MIXING DYNAMICS OF PULP SUSPENSIONS IN CYLINDRICAL VESSELS by KWOK WAI LEO HUI B.A.Sc., The University of British Columbia, Vancouver, Canada, 1996 M.Eng., The University of British Columbia, Vancouver, Canada, 1998 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADUATE STUDIES (CHEMICAL AND BIOLOGICAL ENGINEERING) THE UNIVERSITY OF BRITISH COLUMBIA (Vancouver) April 2011 Kwok Wai Leo Hui, 2011 ii Abstract Cylindrical agitated chests are frequently used to facilitate manufacturing processes in pulp and paper industry and one of their main functions is to attenuate any process disturbances. However, owing to the inherited non-Newtonian nature of pulp suspensions, it is not easy to achieve complete mixing and with the improper chest design, these agitated chests do not always perform ideally or satisfactorily. The cavern formation in incomplete mixing may induce bypassing and dead zones, which significantly affect the chest performance. A study of cavern formation in a cylindrical agitated chest was thus carried out. Also, a dynamic model developed by Soltanzadeh et al. (2009) was used to quantify the mixing dynamics of the cylindrical chest. In addition, using computational fluid dynamics (CFD), the simulated results of the flow in the chest were compared with the experimental results to verify the applicability of the CFD model on the study of pulp suspension agitation. To investigate the cavern formation in a lab-scale cylindrical chest, electrical resistance tomography (ERT) and ultrasonic Doppler velocimetry (UDV) were applied to estimate the cavern shape and size. Both methods gave satisfactory results and as expected, the cavern size was found to increase with impeller speeds. The cavern shape was best described as a truncated right-circular cylinder. Based on this observation, a model considering the interaction between the cavern and chest walls was developed to calculate the cavern volume. With the dynamic model, a series of dynamic tests were carried out to characterize the mixing behavior of the lab-scale cylindrical chest. It was found that the proposed flow configuration with the outlet close to the cavern could minimize the bypassing which affects mixing quality. Also, ERT verified the presence of cavern and dead zone when the chest was not completely agitated in continuous-flow operation. iii Numerical simulations using CFD were compared with the experimental results under different operating conditions. Pulp suspensions are a mixture of water and wood fibres that can entangle each other to form flocs affecting the mixing flow. Owing to this complex rheology, it is not easy to model the agitation precisely in CFD using a homogeneous fluid model. The floc formation and air entrapment observed in experiments were difficult to be numerically taken into account in the simulations. Although the CFD model could not exactly predict the mixing situation of pulp suspensions, it still can be used to estimate the mixing flow patterns, e.g., flow directions, in the proposed chest designs. iv Preface This thesis consists of 6 chapters and there are 2 co-written chapters: 3 CAVERN MEASUREMENT IN PULP SUSPENSIONS 4 DYNAMIC TEST STUDY ON LAB-SCALE CHEST Chapter 3 describes the measurement of cavern formation in pulp suspension agitation. Proposed by my supervisor, Chad Bennington, electrical resistance tomography (ERT) and Ultrasonic Doppler velocimetry (UDV) were applied to estimate the cavern shape and size. I selected the conductive tracer to identify the cavern and designed the measurement method to determine its size. After carrying out the experiments, I analyzed the data, developed a cavern model with the help of Chad Bennington and wrote the manuscript. Finally, my two supervisors, Chad Bennington and Guy Dumont, revised and edited the manuscript. A version of this chapter has been published as “Cavern formation in pulp suspensions using side-entering axial-flow impellers” in Chemical Engineering Science Journal (64) on pages 509-519 in 2009. Chapter 4 illustrates the study of mixing dynamics in a lab-scale cylindrical chest. Using a pseudo-random binary signal (PRBS) to control the addition of a conductive tracer, I carried out a series of tests to analyze the mixing dynamics of the chest. With the input-output data and the dynamic model, I estimated the model parameters using MATLAB and concluded the chest performance. Finally, I wrote the manuscript which was revised and edited by my two supervisors. A version of this chapter will be submitted for publication as “Mixing Dynamics in Cylindrical Pulp Stock Chests” in 2011. v Table of Contents Abstract ........................................................................................................................................... ii Preface............................................................................................................................................ iv Table of Contents ............................................................................................................................ v List of Tables ................................................................................................................................ vii List of Figures .............................................................................................................................. viii List of Symbols and Abbreviations............................................................................................... xii Acknowledgements ..................................................................................................................... xv 1 INTRODUCTION .................................................................................................................. 1 1.1 Thesis background............................................................................................................ 1 1.2 Pulp suspension rheology ................................................................................................. 2 1.3 Agitated chest design ....................................................................................................... 6 1.4 Research objectives .......................................................................................................... 9 2 STUDY METHODS ............................................................................................................. 11 2.1 Electrical resistance tomography (ERT) ........................................................................ 11 2.2 Ultrasonic Doppler velocimetry (UDV) ......................................................................... 14 2.3 Computational fluid dynamics (CFD) ............................................................................ 17 3 CAVERN MEASUREMENT IN PULP SUSPENSIONS ................................................... 23 3.1 Introduction .................................................................................................................... 23 3.2 Experimental set-up and procedure ................................................................................ 26 3.3 Results and discussion .................................................................................................... 30 3.3.1 Selection of cavern measurement technique ........................................................... 30 3.3.2 Effect of impeller speed and pulp mass concentration on cavern volume .............. 37 3.3.3 Effect of pulp type on cavern volume ...................................................................... 38 3.3.4 Impeller offset from the rear wall of the mixing chest ............................................ 39 3.3.5 Comparison of cavern volumes determined by ERT and the cavern models .......... 40 3.3.6 Cavern model including interaction with vessel walls ........................................... 44 3.4 Summary ........................................................................................................................ 52 4 DYNAMIC TEST STUDY ON LAB-SCALE CHEST ....................................................... 53 4.1 Introduction .................................................................................................................... 53 4.2 Experimental set-up and procedures .............................................................................. 56 4.3 Results and discussion .................................................................................................... 61 4.4 Summary ........................................................................................................................ 75 5 CFD SIMULATION OF PULP MIXING ............................................................................ 76 5.1 Introduction .................................................................................................................... 76 5.2 Experimental study of pulp mixing ................................................................................ 79 5.3 CFD modeling of pulp mixing ....................................................................................... 80 5.3.1 Computational geometry ......................................................................................... 80 5.3.2 Modelling suspension rheology .............................................................................. 82 5.3.3 Condition setup for CFD model .............................................................................. 84 5.3.4 Cavern volume determination and dynamic test simulation ................................... 86 5.4 Results and discussion .................................................................................................... 87 5.5 Summary ...................................................................................................................... 108 vi 6 OVERALL CONCLUSIONS AND RECOMMENDATIONS FOR FUTURE WORK ... 109 6.1 Overall conclusions ...................................................................................................... 109 6.2 Recommendation for future work ................................................................................ 112 BIBLIOGRAPHY ....................................................................................................................... 114 APPENDIX A: Details of ERT components .............................................................................. 120 APPENDIX B: MATLAB program for dynamic model parameter estimation ......................... 124 APPENDIX C: Program for cavern volume determination in CFD ........................................... 136 APPENDIX D: Program for tracer analysis in CFD .................................................................. 137 APPENDIX E: ERT data ............................................................................................................ 138 vii List of Tables Table 3.1 Comparison of ERT/UDV cavern measurements (Cm = 2% hardwood pulp, E/D = 0.6, C/D = 0.7 and Z/T = 1.0) ............................................................................................................... 36 Table 4.1 Time delay (Td) measured versus theoretical values for hardwood pulp Cm = 3% (N = 425, 525, 672 and 810 rpm at each flow rate) .............................................................................. 61 Table 4.2 Time constant (2) measured for complete mixing of hardwood pulp at Q = 14 L/min (Theoretical 2 = 149s) .................................................................................................................. 62 Table 4.3 Dynamic test results ...................................................................................................... 64 Table 4.4 Calculation of impeller speed on scale-up using different criteria ............................... 73 Table 4.5 Power predicted for complete mixing of a hardwood pulp suspension in cylindrical stock chests using different scale-up criteria (D/T = 0.43, Z/T = 0.8) .......................................... 73 Table 5.1 Rheological parameters of hardwood and softwood pulps ........................................... 84 Table 5.2 Grid independence study (N = 425rpm, 3% softwood pulp (K1 = 135, n = 0.21, y = 155Pa) ........................................................................................................................................... 91 Table 5.3 Comparison of results from CFD and dynamic tests for Cm = 2% hardwood (Q = 14L/min., Z/T = 0.8) .................................................................................................................... 103 Table 5.4 Comparison of results from CFD and dynamic tests for Cm = 3% softwood (Q = 14L/min., Z/T = 0.8) .................................................................................................................... 103 viii List of Figures Figure 1.1 A generalized stress-rate curve for a fiber suspension (Gullichen and Harkonen, 1981). 4 Figure 1.2 Cross-section of a reduced-bottom chest. ...................................................................... 8 Figure 2.1 Operation of electrical resistance tomography (Malmivuo and Plonsey, 1995). ........ 12 Figure 2.2 Image reconstruction grid (Industrial Tomography Systems, 2007). .......................... 13 Figure 2.3 Principle of UDV showing the measurement of velocity along the measuring line (Takeda, 1995). ............................................................................................................................. 16 Figure 2.4 Cell types used in computational grid (Ferziger and Peric, 1999). ............................. 20 Figure 2.5 2D structured and unstructured grids (Ferziger and Peric, 1999). .............................. 20 Figure 3.1 Cross-section of cylindrical stock chest showing ERT sensor planes and range of impeller positions used. ................................................................................................................ 28 Figure 3.2 Suspension yield stress as a function of mass concentration. ..................................... 29 Figure 3.3 Photograph of conductive tracer particles used in tests. Individual particles range in diameter from 4 to 6 mm (typically 5 mm) and length from 15 to 20 mm (typically 17.5mm). .. 32 Figure 3.4 ERT images obtained for a Cm = 3% hardwood pulp suspension agitated at N = 650 rpm with E/D = 0.4, C/D = 0.7 and Z/T = 1.0. Images on the left were reconstructed using data obtained with tracer particles (Fig. 3.3) and images on the right were reconstructed by adding a conductive fluid of 10.9 mS/cm NaCl (15mL) to a pulp suspension having a background conductivity of 0.2 mS/cm (43.4L). The impeller is located between planes 2 and 3 and electrodes 5 and 6 as shown for reference purposes in the upper left image. ............................... 33 Figure 3.5 Comparison of ERT images reconstructed using the tracer particles (left) and saline solution (right). Uppermost sampling plane (P5). The visual surface view is given in the centre image with the dotted line enclosing the region of active surface motion (determined visually). (Hardwood pulp suspension at Cm = 3%; N = 650 rpm, E/D = 0.4, C/D = 0.7, Z/T = 1.0). ......... 34 Figure 3.6 Probe locations for UDV measurements. The probe is moved around the chest to obtain the velocity profiles needed to map the cavern boundary. The arrows show the direction of the sonic emissions and the numbers give the ERT electrode positions....................................... 34 Figure 3.7 (a) Velocity profile measured by UDV for a representative probe location. The locations where the velocity reaches zero indicate the cavern boundary. (b) Cavern cross-section formed by connecting the boundary points measured from all sample locations. ........................ 35 ix Figure 3.8 Comparison between ERT and UDV caverns made for hardwood pulp at Cm = 2% and N = 200 rpm (E/D = 0.6, C/D = 0.7 and Z/T = 1.0). ..................................................................... 36 Figure 3.9 Cavern shape (shaded volume) determined for a Cm = 2% hardwood at various impeller speeds using ERT data (E/D = 0.6, C/D = 0.7, Z/T = 1.0).............................................. 37 Figure 3.10 Effect of impeller speed (N) and pulp consistency (Cm) on cavern volume (Vc ) for hardwood and softwood pulps (E/D = 0.4, C/D = 0.7 and Z/T = 0.8). ......................................... 39 Figure 3.11 Effect of impeller position (E/D) on cavern volume (Vc) with (a) rotation speed and (b) power (Cm = 2% hardwood, C/D = 0.7, Z/T = 1.0). ................................................................ 41 Figure 3.12 Comparison of measured cavern volumes and predictions made using the equations of Amanullah et al. (1998) and Elson (1990). .............................................................................. 42 Figure 3.13 Axial force number (Nf) (a) and power number (NP) (b) for the D = 140mm Maxflo impeller as a function of impeller speed and suspension mass concentration. ............................. 43 Figure 3.14 Diagram showing intersecting circles with the impeller located at the centre of a virtual cylinder. The overlapping area between the mixed zone and mixing chest is the cavern region. ........................................................................................................................................... 46 Figure 3.15 Rotation speed (a) and wall stress calculated (b) for complete mixing in the vessel with hardwood pulp as a function of suspension mass concentration (E/D = 0.4, C/D = 0.7). .... 47 Figure 3.16 Comparison of wall stresses measured/estimated and suspension yield stress for hardwood pulp as a function of mass concentration. Estimates for w based on the yield stress (y - measured), the point at which the cavern first completely filled the mixing vessel, and friction losses in flowing pulp suspensions. .............................................................................................. 49 Figure 3.17 Cavern volume versus impeller speed for laboratory mixer. Comparison of experimental data (measured by ERT) with model predictions for hardwood pulp. Z/T = 0.8, E/D = 0.4. (a) Cm = 2%: y = 9 Pa ;w =12 Pa ; a = 0; Cm = 3%: y = 38 Pa ;w = 42 Pa ; a = 0; Cm = 4%: y = 69 Pa ;w =90 Pa ; a = 0. (b) (y = w = a) Cm = 2%: y = 9 Pa; Cm = 3%: y = 38 Pa; Cm = 4%: y = 69 Pa. .............................................................................................................................. 51 Figure 4.1Dynamic model for non-ideal flow in agitated pulp stock chests (2>>1). .................. 56 Figure 4.2 Cross-section (a) and photograph (b) of cylindrical stock chest. The arrows in (b) show the direction of suspension flow in continuous operation. .................................................. 59 Figure 4.3 Schematic of apparatus used for the dynamic tests . ................................................... 60 Figure 4.4 Typical signals measured for model parameter identification. ................................... 60 x Figure 4.5 ERT images of a dynamic test at Cm = 3% and N = 425rpm (E/D = 0.6, C/D = 0.7, Z/T = 0.8). The suspension inlet and outlet positions are shown in the images of P4 and P1 at time t = 0 s, respectively. ............................................................................................................................ 65 Figure 4.6 Effect of impeller speed (N) and pulp consistency (Cm) on time constant (2) (E/D = 0.6, C/D = 0.7, Z/T = 0.8). ............................................................................................................ 66 Figure 4.7 Effect of flow rate (Q) on time constant (2) with rotation speed (Cm = 3%, E/D = 0.6, C/D = 0.7, Z/T = 0.8). ................................................................................................................... 67 Figure 4.8 Effect of flow rate (Q) on mixing volume (Vmix) with rotation speed (Cm = 3%, E/D = 0.6, C/D = 0.7, Z/T = 0.8). ............................................................................................................ 67 Figure 4.9 Effect of pulp types (HW: hardwood and SW: softwood) on cavern size obtained in dynamic and batch (ERT) operations at Cm = 3% (E/D = 0.6, C/D = 0.7, Z/T = 0.8). ................. 69 Figure 4.10 Effect of pulp types (HW: hardwood and SW: softwood) with similar yield stress (38Pa) on mixing volume (Vmix) with rotation speed (N) (E/D = 0.6, C/D = 0.7, Z/T = 0.8). .... 69 Figure 4.11 Comparison between cavern size obtained in dynamic and batch (ERT) operations at various hardwood pulp consistencies (Cm) (E/D = 0.6, C/D = 0.7, Z/T = 0.8, Q = 14 L/min for the dynamic tests). .............................................................................................................................. 70 Figure 5.1 Experimental cylindrical chest and computational domain for (a) batch and (b) continuous-flow mixing. ............................................................................................................... 82 Figure 5.2 Locations where velocity samples were determined for comparison of three different meshes. .......................................................................................................................................... 89 Figure 5.3 Velocity profiles calculated for softwood Cm = 3% at two locations (a) and (b) in front of the impeller (as shown in Figure 2) for three different meshing schemes at N = 425rpm. ..... 90 Figure 5.4 Comparison of cavern size obtained from ERT and numerical simulations for a Cm = 2% hardwood (Z/T = 1.0). ............................................................................................................. 91 Figure 5.5 Comparison of cavern size obtained from ERT and numerical simulations for a Cm = 3% softwood (Z/T = 0.8). .............................................................................................................. 93 Figure 5.6 Flow fields of batch mixing for a) Cm = 2% hardwood at N = 200rpm using Bingham plastic model and b) Cm = 3% softwood at N = 475 rpm using Herschel-bulkley model. ........... 94 Figure 5.7 a) Top view and b) side view of probe locations for UDV measurements. The arrows show the direction of the sonic emissions at three heights (12.5mm, 92.5mm and 252.5mm measured from the chest bottom). ................................................................................................. 96 xi Figure 5.8 Comparison of measured and computed velocity profiles at three positions for a Cm = 2% hardwood at N = 200rpm. Positive velocity means the velocity direction is away from the probe. ............................................................................................................................................ 97 Figure 5.9 Comparison of measured and computed velocity profiles at three positions for a Cm = 2% hardwood at N = 250rpm. Positive velocity means the velocity direction is away from the probe. ............................................................................................................................................ 98 Figure 5.10 Comparison of measured and computed cavern volumes. (a) Cm = 2% hardwood pulp using Bingham plastic model for cavern determination (Z/T = 1.0); (b) Cm = 3% softwood pulp using Herschel-Bulkley model for cavern determination (Z/T = 0.8). ............................... 