbia’s, Pulp and Paper Centre. Flow would exit the pump into a 1.5” diameter line where velocity measurement would be made via a venturi with a 0-15pisd differential pressure sensor for high Re measurements and a 0-0.5psid differential pressure sensor for the low Re velocity measurements.  57  Appendix B. Experimental Setup and Methods The actual test section had a frontal projection of 30x30cm and was constructed of acrylic so as to be clear for PIV, and hydrogen bubble observation. The test section contained a recess to accommodated a frame that held cylinders that acted as the model forming fabric.  $ $&           Figure B.1: Cylinder frame equipt with 9.53mm and 19mm diameter cylinders. The frame was fitted with mounting holes that were spaced out by 9.53mm (0.375”) perpendicular to the test section flow and 19mm (0.75”) parallel to the test section flow. It was fitted with cylinders of three diameters, 9.53mm (.375”), 19mm (.75”) and 28.58mm (1.125”). The maximum cylinder diameter was chosen such that the centre of the test section would be as close as possible to 7 diameters from end of the cylinders while keeping with standard cylinder diameters[55]. Since PIV was to be taken upstream of the cylinder bank, and not interstitially, they did not need to be fabricated from acrylic. An image of the frame is shown in Fig. B.1. The test section pressure ports were located upstream and downstream of the cylinder frame so that differential pressure measurements could be made across the bank of cylinders. Ports for the H2 generation wire were installed in the middle of the test section upstream of the cylinder bank so that a 0.002” diameter platinum wire could be suspended across the test section for hydrogen bubble generation. Both of these can be seen in Fig. B.2. 58  Appendix B. Experimental Setup and Methods  $            Figure B.2: Locations of the pressure and hydrogen bubble ports  Figure B.3: The test section fully installed. Flow is from left to right  59  Appendix B. Experimental Setup and Methods  Figure B.4: Schematic of the Test Section Flow Loop The full test section is shown in Fig. B.3. Figure B.4 shows a schematic of the flow loop’s final configuration.  B.3  Test Section Flow Characteristics  In order to have confidence in the data produced, the test section flow uniformity had to be evaluated at different velocities. This was done using both PIV and hydrogen bubble generation. To evaluate the flow uniformity using hydrogen bubbles a high definition digital video camera (Sony HDV Handycam Camcorder, Model HDR-FX1) was placed under the test section and lines of hydrogen bubbles were generated in the flow. Individual images were then captured from the video and hydrogen bubble line was digitized into coordinate data with respect to an origin defined coincident with the hydrogen bubble wire. The distance travelled was normalized by the average distance travelled. This method was not used to find the test section velocity. The images from the test section flow uniformity tests are shown in Figs. B.5 and B.6 and the corresponding digitized data is shown in Fig. B.7  60  Appendix B. Experimental Setup and Methods  Figure B.5: Image of test section flow at 2cm/s. Flow is from bottom to top of image.  61  Appendix B. Experimental Setup and Methods  Figure B.6: Image of test section flow at 4cm/s. Flow is from bottom to top of image.  62  Appendix B. Experimental Setup and Methods  Figure B.7: Data extracted from Figs. B.5 and B.6 The hydrogen bubble generation system showed that the velocity in the test section was even to within 5% over 83% of the test section for both 2cm/s and 4cm/s at a viscosty of 15.17cP. The test section velocity and flow uniformity were also investigated using the PIV system. The setup of the PIV will be detailed in §B.4, however the empty test section results will be discussed here.  $            Figure B.8: Normalized test section velocity with a 135.3mm field of view. 63  Appendix B. Experimental Setup and Methods  $          Figure B.9: Empty test section velocity vectors at location of PIV data collection. The camera was setup below the test section with a field of view of 135.3mm, the largest field of view used in the PIV tests. The velocity was set by the venturi to 3.89cm/s, at a viscosity 13.26cP, corresponding to Red = 65 for the 19mm diameter cylinders. The results were normalized by the set velocity and plotted as a function of position in pixels. Figure B.8 shows the resultant velocity plot and Fig. B.9 shows the vectors plotted on a PIV image. As can be seen in Fig. B.8 the PIV shows the average test section velocity to be 4.05cm/s, 4.1% higher than the 3.89cm/s that was set using the venturi. The velocity was also verified using observations of the flow in the test section. A mark was placed on the side of the test section 10cm downstream from the hydrogen bubble generation wire. Hydrogen bubbles were timed travelling down the test section and the velocity computed. The average time for 10cm was 2.534 seconds with a standard deviation of 0.085. This gives an average velocity of 3.95cm/s. Assuming a position uncertainty of ±2mm, and a time uncertain of ±1σ, the velocity range is Vmax = 4.16cm/s and Vmin = 3.74cm/s. The velocity data is shown in Fig. B.10. The range of velocities from the time/displacement method include both the PIV measurements with an average velocity of 4.05cm/s ± 0.06cm/s 64  Appendix B. Experimental Setup and Methods  4.6 4.5 Accepted Data  4.4  Measured Velocity, cm/s  Rejected Data Average of Good Data  4.3 4.2 4.1 4 3.9 3.8 3.7 3.6  5  10  15  20  25  Run  Figure B.10: Stopwatch measured test section velocity for comparison with PIV. and the venturi pressure drop measurements with an average of 3.89cm/s ± 0.07cm/s.  B.3.1  Test Section Sensor Details  The specific equipment used to determine flow loop conditions were: • High Velocity Pressure Sensor: Sensotec Model FDW Range: 0 – 15 psid Part #: 060-G763-03 Order Code: FDW2BJ,2D5B6Q S/N: 1009322 • Low Velocity Pressure Sensor: Sensotec Model FDW Range: 0 – 0.5 psid Part #: 060-G250-09 Order Code: FDW1AN,2G5A6A S/N: 1094994 Date: 5/12/2006 • Viscosity Measurement: Gilmont Instruments Viscosimeter Size # 2 Model GV-2200 S/N: 24574 65  Appendix B. Experimental Setup and Methods  B.4  PIV Setup  $            Figure B.11: The interrogation areas from the two masked PIV images with a known time separation Particle image velocimetry (PIV) is a method for finding the velocity of a flow field with minimal interference. To calculate PIV velocity profiles the flow was seeded with small (10 micron in this case), neutrally buoyant particles. The the particles were then illuminated by two light sheets that were spatially coincident and temporally separated by a known time. The positions of the seeding particles illuminated by the two flashes were captured as two separate images by a digital camera. These two images were then