100 Figure 5.11 Experimental and computed dynamic responses for Cm = 2% hardwood pulp. ...... 104 Figure 5.12 Experimental and computed dynamic responses for Cm = 3% softwood pulp. ....... 105 Figure 5.13 Path lines of particles (each particle is represented by a different color) simulated by CFD using Cm = 2% hardwood pulp at N = 250rpm and Q = 14L/min. ..................................... 106 Figure 5.14 Experimental and computed determination of mixed volume for (a) Cm = 2% hardwood pulp and (b) Cm = 3% softwood pulp in continuous operation. ................................. 107 xii List of Symbols and Abbreviations a distance between the impeller location and the vessel centre, (m) A2 surface area of cavern, (m 3 ) b perpendicular distance from the intersection point of the vessel wall and the cavern to the distance a, (m) c speed of sound, (m/s) C height between the chest bottom and the impeller centre, (m) C1, C2 constant, dependent on the impeller type and impeller geometry Cm mass concentration or consistency, (%) D, D1, D2 impeller diameter, (m) Dc cavern diameter, (m) Dp pipe diameter, (m) Dr, DR rotor diameter, (m) DT housing diameter, (m) E distance between the chest wall and the impeller centre, (mm) f bypassing parameter f1 factor to correct for temperature and pipe roughness F total force on the cavern boundary, (N) F external body forces Fa axial force, (N) Fd Doppler shift frequency, (Hz) Fe transducer frequency, (Hz) g gravitational acceleration, (m/s 2 ) H rotor height, (m) Hc cavern height, (m) H/L friction loss, (m water/100m pipe) I unit tensor k constant in the Herschel-Bulkley model K numerical coefficient, constant for a given pulp K1 consistency index M, M1, M2 torque, (Nm) Mo, Mo1, Mo2 momentum number (m 4 /s 2 ) Mo level momentum (m 2 /s 2 ) n power-law index N, N1, N2 impeller rotation speed, (rpm) Nf dimensionless axial force number; Fa/N 2 D 4 NP power number; P/N 3 D 5 p static pressure, (Pa) P, P1, P2 power, (W) Q pulp flow rate through the chest, (L/min.) r position vector in the rotating frame rc cavern radius, (m) rv vessel radius, (m) xiii Rey yield stress Reynolds number; N 2 D 2 /y Sa cavern-air surface, (m 2 ) Sb cavern-vessel base surface, (m 2 ) Sm mass added to the continuous phase from the dispersed second phase and any user-defined sources Sp cavern-suspension surface, (m 2 ) Sw cavern-wall surface, (m 2 ) t time, (s) td time delay between the start of the pulse burst and its reception in UDV, (s) T, T1, T2 chest diameter, (m or mm) Td time delay for the dynamic model, (s) Tm maximum torque, (Nm) v velocity, (m/s) v absolute velocity rv relative velocity V, V1, V2 suspension volume in the chest, (m 3 ) Vc cavern volume, (m 3 ) Vmix fully mixed volume in the chest, (m 3 ) Vt suspension volume in the chest, (m 3 ) x distance from the impeller to the chord connecting the intersection points between the cavern and vessel wall (m) X position, (m) Z stock height, (m) Greek symbols angular velocity of the rotating frame , , indices, constant for a given pulp F power dissipation per unit volume, (W/m 3 ) strain rate (s-1) y strain rate at yield stress, (s -1 ) Doppler angle, () 1 angle for determining the cavern-suspension surface, () 2 angle for determining the cavern-wall surface, () fluid density, (kg/m3) dynamic viscosity, (kg/ms) o yielding viscosity, (kg/ms) plastic viscosity, (kg/ms) shear stress, (Pa) stress tensor 1 time constant for bypassing, (s) 2 time constant for mixing zone, (s) a friction force per unit area at the air suspension interface, (Pa) th theoretical time constant of the system, (s) xiv w friction force per unit area at the vessel wall, (Pa) y fluid yield stress, (Pa) Abbreviations CFD Computational fluid dynamics ERT Electrical resistance tomography PIV Particle image velocimetry UDV Ultrasonic Doppler velocimetry xv Acknowledgements I would first like to express my profound gratitude and appreciation to my deceased supervisor Dr. Chad P.J. Bennington for his guidance, encouragement and support throughout the research. I also would like to give special thanks to Dr. Guy A. Dumont for his innumerable assistance and support in the completion of my thesis. I gratefully acknowledge the valuable advice and helpful suggestions of the members of my thesis committee, Dr. Fariborz Taghipour and Dr. James Olson. I am grateful to Mr. Ali Soltanzadeh for his assistance in dynamic model parameter estimation and Dr. Clara Ford for her advice in CFD modeling. I acknowledge the assistance of all the staff in the Pulp and Paper Centre and the Chemical and Biological Engineering Department at UBC. I am especially grateful to Peter Taylor and Tim Patterson for their valuable help in the fabrication, installation and maintenance of the experimental setup. Financial support from the National Sciences and Engineering Research Council of Canada (NSERC) is highly appreciated. xvi To my parents, For their love, encouragement and support. 1 1 INTRODUCTION 1.1 Thesis background Pulp suspension mixing in stock chests is an important operation in pulp and paper industry. It prevents the pulp suspension from dewatering, enhances chemical contact in bleaching stages and blends pulp streams before papermaking. Improper mixing could lead to non-uniform delivery of pulp stock and increase process variability, resulting in unstable production and poor product quality. Although regulatory control loops are employed to minimize the effects of these process upsets and to maximize product uniformity, they are only effective at regulating out the disturbances at low frequencies (Bialkowki, 1992). Thus, agitated chests are required to remove high frequency variability, complementing the action of control loops. However, these chests do not perform ideally because of the complex rheology of pulp suspensions and the ineffective design of feed flow in and out of the chests. The suspension yield stress resulted from the mechanical strength of the wood fibre networks may lead to the formation of mixing and stagnant zones, creating undesired flows in the agitated chests. Ein- Mozaffari et al. (2003) analyzed the output signal of an industrial pulp stock chest to a step input change and found that the non-ideal flows, i.e., bypassing and dead zones, deteriorated the response signal. To identify what parameters affecting the responses of the agitated pulp chests, Ein-Mozaffari et al. (2005) studied the mixing dynamics in a rectangular pulp chest using two different pulp suspensions: a short-fibred hardwood pulp and a long-fibred softwood pulp over a range of fibre mass concentration and addressed the importance of inlet and outlet locations relative to the mixing zone. Other studies have also been carried out to investigate the flow patterns and dynamic behavior of pulp suspensions in agitated stock chests and most of these studies were conducted 2 in rectangular chests (Bakker and Fasano, 1993; Ein-Mozaffari et al., 2003; Ford, 2004). Very little research (Wilstrom and Rasmuson, 1998a) about pulp agitation has been carried out in cylindrical chests, which are widely used, particularly as controlled mixing zone in bleaching towers and as dilution zone for high density pulp storage chests. Therefore, the purpose of this study is to investigate the mixing performance of cylindrical chests using electrical resistance tomography (ERT), ultrasonic Doppler velocimetry (UDV) and dynamic tracer tests. In addition, computational fluid dynamics (CFD) was used to simulate the mixing behavior of pulp suspensions in cylindrical chests. To achieve an effective mixing of a fluid, an understanding of the fluid properties, especially rheology, is important and essential. Rheology governs fluid motion during agitation. Pulp suspensions, as a non-Newtonian fluid, have caverns (the turbulent mixed regions) formed around impellers when they are not in complete mixing. The cavern size would significantly affect the mixing efficiency of the stock chests which are designed, based on past experience and semi-empirical techniques, with their shape and size to facilitate the agitation of pulp suspension. The details of cavern formation are discussed in Chapter 3. 1.2 Pulp suspension rheology Fluid rheology plays an important role in determining the fluid behavior in stirred vessels. A pulp suspension is a heterogeneous mixture of water and wood fibres. The fibres interact and entangle with each other to form a viscoelastic network consisting of fibre aggregates (flocs) (Gullichsen and Harkonen, 1981). The longer the fibers, the easier the floc formation. This fibre network can be described with two levels: MACRO (inter-network) and MICRO (intra-network) scales (Wikstrom and Rasmuson, 1998b). The MACRO scale represents the network between the 3 fibre flocs and the MICRO is described as a 3-D fibre network within a fibre flock. This complex structure determines the rheological behavior of the pulp suspension, which initially acts as a non-Newtonian fluid having a significant yield stress (y) (Figure 1.1). When the applied shear exceeds the suspension yield stress, the fibre network is disrupted and fluid-like motion can be created within the suspension. At low shear stresses, the MACRO structure of the fibre network is first broken and the flow of pulp suspension is dominated by the break-up between the fibre flocs. The fibre flocs flow as fibre flock spheres. When the shear becomes higher and exceeds a certain value (d) at the fluidization point, the MICRO network is ruptured into individual fibres, i.e., fluidization and the suspension behaves nearly as a turbulent Newtonian fluid. The simplest rheological model for a fluid possessing a yield stress is the Bingham model, which is a linear equation for the shear rate (Silvester, 1985; Barnes et. al, 1989; Chhabra and Richardson, 1999): (1.1) where is the shear stress,y is the yield stress, is the shear rate and is the plastic viscosity or coefficient of rigidity. However, based on the generalized stress-rate curve of the pulp suspension in Figure 1.1, at the low shear rates where the shear stress is between y and d (the flow is laminar), shear- thinning behavior is observed after the yield stress point. The Herschel-Bulkley model (Macosko, 1993; Chhabra and Richardson, 1999), which contains a yield stress and a shear- thinning parameter, can be better suited for describing the non-Newtonian nature of pulp suspensions. 4 Figure 1.1 A generalized stress-rate curve for a fiber suspension (Gullichen and Harkonen, 1981). y nk (1.2) The shear-thinning behavior is obtained when the power index, n, is set to less than 1 and k is a constant in the model. For pulp suspensions, Ford (2004) estimated k and n to be 0.001 and 0.25 respectively. To create motion or turbulence in the pulp suspension, the yield stress must be exceeded and the required stress has been correlated with the suspension mass concentration by the following equation (Bennington et al. 1990): y mC (1.3) where y is the yield stress, Cm is the suspension mass concentration in percent, and are fitted constants. Ranges of and were reported to be 1.18 - 24.5 and 1.25 - 3.02 respectively. For the mass concentration between 0.6% and 30%, the yield stress will vary from 0.3 Pa to 3104Pa. Shear rate, S h ea r s tr es s, Laminar Turbulent y d Newtonian fluid Bingham model 5 Pulp suspensions in a turbulent state are often referred as “fluidized”, which implies fibre motion leading to energy dissipation (Bennington and Kerekes, 1996). Fluidization can be quantified through power and energy expenditure, and Wahren (1980) applied this concept to estimate the power dissipation per unit volume of pulp suspension for the onset of fluidization, F, using the yield stress of the fibre network: 3.54 2 102.1 m y F C (1.4) where F is in W/m 3 and is the viscosity of water. However, the use of the water viscosity to estimate the power required to fluidize a pulp suspension is inappropriate because the apparent viscosity of the suspension is larger than that of water (Bennington and Kerekes, 1996). Thus using the criterion chosen by Gullichsen and Harkonen (1981) for the onset of fluidization, Bennington and Kerekes (1996) developed an expression to estimate the energy dissipation necessary for fluidization: F m T R C D D 4 5 104 2 5 2 3 . . . ; (1.5) 13 3110 12 6. . ; . . D D CT R m where DT is the diameter of the housing and DR is the diameter of the rotor. When the gap size is extrapolated to zero, the equation becomes: 5.24105.4 mF C (1.6) Owing to the complexity of the rheological behavior of pulp suspension, it is not easy to agitate the suspension thoroughly. The shear stress provided by the agitators must exceed the 6 yield stress everywhere in the suspension to create motion through the whole chest. In actual mixing situations, however, the applied shear stress may not be strong enough to create thorough mixing in the suspension, leading to the formation of mixing regions surrounded by stagnation zones. 1.3 Agitated chest design To facilitate the bulk motion and the cavern generated by the impeller, the chest shape and size are important in achieving thorough agitation in the suspension. Generally, the agitated pulp stock chests are designed based on experience. One of the methods for designing these chests was developed by Yackel (1990). Derived from the concept of “conservation of momentum”, the method is based on the impeller momentum number and the level momentum. Each particular impeller has a momentum number (Mo) which is defined as: (1.7) where N is impeller rotation speed, D is impeller diameter and C1 is a constant that depends on the suspension rheology, impeller type and impeller geometry. By referring to sets of diagrams and tables relating chest diameter or width, consistency, stock level and retention time to obtain a process number which is converted to the corresponding momentum number, the required impeller size and horsepower for agitation could be determined. However, there is no direct reference to suspension rheology in Yackel‟s method. The non-ideal flows like bypassing and dead zone are not considered in the chest design and the degree of motion generated in the suspension is not given. 42 1 DNCMo 7 In terms of chest geometry, rectangular chests are widely used in pulp mixing because they save space when they are grouped together using common walls. The optimum shape for a rectangular chest to achieve complete motion in a pulp suspension is a cube (Yackel, 1990). Owing to space limitation or restriction, all chests cannot be perfect cubes and so a general rule of thumb for rectangular chests is that the length to width ratio should not be over 1.5. Besides rectangular ones, cylindrical chests are used in pulp and paper industry. Since cylindrical chests take advantage of hoop stress design (tension), they generally have thinner walls than rectangular chests of the same height (Yackel, 1990; Reed, 1995). Thus, it could be economical to have a cylindrical chest rather than a rectangular one. To achieve complete motion of a pulp suspension with the minimum power in a cylindrical chest, the ideal ratio of stock depth (Z) to the chest diameter (T) should be 0.8 (Z/T = 0.8) (Yackel, 1990). The reduced-bottom chest is another chest design that is employed to save energy (Figure 1.2). It is like a high-density storage chest placed on top of a dilution chest where controlled zone agitation is maintained. The top of the chest operates in the range of 5% to 12% stock consistency and the stock flows slowly down into the bottom of the chest where dilution water is added to reduce the consistency to 3 - 4%. The lower section is a cylindrical chest with a height equal to one-half of the diameter (Yackel, 1990). The diameter of the upper section where plug flow occurs to prevent any stagnant zones is usually designed to be 1.6-1.8 times the diameter of the lower section. The overall height is three or more times the lower-section diameter. Depending on the diameter of the lower section, a fillet or a solid horseshoe structure made of metal or concrete may be used to reduce the volume of the dilution chest. It allows a smaller mixing zone for decreasing the agitator horsepower without the build-up of unwanted stagnant pulp zones (dead zones). A general guideline for maximum fillet size is one-third chest diameter 8 but owing to material cost, normally fillets are 25% or less of chest diameter (Devries and Doyle, 1995). Figure 1.2 Cross-section of a reduced-bottom chest. Even though these design rules are followed, channeling (bypassing) of pulp through the reduced-bottom chest may still occur because no consideration for cavern formation in the controlled mixing zone is included in the design. The flows induced by the cavern may create channeling in the upper chest portion where the high-density pulp suspension flows downward. Also, if the cavern is not large enough, the resultant non-mixing regions will create some shortcuts for the downward pulp stock to flow directly to the exit, without entering the mixing zone. This channeling can reduce the residence time and create operation problems like poor consistency control and increased chemical usage. Chest specifications: - T2 = T1(1.6-1.8); - Zm = 0.5 T1; - Z/T1 = or 3 Zm Z T2 Mixing zone T1 9 1.4 Research objectives To analyze the mixing dynamics in cylindrical vessels, the investigation was divided into two parts: experimental study and numerical simulation. Chapter 2 first describes two methods used in experimentation: electrical resistance tomography (ERT) and ultrasonic Doppler velocimetry (UDV), and the numerical simulation method - computational fluid dynamics (CFD). Then the following two chapters show the experimental results of a lab-scale cylindrical chest. In chapter 3, the mixing behavior of the chest in batch operation was characterized using ERT and UDV, and the experimental results of the chest operated in continuous mode are given in chapter 4. After the experimental study, a CFD model was developed to simulate the mixing situations for verifying its ability to predict the experimental data and the evaluation is described in chapter 5. Finally, the overall conclusions of the research and recommendations for future work are given in chapter 6. The objectives of this research are summarized as: I. To characterize the batch mixing of pulp suspensions in a cylindrical chest with a side entering agitator by studying the cavern formation and the flow fields with the aid of electrical resistance tomography (ERT) and ultrasonic Doppler velocimetry (UDV). II. To carry out dynamic tests under scaled industrial operating conditions for identifying and determining the response of an experimental chest. The data will be used to estimate the channeling parameter and the time constant (for the mixed zone) in the dynamic model of chest dynamics. III. To simulate the mixing situation by means of computational fluid dynamics (CFD) and compare the results with the experimental data. 10 The achievement of the objectives can enhance the understanding of mixing dynamics of pulp suspensions in cylindrical vessels under batch and continuous operations. The study can provide information about the properties of cavern (mixing zone) in batch mixing. The cavern shape and size directly affect the mixing capacity of agitated pulp stock chests. In continuous operation, the dynamic tests can estimate the chest performance in terms of some quantitative parameters. These parameters can provide some insights whether the existing chest is operated at its full capability. Finally, the numerically simulated results can verify the predictability of the CFD model for pulp suspension agitation, evaluating its potential as a tool for chest design. 11 2 STUDY METHODS 2.1 Electrical resistance tomography (ERT) With its mathematical concept introduced in the nineteenth century, tomography is a technique for reconstructing the internal distribution of 2D and 3D objects from multiple external viewpoints to provide cross-sectional slice images through the object (York, 2001; Hoyle et al., 2005). Among different types of process tomography, electrical resistance tomography (ERT) is increasingly used by many researchers to study industrial processes because of its advantages including non-intrusive flow visualization, high speed measurement and no radiation hazard (Mann et al., 1997; Ma et al., 2001; Kim et al., 2006). It is used to determine the resistance distribution in the domain of interest by obtaining a set of measurements using sensors that are distributed around the periphery without affecting the flow or movement of materials. The main continuous phase in the domain must be slightly conductive and the other phases (components) have different values of conductivity. The resistance distribution in a cross region is obtained by injecting electrical currents on the domain and measuring voltages on it through a number of spaced electrodes which are mounted on its periphery. The current is injected by using a pair of neighboring electrodes and the voltage differences are measured by using all other pairs of neighboring electrodes (Figure 2.1a). This process is then repeated by injecting current using different pairs of neighboring electrodes and the spatial gradients of electrical conductivity are measured (Figure 2.1b). Since ERT can detect local changes in resistivity/conductivity, it can be used to study the mixing dynamics of a system if the fluids to be blended have different conductivities. An injection of a fluid tracer with high conductivity can help to examine the flow 12 patterns of those non-conductive mixing components. Also, the use of multi-plane electrical sensors in ERT can provide a pseudo three-dimensional description of the mixing process. Figure 2.1 Operation of electrical resistance tomography (Malmivuo and Plonsey, 1995). An electrical tomography system is composed of a sensor system, a data acquisition system (DAS) and an image reconstruction system (Dickin and Wang, 1996). For a cylindrical chest, the sensor system is usually a set of 16 electrodes arranged in rings, across which an electrical current is applied and the resulting voltage is measured. The DAS, mainly consisting of a signal source, voltmeters and a control circuit unit, coordinates the current injection and the voltage measurement to obtain the required data for image reconstruction. The image reconstruction system, which is simply an algorithm, is used to determine the distribution of regions of different conductivities within the mixing vessel by processing the collected data and displaying the resultant distribution image. Further information about the three ERT components is given in Appendix A. Although the ERT process is fast for obtaining results, the image resolution is limited because only 316 pixels (the default square grid has 20 20 = 400 pixels) are available for representing the cylindrical chest interior cross section (Figure 2.2). a) b) 13 Figure 2.2 Image reconstruction grid (Industrial Tomography Systems, 2007). ERT has been be used as a measurement technique in many applications like level monitoring in a horizontal pipe (Ma et al., 2001), flow visualization (Bolton et al., 2004; Wang, 2005) and jet mixing within pipelines (Stephenson, 2007). Ma et al. (2001) applied ERT to monitor air/water two-phase flow in horizontal pipes and developed a Liquid Level Detection method to check the water surface level. In the analysis of jet mixing within pipelines, Stephenson et al. (2007) demonstrated the capability of ERT to visualize tracer plume development exiting a coaxial or side entry jet in mains water and to allow a measure of mixing progress under different operating conditions. ERT is also a popular analysis method in mixing processes in agitated vessels. Owing to the slurry conductivity, William et al. (1996) applied ERT for the three dimensional imaging of the concentration of solids in a slurry mixer as a function of key process variables (particle size, impeller type and agitation speed) and for information collection to formulate the improved mixing models and mixer design. Holden et al. (1998) demonstrated the ability of the ERT system to distinguish the different flow patterns from two types of impellers and to examine the mixing processes by monitoring the dispersion of a 14 brine pulse tracer. Since the resistivity would be varied by the void spaces due to bubbles in the fluid, Wang et al. (2000) detected and constructed the pseudo-stationary pattern of gas-liquid mixing in 3-D using an 8-plane 16-ring elements ERT system. Hosseini et al. (2010) correlated the ERT measurements in solid-liquid mixing to solid concentration profiles to quantify the degree of homogeneity and studied the effects of impeller and particle characteristics on the mixing quality. Similarly, by introducing a saline solution into the region near the impeller, ERT can be applied to study the cavern size formed in pulp suspension mixing. 