masked to maximize calculation efficiency and divided into small interrogation areas which were then compared on a one-to-one basis between the two images. The result was an average velocity for each interrogation area. This process is shown in Figs. B.11 and B.12. The PIV system was setup with the laser sheet directed from the side of the test section, aimed to illuminate the area upstream of the cylinder bank. Imaging was done from below using a 1 mega-pixel, 8-bit digital camera. A drawing of the setup is shown if Figure B.13. To determine if there was a consistent offset error in the PIV system, the velocity in the empty test section was evaluated by PIV and then 66  Appendix B. Experimental Setup and Methods  $            Figure B.12: The two interrogation areas are then compared by cross-correlation to find the average particle displacement and hence velocity. compared to velocities measured by a manual time/displacement method. The results of this analysis were discussed in §B.3.  B.4.1  PIV Equipment Details  The specific equipment used in the PIV analysis was: • Signal Generator: Berkeley Nucleonics Corporation Model 500D S/N: 22659 • Lasers: New Wave Research Gemini PIV 15Hz Laser-1 S/N: 10288, Aug 2000 Laser-2 S/N: 10289, Aug 2000 • Camera: Roper Scientific MEGAPLUS Model ES 1.0 S/N: 93036000CS3NA  B.5  Pressure Drop Setup  The pressure drop measurements were made directly. The wet/wet differential pressure sensor had a range of 0-0.1” water with a 10V DC output,  67  Appendix B. Experimental Setup and Methods  Figure B.13: PIV arrangement. and was manufacturer calibrated to an accuracy of ±0.25%FS. The pressure was measured across the bank of cylinders and the output was read off an occiliscope after a consistent reading for a minimum of 60 seconds. The locations of the pressure ports are visible in Fig. B.2 on page 59.  B.5.1  Pressure Drop Equipment Details  The specific equipment used for the pressure drop measurements were: • Pressure Drop Sensor: Validyne Engineering Model DP103-06-10306N1S4D S/N: 127315 • Carrier Demodulator: Validyne Engineering Model CD15-A-1-A-1 S/N: 118923  B.6  Hydrogen Bubble Generation System Setup  The hydrogen bubble generation system was implemented to provide the capability to visualize time dependent variations in the flow field and provide quick, qualitative flow field visualization. It also provided a means to evaluate the flow conditions of the test section, making sure that flow was even and steady at multiple velocities. The bubbles were generated off of a 0.002” diameter platinum wire. Hydrogen bubbles released off of the wire can be expected to have a diameter of about half that of the generating wire, so the bubbles produced were  68  Appendix B. Experimental Setup and Methods nominally 0.001” in diameter[53]. It was necessary to find the rise velocity of the bubbles in order to show that they would follow the test section flow. For Stokes flow the drag coefficient is known to be[54], 24 for Re ≤ 1 Re FD CD = 1/2 · ρ · U 2 · l2 CD =  (B.1) (B.2)  To find the terminal velocity of the hydrogen bubbles in water the buoyancy equation reduces to, FD = (ρf luid − ρbubble ) · V · g FD = 7.20693E − 11N  (B.3)  Combing equations B.1 and B.2 and rearranging for velocity we get, FD 12 · µ · d 7.21E − 11 U = 12 · 15E − 5 · 2.54E − 5 U = 1.31E − 5m/s  U =  (B.4)  The running velocity of the test section is variable, but is of the order of 10−2 m/s, or three orders of magnitude greater than the rise velocity of the hydrogen bubbles. It is therefore possible to neglect the bubble’s perpendicular velocity.  