2.2 Ultrasonic Doppler velocimetry (UDV) Ultrasonic Doppler velocimetry (UDV) is a real-time and non-intrusive fluid flow measurement method to obtain a spatial distribution of the velocity field (Takeda, 1995; Shekarriz and Sheen, 1998). Initially developed for medical applications, it has been increasingly used as a tool to study the physics and engineering of fluid flow (Takeda, 1999). Besides providing spatiotemporal information, UDV could be an efficient flow mapping process for comparison with numerical simulation. Unlike laser Doppler anemometry, it is also applicable to opaque fluids like pulp suspensions because it is unaffected by the optical properties of the fluid. UDV is based on pulsed ultrasound echography and relies on the principle of the Doppler frequency shift of moving particles or scatters within the flowing fluid. An ultrasonic pulse (0.45 - 10.5MHz) is emitted from the transducer along the measuring line and the same transducer receives the echo reflected form the surface of the particles suspended in the liquid (Figure 2.3). The velocity field information is contained in the echo. Information on the position, X, from which the ultrasound is reflected is extracted from the time delay, td, between the start of the pulse burst and its reception as 15 2 dctX (2.1) where c is the velocity of sound in the medium. At the same time, the flow velocity (v) is determined from the Doppler shift frequency at that instant: cos2 e d F cF v (2.2) where Fd and Fe are Doppler shift frequency and basic frequency of ultrasound respectively. Thus a velocity profile can be obtained by analyzing the echo signal to derive instantaneous frequencies at each instant. The measurement accuracy of UDV stated by Takeda (1991) is good: 5% for velocity and 1% for position. In the case of velocity measurement in pulp suspensions, the sonic velocity in the suspension would be affected by suspension temperature and fibre concentration. At room temperature (20C), the sound velocity in water is 1482 m/s. When there are fibres in the suspension, the sound velocity would increase, but the change would not be significant (Xu, 2003). In a recent study of measuring pulp fiber suspension flow in a rectangular channel, Xu used a commercial UDV instrument (DOP 1000, Signal Processing, Switzerland) and determined the error of the measured velocity to be about 2.4%. 16 Figure 2.3 Principle of UDV showing the measurement of velocity along the measuring line (Takeda, 1995). Besides the normal instantaneous velocity measurement, UDV has been successfully applied in some special flow configurations like oscillating pipe flow and T-branching flow of mercury (Takeda, 1995). Also, in a stirred tank, Bouillard et al. (2001) showed that ultrasonic techniques can be used to identify fluid recirculation patterns, cavern regions and velocity fluctuations (turbulence) although their reliability are hampered by the presence of bubbles in the liquid phase. In addition, Wang et al. (2004) studied how the temperature affected UDV measurements and concluded that the temperature effect may not be a significant issue for measurements in aqueous solution. However, corrections are recommended for measurements in oil flows when non-isothermal condition exists. With regard to pulp suspensions, a number of studies have been carried out in pipe flows and stirred tanks. Dietemann and Rueff (2004) analyzed the flow of a pulp suspension at three consistencies up to 2.1 % in a transparent 80-mm duct using UDV and different flow regimes, i.e., plug flow, mixed flow and turbulent flow, were clearly observed. Xu and Aidun (2005) 17 employed the pulsed ultrasonic Doppler velocimetry to measure the velocity profiles of fibre suspension flow in a rectangular channel and a correlation was derived for the velocity profile of fibre suspension in turbulent flow. In addition, Ein-Mozaffari et al. (2006) identified the flow characteristics, i.e., flow number and pumping rate, of a Maxflo impeller by using the velocity profiles measured by UDV. Since the flow velocity inside the mixing cavern is obviously higher than that in the surrounding regions, UDV can be applied to detect the location and the dimension of the cavern by measuring the velocity profile across the mixing zone. This method could provide an accurate and well-defined picture of the cavern but it requires more time to get the result since a lot of measurements have to be made. 2.3 Computational fluid dynamics (CFD) In engineering analysis or design, many flows of practical interest cannot be easily described or solved analytically. The approximate solutions of these flows and related phenomena can only be obtained numerically. Computational fluid dynamics (CFD) is the numerical simulation of fluid motion by using computers. It estimates and predicts the flow field by solving a set of conservation equations for fluid flow, turbulence, chemical species transport and heat transfer. These equations form a system of coupled non-linear partial differential equations (PDEs) which describe the continuous movement of the fluid in space and time. The two important governing equations for fluid flow are the conservation of mass and momentum. For mass, which is neither created nor destroyed in the flows of engineering interest, the conservation equation is (Ferziger and Peric, 1999; Bird et al., 2002): 0 v t (2.3) 18 where for fluid density, t for time and v for velocity vector. In the case of momentum conservation, besides the transport by convection (the left-hand side of the Eqn. (2.4)), several momentum sources like the pressure gradient, diffusion and other forces are also involved. If gravity is the only body force with no other forces, the equation can be written as (Ferziger and Peric, 1999; Bird et al., 2002): gpvvv t (2.4) where p is the static pressure, is the stress tensor, and g is the gravitational body force. The first term is the rate of change of momentum, the second is the convection of momentum and the fourth is the diffusion of momentum. The stress tensor is described by Ivvv T 3 2 (2.5) where is the dynamic viscosity, I is the unit tensor and the second term on the right hand side is the effect of volume dilation. To solve these equations numerically, all aspects of the process must be discretized, i.e., a change from a continuous to a discontinuous formulation. The domain where the fluid flows needs to be described by a series of connected control volumes, or computational cells and the governing equations need to be written in an algebraic form. The process to decompose the domain into a set of discrete sub-domains, or computational cells is called grid generation. In two-dimensional domains, the cells are triangular or quadrilateral. In three-dimensional domains, the elements can be tetrahedra, wedge-shaped prisms, pyramids or hexahedra (Figure 2.4). Also, the way of dividing the domain into a finite number of cells can be classified into two main groups: structured and unstructured (Figure 2.5). In structured grids, cells of a single family occupy the whole domain and the positions of grid points are uniquely identified in a way 19 logically equivalent to a Cartesian grid. This simple and regular cell connectivity creates a standard matrix of the algebraic equation system, which can be easily solved by a number of methods. The unstructured grid, on the other hand, can consist of cells of any shape, without any restriction on the number of neighbor elements or nodes. It is more flexible than the structured grid and applicable to very complex geometries. However, the node locations and neighbor connections need to be specified explicitly, so the matrix of the algebraic equation system is irregular and it takes more time to solve it than that for the structured grid. Generally, the density of cells in a computational grid needs to be fine enough to capture the flow details, but not so fine because large numbers of cells require more time to solve. For discretization of the equations, the well known processes used in CFD are the finite difference (FD) method and the finite volume (FV) method. In the finite difference method, derivatives in the PDEs are written in finite differences evaluated at the grid nodes using Taylor‟s series expansion or polynomial fitting (Ferziger and Peric, 1999). The FD method can be applied to any grid type but generally it is mostly used in structured grids. It is also simple, effective and very easy to obtain higher-order schemes on regular grids. In the finite volume approach, the integral form of the conservation equations is applied to the control volumes (CVs) defined by cells (Ferziger and Peric, 1999). The variables are approximated at the centroid of each CV. This method is suitable for complex geometries and conservative by construction provided that surface integrals are the same for the CVs sharing the boundary. 20 Figure 2.4 Cell types used in computational grid (Ferziger and Peric, 1999). Figure 2.5 2D structured and unstructured grids (Ferziger and Peric, 1999). The finite set of coupled algebraic equations resulting from the discretization process should be solved simultaneously in every cell in the solution domain. Owing to the non-linearity of the equations that govern the fluid flow and related processes, an iterative solution procedure is required and two methods are commonly used. A coupled solution approach is the one where all variables, or at a minimum, momentum and continuity, are solved simultaneously in a single cell before the solver moves to the next cell, where the process is repeated. This approach is excellent for equations to be linear and tightly coupled. On the other hand, when the equations are complex and non-linear, a segregated solution approach is preferred. In this method, one Triangle Quadrilateral Tetrahedron Wedge Pyramid Hexahedron 21 variable at a time is solved throughout the entire domain. Each equation is solved for its dominant variable, temporarily treating the other variables as known, using the best available values for them. This process is iterated through the equations until all equations are satisfied. The segregated solution approach is popular for incompressible flows with complex physics, typical of those found in mixing applications. CFD is a useful tool for studying the flow dynamics in fluid mixing. Many literatures show that CFD can predict velocity distribution, evaluate industrial stirred equipments like fermenters and solid suspension vessels, and optimize impeller design (Armenante et al., 1997; Bhattacharya and Kresta, 2002; Paul et al., 2004; Li et al., 2005; Kasat et al., 2008; Ankamma Rao and Sivashanmugan, 2010). Ein-Mozaffari and Upreti (2009) applied CFD to study the performance of three impellers in mixing of pseudoplastic fluids (xanthum gum solutions) which were treated as a Herschel-Bulkley fluid. The computed velocities were found to agree reasonably well with the measured results. The validated model was then used to simulate the mixing time and the results showed the fluid yield stress and the clearance of the impeller had considerable effects on the mixing time. Using an experimentally validated CFD model, Bakker et al. (2010) numerically determined cavern shapes in a pilot-scale flotation cell for a range of mineral slurries. The combined Herschel-Bulkley and Bingham plastic rheology models were used to describe the slurries. A model was developed to estimate cavern height, which was found to be inversely proportional to the slurry yield stress, in the flotation cell and this would be useful for an engineering approximation of the cell size in the preliminary design. Gomez et al. (2010) investigated the flow field in a rectangular vessel filled with glycerin solution and equipped with a side-entering agitator, using CFD and particle image velocimetry (PIV). The CFD predicted 22 velocities agreed very well with the PIV measurements and this confirmed CFD as a useful tool for optimization and design of mixing systems. CFD has also been used in the study of pulp mixing. With the rheological data from Gullichsen and Harkonen (1981) for the pulp suspension, Bakker and Fasano (1993) applied CFD to model the flow in a rectangular pulp chest with a side-entering impeller and the simulated results were in satisfactory agreement with the flow field visualizations. Wikstrom and Rasmuson (1998a) assumed the pulp suspension as a Bingham fluid and studied the effect of fibre suspension rheology on flow behavior in a laboratory tank equipped with a jet nozzle agitator. The comparison of experimental results with theoretical CFD-calculations showed that the applied Bingham model could not fully describe the rheology of the pulp suspension since the calculated flow field deviated increasingly from the measurements as the distance from the impeller increased and so a new rheology model must be developed to predict the entire flow field correctly. In the CFD modeling of a rectangular pulp stock chest with a side-entry agitator, Ford (2004) described the pulp rheology using a modified Bingham plastic model and showed that the velocity vectors obtained from the CFD simulations agreed qualitatively with the flow patterns observed in the experiment. The non-desirable flows like bypassing and recirculation within the chest could be described by the predicted flow fields. Using a Bingham approximation for the suspension rheology, Bhole et al. (2009) used CFD to model impeller flow and the computed power and axial thrust numbers were found to be very close to the experimental measurements. Also, treating pulp suspension as a modified Herschel-bulkley fluid, Bhattacharya et al. (2010) studied two industrial pulp chests by simulating the process conditions to calculate the steady-state flows using CFD. The simulated flows were compared with the experimental results and found to agree reasonably well. 23 3 CAVERN MEASUREMENT IN PULP SUSPENSIONS1 3.1 Introduction Agitated pulp stock chests play an important role in pulping and papermaking operations. They are used to prevent pulp suspensions from dewatering, control pulp consistency (mass concentration) prior to other processing steps, and for blending pulp streams ahead of papermaking operations. Pulp suspensions display non-Newtonian rheology, including a yield stress, which under certain conditions allows creation of caverns (regions of active mixing) around impellers. Ein-Mozaffari et al. (2003) showed that industrial pulp chests were not ideally mixed and developed a dynamic model which included the possibility of stock channeling to quantify the mixing quality attained. Channeling was exacerbated by the size and location of the cavern relative to the stock exit (Ein-Mozaffari et al., 2005) and consequently to fully understand pulp mixing, the cavern location, size and shape should be characterized and correlated with the operating conditions of the chest. A number of studies have examined cavern formation in a range of non-Newtonian fluids as a function of fluid properties and mixing conditions (Wichterle and Wein, 1981; Solomon et al., 1981; Silvester, 1985; Elson, 1990; Amanullah et al., 1998; Wilkens et al., 2005). Solomon et al. (1981) showed the presence of caverns in both shear thinning and yield stress fluids (CMC, Carbopol and Xanthan gum) agitated with top-entering radial flow impellers in conventionally stirred baffled tanks. The caverns formed were spherical and increased in size as impeller speed was increased. Elson (1990) examined the effect of impeller type and speed on cavern development using 1% Xanthan gum solutions in water in a similar mixing configuration. Prior 1 A version of this chapter has been published. Leo K. Hui, C.P.J. Bennington and G.A. Dumont (2009) Cavern formation in pulp suspensions using side-entering axial-flow impellers, Chem. Eng. Sci., 64, 509-519. 24 to the cavern reaching the vessel wall, cavern volume per unit power input was greatest for the axial flow impeller studied when compared with a Rushton turbine (radial flow impeller). However, once the cavern interacted with the baffles vertical expansion of the cavern was only slightly affected by impeller type with the height of the cavern increasing with the impeller rotational speed raised to a power of about 0.7. Several mathematical models have been developed to predict cavern size as a function of mixing conditions and fluid properties. In general, these models specify a shape for the cavern (based on observation) and then balance the shear force transported to the cavern surface with the yield stress of the fluid. In all cases the models were developed for isolated caverns that do not interact with the vessel walls. Soloman et al. (1981) assumed a spherical cavern with the torque induced by the impeller acting tangentially at the cavern boundary. This gives an equation for the cavern diameter, Dc y P c DNN D D 22 3 3 4 (3.1) where , N, D and y are the fluid density, impeller rotation speed, impeller diameter and fluid yield stress, respectively. The impeller power number, NP = P/(N 3 D 5 ), must be known or measured and the yield stress Reynolds number, Rey = N 2 D 2 /y, calculated. Elson et al. (1986) used an x-ray technique to image caverns produced in Xanthan gum solutions with Rushton impellers. The cavern was described by a right circular cylinder of height Hc and diameter Dc with the ratio of Hc/Dc set between 0.35 and 0.45 for the Ruston impeller. y P cc c DNN DHD D 22 2 3 31 1 (3.2) 25 Amanullah et al. (1998) considered the total force transported to the cavern boundary by an axial flow impeller as being the sum of the tangential and axial force components. The model was applied to a spherical cavern, giving 2 2 222 3 41 P f y c NN DN D D (3.3) where Nf =Fa/(N 2 D 4 ) is the dimensionless axial force number that must be measured or determined from correlations. Amanullah et al. extended the model to shear thinning power-law fluids, and located the cavern boundary as the surface having a specified (and low) velocity. The model by Elson et al. (1986) predicted the cavern (right-circular cylindrical cavern) diameter within 9% for disc turbines. The model by Amanullah et al. (1998) predicted the cavern (spherical) diameter for an axial flow impeller with Re > 20 to within 5 – 15%. As volume scales with the cube of diameter, the error expressed on a volume basis ranged from 16 to 53%. Application of the above equations to pulp mixing operations is anticipated to be difficult. First, in most industrial pulp mixer configurations the impellers are side-entering. This will impose interaction between the cavern and the vessel floor and side walls, which has not been considered in past work. The yield stress of pulp suspensions can be measured and characterized, but the preferred method of Amanullah (which requires fitting the rheological curve to a power law relationship) would be difficult as the flow curve of a fibre suspension beyond the yield point is not well characterized or readily measured. In this chapter, cavern size and shape was measured as a function of mixer operating conditions (impeller speed, impeller offset from the wall, suspension height in the chest) for several pulp fibre suspensions (hardwood and softwood pulp with suspension mass 26 concentrations from 1 to 5%). A scaled version of a commercial axial flow impeller designed for pulp suspension agitation was used in the standard side-entering configuration in a cylindrical chest. Due to the opaque nature of the suspensions it was impossible to use direct optical techniques to measure cavern size. Two methods, electrical resistance tomography (ERT) and ultrasonic Doppler velocimetry (UDV) were evaluated to obtain the needed data. Measured cavern sizes and volumes were then compared against the cavern models available in the literature, and a new model that accounts for interaction with the vessel boundaries was developed. 