69  Bibliography [53] Richard J. Goldstein, editor. Fluid Mechanics Measurements. Taylor and Francis, Washington, DC, second edition, 1996. [54] B.R. Munson, D.F. Young, and T.H. Okiishi. Fundamentals of Fluid Mechanics. John Wiley and Sons, Inc., New York, New York, 1998. ¨ [55] D. Sumner, S.S.T. Wong, S.J. Price, and M.P. Padoussis. Fluid behaviour of sideby-side circular cylinders in steady cross-flow. Journal of Fluids and Structures, 13:309–338, 1999.  70  Appendix C Uncertainty Analysis C.1  Single Valued, Multi-Variable Functions  The uncertainty analysis used in this project followed the general engineering method. A single valued, multi-variable function, such as the Reynolds number, is differentiated by each of its variables. The total uncertainty associated with a given function is then the vector sum of the changes created by each variable independently. The equations for the uncertainty of the various functions are given along with screen shots of the spread sheets used to calculate them. Density uncertainty was found from: ρ =  m V  (C.1)  ⎡  ∂ρ δρ = ⎣ δm ∂m  2  ∂ρ + δV ∂V  ⎤ 2 1/2 ⎦  (C.2)  Where, 1 ∂ρ = ∂m m m ∂ρ = − 2 ∂V V Inlet pipe velocity uncertainty was found from:  Upipe = k  ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ρ  ⎡  δUpipe Where,  2 ∆P D1 D2  4  ∂Upipe = ⎣ δk ∂k ⎧ ⎪ ⎪ ⎨  ∂Upipe = ⎪ ∂k ⎪ ⎩ρ  ⎫1/2 ⎪ ⎪ ⎬  2 ∆P D1 D2  4  (C.3)  ⎪ ⎭ −1 ⎪ 2  ∂U + δP ∂∆P  2  ∂U + δρ ∂ρ  ⎫1/2 ⎪ ⎪ ⎬  ⎪ ⎭ −1 ⎪  71  2  ∂U + δD1 ∂D1  2  ⎤ 2 1/2  ∂U ⎦ + δD2 (C.4) ∂D2  Appendix C. Uncertainty Analysis ⎧ ⎪ ⎪ ⎨  ∂Upipe 2 = k⎪ ∂∆P ⎪ ⎩ 4 ∆P ρ D1 D2  ⎫1/2 ⎪ ⎪ ⎬ 4  ⎪ ⎭ −1 ⎪  ⎧ ⎪ ⎪ ⎨  ⎫1/2 ⎪ ⎪ ⎬  2 ∆P ∂Upipe = −k ⎪ 4 ⎪ ∂ρ ⎪ ⎩ 4ρ3 D1 ⎭ −1 ⎪ D2 √ −2D13 D22 ∆P 2 ∂Upipe = k 2 ∆P ρ ∂D1 (D14 − D24 ) (D14 −D24 )ρ ⎧ ⎫ ⎪ √ ⎪ ⎪ ⎪ ⎨ ∂Upipe 2D25 ∆P 2 −2∆P ⎬ = k⎪ + 2D2 2 −∆P ρ ∂D2 (D24 − D14 ) ρ ⎪ ⎪ ⎪ ⎩ (D24 − D14 ) ⎭ 4 4 (D2 −D1 )ρ Test section velocity uncertainty was found from: UT S =  πUpipe D2 4HW  (C.5)  ⎡  ∂UT S δUpipe = ⎣ ∂Upipe  δUT S Where, ∂UT S ∂Upipe ∂UT S ∂D ∂UT S ∂H ∂UT S ∂W  2  ∂UT S + δD ∂D  2  ∂UT S + δH ∂H  2  ∂UT S + δW ∂W  ⎤ 2 1/2  ⎦ (C.6)  πD2 4HW Upipe πD = 2HW Upipe πD2 = − 4H 2 W Upipe πD2 = − 4HW 2 =  Viscosity uncertainty was found from: µ = k (ρf luid − ρball ) t ⎡  δµ = ⎣  ∂µ δk ∂k  2  ∂µ + δt ∂t  (C.7)  2  ∂µ + δρf luid ∂ρf luid  Where, ∂µ = (ρf luid − ρball ) t ∂k ∂µ = k (ρf luid − ρball ) ∂t ∂µ = kt ∂ρf luid 72  2  ∂µ + δρball ∂ρball  2  ⎤1/2  ⎦ (C.8)  Appendix C. Uncertainty Analysis ∂µ = −kt ∂ρball Reynolds Number uncertainty was found from: Re =  ρU d µ  (C.9)  ⎡  ∂Re δρ δRe = ⎣ ∂ρ Where, ∂Re ∂ρ ∂Re ∂U ∂Re ∂d ∂Re ∂µ  2  ∂Re + δU ∂U  2  ∂Re + δd ∂d  2  ∂Re + δµ ∂µ  ⎤ 2 1/2 ⎦  (C.10)  Ud µ ρd = µ ρU = µ ρU d = − 2 µ =  Pressure drop coefficient uncertainty was found from: k =  ∆P  (C.11)  1 ρUT2 S 2  ⎡  ∂k δ (∆P ) δk = ⎣ ∂∆P  2  ∂k + δρ ∂ρ  2  ∂k + δUT S ∂UT S  ⎤ 2 1/2 ⎦  (C.12)  Where, ∂k 1 = 1 2 ∂∆P ρUT S 2 ∆P ∂k = −1 2 ∂ρ ρUT S 2 2∆P ∂k = −1 3 ∂UT S ρUT S 2 These equations were entered into a Microsoft Excel spread sheet and the uncertainty was calculated. A screen shot of the spread sheet is shown in Fig. C.1. The venturi constant uncertainty was found using time displacement data from the test section H2 bubble generation system. Since it only needed to be evaluated once, MatLab was used to find the derivatives of the function for k and Microsoft Excel was used to evaluate the functions. The MatLab command window is give below. 