3.2 Experimental set-up and procedure A transparent 1/10 scale-model of a cylindrical stock chest (T = 38.1 cm) was built for studying mixing in pulp suspensions. The chest was equipped with a side-entering Maxflo impeller (Chemineer Inc., Dayton, OH) having D = 16.5 cm mounted 12.0 cm (C) from the bottom (i.e., C/D = 0.7. C/D is typically in the range of 0.5 to 1.5 (AGIMIX International, Uddevalla, Sweden)). The impeller assembly could be moved horizontally to study the effect of impeller position on cavern formation and two offsets, E = 7.0 and 9.6 cm (giving E/D = 0.4 and 0.6), were used. The laboratory chest was fitted with 6 equally-spaced sensor planes for electrical resistance tomography (ERT) with each plane having 16 square (2.52.5 cm2) stainless-steel electrodes flush-mounted around the vessel periphery at 22.5 o intervals (Fig. 3.1). Three stock height to chest diameter ratios (Z/T = 0.8, 1.0 and 1.2) were examined (based on recommendations by Yackel, 1990). Equations 3.1 through 3.3 demonstrate the predicted dependences of cavern diameter on yield stress, which for a pulp suspension depends primarily on fibre (pulp) type and suspension 27 mass concentration. Two bleached kraft pulps (hardwood and softwood) from Domtar Inc. (Windsor, ON) were used in the study. The hardwood pulp had a length-weighted mean fibre length of 1.28 mm and the softwood pulp had a length-weighted mean fibre length of 2.96 mm. For the hardwood pulp, detailed cavern development was studied for three different suspension mass concentrations (Cm = 2, 3 and 4%) while for the softwood pulp, only Cm = 3% was used. The point at which the cavern completely filled the vessel was determined over a wider concentration range for both pulps. The suspension yield stress was measured using a Haake RV12 Rotovisco concentric cylinder viscometer (Thermo Fisher Scientific, Waltham, US). A four-bladed vaned rotor (with a diameter of 19 mm and a height of 38 mm) was immersed in the suspension and the rotor accelerated slowly from rest to a maximum speed of 0.2 min -1 . The peak value of torque was measured and used to calculate the yield stress 3 1 2 3 r r m y D H D T (3.4) where Tm, Dr, H and y are maximum torque, rotor diameter and height, and suspension yield stress, respectively (Macosko, 1994). This calculation assumes that the yielding surface is defined by the outer surface of the rotor blades. Figure 3.2 summarizes the yield stress as a function of suspension mass concentration for the pulps studied in this work. For the hardwood pulp suspension, the yield stress varied to the power 3.0 0.1 of suspension mass concentration; for the softwood suspension the dependence was to the power 2.0 0.1. These dependences are similar to those reported in the literature (Bennington et al., 1990). Two methods were evaluated for measuring the cavern size in our tests: electrical resistance tomography (ERT) and ultrasonic Doppler velocimetry (UDV). ERT is a non-invasive technique that can reconstruct the conductivity distribution within a region of interest from 28 50.8mm w = 5.1mm 381mm 625mm Electrodes Sensor plane 120mm P5 P3 P2 P1 P4 80mm electrical measurements made through a series of electrodes placed uniformly around the periphery of a vessel. Electrical current is injected through a pair of neighboring electrodes with voltage measurements taken between all remaining pairs of neighboring electrodes. The process is repeated by rotating current injection around all the electrodes. The aggregate data is then used to reconstruct the original conductivity distribution within the vessel (Mann et al., 1997; York, 2000). A P2000 ERT tomography instrument (ITS, Manchester, UK) was used to obtain the data with the image reconstructed using the linear back projection algorithm. A suitable conductive tracer was added to the cavern, with ERT used to locate the region of higher conductivity and thus the cavern size and shape. Data collection using the ERT process is computerized and rapid, with about 45 minutes required to obtain cavern dimensions for a given test. However, the spatial resolution of the cavern boundary is limited due the characteristics of the electric field and the algorithm and grid size used for image reconstruction. The measurement error of the cavern boundary (based on the resolution of the image grid) is typically given as 5 to 10% of the chest diameter – about 1.9 to 3.8 cm for our vessel. Figure 3.1 Cross-section of cylindrical stock chest showing ERT sensor planes and range of impeller positions used. 29 Figure 3.2 Suspension yield stress as a function of mass concentration. Ultrasonic Doppler velocimetry (UDV) was used to measure velocity profiles across the mixer and has been used successfully on pulp suspensions in other flow configurations (Dietemann and Rueff, 2004; Xu and Aidun, 2005; Ein-Mozaffari et. al., 2006). A single probe is used to transmit and receive the ultrasonic signals. An ultrasonic pulse, emitted at a set frequency, travels through the suspension and is reflected back to the probe by the fibres suspended in the flow. The time delay (t) and frequency shift (Fd) measured by the probe allow the velocity (v) of the particles (fibres) a distance (X) along the probe sampling direction to be determined X ct 2 (3.5) v F c F d e 2 cos (3.6) 0.1 1 10 0.1 1 10 100 1000 hardwood softwood y ( P a ) C m (%) 30 where c is the speed of sound in the suspension, Fe is the frequency emitted from the transducer and is the Doppler angle (the angle the particle trajectory makes with the propagation direction of the ultrasonic wave). A commercial UDV instrument (DOP2000, Signal Processing, Switzerland) was used in our study. The probe was placed at various locations around the vessel periphery to locate the points at which the suspension velocity fell to zero and hence the cavern boundaries. The measurement accuracy of the UDV technique is good: 5% for velocity and 1% for position (Takeda 1991) which is equivalent to a spatial resolution of about 4 mm in this study. However, reconstruction of the cavern boundary for one mixing condition takes about 2 days because data collection and analysis are time consuming. Results obtained for selected tests made with UDV and ERT were compared to determine the most suitable technique for our study. The point at which the cavern completely filled the vessel was readily determined by visual inspection as the laboratory chest is fabricated from Plexiglas and is transparent. In addition, video clips were taken of selected tests to confirm the cavern shapes/dimensions by locating the cavern boundary at the vessel wall and suspension surface. Image orientation between visual, ERT and UDV data was accomplished easily using the ERT electrodes as reference points. 3.3 Results and discussion 3.3.1 Selection of cavern measurement technique Electrical resistance tomography (ERT) was able to image caverns formed in the pulp suspensions once a conductive tracer was added to the well-mixed region around the impeller and dispersed throughout the cavern. Initially, a concentrated saline solution was used as the 31 tracer. However, the reconstructed images did not agree with other measurements (visual inspection of the cavern surface in contact with the chest wall or with UDV measurements). Other difficulties occurred. For example, when the saline solution was injected into the mixed region, although the signal (the conductivity contrast between traced and non-traced regions) was initially strong, the reconstructed image became blurred or completely disappeared as tracer spread and was diluted through the cavern. To overcome the difficulties with suboptimum conductivity contrast, we attempted to continuously add saline solution to the cavern or to increase the initial concentration/dosage. In both cases the imaged cavern was larger than that observed by other techniques which may be due to convection or diffusion of the aqueous tracer beyond the cavern boundary. To create reliable cavern images and avoid concerns about potential tracer diffusion, conductive tracer particles were used. These particles were made by covering small pieces of sponge with aluminum foil, with a typical particle being cylindrical, 5 mm in diameter and 17.5 mm in length, and having a volume of approximately 0.29 cm 3 – a bit larger than that of a typical fibre floc. The tracer particle density ranged from 0.90 to 1.06 g/cm 3 (they could absorb water and change density with prolonged use) and would move inside the cavern with the fibre suspension. The number of particles deployed in a test depended on the volume of the cavern to be measured and was optimized to obtain a good ERT image. Typically 100 to 120 particles were introduced into the mixing zone near the impeller – corresponding to about 0.2% of the cavern volume. The particles are shown in Figure 3.3. Figure 3.4 compares ERT images reconstructed for a one set of test conditions (hardwood pulp at Cm = 3% with N = 650 rpm) made using both the saline solution and the conductive particles as tracers. The cavern shape in each measurement plane is represented (P1 is at the base of the chest and the impeller is located 32 between electrodes 5 and 6 and imaging planes 2 and 3, see Figure 3.1). In all cases the cavern size imaged is larger when the saline tracer solution is employed. Figure 3.5 shows that the ERT image obtained using tracer particles (on the right) agreed best with visual inspection of the top surface of the suspension (note that P5 is 4.85 cm below the suspension surface). Figure 3.3 Photograph of conductive tracer particles used in tests. Individual particles range in diameter from 4 to 6 mm (typically 5 mm) and length from 15 to 20 mm (typically 17.5mm). UDV was used to measure fibre suspension velocity profiles at eight different positions spaced evenly between the ERT sensors on each sample plane as illustrated in Figure 3.6. The cavern boundary was located at the points where suspension velocity fell to zero along this measurement chord. By rotating the probe around the vessel, an image of the cavern was created (Figure 3.7). Repeating these measurements for the each imaging plane allowed a three dimensional reconstruction of the cavern to be created that was directly comparable with the ERT measurements and from which the cavern volume could be calculated. 33 Tracer particle Saline solution Plane 1 (the lowest plane) Plane 2 Plane 3 Plane 4 Figure 3.4 ERT images obtained for a Cm = 3% hardwood pulp suspension agitated at N = 650 rpm with E/D = 0.4, C/D = 0.7 and Z/T = 1.0. Images on the left were reconstructed using data obtained with tracer particles (Fig. 3.3) and images on the right were reconstructed by adding a conductive fluid of 10.9 mS/cm NaCl (15mL) to a pulp suspension having a background conductivity of 0.2 mS/cm (43.4L). The impeller is located between planes 2 and 3 and electrodes 5 and 6 as shown for reference purposes in the upper left image. Low conductivity High conductivity 34 Figure 3.5 Comparison of ERT images reconstructed using the tracer particles (left) and saline solution (right). Uppermost sampling plane (P5). The visual surface view is given in the centre image with the dotted line enclosing the region of active surface motion (determined visually). (Hardwood pulp suspension at Cm = 3%; N = 650 rpm, E/D = 0.4, C/D = 0.7, Z/T = 1.0). Figure 3.6 Probe locations for UDV measurements. The probe is moved around the chest to obtain the velocity profiles needed to map the cavern boundary. The arrows show the direction of the sonic emissions and the numbers give the ERT electrode positions. 1 2 4 6 3 7 8 5 9 10 Tracer particle Saline solution 7 13 UDV probe 14 15 11 12 8 3 5 4 6 1 2 16 10 9 measuring line 35 Figure 3.7 (a) Velocity profile measured by UDV for a representative probe location. The locations where the velocity reaches zero indicate the cavern boundary. (b) Cavern cross-section formed by connecting the boundary points measured from all sample locations. The cavern areas for each sample plane and the total cavern volume are summarized in Table 3.1 for a representative set of tests where both the ERT (using tracer particles) and UDV techniques were used. A visual comparison of the cavern images measured in each plane for one test condition is also presented in Figure 3.8. There is no comparison of images for P3 because the design of the chest could not allow the velocity measurement using UDV probe at that plane. Cavern images reconstructed using both ERT and UDV were similar, although the ERT data consistently gave a larger cavern volume which we attribute to the image reconstruction technique used. The UDV data underestimates the location of the cavern boundary as it measures only the suspension velocity component in the direction of the probe. For example, in planes P4 and P5 (Figure 3.8), the cavern area measured by UDV does not extend to the vessel wall although suspension flowed vertically (observed visually) there. Despite these differences, the measured cavern volumes are within 10% of each other, with the ERT technique predicting 0 100 200 300 400 -10 0 10 20 30 40 50 60 70 Cavern region V e lo c it y ( m m /s ) Distance (mm) x1 x2 14 13 12 11 15 10 9 8 7 1 16 2 3 4 5 6 x1 x2 a) b) 36 slightly larger cavern volumes. The uncertainty in cavern volume is estimated to be 16% for ERT and 7% for UDV measurements (based on the accuracy to which the cavern boundary can be located by each technique). ERT, with conductive particles used as tracer particles, was used for all subsequent tests due to the considerable reduction in time required for data acquisition and analysis. Table 3.1 Comparison of ERT/UDV cavern measurements (Cm = 2% hardwood pulp, E/D = 0.6, C/D = 0.7 and Z/T = 1.0) Cavern area or volume Impeller Speed, N (rpm) 250 200 150 ERT UDV ERT UDV ERT UDV P1 (dm 2 ) 6.42 6.39 5.59 4.83 5.19 4.32 P2 (dm 2 ) 6.28 7.08 5.84 4.94 5.44 5.01 P4 (dm 2 ) 7.11 5.62 5.37 3.88 3.67 3.48 P5 (dm 2 ) 6.02 6.24 2.21 4.06 2.21 2.14 Volume (dm 3 ) 24.6 24.1 18.1 16.9 15.7 14.2 P1 P2 P4 P5 Figure 3.8 Comparison between ERT and UDV caverns made for hardwood pulp at Cm = 2% and N = 200 rpm (E/D = 0.6, C/D = 0.7 and Z/T = 1.0). 13 14 15 16 1 2 3 4 5 7 6 8 9 10 11 12 13 14 15 16 1 2 3 4 5 12 11 10 9 8 7 6 13 14 15 16 1 2 3 4 5 12 11 10 9 8 7 6 13 14 15 16 1 2 3 4 5 12 11 10 9 8 7 6 37 The cavern geometry produced in a 2% hardwood pulp suspension is reconstructed in Figure 3.9 from ERT data at three impeller speeds. The cavern is cylindrical in shape, although a pronounced taper is evident towards the suspension surface at lower rotational speeds. As the impeller speed is increased, the cavern enlarges, particularly at the suspension surface, with the cavern extending to the vessel surface. N = 150 rpm 200 rpm 250 rpm Figure 3.9 Cavern shape (shaded volume) determined for a Cm = 2% hardwood at various impeller speeds using ERT data (E/D = 0.6, C/D = 0.7, Z/T = 1.0). 3.3.2 Effect of impeller speed and pulp mass concentration on cavern volume The cavern volume increased with increasing impeller speed as shown in Figure 3.10. The growth in cavern volume is fairly uniform until the cavern boundary begins to interact significantly with the vessel walls (about half the suspension volume is agitated at this point). Any further increase in impeller speed rapidly causes the cavern to fill the vessel. As suspension mass concentration (and consequently the suspension yield stress) was increased, higher impeller line of rotation 38 speeds were required to create a given cavern volume (in agreement with Eqns. 3.1 through 3.3). However, in all cases once the cavern had grown to about half the vessel volume a more dramatic increase in cavern volume was created for a modest change in impeller speed. This step change is similar to the dramatic change in channeling that was observed for small changes in impeller speed in dynamic mixing tests made previously (Ein-Mozaffari et al., 2005). Perhaps we have found a physical reason for the sudden transition to channeling. 3.3.3 Effect of pulp type on cavern volume Figure 3.10 also includes a plot of cavern development for a softwood pulp suspension at Cm = 3%. Here the suspension yield strength is significantly larger than that of the corresponding hardwood pulp, which results in a reduced cavern volume for a given impeller speed. Pulp fibre suspensions display unique flow behavior that depends on many aspects of the suspension, primarily the suspension mass concentration but also the average fibre length and fibre length distribution, fibre surface properties and the suspending medium. Although the suspension yield stress is readily characterized, suspension behavior in flow has not yet been adequately described by a rheological model. 39 Figure 3.10 Effect of impeller speed (N) and pulp consistency (Cm) on cavern volume (Vc ) for hardwood and softwood pulps (E/D = 0.4, C/D = 0.7 and Z/T = 0.8). 3.3.4 Impeller offset from the rear wall of the mixing chest The location of the impeller in an industrial vessel is fixed, with the typical offset clearance from the rear wall being from E/D = 0.3 to 1.0 (AGIMIX International, Uddevalla, Sweden). Proximity to the wall can restrict suspension flow to the impeller and hence reduce the pumping ability of the impeller. The impact of impeller clearance on cavern volume is shown in Figure 3.11 for our laboratory vessel. Here a Cm = 2% hardwood kraft pulp suspension was agitated using two different impeller clearances. When the off-wall clearance was small (E/D = 0.4) the cavern volume was smaller than when the off-wall clearance was larger (E/D = 0.6) for a fixed impeller speed. As the power required by the impeller (at a fixed speed) was about the same (Figure 3.11b) the larger offset resulted in slightly more energy-efficient mixing. However, any 0 200 400 600 800 1000 0.000 0.005 0.010 0.015 0.020 0.025 0.030 0.035 0.040 suspension volume = 0.0347m 3 V c ( m 3 ) N (rpm) C m (%) y (Pa) HW 2 9 HW 3 38 HW 4 69 SW 3 119 40 increase in impeller offset may increase stresses on the impeller shaft and a balance between increased pumping action and the mechanical installation must be achieved. 3.3.5 Comparison of cavern volumes determined by ERT and the cavern models The predictions for cavern volume made with the available literature correlations are compared with our experimental data in Figure 3.12. Here cavern volume is plotted against the yield stress Reynolds number, with predictions made using the Amanullah et al. (1998) model for a spherical cavern and the Elson (1990) model for a right circular cylinder. In both cases the specific power number and/or axial force number measured for the impeller under the test conditions (identical impeller geometry, impeller speed, pulp type and suspension mass concentration, see data in Figure 3.13) was used. 41 a) b) Figure 3.11 Effect of impeller position (E/D) on cavern volume (Vc) with (a) rotation speed and (b) power (Cm = 2% hardwood, C/D = 0.7, Z/T = 1.0). 100 200 300 400 0.00 0.01 0.02 0.03 0.04 0.05 E/D = 0.6 E/D = 0.4 suspension volume = 0.0434m 3 V c ( m 3 ) N (rpm) 0 2 4 6 8 10 0.00 0.01 0.02 0.03 0.04 0.05 suspension volume = 0.0434m 3 V c ( m 3 ) Power (W) E/D = 0.6 E/D = 0.4 42 Figure 3.12 Comparison of measured cavern volumes and predictions made using the equations of Amanullah et al. (1998) and Elson (1990). The predictions of the Amanullah et al. (1998) model show the best agreement with the experimental data, although the cavern formed is clearly cylindrical in geometry (Figure 3.9). Amanullah‟s model predicts more rapid cavern growth with increasing Rey than found. Also, the rotational speed required for the cavern to fill the vessel is under predicted. The model by Elson (1990) significantly under-predicts cavern volumes for all test conditions despite modeling the cavern as a cylinder. The main explanation for this is that axial thrust is not explicitly included in the model. Elson also noted that once the cavern reached the baffles a further increase in impeller speed increased cavern size at a slower rate than prior to contact with the vessel boundary, opposite to that found here. As both models assume no interaction with vessel walls the successful prediction of cavern volume in a side-entering geometry is not expected. 0 20 40 60 80 100 0.000 0.005 0.010 0.015 0.020 0.025 0.030 0.035 0.040 Elson (1990) suspension volume = 0.0347m 3 V c ( m 3 ) Re y Measured Predicted HW 2% HW 3% HW 4% SW 3% Amanullah et al. (1998) 43 Consequently, we sought to develop a model where cavern interaction with the vessel was accounted for. a) b) Figure 3.13 Axial force number (Nf) (a) and power number (NP) (b) for the D = 140mm Maxflo impeller as a function of impeller speed and suspension mass concentration. 0 200 400 600 800 1000 1200 0.1 1 10 100 Hardwood C m (%) 0 1 2 3 4 N f N (rpm) 0 200 400 600 800 1000 1200 0.1 1 10 100 Hardwood C m (%) 0 1 2 3 4 N P N (rpm) 44 3.3.6 Cavern model including interaction with vessel walls A model for cavern development with interaction between the cavern and vessel walls due to impeller location is developed below. We begin with the equation developed by Amanullah et al. (1998) to estimate the force transferred by an axial flow impeller to a spherical surface 2 242 3 4 Po NDNF f (3.7) This force is assumed to be balanced with the total force resisting motion at the cavern surface regardless of the cavern geometry. For unbounded caverns this is the suspension yield stress at the cavern-suspension interface. For bounded caverns, as considered here, the total resistive force can arise from interactions with three surfaces: the cavern-suspension surface, Sp, the cavern-wall surface, Sw, and the cavern-air surface, Sa. Equating the force transferred by the impeller (Eqn. 3.7) with the total contribution from these three surfaces gives aawwPy SSSF (3.8) where y is the pulp suspension yield stress (measured as discussed previously), w is the friction force at the vessel wall (which must be measured or estimated as detailed later), and a is the friction force per unit area at the air suspension interface, taken as zero. To proceed further we must specify the shape of the cavern. To a good approximation the cavern is a truncated (due to interaction with the side of the mixing chest) right circular cylinder (see Fig. 3.9) of radius rc (here the cavern “radius” is measured from the centre of the impeller to the cavern boundary in front of the impeller) and height Z (extending from the base of the vessel to the suspension surface). Once the cavern shape is specified the appropriate surface areas SP, Sw and Sa must be calculated to use in Eqn. 3.8. Analytical equations for the area and arc lengths of two intersecting circles are given by Martin (1874) and adapted to the cavern-vessel geometry as shown in Figure 3.14. The cavern- 45 vessel surface area, Sw, is given by two components – the interaction at the base of the vessel, Sb, and interaction along the side walls. The surface area where motion occurs at the base of the vessel is given by 2 222 2 222 12 222 12 22 cos 2 cos a rra ra ar rra r ar rar rS vcc c vc c v cv vb (3.9) where the impeller is located at the centre of a virtual cavern. Here rv is the vessel radius (rv = T/2), rc is the cavern radius and a is the distance between the centre of a virtual cylindrical cavern (the impeller location) and the centre of the mixing vessel (a = (T/2-E)). To find the cavern-side wall and cavern-suspension surface areas, the distance, b, perpendicular to the line running from the impeller to the centre of the vessel and the point where the cavern intersects the vessel wall must be determined. We calculate this using Pythagoras‟ theorem, giving 222 bxrc (for the cavern) and 222 )( bxarv (for the chest) 22 xrb c where a arr x vc 2 222 (3.10) allowing the total cavern-wall surface area, Sw, bv v bvw SZr r b SZrS 12 sin22 where vr b1 2 sin (3.11) and the cavern-suspension surface area, Sp Zr r b ZrS c c cp 11 sin22 where cr b1 1 sin (3.12) to be determined. Z is the height of suspension in the chest (the height of the cavern). The cavern volume was calculated using an iterative procedure. The three surface areas, Sa = Sb, Sp and Sw were determined as a function of rc for the chosen mixer geometry (given D, T, E). The resistive force contributed by these interfaces (Eqn. 3.8) was then balanced with the 46 force generated by the impeller (Eqn. 3.7) to determine the cavern radius associated with a given impeller speed. The cavern volume was then computed by multiplying the floor surface area Sb at this point by the stock height, Z. Figure 3.14 Diagram showing intersecting circles with the impeller located at the centre of a virtual cylinder. The overlapping area between the mixed zone and mixing chest is the cavern region. To use Eqn. 3.8 we must determine the resistive force at the suspension-wall interface, w, which we did by measuring the rotational speed at the point where the cavern first completely filled the vessel. The impeller power number and flow number were determined using the data in Figure 3.13 which permitted the force at the interface to be calculated using Eqn. 3.7. (The power numbers measured during the cavern tests and those determined using the data from Figure 3.13 agreed within 3%). At this point Sp = 0, a = 0. Sw is given by the vessel geometry and w was obtained by dividing the total applied impeller force by Sw (shown in Figure 3.15(b) for hardwood pulp as a function of suspension mass concentration). Cavern region Impeller Chest wall x rc b rv 1 2 a 47 a) b) Figure 3.15 Rotation speed (a) and wall stress calculated (b) for complete mixing in the vessel with hardwood pulp as a function of suspension mass concentration (E/D = 0.4, C/D = 0.7). 