73  Appendix C. Uncertainty Analysis  Figure C.1: Screen shot of the spread sheet used to calculate uncertainty.  74  Appendix C. Uncertainty Analysis >> syms k s h w pi t d1 p r d2 >> k = (s*h*w)/(pi/4*t*d1^2*((2*p)/(r*((d1/d2)^4-1)))^0.5) k = 2*s*h*w/pi/t/d1^2*2^(1/2)/(p/r/(d1^4/d2^4-1))^(1/2) >> diff(k,s) ans = 2*h*w/pi/t/d1^2*2^(1/2)/(p/r/(d1^4/d2^4-1))^(1/2) >> diff(k,h) ans = 2*s*w/pi/t/d1^2*2^(1/2)/(p/r/(d1^4/d2^4-1))^(1/2) >> diff(k,w) ans = 2*s*h/pi/t/d1^2*2^(1/2)/(p/r/(d1^4/d2^4-1))^(1/2) >> diff(k,t) ans = -2*s*h*w/pi/t^2/d1^2*2^(1/2)/(p/r/(d1^4/d2^4-1))^(1/2) >> diff(k,d1) ans = -4*s*h*w/pi/t/d1^3*2^(1/2)/(p/r/(d1^4/d2^4-1))^(1/2)+4*s*h*w/pi/t* >d1*2^(1/2)/(p/r/(d1^4/d2^4-1))^(3/2)*p/r/(d1^4/d2^4-1)^2/d2^4 >> diff(k,d2) ans = -4*s*h*w/pi/t*d1^2*2^(1/2)/(p/r/(d1^4/d2^4-1))^(3/2)*p/r/(d1^4/d2^ >4-1)^2/d2^5 >> diff(k,p) ans = -s*h*w/pi/t/d1^2*2^(1/2)/(p/r/(d1^4/d2^4-1))^(3/2)/r/(d1^4/d2^4-1) >> diff(k,r) ans = s*h*w/pi/t/d1^2*2^(1/2)/(p/r/(d1^4/d2^4-1))^(3/2)*p/r^2/(d1^4/d2^4-1) Where, k s h w t d1 d2 p r  = = = = = = = = =  Venturi constant H2 bubble line displacement Test section height Test section width H2 bubble line time Venturi inlet diameter Venturi throat diameter Pressure drop Fluid density  The spread sheet for calculation of the venturi constant and its uncertainty is shown in Fig. C.2. 75  Appendix C. Uncertainty Analysis  Figure C.2: Screen shot of the spread sheet used to find the venturi constant and uncertainty.  C.2  PIV uncertainty  The PIV uncertainty is a two part process. First it must be determined if there is a consistent offset in the PIV data. This analysis was discussed in §B.3. It must also be determined whether a sufficient number of images pairs were taken to have confidence in the PIV values. Using the data from the empty test section test shown in Fig. B.9 the standard deviation for each velocity vector was examined. It was found that the maximum standard deviation after averaging over 70 image pairs was 14% of the average value. While this is quite high, it was only the case for 3 out of 1476 vectors. The averaga standard deviation was 2.42% of the average value over all cross-correlated vectors. This provides confidence in the avaerage values of the vectors found via PIV. The uncertainty in each average vector was found using a Student’s tdistribution. The PIV software returned the average, the standard deviation, and the number of valid vectors used to find these quantities. These quantities were then placed into equation C.13 to find the 80% confidence uncertainty. V elocity = Vmean ±  76  t0.1 · σV √ N  (C.13)  Appendix C. Uncertainty Analysis The one-sided value of t0.1 = 3.078 gives a ± confidence of (1 − 2 · 0.1) = 0.8.  The representations of the typical uncertainty, such as the bars shown on the plots in §5.4, were found using a two part process. First, the uncertainties for the regions of high velocity gradient, such as the vicinity of y/G1 = 2 in Fig. 5.4 were neglected. These uncertainties tended to be quite high due to the different velocities that existed within a given interrogation area and were not considered representative of the plot’s general uncertainty. Second, the maximum uncertainty in regions of low velocity gradient, such as in the vicinity of y/G1 = 1.75 in the same figure, was taken as typical of the plot uncertainty.  77  

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