0 1 2 3 4 5 0 50 100 150 200 Z/T 0.8 1.0 1.2 w ( P a ) C m (%) 0 1 2 3 4 5 0 400 800 1200 1600 Z/T 0.8 1.0 1.2 N ( rp m ) C m (%) 48 w can also be estimated from published pipe friction data. We used the correlation presented in the TAPPI (1998) technical information sheets to do this. Here the head loss in a piping system is correlated by H/L = f1 K V Cm T (3.13) where the factors f1, K,, and are given for a range of pulp types for a hardwood pulp: f1 = 1.15, K = 236, = 0.27, = 1.78 and = -1.08 (TAPPI TIS 0410-14). The chest diameter is T = 381mm. The velocity at which the head loss is determined must be specified and we chose values of V = 0.001, 0.01 and 0.1 m/s for our calculations although the suspension velocity along the cavern-wall interfaces will vary with position throughout the chest and with impeller speed. Cm is the suspension mass concentration (%). Figure 3.16 compares these „pipe flow‟ estimates of w with those determined using the point at which the cavern first filled the vessel and the measured values of the suspension yield stress, y. The pipe flow estimates of w were much lower than the other values. We are now in a position to predict cavern development as a function of impeller speed in our mixing vessel. The axial force and power number at a given impeller speed and suspension consistency were determined using the data in Figure 3.13. The force generated by the impeller was iteratively balanced with the forces at the cavern interfaces, as described previously. We examined two scenarios. In the first (Figure 3.17a) we use the suspension yield stress as measured in the rheometer for y (data in Figure 3.2), w determined at the point where the cavern first filled the vessel (using the procedure described above), and a = 0. Figure 3.17a compares model predictions (solid lines) with the experimental data (ERT) for hardwood pulp at three mass concentrations. The dashed lines show the predictions with the suspension yield stress varied by 20% (the average coefficient of variation for the yield stress measurements was 22%, 49 with a 20% variation in y giving a confidence interval of approximately 95%). The error bars shown for selected experimental data represent 16% uncertainty in the cavern volume measurements made using ERT. The figure shows that the model predicts the trend in cavern development well, particularly for the suspensions at Cm = 3 and 4%. In general the experimental points are reasonably close to those predicted by the model (volume predicted to an average of 21%). Note that because the model was used to evaluate w each curve intersects the experimental point where the cavern first filled the vessel. Figure 3.16 Comparison of wall stresses measured/estimated and suspension yield stress for hardwood pulp as a function of mass concentration. Estimates for w based on the yield stress (y - measured), the point at which the cavern first completely filled the mixing vessel, and friction losses in flowing pulp suspensions. 0 1 2 3 4 5 0 50 100 150 200 w , y ( P a ) C m (%) Measured y (HW) Experimental Calculated TAPPI) v=0.001m/s v=0.01m/s v=0.1m/s 50 From Figure 3.16 we see that the values for w measured using the onset of complete motion in the vessel are almost identical to the values of y measured by rheometry. Indeed, it seems that this estimate of w evaluates y. In any case, an argument can be made for requiring that the suspension yield stress must be exceeded at all cavern interfaces. This is because the suspension yield stress must be exceeded at all suspension-suspension interfaces just prior to the cavern reaching the vessel wall or suspension-air interface. Thus for the second scenario, given in Figure 3.17(b) the computations were repeated with a = w = y with y determined by rheometry. These predictions agree better with the experimental data, and require that only one suspension measurement – the suspension yield stress – be made. For both scenarios, cavern development (shape of the volume vs. impeller speed curves) is largely determined by cavern geometry and its interaction with the vessel walls. However, the model (using a = w = y) was found to predict the cavern volume to an average of 13%, which is better than that found for previous models. This accuracy depends primarily on the yield stress which was determined to 22%. In the future, computational fluid dynamics may improve cavern estimation by calculating the cavern geometry directly; however, the specification of suspension rheology is still an issue and subject to measurement inaccuracies. 51 a) b) Figure 3.17 Cavern volume versus impeller speed for laboratory mixer. Comparison of experimental data (measured by ERT) with model predictions for hardwood pulp. Z/T = 0.8, E/D = 0.4. (a) Cm = 2%: y = 9 Pa ;w =12 Pa ; a = 0; Cm = 3%: y = 38 Pa ;w = 42 Pa ; a = 0; Cm = 4%: y = 69 Pa ;w =90 Pa ; a = 0. (b) (y = w = a) Cm = 2%: y = 9 Pa; Cm = 3%: y = 38 Pa; Cm = 4%: y = 69 Pa. 0 200 400 600 800 1000 0.00 0.01 0.02 0.03 0.04 C m (%) 2 3 4 Model +/-20% y suspension volume = 0.0347m 3 V c ( m 3 ) N (rpm) 0 200 400 600 800 1000 0.00 0.01 0.02 0.03 0.04 C m (%) 2 3 4 Model +/-20% y suspension volume = 0.0347m 3 V c ( m 3 ) N (rpm) 52 3.4 Summary Since pulp suspensions are opaque, cavern size must be imaged using indirect techniques. We evaluated two methods, ERT and UDV, and found that both techniques gave satisfactory measurements from which cavern shape and volume could be determined. However, ERT was chosen due to the marked decrease in the time required to acquire data. The shape of the cavern was best approximated as a truncated right-circular cylinder, and as expected, increasing impeller speed increased cavern volume. However, development of cavern volume with increasing impeller speed was not uniform, which was attributed to interaction between the cavern and the vessel walls. Current models for predicting cavern development in yield stress fluids treat the cavern as being isolated with no interaction with the vessel walls and do not describe our experimental results well. This prompted us to develop a model that included interaction with vessel boundaries which required specification of cavern shape (based on ERT imaging) and development of equations to balance the force provided by the impeller with the net resistive force at the cavern boundaries. The model proposed predicts the trend in cavern volume with increasing impeller speed well although the absolute cavern volume is only predicted to within 13%. These predictions are sensitive to the values of suspension yield stress which was measured to 22%. 53 4 DYNAMIC TEST STUDY ON LAB-SCALE CHEST2 4.1 Introduction Agitated stock chests are used in conjunction with process control algorithms to maintain process and product uniformity during the manufacture of pulp and paper products. Process control is effective at attenuating process disturbances below the cut-off frequency of the control loop, which is determined by process dynamics and the type of control algorithm and tuning employed (Bialkowki, 1992). Aggressive tuning increases the cut-off frequency, although over aggressive tuning can create oscillations which amplify variability. The agitated stock chest acts as a low- pass filter to remove the high-frequency variability and extend the range of disturbance attenuation. In most cases stock chests are designed based on past experience and semi-empirical techniques (Yackel, 1990; Reed, 1995). Two common chest geometries are employed for pulp suspension mixing: rectangular and cylindrical. Rectangular chests save space when grouped together using common walls (as in the basement of a papermachine) while cylindrical chests can take advantage of hoop stress design which allows the use of thinner walls for vessels of the same height. For cylindrical pulp stock chests, complete motion across the suspension surface is most readily achieved when the stock height (Z) to chest diameter (T) ratio is 0.8 (Z/T = 0.8) (Yackel, 1990). An axial flow impeller mounted in a side-entering mode is typically used in this application. Guidelines for impeller installation, including impeller offset from the rear chest wall (E) and height from the chest bottom (C) are typically E/D = 0.3 – 1.0 and C/D = 0.5 – 1.5 (where D is the impeller diameter) (AGIMIX International, Uddevalla, Sweden). A common 2 A version of this chapter will be submitted for publication. Leo K. Hui, C.P.J. Bennington and G.A. Dumont (2011) Mixing Dynamics in Cylindrical Pulp Stock Chests. 54 design strategy matches the momentum required to agitate a pulp suspension in a given application with that generated by the impeller (Yackel, 1990). Cylindrical chests are widely used in the pulp and paper industry, particularly for creating controlled mixing zones in bleaching towers and in high-density pulp storage chests. Agitated stock chests often do not behave ideally (Ein-Mozaffari et al., 2003). When a non-Newtonian fluid, such as a pulp suspension, is agitated, a cavern (a region of active motion around the impeller) is created (Wichterle and Wein, 1981; Solomon et al., 1981; Silvester, 1985; Elson, 1990; Amanullah et al., 1998; Wilkens et al., 2005; Hui et al., 2009). In batch operations there is limited or no fluid flow outside the cavern; in continuous operations, suspension flow but limited mixing occurs outside the cavern. Ein-Mozaffari et al. (2003) measured the efficiency of several industrial pulp chests and found significantly reduced attenuation efficiency. The non- ideal flows identified included bypassing or channeling (where part of the feed directly flowed to the exit without passing through the impeller zone) and the formation of dead zones (regions of stagnant suspension that were disconnected from or moved significantly slower than the bulk flow). These phenomena were exacerbated by the non-Newtonian nature of the pulp suspensions, reducing chest mixing capacity and the ability to attenuate process disturbances. To identify the factors that affected mixing quality, Ein-Mozaffari et al. (2005) studied mixing dynamics in a laboratory-scale rectangular stock chest using two pulp suspensions, a shorter-fibred hardwood pulp and a longer-fibred softwood pulp, over a range of fibre mass concentrations. It was found that the cavern volume, which was affected by the suspension yield stress and the impeller rotational speed, played a significant role in determining mixing dynamics. As the yield stress decreased or impeller speed increased, the extent of bypassing and dead zone formation was reduced and the degree of disturbance attenuation improved. In addition, the 55 imposed flow configuration, determined by the position of the stock inlet and outlet points relative to the cavern, affected mixing performance. Bypassing was minimized when the pulp outlet was located within the cavern zone. Other studies with pulp suspensions (the majority in rectangular chests) have confirmed these findings (Bakker and Fasano, 1993; Ein-Mozaffari et al., 2003; Ford, 2004; Wilstrom and Rasmuson, 1998a). In this chapter, the dynamic response of a cylindrical chest was examined as a function of impeller speed for several pulp suspensions. The results were also compared with previous findings for a rectangular chest configuration. A dynamic mixing model (Soltanzadeh et al., 2009) was used to quantify mixing dynamics. Figure 4.1 shows the continuous-time dynamic model used to represent the observed behavior of a pulp chest. The model incorporates two parallel suspension flow paths through the vessel: one that enters the cavern created by the impeller and one that bypasses this mixed zone and exits the chest with minimal mixing. f is the fraction of pulp stock that channels through the chest while the remaining stock (1-f) enters the mixing zone. 1 and 2 are the time constants for each zone, respectively (with 1 2). Note that it is possible to have transport delay, Td, in the step response signal (due primarily to flow delay in the process piping prior to and following the chest). Using input-output data obtained when the chest is excited with an appropriately designed signal allows the model parameters to be identified. For the model chosen (Fig. 4.1), ideal mixing is approached when the extent of bypassing approaches zero and the time constant for the mixing zone approaches the theoretical time constant of the system, i.e., th = Vt/Q, where Vt and Q are the suspension volume and flow rate. 56 Figure 4.1Dynamic model for non-ideal flow in agitated pulp stock chests (2>>1). 4.2 Experimental set-up and procedures The transparent 1/10 scale-model of a cylindrical stock chest (T = 38.1 cm) with a laboratory- scale (D = 16.5 cm) side-entering axial-flow impeller (Maxflo, Chemineer Inc., Dayton, OH) used in this study is shown in Fig. 4.2. The impeller was located 7.0 cm from chest wall (E/D = 0.4) and 12.0 cm from the bottom of the chest (C/D = 0.7). The pulp suspension was pumped from a feed tank, through the chest and then to a discharge tank. The suspension was added to the pulp chest through an inlet pipe placed at the suspension surface near the wall opposite the impeller (a thin plastic sheet was attached to the pipe opening to avoid pulp splashing as it dropped into the chest). The pulp exit was located at the wall behind and below the impeller. This stock entrance/exit configuration is optimal for forcing the suspension feed through the cavern (Ein-Mozaffari et al., 2005). The mixing dynamics were measured by adding a conductive tracer (saline solution) to the pulp feed using a pseudo-random binary signal (PRBS) to control tracer addition (Fig. 4.3) f 1 - f u + + y 1+2s 1 1+1s 1 57 and excite the chest dynamics at the appropriate frequencies. The conductivity of the suspension entering and leaving the chest was measured (Fig. 4.4 shows one typical data set) with the first portion of the experimental data used to estimate the model parameters and these parameters were then used to model the second part of the data. The MATLAB program used for this model parameter estimation is shown in Appendix B. The model output for the latter portion of the test was compared with the measured data and the extent of agreement expressed as: n i measuredmeasured measuredcomputed ymeany yy n fit i ii 1 )( 1 1%100% (4.1) where ycomputed is the output calculated from the model and ymeasured is the actual output. The effectively mixed volume (Vmix) was estimated from the identified parameters using Vmix = Q2(1- f). Dynamic tests were performed with hardwood and softwood bleached kraft pulps from Domtar Inc. (Windsor, QC). The hardwood pulp had a length-weighted mean fibre length of 1.28 mm and the softwood pulp had a length-weighted mean fibre length of 2.96 mm. Suspension concentrations of Cm = 2, 3 and 4% were used for the hardwood pulp and Cm = 1.8 and 3% for the softwood pulp. The suspension yield stress for the pulps were measured using a Haake RV12 Rotovisco concentric cylinder viscometer (Thermo Fisher Scientific, Waltham, US) and a vaned measuring bob, as described by Hui et al. (2009). The pulp flow rates through the chest were 7, 14 and 20 L/min, with the mean pulp residence time greater than 4 times the blend time (Yackel, 1990). The batch blend time was determined for a typical test condition (Cm = 3% hardwood pulp at N = 500 rpm (the median impeller rpm used for the tests)) by measuring the response to a tracer addition above the impeller using a conductivity probe located on the opposite wall, giving 58 a blend time of 6 s. The residence times at the highest flow rate of 20 L/min and at the lowest flow rate of 7 L/min were 104 s and 298 s respectively (both > 24 s). The net power required to create complete motion in the chest was determined using P = 2NM, where the impeller speed (N) and net torque (M) were measured using an inductive-rotary torque transducer (model 0411IE50, Staiger Mohilo, Germany) mounted on the impeller shaft, with complete motion determined by visually inspecting the vessel through the clear walls (i.e., the entire suspension was moving at all the vessel boundaries, not just on the suspension surface). For selected tests, electrical resistance tomography (ERT) was used to image flow through the chest under conditions where a cavern existed. ERT is a non-invasive imaging technique that reconstructs the conductivity distribution inside the mixing vessel using voltage measurements made at the chest periphery. A step change of saline concentration was introduced to the suspension feed with the vessel imaged using a P2000 ERT tomography instrument (ITS, Manchester, UK) as a function of time. Four sensor planes (P1 - P4), each with 16 stainless steel electrodes installed on the chest periphery, were used to measure the conductivity changes within the chest and allowed the temporal reconstruction of the tracer‟s progress through the vessel. ERT was also used under batch mixing conditions to compare the agitated cavern volume with that measured using the dynamic tests (Hui et al., 2009). 59 a) b) Figure 4.2 Cross-section (a) and photograph (b) of cylindrical stock chest. The arrows in (b) show the direction of suspension flow in continuous operation. Feed Exit Impeller shaft 50.8mm w = 5.1mm 381mm (15”) 625mm (24.6”) Outlet 120mm 70mm Inlet P1 P2 P3 P4 60 Figure 4.3 Schematic of apparatus used for the dynamic tests. Figure 4.4 Typical signals measured for model parameter identification. Stock chest Discharge tank Feed Tank 3 4 6 6 7 7 8 1 : Transmitter 2 : Indicator 3 : Tachometer 4 : Torque transducer 5 : Magnetic flowmeter 6 : Conductivity sensor 7 : Pump 8 : Solenoid valve 9 : Pulsation dampeners Liquid Tracer 7 5 5 2 1 1 1 1 9 Estimation Verification 0 600 1200 1800 2400 2 3 4 5 6 7 C o n d u c ti v it y ( m S ) Time (s) Input Output 61 4.3 Results and discussion The accuracy of the dynamic model was verified by measuring the time delay (Td) and mixing time constant (2) under conditions of complete vessel agitation and comparing them with theoretical values. Under conditions of ideal mixing, the volume of the suspension in the piping between the locations of conductivity measurement and the chest inlet and outlet should account for the delay. The time delay (and its uncertainty) at several flow rates was estimated by averaging the results of four independent tests made using hardwood pulp at Cm = 3% under conditions where different extents of mixing existed in the vessel. Different impeller speeds were used, but no trend in delay time with impeller speed was observed. The delay estimated using the model was within experimental error of that calculated, as shown in Table 4.1 (the uncertainty in the theoretical time delay was estimated using the accuracy of pipe and flow rate measurements). Also, it was found that the 2 estimate (for complete mixing) was always slightly larger than the theoretical value (by 6% on average, which is close to the error (7%) in parameter estimation) (Table 4.2). As the preparation for each test is time-consuming, only one trial was conducted at each test condition given in Table 4.3. Table 4.1 Time delay (Td) measured versus theoretical values for hardwood pulp Cm = 3% (N = 425, 525, 672 and 810 rpm at each flow rate) Flow rate (Q) (L/min.) Td estimated from dynamic tests (s) Td based on flow and system geometry (s) 7 79.6 9.5 81.9 6.4 14 38.4 2.0 40.9 2.9 20 31.7 4.4 28.7 2.1 62 Table 4.2 Time constant (2) measured for complete mixing of hardwood pulp at Q = 14 L/min (Theoretical 2 = 149s) Cm (%) N (rpm) 2 (s) Percentage of fitting Eqn. (4.1) (%) 2 380 1557 91 460 14210 91 3 672 1634 92 810 16121 78 4 830 16311 93 1000 16413 91 The data from all the tests are reported in Table 4.3. On average, the identified model parameters were found to fit the experimental data to 88%, with all cases fitted above 78% (Eqn. 4.1). The flow configuration used in the tests forces the pulp feed through the cavern before leaving the chest and consequently, the parameters f and 1 were expected to be low or negligible. This was found in all but six cases. In the cases where the bypassing was identified it was found to involve less than 10% of the flow. To verify the presence of a cavern and a poorly agitated zone in a partially agitated chest under continuous operation conditions, ERT was used to image the mixing of a pulse of tracer (220 s in duration) through the chest under typical operating conditions – agitation of a hardwood pulp suspension at Cm = 3%, with the standard vessel geometry (E/D = 0.6, C/D = 0.7 and Z/T = 0.8) using an impeller speed that would lead to cavern formation (N = 425 rpm and Q = 14 L/min). Under the test condition 2 was found to be 70s (Table 4.3), implying that the cavern occupied about 47% of the vessel volume since the theoretical 2 of complete vessel agitation was 149s. 63 The step change in tracer was followed as it entered the vessel at the suspension surface and flowed towards the side exit located between ERT sensors 3 and 4 above plane 1 (P1), as shown in the t = 0 image in Fig. 4.5. Note that before the injection of tracer, the conductivity in the chest was low (0.6 mS/cm) as indicated by the blue color in the tomographic images. As the saline solution entered the chest the local conductivity began to increase (to green in the images). At t = 30 s the tracer moved to the impeller-side of the chest along the suspension surface by the upper circulation loop and began to fill the top part of the vessel. The pulse then moved downward and spread out to the lower planes (t > 30s). As more tracer accumulated in the chest, the conductivity increased (indicated by the red colour in the images for t = 100s and 170s) and the truncated cylindrical shape of the cavern can be clearly seen. Tracer injection stopped at t = 220 s, and the tracer began to be washed from the chest (t = 255 to 480s). Towards the end of the test, the presence of tracer was observed to linger in regions of the vessel (note the green regions in the top image plane, P4, of the t = 360 s image) which indicates regions of slower mixing. No obvious sign of direct passing of tracer to the exit could be seen, which supports the “negligible bypassing” results obtained from the dynamic tests. In addition, the ERT images show that outside the cavern there is a stagnant region the tracer does not enter. This dead zone was also observed visually during the experiment (lack of flow adjacent to the vessel wall). This suggests that a fillet (the insertion of solid material in this region) could be used to reduce this unwanted dormant volume. 64 Table 4.3 Dynamic test results Hardwood Pulp Suspension Mass Concentration (%) (%) Flow rate (L/min) Rotation speed (rpm) f 1 (s) 2 (s) Percentage of fitting (%) 2.0 14 125 0 0 47 89 250 0 0 106 85 380 0 0 155 91 460 0 0 142 91 3.0 7 425 0.031 0 168 92 525 0 0 256 92 672 0 0 301 92 810 0.065 1.07 304 86 14 425 0.087 5.93 70 89 525 0.017 0 113 89 672 0 0 163 92 810 0 0 161 78 20 425 0 0 56 88 525 0 0 93 85 672 0 0 115 89 810 0 0 106 86 4.0 14 550 0 0 68 91 675 0 0 104 90 830 0 0 163 93 1000 0 0 164 91 Softwood Pulp Suspension Mass Concentration (%) Flow rate (L/min) Rotation speed (rpm) f 1 (s) 2 (s) Percentage of fitting (%) 1.8 14 425 0 0 116 85 525 0.099 0 133 83 680 0 0 151 92 750 0.013 0 171 87 3.0 14 425 0 0 69 86 525 0 0 97 82 1020 0 0 160 91 1100 0 0 156 80 65 Time(s) P1 P2 P3 P4 Side view along the dotted line 0 30 40 100 170 220 255 290 360 480 Figure 4.5 ERT images of a dynamic test at Cm = 3% and N = 425rpm (E/D = 0.6, C/D = 0.7, Z/T = 0.8). The suspension inlet and outlet positions are shown in the images of P4 and P1 at time t = 0 s, respectively. 5 6 7 8 9 4 3 2 1 10 11 12 13 14 15 16 5 6 7 8 9 4 3 2 1 10 11 12 13 14 15 16 5 6 7 8 9 4 3 2 1 10 11 12 13 14 15 16 5 6 7 8 9 4 3 2 1 10 11 12 13 14 15 16 inlet outlet 5 6 7 8 9 4 3 2 1 10 11 12 13 14 15 16 5 6 7 8 9 4 3 2 1 10 11 12 13 14 15 16 5 6 7 8 9 4 3 2 1 10 11 12 13 14 15 16 5 6 7 8 9 4 3 2 1 10 11 12 13 14 15 16 5 6 7 8 9 4 3 2 1 10 11 12 13 14 15 16 5 6 7 8 9 4 3 2 1 10 11 12 13 14 15 16 5 6 7 8 9 4 3 2 1 10 11 12 13 14 15 16 5 6 7 8 9 4 3 2 1 10 11 12 13 14 15 16 5 6 7 8 9 4 3 2 1 10 11 12 13 14 15 16 5 6 7 8 9 4 3 2 1 10 11 12 13 14 15 16 5 6 7 8 9 4 3 2 1 10 11 12 13 14 15 16 5 6 7 8 9 4 3 2 1 10 11 12 13 14 15 16 5 6 7 8 9 4 3 2 1 10 11 12 13 14 15 16 5 6 7 8 9 4 3 2 1 10 11 12 13 14 15 16 5 6 7 8 9 4 3 2 1 10 11 12 13 14 15 16 5 6 7 8 9 4 3 2 1 10 11 12 13 14 15 16 5 6 7 8 9 4 3 2 1 10 11 12 13 14 15 16 5 6 7 8 9 4 3 2 1 10 11 12 13 14 15 16 5 6 7 8 9 4 3 2 1 10 11 12 13 14 15 16 5 6 7 8 9 4 3 2 1 10 11 12 13 14 15 16 5 6 7 8 9 4 3 2 1 10 11 12 13 14 15 16 5 6 7 8 9 4 3 2 1 10 11 12 13 14 15 16 5 6 7 8 9 4 3 2 1 10 11 12 13 14 15 16 5 6 7 8 9 4 3 2 1 10 11 12 13 14 15 16 5 6 7 8 9 4 3 2 1 10 11 12 13 14 15 16 5 6 7 8 9 4 3 2 1 10 11 12 13 14 15 16 5 6 7 8 9 4 3 2 1 10 11 12 13 14 15 16 5 6 7 8 9 4 3 2 1 10 11 12 13 14 15 16 5 6 7 8 9 4 3 2 1 10 11 12 13 14 15 16 5 6 7 8 9 4 3 2 1 10 11 12 13 14 15 16 5 6 7 8 9 4 3 2 1 10 11 12 13 14 15 16 5 6 7 8 9 4 3 2 1 10 11 12 13 14 15 16 5 6 7 8 9 4 3 2 1 10 11 12 13 14 15 16 5 6 7 8 9 4 3 2 1 10 11 12 13 14 15 16 5 6 7 8 9 4 3 2 1 10 11 12 13 14 15 16 5 6 7 8 9 4 3 2 1 10 11 12 13 14 15 16 low conductivity high conductivity 66 Figures 4.6 and 4.7 show that 2, and hence the mixing quality, increases with increasing impeller speed until a plateau is attained. The error in determining 2 was ~7% (the average standard deviation calculated in model parameter estimation using MATLAB for three hardwood mass concentrations at Q = 14L/min). As impeller speed increased, the cavern enlarged increasing the active mixed volume and hence 2. As suspension mass concentration increased, the suspension yield stress increased which decreased cavern volume at a fixed impeller speed. Thus a higher impeller speed was needed to maintain the same 2. At a fixed impeller speed, 2 decreased as the flow rate was increased because it is inversely proportional to Q (Fig. 4.7). When Vmix was calculated using 2 (Fig. 4.8), it was found that for a fixed suspension concentration, changing the flow rate did not significantly affect the mixed volume which depended mainly on the impeller speed. The uncertainty of Vmix was estimated to be 10% based on the error in measuring 2 and Q. Figure 4.6 Effect of impeller speed (N) and pulp consistency (Cm) on time constant (2) (E/D = 0.6, C/D = 0.7, Z/T = 0.8). 200 400 600 800 1000 0 50 100 150 200 th =149s (14L/min) s N (rpm) Hardwood C m (%) 2 3 4 67 Figure 4.7 Effect of flow rate (Q) on time constant (2) with rotation speed (Cm = 3%, E/D = 0.6, C/D = 0.7, Z/T = 0.8). Figure 4.8 Effect of flow rate (Q) on mixing volume (Vmix) with rotation speed (Cm = 3%, E/D = 0.6, C/D = 0.7, Z/T = 0.8). 400 500 600 700 800 900 0 100 200 300 th =104s (20L/min) th =149s (14L/min) th =298s (7L/min) s N (rpm) Q(L/min) 7 14 20 400 500 600 700 800 900 0.00 0.01 0.02 0.03 0.04 V m ix ( m 3 ) N (rpm) Q (L/min) 7 14 20 V t = 0.0347m 3 68 Figure 4.9 shows the change in mixed volume (cavern size) with impeller speed for the hardwood and softwood pulps at Cm = 3%. Vmix for the hardwood suspension is larger than that for the softwood for all impeller speeds. However, comparison on the basis of suspension concentration is not appropriate as cavern size is determined primarily by the yield stress of the suspension (Hui et al., 2009). A better comparison can be made using hardwood and softwood suspensions of the same yield stress. In Fig. 4.10, the cavern volume of a Cm = 1.8% softwood suspension is compared with that of the Cm = 3% hardwood suspension (both have a yield stress of 38 Pa). We see that the mixed volume is very similar, except at the lowest impeller speeds tested where the softwood pulp had the higher mixed volume. This same trend was also reported by Ein-Mozaffari et al. (2005) for tests in rectangular chests. Thus, in addition to the suspension yield stress other suspension factors (which we can only categorize in the broadest sense of „pulp type‟) also affect mixing dynamics. As impeller speed is increased to approach the fully mixed volume, the differences in cavern volume are reduced within our ability to measure them. Other factors may account for these observations, including the uncertainty with which the yield stress can be determined experimentally or the effect of the fibre type on the impeller flow and power number, which have not been widely studied. Finally, mixing volumes calculated using the dynamic tests were compared with those measured using ERT under batch conditions as shown in Fig. 4.11. For the batch tests, the cavern was imaged using conductive particle tracers added to the impeller region (see Hui et al., 2009). It was found that the mixed volume determined using the dynamic model, Vmix, was slightly larger than that determined by ERT under batch conditions, Vc. This may be due to the location of the feed pipe and the momentum added to the mixed suspension by the entering flow. 69 However, changes in flow rate (Q = 7, 14, 20 L/min) did not significantly change the mixed volume. Figure 4.9 Effect of pulp types (HW: hardwood and SW: softwood) on cavern size obtained in dynamic and batch (ERT) operations at Cm = 3% (E/D = 0.6, C/D = 0.7, Z/T = 0.8). Figure 4.10 Effect of pulp types (HW: hardwood and SW: softwood) with similar yield stress (38Pa) on mixing volume (Vmix) with rotation speed (N) (E/D = 0.6, C/D = 0.7, Z/T = 0.8). 400 600 800 1000 1200 0.000 0.005 0.010 0.015 0.020 0.025 0.030 0.035 0.040 0.045 V t = 0.0347m 3 V m ix , V c ( m 3 ) N (rpm) Continuous Batch (Dynamic) (ERT) HW 3% SW 3% 400 500 600 700 800 0.00 0.01 0.02 0.03 0.04 V t = 0.0347m 3 V m ix ( m 3 ) N (rpm) HW 3% SW 1.8% 70 Figure 4.11 Comparison between cavern size obtained in dynamic and batch (ERT) operations at various hardwood pulp consistencies (Cm) (E/D = 0.6, C/D = 0.7, Z/T = 0.8, Q = 14 L/min for the dynamic tests). Scaling procedures are used to design mixers based on data obtained in geometrically identical mixers at a different, usually smaller, scale. Yackel‟s (1990) design procedure for pulp and paper chests uses the concept of level momentum. The momentum number (Mo) and level momentum ( Mo ) are defined as: 42 1 DNCMo (4.2) 3 2 42 1 3 2 V DNC V Mo Mo (4.3) where C1 is a constant that depends on impeller type and geometry. The V 2/3 term is due to the proportionality between chest surface area and chest volume, which also fits the experimental criteria Yackel uses (of having complete motion across the surface) to determine good mixing. According to Yackel, designing for equal level momentum produces equal bulk velocities in the scaled vessel and thus similar mixing results: 0 200 400 600 800 1000 0.00 0.01 0.02 0.03 0.04 V t = 0.0347m 3 V m ix , V c ( m 3 ) N (rpm) Continuous Batch (Dynamic) (ERT) HW 2% HW 3% HW 4% 71 3 2 2 4 2 2 22 3 2 1 4 1 2 11 V DNC V DNC (4.4) where N1, D1 and V1 are the impeller speed, impeller diameter and fluid volume used in the smaller chest and V2 is the volume of the larger chest having an impeller diameter of D2. Assuming C1 = C2 (geometric similarity), the corresponding impeller speed (N2) for a cylindrical chest having a larger fluid volume (V2) is calculated using 2 1 3 2 1 4 2 3 2 2 4 1 2 1 2 VD VDN N (4.5) Since V1 and V2 are proportional to T1 3 and T2 3 , respectively, and that D1/T1 = D2/T2 (geometric similarity), N2 is given as: 2 1 1 2 1 2 1 4 2 2 2 4 1 2 1 2 D D N TD TDN N (4.6) Other scale-up methods are used in literature, including constant power/volume (P/V) and constant torque/volume (M/V). Constant P/V is the most commonly recommended scaling criteria for mixing operations (Oldshue, 1983; Wilkens et al., 2003; Paul et al., 2004). This method correlates well with mass-transfer characteristics in a mixer and is suitable for calculating the power needed for gas dispersion or dispersion of immiscible liquids (Zlokarnik, 2006). Equal M/V is also a practical and common scale-up criterion because it relates directly to the overall size, torque capability and cost of the mixer (Paul et al., 2004). It is the same as scaling for constant tip speed and is usually applied when flow velocities in the impeller region need to be the same in both vessels. The cavern model developed by Hui et al. (2009) can also be used to scale pulp mixers. This model takes into account the interactions between the actively mixed cavern and the vessel walls by balancing the forces produced by the impeller with those 72 acting at the cavern boundary. To create complete mixing in the vessel, the cavern boundaries must extend to fill the mixer volume. Table 4.4 gives the dependence of N2 on N1 for four scale- up methods considered and the calculation of N2 using the cavern model. Table 4.5 gives the power predicted by scaling three- and ten-fold the volume of our laboratory cylindrical chest using the criteria described above. Geometric similarity was maintained (D/T = 0.43, Z/T = 0.8) with the base conditions measured in the 34.7L vessel for complete motion. Of the methods considered, the constant P/V criterion gives the highest power consumption. The constant M/V and level momentum scaling criteria give identical predictions because, for geometric similarity, they both scale as N2=N1(D1/D2) as shown in Table 4.4. Determining N2 using the cavern model does not require knowing N1, just the impeller characteristics (Nf and Np) and suspension yield stress (y). The model predicts power levels close to those calculated using the constant M/V and level momentum scaling methods. This is expected as the cavern model also scales as N2=N1(D1/D2) for geometric similarity and constant impeller behavior. In industrial processes agitated pulp chests are usually rectangular or cylindrical. To compare which shape is more efficient, the specific power to create complete mixing in a rectangular chest (measured by Ein-Mozzaffari, 2002) was compared with that for a cylindrical chest. As the cylindrical chest used in our study had a lower volume than that used by Ein- Mozaffari, the base data was scaled to that of a chest having equal volume and allowing for the different suspension height in the rectangular chest, Z/T = 1.2, and the fact that D/T changed between chests. The power for the scaled cylindrical chest was found to be 1.4 W/kg and 4.8 W/kg for Cm = 1.8% and 3% softwood pulp suspensions, respectively. Both these predictions are close to the values measured for the rectangular chest (1.4 W/kg at Cm = 1.8% and 5.4 W/kg at 73 Cm = 3%). Due to the use of scaling and the corrections needed to make this comparison we can only state that vessel geometry, cylindrical or rectangular, requires approximately the same amount of energy to create a well-mixed state. Table 4.4 Calculation of impeller speed on scale-up using different criteria Criteria Relationship Scaled impeller speed Constant power per unit volume* 2 2 1 1 V P V P 53DNNP p 3 2 2 1 12 D D NN Constant torque per unit volume* 2 2 1 1 V M V M N P M 2 2 1 12 D D NN Level momentum* 3 2 2 2 3 2 1 1 V Mo V Mo 2 1 12 D D NN Constant tip speed 2211 DNDN 2 1 12 D D NN Cavern model 2AF y 2 24 2 2 2 3 4 p f N NDNF (A2: surface area of cavern; Nf: axial force number of impeller) 2 1 224 2 2 2 34 pf y NND A N *Assumptions: D/T = constant; Np = constant, geometric similarity Table 4.5 Power predicted for complete mixing of a hardwood pulp suspension in cylindrical stock chests using different scale-up criteria (D/T = 0.43, Z/T = 0.8) Cm (%) y (Pa) Scale-up volume ratio Power (W) Constant power/volume (Wilkens et al., 2003) Constant torque/volume (Wilkens et al., 2003) Level momentum (Yackel, 1990) Cavern model (Leo et al., 2009) 2 9 1 : 3 22 18 18 12 1 : 10 76 41 41 28 3 38 1 : 3 142 98 98 105 1 : 10 497 227 227 243 4 69 1 : 3 374 264 264 258 1 : 10 1310 593 593 579 74 Other factors should be considered when selecting a pulp mixer. Rectangular chests are more susceptible to the creation of stagnant or dead zones, especially near the chest corners (Ford, 2004). Also, a sudden transition to non-ideal behavior has been observed in a pilot-scale rectangular chest under certain operating conditions (e.g., a dramatic increase of bypassing could occur when the impeller speed was decreased slightly, all other operating conditions remaining fixed) (Ein-Mozaffari, 2005). This was attributed to the location of the suspension feed and exit relative to the impeller. This undesirable flow bifurcation could create fluctuations and/or instability in the performance of an agitated pulp chest but was not observed in the cylindrical chest. It can be avoided by ensuring that entrance and exit locations are optimized. 75 4.4 Summary Dynamic tests were carried out in a laboratory-scale cylindrical chest to identify and characterize the mixing quality. The dynamic model chosen, which included an actively mixed region with the possibility of bypassing, was found to accurately describe the dynamics measured. The insignificant bypassing measured indicated that the flow configuration used (with the exit located within the active cavern, close to and below the impeller) is an effective configuration for avoiding stock bypassing. Imaging, using electrical resistance tomography (ERT), verified the size and shape of the caverns created when the chest was not fully agitated. Cavern size could be increased at a given suspension concentration by increasing impeller speed, which improves mixing quality. The power required to completely agitate a cylindrical chest was similar to that required for a rectangular chest of the same volume. 76 5 CFD SIMULATION OF PULP MIXING 5.1 Introduction Agitated stock chests are important in pulp and paper processing. They keep pulp suspensions in motion to avoid dewatering and blend different pulp stocks or blend pulp with other chemicals. They also act buffers between processes and act as low-pass filters to remove high frequency variability and to complement the action of control loops. However, pulp chests do not always function as expected. Ein-Mozaffari et al. (2003) studied the mixing performance of industrial chests and noticed the non-ideal dynamic response of rectangular pulp stock chests. This abnormality was attributed to improper chest design and the non-Newtonian rheology of pulp suspension, which has a yield stress promoting the formation of caverns (regions of active motion) around impellers. Under certain flow configurations, these caverns can induce undesirable flow, like bypassing (part of the feed flows directly to the exit without entering the cavern) and dead zones (stagnant regions outside the cavern) which degrade mixing and reduce the ability of the chest to attenuate variability. Poor chest performance can lead to unstable downstream processes and low product quality, resulting in high production costs. Pulp stock chests are typically rectangular or cylindrical. Their design is based on the use of semi-empirical methods and experience with a design strategy of matching the momentum generated by an impeller with that required to provide complete suspension motion across the surface of the chest (Yackel, 1990). This method does not specifically take the locations of pulp inlet and outlets into consideration, although heuristics are provided. Laboratory studies using tracer analyses, ultrasonic Doppler velocimetry (UDV) and electrical resistance tomography (ERT) have investigated mixing and suspension flow through agitated stock chests. Ein- 77 Mozaffari et al. (2005) used dynamic test results to show that cavern size and its position relative to inlet and outlet locations played an important role in determining the amount of feed entering the cavern which affected the mixing dynamics of the chests. Bypassing was minimized when the stock outlet was located within the cavern zone. In addition, Ein-Mozaffari et al. (2007) measured the flow characteristics (impeller flow number and pumping rate) of a Maxflo impeller and located the cavern using the velocity profiles measured by UDV. Hui et al. (2009) applied ERT and UDV to measure the cavern volume in a cylindrical chest. A model was developed to estimate the cavern size using a force balance with consideration of interaction between the cavern and chest walls. Although experimental studies provide some information about the mixing dynamics inside stock chests, obtaining this data is very difficult due to the opaqueness of the suspensions. Computational fluid dynamics (CFD) has been used to obtain information about the velocity fields inside these and other vessels, although there are difficulties associated with modeling the suspension and validating the simulations made. Many studies have shown that CFD can predict the velocity distribution, evaluate industrial stirred equipment, and optimize impeller design (Armenante et al., 1997; Bhattacharya and Kresta, 2002; Paul et. al., 2004; Li et al., 2005; Ankamma Rao and Sivashanmugan, 2010). For pulp suspensions, a number of studies using CFD have been carried out to investigate the flow in agitated pulp chests. Bakker and Fasano (1993) applied CFD and used the rheological data from Gullichsen and Harkonen (1981) for pulp suspension to model the flow in a rectangular pulp chest with a side-entering impeller. The numerically simulated results were found to agree with visual observation of the suspension surface. In the study of the impact of pulp suspension rheology on flow field in an agitated pilot tank with a jet nozzle mixer, Wikstrom and Rasmuson (1998a) compared the CFD-calculations 78 with the measured results and concluded that a more sophisticated rheology model should be used instead of the Bingham model because of the increasing deviation between the computed and measured flow field as the distance from the impeller increased. Ford et al. (2006) developed a CFD model of a rectangular pulp mixing chest and used the Bingham plastic model as the pulp suspension rheology. The model captured the mixing dynamics of the chest fairly well but it overestimated the extent of mixing in the bypassing flow, especially in flow situations with significant bypassing. This deviation was attributed to the inability of the Bingham model to fully describe the suspension rheology. In the CFD modeling of a cylindrical tank agitated with an axial-flow impeller, Saeed et al. (2007) treated pulp suspension as a Herschel-Bulkley fluid. The CFD simulations picked up the features of the flow field and the computed velocities agreed satisfactorily with the measured results. The discrepancy between the computed and measured velocities was believed to be due to the suspension rheology. Bhattacharya et al. (2010) modeled two industrial pulp stock chests using CFD and described the pulp rheology as a modified Herschel-Bulkley model. The computed dynamic response of the chests was found to agree reasonably well with the experimental measurements and the simulated flow fields provided insight about cavern formation, stagnant regions and bypassing zones created in the chests. However, these studies have not provided a detailed examination of the cavern flow, which is important in determining the non-ideal flow in pulp mixing chests. The extent of cavern growth could determine the degree of bypassing and the size of dead zone in agitated chests. In this chapter, the commercial CFD software (FLUENT 6.2, Fluent Inc, Lebanon, NH) was used to model a cylindrical chest equipped with a scaled version of a commercial axial-flow Maxflo impeller (Chemineer Inc., Dayton, OH) in the standard side-entering configuration for simulating the mixing of two pulp suspensions (hardwood and softwood) over a range of 79 impeller rotational speeds (N) and suspension concentrations (Cm). Ford et al. (2006) have presented a computational approach to simulate pulp suspension in a rectangular agitated chest equipped with a side-entering agitator for a range of operating conditions. This similar approach was used to simulate the pulp mixing in the cylindrical chest. Two non-Newtonian fluid models were used to describe suspension rheology (based on rheological measurements). The simulation results were then used to calculate the cavern size in batch mixing mode and to estimate the dynamic response of the chest in continuous operation. The computed results were then compared with the experimental results obtained in the studies of Hui et al. (2010). By determining the degree to which the CFD model can predict experimental data, its usefulness as a tool for chest design can be evaluated. If the evaluation is positive, the CFD model can be used for optimization and economical design of pulp chests for a range of objectives, including mixing efficiency, energy consumption and equipment manufacturing costs. 5.2 Experimental study of pulp mixing A number of experimental methods were used to study the mixing behavior of pulp stock chests. For batch operation, electrical resistance tomography (ERT) and ultrasonic Doppler velocimetry (UDV) were applied to study the cavern shape and size. ERT is a non-invasive imaging technique that can image the conductivity distribution inside a region of interest using voltage measurements made through a series of electrodes placed at the chest periphery. With the addition of a conductive tracer to the cavern, a P2000 ERT tomography instrument (ITS, Manchester, UK) was used to locate the region of higher conductivity and thus the cavern shape and size (Hui et al., 2009). The measurement error of the cavern boundary is typically about 5- 80 10% of the chest diameter. Also, the cavern results obtained from ERT were validated by visual observations of motion at the vessel walls and the suspension surface. Another method used to study caverns was UDV, a real-time fluid flow measurement based on the principle of Doppler frequency shift. An ultrasonic pulse is emitted from the UDV probe (DOP2000, Signal Processing, Switzerland) and the echoes reflected from the particles in the fluid are used to determine the velocity profile along the measurement line with the velocity direction parallel to the pulse path. The locations where the suspension velocity falls to zero are the cavern boundaries. The measurement accuracy of UDV is about 2.4% for velocity (Xu, 2003). For continuous operation, a dynamic mixing model (Fig. 4.1) was used to quantify the mixing dynamics in the cylindrical chest. f is the fraction of pulp stock that bypasses the mixed zone with minimal mixing and (1-f) is the remaining stock entering the mixed zone. 1 and 2 are the time constants for each portion, respectively and Td is the transport delay due to the flow delay in the process piping prior to and following the chest. The model parameters were determined by adding a conductive tracer (saline solution) to the pulp feed using a pseudo- random binary signal (PRBS) to control tracer addition and excite the chest dynamics. The conductivity of the suspension entering and leaving the chest was measured and this data set was used to determine the parameters in the dynamic model. 5.3 CFD modeling of pulp mixing 5.3.1 Computational geometry The configuration of the mixing domain, identical to the lab-scale cylindrical chest was modeled using GAMBIT, a geometry and grid generation software bundled with FLUENT (Figure 5.1(a)). 81 It was used for both batch and continuous-flow systems; except that for the dynamic simulations, an inlet was created at the top of the surface and an outlet at 45 with respect to the impeller as shown in Fig. 5.1(b). The flow in the mixing chest is not steady in an inertial frame of reference because the impeller blades sweep the domain periodically. However, since no baffles or stators were involved in the mixing chest, the flow could be viewed as steady relative to the rotating (non-inertial) frame (Ford, 2004). Thus, a multiple reference frame approach, including a rotating frame and a stationary frame, was used to model the mixing domain. The rotating frame is a cylindrical region comprising the impeller with the impeller shaft acting as its axis. The rest of the vessel is in the stationary frame. To obtain reliable computational results, it is desirable to balance a fine mesh structure with the need of having a reasonable computational time. A mesh of 700,000 cells took 2 to 3 days to converge (on a Pentium(R) 4 CPU 3.00GHz) while meshes of 200,000-300,000 cells took about one day. To compromise between the running time and computational reliability, a finer mesh was only used in the rotating domain and a check of mesh independency was also carried out. For the batch simulation, the mixing domain was divided into six blocks for meshing. Unstructured mesh, using tetrahedral elements connected one other, was applied to the rotating domain and its neighbouring block because of their irregular geometries. To capture flow details near the impeller, a finer mesh was created in the vicinity of the impeller blades. A conformal mesh (having element nodes at identical locations at the block interfaces) was created using a size function in the rotating domain to allow mesh elements to grow slowly as a function of the distance from the impeller hub and blades so that the fine mesh in the rotating domain were logically linked to the surrounding stationary domain which was not highly meshed (Ford, 2004). The other four regular blocks were discretized using a structured mesh, comprising hexahedral 82 elements logically connected. To ensure the simulated results were not dependent on the grid density, three different mesh sizes (198,084, 407,592 and 633,013 cells) were used to check grid independence under the same simulation conditions (i.e., N = 425 rpm and Cm = 3% softwood). a) b) Figure 5.1 Experimental cylindrical chest and computational domain for (a) batch and (b) continuous-flow mixing. 5.3.2 Modelling suspension rheology A pulp suspension is a heterogeneous fluid (fibres and water) that is not easily modeled. As it possesses a yield stress, the first comparable model that can be used to describe its rheology is the Bingham model. Wikstrom and Rasmuson (1998) used this approach to model the agitation Impeller Rotating frame Stationary frame Impeller Inlet Outlet Impeller shaft Rotating frame Outlet Inlet Stationary frame Impeller 83 of pulp suspensions with a jet nozzle agitator and found that the flow field far away from the boundary conditions was underestimated due to the shear-thinning property of pulp suspension. The Herschel-Bulkley model, which contains a shear-thinning parameter in addition to a yield stress, can also be used to model the pulp suspension rheology. Both models are expressed in Fluent, as n o yn y K1 (5.1) where y, K1, n, o and are yield stress threshold, consistency index, power-law index, yielding viscosity and strain rate respectively. All these parameters depend on the mass concentrations (Cm) of pulp suspensions and can be determined from rheological measurements of suspensions. The rheological parameters for the Herschel-Bulkley model were determined by fitting the flow curves of pulp suspensions in the study of Gomez et al. (2010) to the model equation. For the Bingham approximation, the pulp suspension is described with K1 = 0.001kg/ms (equal to the viscosity of water) and n = 1 since the pulp and paper industry often uses the water viscosity to design equipment where the suspension is exposed to high shear rates (Ford, 2004). Both rheological models were applied in the CFD simulations of pulp agitation. The rheological parameters for hardwood pulp (Cm = 2%) and softwood pulp (Cm = 3%) are summarized in Table 5.1. 84 Table 5.1 Rheological parameters of hardwood and softwood pulps Cm = 2% hardwood Cm = 3% softwood Model Herschel-Bulkley Bingham Herschel-Bulkley Bingham K1 (kgs n-2 /m) 4.97 0.001 135 0.001 n 0.328 1 0.21 1 y (Pa) 9.44 9.44 155 155 o (kg/ms) 9.6 9.6 10400 10400 5.3.3 Condition setup for CFD model In the rotating domain, an angular velocity was specified equal to the impeller rotation rate used in the experimental test. The rotation axis was set at the centre of the impeller hub, with the positive Y direction extending into the vessel (Figure 5.1). The impeller hub and blades were modeled as rotating walls and moved with zero relative velocity with respect to the rotating frame. The continuity and momentum equations governing the rotating domain are given by: mr Sv t (5.2) where is the fluid density, t is time, rv is the relative velocity and Sm is the mass added to the continuous phase from the dispersed second phase and any user-defined sources. Fgpvvvv t r (5.3) where v is the absolute velocity, p is the static pressure, is the stress tensor, g is the gravitational body force and F is the external body forces. The relative velocity, rv , is related to the absolute velocity, v , using the following equation: 85 rvvr (5.4) where is the angular velocity of the rotating frame, r is the position vector in the rotating frame. In the stationary frames, the chest walls were set with a velocity of zero in the absolute reference frame. The upper surface of the suspension was treated as a stationary wall with zero shear stress. Since no rotation is involved in the stationary domain, the continuity and momentum equations applied in this domain are written as follows: mSv t (5.5) Fgpvvv t (5.6) At the boundaries between the two domains, the continuity of absolute velocity is enforced to provide the correct neighbor values of velocity for the corresponding domain. In the pulp mixing experiments, the observed suspension flow was generally laminar, so the continuity and momentum equations were solved in laminar regime using a second-order upwind scheme. Near the impeller, the flow might be turbulent, but the fluctuation velocities would fade out quickly in the regions outside of the impeller zone because of the fibrous structure of the heterogeneous suspensions. Also, the non-Newtonian Reynolds number used by Gibbon et al. (1962) and Blansinski et al. (1972) for mixing pulp suspensions was calculated and found to be below 10 3 (i.e., in the laminar regime), so treating the suspension flow as laminar in the entire domain would be fine for CFD simulation. 86 When continuous operation was simulated, inlet and outlet pipes were added to the chest geometry. The lengths (L) and diameters (Dp) of the pipes in the simulation followed the dimensions of the experimental set-up, i.e., L = 215 cm and Dp = 5cm for the inlet and L = 257cm and Dp = 5cm for the outlet. The inlet was defined in the absolute frame as having a flat profile with a velocity magnitude ( 2 4 pDQv ). Since L/Dp for the outlet was greater than 10, the exit was defined as an outflow boundary condition under the fully-developed flow assumption. The boundary conditions at the outflow boundary are a zero diffusion flux for all flow variables and an overall mass balance correction. The zero diffusion flux condition means that the conditions of the outflow plane are extrapolated from within the domain and have no effect on the upstream flow. 5.3.4 Cavern volume determination and dynamic test simulation Several researchers have determined the cavern boundary by assuming that the shear stress at the cavern surface is equal to the yield stress of the suspension (y) (Solomon et al., 1981; Elson, 1990; Amanullah et al., 1998; Hui et al., 2009). Thus, this criterion was used to determine the cavern boundary in the CFD simulations. Since the shear stress was not given in Fluent, the strain rate at the yield stress (y) was used to define the cavern boundary o y y (5.7) A User Defined Function (UDF) was used to determine those cells included in the cavern criteria chosen to calculate the total cavern volume (Appendix C). The flow results calculated from the CFD simulations were also used to determine virtual dynamic responses to the experiments conducted by Hui et al. (2010). Since the properties of the 87 tracer and the modeled pulp suspension are similar, the converged results for the flow field of the suspension were used to perform transient species calculation for the tracer (Ford, 2004). Here the species conservation equation was solved in time-dependent form, discretized in both space and time. A UDF (Appendix D) specifying the tracer injection sequence at the inlet was linked to the FLUENT solver for the continuous-flow system simulation and the concentration of the tracer „measured‟ at the outlet. The inlet-outlet concentration profiles were then analyzed using the dynamic model of Soltanzadeh et al. (2009) to quantify the mixing and compare them directly with the experimental results. The model parameters identified included the transport delay (Td) arising mainly due to flow in the process piping before and after the chest, the bypassing (channeling) fraction (f) of suspension which flows through the chest with minimal mixing, and the time constant for the mixing zone (2). The data were divided into two parts: one for model parameter estimation and one for parameter validation. The extent of agreement between the model output and the simulated data was expressed using %100 )( 1 1% 1 n i simulatedsimulated simulatedcomputed ymeany yy n fit i ii (5.8) The time constant (2) was then used to estimate the effectively mixed volume (Vmix) in the chest using the equation Vmix = Q2(1-f). 5.4 Results and discussion The effect of the meshing scheme used in the CFD calculations was first examined. The velocity profiles calculated at two locations in front of the impeller (as shown in Figure 5.2) for the three 88 meshing schemes are given in Figure 5.3. There is no significant difference among the three schemes. Thus, the lowest density mesh was used for the simulations because of the shorter computer running time required. Table 5.2 shows cavern volumes determined using the three meshes described earlier, with cavern size decreasing slightly as the cell density increases (and consequently the average cell size decreases). The volume difference is small, less than 2.5%, and is attributed to the finer resolution of the cavern as cell density increases. CFD simulations were made for a range of operating conditions (two pulp types, a range of suspension concentrations and a number of impeller speeds). Figures 5.4 and 5.5 compare cavern shape and volume measured using ERT and computed using CFD for Cm = 2% hardwood and Cm = 3% softwood at three different impeller speeds. For the hardwood pulp (Figure 5.4), when the stock height (Z) was equal to the chest diameter (T), all the simulations were unable to predict the suspension surface motion observed visually during the experiments. The possible reason for this deviation could be that the yield stress may not be consistent throughout the pulp suspension during agitation. Since pulp suspension is a two-phase fluid, the pulp fibres would be redistributed in the suspension by the vigorous action of the impeller in actual situation. Fewer fibres would be in the region around the impeller and so the yield stress in the cavern could be slightly lower than the surrounding, promoting the cavern growth. However, in CFD simulations, the pulp suspension is assumed to be homogenous, with no localized difference in yield stress and so the predicted cavern volume could be smaller. The best way to justify this explanation is to repeat the experiments using a homogeneous fluid with known rheology (possibly similar to the properties of pulp suspension) and then compare with the results of heterogeneous pulp suspensions. Also, in actual mixing situation, the top surface of the agitated chest was in motion and open to the air, allowing air to enter the chest and promoting the cavern growth which 89 enhances the surface motion. However, in CFD simulations, the top surface was modeled as a flat stationary wall with no air entering and so the actual situation may not be easily simulated. The two rheology models examined did not make a significant difference in the cavern shape determined for hardwood pulp. Compared with the ERT results, the average percentage differences in cavern volumes for both models are also similar: 53% for the Herschel-Bulkley model and 49% for the Bingham model. However, the Bingham model predicted the cavern volume better at higher impeller speeds. When the impeller speed was increased to the situation close to complete mixing (i.e., 397rpm - experimental value for 2% hardwood pulp), the Bingham plastic model using the viscosity of water for K1 and n = 1 gave the cavern volume more comparable to the experimental result. This observation agrees with the study of Gullichsen and Harkonen (1981), showing that pulp suspensions behave more like water at high rotational speeds. Figure 5.2 Locations where velocity samples were determined for comparison of three different meshes. Impeller Chest wall Locations where velocity profiles are calculated at z = 12cm (measured from the chest bottom) (a) (b) 90 a) b) Figure 5.3 Velocity profiles calculated for softwood Cm = 3% at two locations (a) and (b) in front of the impeller (as shown in Figure 2) for three different meshing schemes at N = 425rpm. -0.20 -0.15 -0.10 -0.05 0.00 0.05 0.10 0.15 0.20 0.0 0.2 0.4 0.6 0.8 1.0 V e lo c it y m a g n it u d e ( m /s ) Distance from the shaft centre line (m) low mesh medium mesh high mesh -0.20 -0.15 -0.10 -0.05 0.00 0.05 0.10 0.15 0.20 0.00000 0.00005 0.00010 0.00015 0.00020 0.00025 0.00030 0.00035 V e lo c it y m a g n it u d e ( m /s ) Distance from the shaft centre line (m) low mesh medium mesh high mesh 91 Table 5.2 Grid independence study (N = 425rpm, 3% softwood pulp (K1 = 135, n = 0.21, y = 155Pa) No. of element cells Vc (dm 3 ) 198,084 9.2 407,592 9.0 633,013 8.9 Method ERT CFD Rheology Model Herschel-Bulkley Bingham K1 (kgs n-2 /m) 4.965 0.001 n 0.328 1 y (Pa) 9.443 9.443 Cavern shape (150rpm) Vc (dm 3 ) 15.7 6.2 5.4 Cavern shape (200rpm) Vc (dm 3 ) 18.1 9.3 9.2 Cavern shape (250rpm) Vc (dm 3 ) 24.6 12.2 16.7 Figure 5.4 Comparison of cavern size obtained from ERT and numerical simulations for a Cm = 2% hardwood (Z/T = 1.0). 92 The CFD simulations for softwood pulp (Cm = 3%) are similar to those made for the hardwood pulp. The two rheology models did not make significant differences in cavern shape although mild surface motion was predicted in the cases of Herschel-Bulkley model, which also provided cavern volumes closer to the experimental results, with an average of 13% lower than the ERT results (Figure 5.5). Since the fibres of softwood pulp are longer than those of hardwood pulp, they easily tangle together, especially at low impeller speeds, but they gradually align more substantially than shorter hardwood fibres in the direction of increasing shear, producing less resistance to flow. This makes softwood pulp behave more like a shear-thinning fluid, which can be further supported by the value of the power-law index (n = 0.21 for Cm = 3% softwood whereas 0.328 for Cm = 2% hardwood with n < 1 for shear-thinning fluid and n = 1 for Newtonian fluid). Also, the impeller speeds used in the comparison are not close to the point of complete mixing (i.e., 1020rpm for 3% softwood pulp) and so it is unlikely that the suspension would behave like water (Bingham model with n = 1). With favorably tangled long-fibred structure, the 3% softwood pulp would be better described by Herschel-Bulkley model in mixing simulation at low impeller speeds. Figure 5.6 shows the flow field computed at the measurement location along the cross- section of the impeller hub (Fig. 5.2) for the simulations of two pulp types using their most comparable rheology models mentioned above. Both simulated flow fields of hardwood and softwood pulps were found to match the flows expected for an axial-flow impeller. The impeller discharges the suspension along the rotation axis and up to the surface for return to the suction side of the impeller, creating a large upper circulation loop which was observed during experiments. The smaller loop generated under the impeller is also shown in the two simulations. 93 Method ERT CFD Rheology Model Herschel-Bulkley Bingham K1 (kgs n-2 /m) 135 0.001 n 0.21 1 y (Pa) 155 155 Cavern shape (425rpm) Vc (dm 3 ) 9.8 9.2 6.5 Cavern shape (475rpm) Vc (dm 3 ) 10.6 9.6 7.0 Cavern shape (525rpm) Vc (dm 3 ) 13.3 10.0 7.4 Figure 5.5 Comparison of cavern size obtained from ERT and numerical simulations for a Cm = 3% softwood (Z/T = 0.8). 94 a) b) Figure 5.6 Flow fields of batch mixing for a) Cm = 2% hardwood at N = 200rpm using Bingham plastic model and b) Cm = 3% softwood at N = 475 rpm using Herschel-bulkley model. 95 In addition to the observed flow from the sides of the vessel, the velocity profiles calculated from CFD simulations were compared with the ones measured using UDV at the specified location shown in Figure 5.2 at three different heights (Figure 5.7). The velocity profile along the measurement line (the dotted arrow lines) was measured with the velocity direction parallel to the pulse path. The positive velocity means the velocity direction is away from the UDV probe. Figures 5.8 and 5.9 compare the measured velocities with the ones calculated using two rheology models for Cm = 2% hardwood at two impeller speeds. The sudden irregular ups and downs shown on the UDV-measured velocity curves are possibly due to the interference of bubbles introduced by air entrapment at the free suspension surface during agitation. The bubble movement and coalescence-breakup behavior induce turbulent fluctuations to the flow field in both direction and amplitude, and this would distort the ultrasonic signals and affect the velocity measurement (Bouillard et al., 2001; Wang et al. 2003). Since UDV measured the velocity of pulp fibres whereas CFD computed the fluid velocity, it would be expected that the computed values would be slightly different from the measured values. At N = 200 rpm, neither model can exactly predict the measured velocities but the velocity profiles obtained from the Bingham plastic model are closer to the measurements, e.g., at position 2, the velocity profile computed using the Bingham model follows the direction change of the measured profile (from positive to negative) at about +0.05m from the chest centre. At N = 250rpm, the velocity profile shape of the Bingham model is very close to the measured one. The peaks and dips of both profiles occur almost at the same locations and so it is obvious that the Bingham model would be preferred for hardwood pulp mixing simulation. 96 a) b) Figure 5.7 a) Top view and b) side view of probe locations for UDV measurements. The arrows show the direction of the sonic emissions at three heights (12.5mm, 92.5mm and 252.5mm measured from the chest bottom). UDV probe Position 2 Position 1 Position 3 UDV probe Impeller Line of the impeller shaft 97 a) Position 1 b) Position 2 c) Position 3 Figure 5.8 Comparison of measured and computed velocity profiles at three positions for a Cm = 2% hardwood at N = 200rpm. Positive velocity means the velocity direction is away from the probe. -0.2 -0.1 0.0 0.1 0.2 -0.04 -0.03 -0.02 -0.01 0.00 0.01 0.02 0.03 0.04 V e lo c it y ( m /s ) Distance from the chest centre (m) UDV Herschel-Bulkley Bingham plastic -0.2 -0.1 0.0 0.1 0.2 -0.03 -0.02 -0.01 0.00 0.01 0.02 0.03 0.04 0.05 V e lo c it y ( m /s ) Distance from the chest centre (m) UDV Herschel -Bulkley Bingham plastic -0.2 -0.1 0.0 0.1 0.2 -0.08 -0.06 -0.04 -0.02 0.00 V e lo c it y ( m /s ) Distance from the chest centre (m) UDV Herschel-Bulkley Bingham plastic 98 a) Position 1 b) Position 2 c) Position 3 Figure 5.9 Comparison of measured and computed velocity profiles at three positions for a Cm = 2% hardwood at N = 250rpm. Positive velocity means the velocity direction is away from the probe. -0.2 -0.1 0.0 0.1 0.2 -0.04 -0.02 0.00 0.02 0.04 0.06 0.08 0.10 V e lo c it y ( m /s ) Distance from the chest centre (m) UDV Herschel-Bulkley Bingham plastic -0.2 -0.1 0.0 0.1 0.2 -0.15 -0.10 -0.05 0.00 0.05 0.10 0.15 0.20 V e lo c it y ( m /s ) Distance from the chest centre (m) UDV Herschel-Bulkley Bingham plastic -0.2 -0.1 0.0 0.1 0.2 -0.07 -0.06 -0.05 -0.04 -0.03 -0.02 -0.01 0.00 0.01 V e lo c it y ( m /s ) Distance from the chest centre (m) UDV Herschel-Bulkley Bingham plastic 99 CFD was also used to predict the point where the suspension was in complete motion throughout the mixing chest (i.e., where the cavern volume equaled the suspension volume). The impeller speeds for complete mixing predicted by CFD for both pulps are about twice the experimental one. This deviation may be due to the complex pulp suspension rheology, which is not easily described by CFD, e.g., floc formation easing the flow of pulp suspension. Also, for both the hardwood and softwood pulps, the cavern volumes determined using CFD and ERT increased with impeller speed (Fig. 5.10). The increasing trend of cavern volume observed in ERT results was predicted satisfactorily for the hardwood pulp but not for the softwood pulp, which has CFD results increasing more steadily and slowly than those of ERT. The possible reason for this difference is the difficulty of using CFD to describe the complex behavior of pulp fibres in agitation, i.e., softwood pulp forms flocs more readily than hardwood pulp in actual situation. With the same amount of fibres, the total surface area of flocs is smaller than the sum of individual fibre surface area. Thus, the resistance to flow in softwood pulp suspension becomes lower than the expected because flocs would move more easily than long pulp fibres. In CFD, no simulation of floc formation was involved and so the resistance to flow would not be changed, leading to relatively smaller cavern volume when compared with the experimental results. In addition, during experiments, it was observed that air entrapment was significant in agitation of softwood pulp. This would enhance the cavern growth, which cannot be easily simulated in CFD. 100 a) b) Figure 5.10 Comparison of measured and computed cavern volumes. (a) Cm = 2% hardwood pulp using Bingham plastic model for cavern determination (Z/T = 1.0); (b) Cm = 3% softwood pulp using Herschel-Bulkley model for cavern determination (Z/T = 0.8). Figures 5.11 and 5.12 compare the responses simulated from CFD with those obtained from experiments for dynamic tests. It seems that both the CFD and experimental results are reasonably similar and increasingly close to each other with increasing impeller speed. Both simulations and experiments show no bypassing was detected in the studied mixing situations. Figure 5.13 shows the simulated path lines of particles released into the chest using CFD. It is 400 800 1200 1600 2000 2400 10 15 20 25 30 35 40 V c ( d m 3 ) N (rpm) ERT CFD Suspension volume = 34.7dm 3 100 200 300 400 500 600 0 5 10 15 20 25 30 35 40 45 50 Suspension volume = 43.4dm 3 V c ( d m 3 ) N (rpm) ERT CFD 101 clear that the particles enter the region (cavern) around the impeller before exiting the chest and this agrees with the experimental result of no channeling. Tables 5.3 and 5.4 shows the dynamic parameters (Td – time delay and 2 – time constant for mixing zone) obtained from CFD and experimental responses. For both pulps, Td predicted by CFD is larger than that of dynamic test, especially for softwood pulp. The possible explanation for this is the air entrapped in the pipe flow in actual situation. During experiments, the feed tank was open and agitated by a top-entry mixer. Air could then be drawn into the pulp suspension and flowed with the suspension. Thus, besides pulp suspension, there was air in the piping. This air occupancy would create “shortcut” in the pipes and shorten the time delay. However, no such phenomenon was considered in CFD simulations and so the CFD Td was found to be larger than the experimental one. Since air entrapment is more significant for softwood pulp than for hardwood pulp as observed during experiments, the deviation in Td is higher for softwood pulp. For the 2 of 2% hardwood pulp, excluding the value at low impeller speed (i.e., 125rpm), the average difference between simulated and experimental values is about 16%. This result is in accord with that of batch mixing in Figure 5.10a where CFD predicted fairly well at high speeds. For 3% softwood pulp, the difference in 2 becomes higher as the impeller speed increases. The predicted time constants of the last two impeller speeds (i.e., 1020 and 1100rpm), which show complete mixing in experiments, are much lower than the experimental ones. The reason is that under CFD simulations, these two speeds are not at complete mixing, as shown in Figure 5.10. The impeller speed calculated from CFD for complete mixing of 3% softwood pulp is over 2300 rpm and so that‟s why the predicted time constants are lower than the experimental ones. Figure 5.14 also shows the effectively mixed volume calculated from 2 for Cm = 2% hardwood and Cm = 3% softwood. It is clear that CFD predicted better for hardwood pulp than softwood pulp and 102 the reason would possibly be the higher tendency of long-fibred softwood pulp to form flocs, which offer less resistance to flow and this phenomenon could not be easily simulated by CFD. This also agrees with the CFD results of complete mixing in batch operation (Figure 5.10), showing that CFD approximated better for hardwood than softwood. In addition, the percentages of fitting for CFD are generally higher than those for dynamic tests because unlike the experimental outputs, the simulated responses generally do not contain any noises that would affect the fitting process. To conclude the study, the possible reasons for the difference between experimental results and numerical simulations are the heterogeneous nature of pulp suspensions and the air entrapment during experiments. To verify these explanations, this study should be repeated with a homogeneous fluid with known rheology (possibly similar to the properties of pulp suspensions) in a closed system. The approach using glycerin solution in the study of the flow field in a rectangular vessel by Gomez et al. (2010) can be used and then the outcome can be compared with the results of this study. In addition, their study showed very good agreement between the experimental and computational results. Since the CFD modeling in this study is similar to that of Gomez‟s study, the deviation between the modeling and experimental results is most likely due to the special rheology of pulp suspension, which is not easy to be formulated and implemented in the CFD model. 103 Table 5.3 Comparison of results from CFD and dynamic tests for Cm = 2% hardwood (Q = 14L/min., Z/T = 0.8) N (rpm) Td (s) 2 (s) Percentage of fitting (%) CFD Dynamic test CFD Dynamic test CFD (Eqn (5.8)) Dynamic test (Eqn. (4.1)) 125 65.0 58.8 124 47 94 89 250 55.0 40.0 116 106 95 85 380 49.0 34.8 178 155 90 91 460 41.6 38.7 177 142 89 91 Table 5.4 Comparison of results from CFD and dynamic tests for Cm = 3% softwood (Q = 14L/min., Z/T = 0.8) N (rpm) Td (s) 2 (s) Percentage of fitting (%) CFD Dynamic test CFD Dynamic test CFD (Eqn.(5.8)) Dynamic test (Eqn.(4.1)) 425 60.0 41.4 70 69 88 86 525 56.0 40.0 72 97 85 82 1020 50.0 36.2 68 160 90 91 1100 55.0 40.3 75 156 83 80 104 125rpm 250rpm 380rpm 460rpm Figure 5.11 Experimental and computed dynamic responses for Cm = 2% hardwood pulp. 0 240 480 720 960 1200 1440 1680 1920 2160 2400 8 9 10 11 12 C o n d u c ti v it y ( m S ) Flow time (s) CFD Experimental 0 240 480 720 960 1200 1440 1680 1920 2160 2400 8.5 9.0 9.5 10.0 10.5 11.0 11.5 12.0 C o n d u c ti v it y ( m S ) Flow time (s) CFD Experimenta 0 240 480 720 960 1200 1440 1680 1920 2160 2400 4.0 4.5 5.0 5.5 6.0 6.5 7.0 7.5 C o n d u c ti v it y ( m S ) Flow time (s) CFD Experimental 0 240 480 720 960 1200 1440 1680 1920 2160 2400 6 7 8 9 10 C o n d u c ti v it y ( m S ) Flow time (s) CFD Experimental 105 425rpm 525rpm 1020rpm 1100rpm Figure 5.12 Experimental and computed dynamic responses for Cm = 3% softwood pulp. 0 240 480 720 960 1200 1440 1680 1920 2160 2400 0.5 1.0 1.5 2.0 2.5 3.0 3.5 C o n d u c ti v it y ( m S ) Flow time (s) CFD Experimental 0 240 480 720 960 1200 1440 1680 1920 2160 2400 2.0 2.5 3.0 3.5 4.0 4.5 5.0 C o n d u c ti v it y ( m S ) Flow time (s) CFD Experimental 0 240 480 720 960 1200 1440 1680 1920 2160 2400 0.0 0.5 1.0 1.5 2.0 2.5 C o n d u c ti v it y ( m S ) Flow time (s) CFD Experimental 0 240 480 720 960 1200 1440 1680 1920 2160 2400 2.5 3.0 3.5 4.0 4.5 5.0 5.5 C o n d u c ti v it y ( m S ) Flow time (s) CFD Experimental 106 Figure 5.13 Path lines of particles (each particle is represented by a different color) simulated by CFD using Cm = 2% hardwood pulp at N = 250rpm and Q = 14L/min. Outlet Inlet 107 a) b) Figure 5.14 Experimental and computed determination of mixed volume for (a) Cm = 2% hardwood pulp and (b) Cm = 3% softwood pulp in continuous operation. 400 600 800 1000 1200 0 5 10 15 20 25 30 35 40 V m ix ( d m 3 ) N (rpm) Dynamic test CFD 100 200 300 400 500 0 5 10 15 20 25 30 35 40 45 V m ix ( d m 3 ) N (rpm) Dynamic test CFD 108 5.5 Summary A CFD model of a lab-scale cylindrical mixing chest was developed using FLUENT 6.2. To verify the ability of the CFD model for flow estimation in agitated pulp stock chests, a number of numerical simulations were carried out to compare with the experimental results in terms of cavern volumes and dynamic model parameters. Although the CFD model cannot precisely depict the cavern shape and the surface motion, it can predict the increasing trend of cavern size with impeller speed. Based on the cavern comparison, it seemed that Bingham plastic model would be preferred for the hardwood pulp at impeller speeds close to complete mixing whereas Herschel-Bulkley model predicted better for the softwood pulp at low impeller speeds. The suitability of the Bingham model for the hardwood pulp was further supported by the UDV measurement. The reason why the CFD model cannot fully predict the actual mixing situations is that the actual behavior of pulp fibres, like floc formation of softwood pulp in agitation, and the air entrapment during agitation cannot be easily described by CFD. The best way to justify the explanation is to repeat the mixing experiments using a homogeneous fluid with no known rheology similar to that of pulp suspensions in a closed system and then compare the outcome with the results of this study. Actually, a major difficulty in using CFD for simulation of pulp mixing is the lack of good description of rheology in CFD code. However, the agreement of simulated results of particle path lines with the experiment result of no passing suggests that the CFD model can provide valuable information about the flow patterns in agitated pulp chests. 109 6 OVERALL CONCLUSIONS AND RECOMMENDATIONS FOR FUTURE WORK 6.1 Overall conclusions Since pulp suspensions possess a complex rheology, it is not easy to achieve an effective agitation in cylindrical chests. Caverns form when the chests are not completely agitated and this may lead to bypassing and dead zone formation. In order to avoid undesirable flows affecting chest performance, a study of pulp mixing in batch and continuous operations was carried out. Electrical resistance tomography (ERT) and Ultrasonic Doppler velocimetry (UDV) were used to study the cavern shape and size in batch operation. Both methods gave comparable results but ERT was preferred due to less time required to acquire data. The cavern shape was best described as a truncated right-circular cylinder and its volume was found as expected to increase with impeller speed but the increase was non-uniform due to interaction between the cavern and the vessel walls. There are models for estimating cavern volumes in fluids with yield stresses but without including the interaction between the cavern and the vessel walls. Like the situations in the study, they do not describe the experimental results well. Thus, based on the observed cavern shape in ERT imaging, a model was developed to estimate the cavern volume by balancing the force provided by the impeller with the resistive force at the cavern boundaries. Although the proposed model could not accurately calculate the cavern volume, it could predict the increasing trend in cavern volume with impeller speed. In continuous operation, dynamic tests were used to characterize the mixing quality in a cylindrical chest. A model including a mixed region (cavern) and the possibility of bypassing 110 was shown to precisely describe the mixing dynamics of the chest because the time delay estimated using the model agreed closely with the calculated result based on flow and system geometry. The measured bypassing was found to be insignificant, indicating that the flow configuration used in the study (with the exit within the cavern, close to and below the impeller) is effective for avoiding feed bypassing. Also, ERT further confirmed the presence of cavern when the chest was not fully agitated in continuous operation with feed and the cavern volume increased with impeller speed. In addition, the power for complete agitation in a cylindrical chest was similar to that in a rectangular chest of the same volume. Besides the experimental study of pulp mixing in a cylindrical chest, the ability of computational fluid dynamics (CFD) to estimate the flow in agitated pulp chests was examined. Numerical simulations were carried out to compare with the experimental results. Owing to the complex properties of pulp suspension, it is not easy to describe its unique behavior like floc formation in agitated chests using CFD. The application of both Bingham and Herschel-Bulkley models could not exactly portray the cavern, but the increasing trend of cavern volume with the impeller speed was still predicted. The CFD simulations also picked up the features of the flow field measured by UDV at the high impeller speed. The calculated path lines of the simulated particle feed matched with the experimental results of no bypassing. Therefore, in spite of its limitation to model pulp suspensions, the CFD model can still provide useful flow information in pulp agitated chests. The outcome of this research can aid the design of industrial cylindrical chests. The cavern model can estimate the mixing volume in proposed agitated pulp chests, aiding the sizing of chests. The dynamic test study provides insight about the positions of in/out piping of new chests, e.g., the outlet should be placed close to the cavern, to avoid non-ideal flows which affect 111 the chest performance. Besides new chests, this test method can also be used to estimate the performance of existing pulp chests, trying to improve or enhance their mixing efficiency. In addition, the flow configuration of the proposed chest design can be checked using CFD. Saving the need of building a lab-scale chest and doing experiments, the CFD model can provide information about the flow pattern in the proposed chest, helping chest design and optimization. Finally, in general, the contributions of this research are: The experimental results about the cavern geometry in cylindrical vessels can provide insights for the design of chest geometry to facilitate pulp mixing, e.g., installation of fillets is needed to minimize dead zone. The proposed cavern model could be used to estimate the mixing volume in existing and proposed pulp chests to see whether they meet the requirement. The results of dynamic tests show that the flow configuration with the exit installed near the cavern can avoid bypassing. This information confirms the importance of exit location in an agitated cylindrical chest and helps the design of efficient-mixing chests. The results of the CFD model can provide valuable information about the flow pattern, e.g., the presence of bypassing, inside the existing and proposed chests. This also helps the design of flow configuration of agitated pulp chests, like the locations of the impeller, inlet and outlet. 112 6.2 Recommendation for future work The results of this study provided some insights for future consideration as follows: The presence of dead zone under incomplete mixing implies the possibility of disturbance to normal operation (e.g., the oozing of undesirable materials from the dead zone into the cavern). The installation of fillets can avoid this hazard and its effect on the cavern formation should be examined. Based on the experimental results of ERT and UDV, a model for determining the cavern volume in a cylindrical chest was developed. Similar modeling should be carried out in rectangular chests because they are also widely used in the industry. This model can be modified to extend its application on rectangular chests. For the agitation of softwood pulp, entrapment of air into the suspension is significant at high impeller speeds and this will affect the cavern size. Thus, the effect of air entrapment on the cavern formation should be studied. Owing to the poor knowledge of pulp suspension rheology, it was proposed that the heterogeneous structure of pulp suspensions could cause the discrepancies between the experimental results and CFD simulations. To validate this proposition, a study of the same mixing system using a homogeneous fluid with known rheology should be carried out. In this study, the simulated results of CFD in cavern size are not in complete agreement with the experimental results and one of the possible reasons is the incomplete quantitative description of pulp suspension rheology, e.g., the interactions of flocs. 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Sensor system The size and the material of the electrodes are both important in producing and sensitively measuring the electrical field distribution (Mann et al., 1997). Wang et al. (1995) found that the smaller the size of electrodes, the higher the sensitivity of the voltage measurement electrodes and the higher common voltage at current driven electrodes, when measurement and current excitation use the same electrode system. However, the higher common voltage will produce a poor signal-to-noise ratio. Also, current-injecting electrodes should have large surface areas in order to generate an even current density (Dickin and Wang, 1996). Thus, the electrode size should not be too large or too small. In addition, the electrodes must be made of a material more electrically conductive than the process fluid (Ricard, 2005). Metallic electrodes are thus often used for process applications, e.g., stainless steel, silver, gold or platinum. B. Data acquisition system (DAS) In order to track the small changes of resistivity in real-time, the data collection has to be quick and accurate, enabling the reconstruction algorithm to provide a precise indication of the true resistivity distribution. The DAS is composed of a signal source, an electrode multiplexer array, voltmeters, signal demodulators and a system controller (Mann et al., 1997). This complexity is required because of the low amplitude of measurements at the boundary, the small responses of dynamic change, the large number of electrode channel operations, the high common voltages and the large stray capacitance of coaxial cable. The signal source consists of a master oscillator and a voltage-to-current converter (Plaskowski et al., 1995). The oscillator, serving as a frequency and amplitude reference for all of the current sources channels and as a switching 121 function for the demodulator stage, generates a harmonically pure sine-shaped waveform signal in order to “probe” the material under investigation. The sine-wave voltage output from the oscillator is fed into a voltage-to-current converter. Current is used in preference to voltage as the electrical “probe” due to the variation of contact impedance between the electrode and the fluid inside the vessel. The multiplexers are used to “share” the current source and voltage measurement stages between any numbers of electrodes. They must exhibit a number of properties: low on-resistance, fast switching speed, low inter-switch cross talk and low power consumption. The signal demodulators are used to “decode” the voltage signal and to optimize the signal-to-noise ratio by recovering the amplitude attenuation and phase shift of the sine wave signal as a result of passing through a resistive medium. C. Image reconstruction system The image reconstruction algorithm can be thought of simply as a series of procedures performed repeatedly on digitized measurement data to determine the distribution of regions of different resistivities (e.g., component concentrations) within the process vessel. There are two different reconstruction algorithms: “qualitative” and “quantitative”. The qualitative algorithm produces images depicting a change in resistivity relative to an initially acquired set of “reference” data and the quantitative one creates images depicting values of resistivity or conductivity for each pixel (Dickin and Wang, 1996). The technique used in the qualitative algorithm is referred as backprojection between equipotential lines. The potential difference, calculated by the forward solver, between two equipotential lines on the boundary is back-projected to a resistivity value in the area enclosed by the two lines for all possible injection/measurement combinations. The main advantage of this 122 algorithm is that it can be performed in a single step using a pre-calculated pixel sensitivity matrix and the image is simply reconstructed via a matrix/vector multiplication. The quantitative algorithm is an iterative Newton-Raphson-based algorithm specifically developed for nonlinear problems. It is intended to quantify the variation of the conductivity in the region of interest during the process. The reconstruction process is initiated when a set of resistivities for the region of interest is fed into the forward problem solver. When the least- squares error between the calculated boundary voltages and the data acquisition voltages is less than the pre-defined one, the reconstruction process is halted and the final updated set of resistivities will be the solution. 123 References Dickin, F. and M. Wang, “Electrical Resistance Tomography for Process Applications”, Meas. Sci. Technol., 7, 247-260 (1996). Mann, R., F.J. Dickin, M. Wang, T. Dyakowski, R.A. Williams, R.B. Edwards, A.E. Forrest and P.J. Holden, “Application of Electrical Resistance Tomography to Interrogate Mixing Processes at Plant Scale”, Chemical Engineering Science, 52, 2087-2097 (1997). Plaskowski, A., M.S. Beck, R. Thorn and T. Dyakowski, Imaging Industrial Flows, Applications of Electrical Process Tomography, Institute of Physics Publishing (1995). Ricard, F., C. Brechtelsbauer, Y. Xu, C. Lawrence and D. Thompson, “Development of an Electrical Resistance Tomography Reactor for Pharmaceutical Process”, The Canadian Journal of chemical Engineering, 83(2), 11-18 (2005). Wang, M., F.J. Dickin and R.A. Williams, “The Grouped-node Technique as a Means of Handling Large Electrode Surfaces in Electrical Impedance Tomography”, Physiological Measurement, 16, 219-226 (1995). 124 APPENDIX B: MATLAB program for dynamic model parameter estimation Program “procmodeliden.m”; this program is used to estimate the dynamic model parameters from the input-output data in M-file format. clc clear for i=1:1 filen=strcat('lab','1') load(filen); Ts=ts; Tres=3; datac=iddata(cout',cin',Ts) % datac=resample(datac,1,Tres) % datac=datac(10:end); datac.int='foh'; % ze =datac(1:floor(length(datac.u)/2)); % defining data set ze=datac(1:floor(length(datac.u)/2)); %removing mean ze=detrend(ze,'constant') advice(datac) % identification command m = pem(ze,'P1D','kp',{'max',2},'kp',{'min',0.5},'Td',{'max',100},'Td',{'min',2}, 'Tz',{'min',0},'dist','arma2') m = pem(ze,m) present (m) % zv=datac(floor(length(datac.u)/2):length(datac.u)); zv=datac(floor(length(datac.u)/2):length(datac.u)); zv=detrend(zv,'constant'); Kp(i)=m.kp.value; Td(i)=m.td.value%*Tres*Ts; Tz(i)=m.tz.value%*Tres*Ts; Tp1(i)=m.tp1.value%*Tres*Ts; Tp2(i)=m.tp2.value%*Tres*Ts; tu1(i)=min(Tp1(i),Tp2(i)); tu2(i)=max(Tp1(i),Tp2(i)); f(i)=(Tz(i)-tu1(i))/(tu2(i)-tu1(i)); figure(i) compare(zv,m); resid(zv,m); advice(m); % figure(i+1) [yo,fit(i)]=compare(zv,m); ou=yo{1,1}; subplot(3,1,1) plot(ou.sa,ou.y,'b',zv.sa,zv.y,'k') title(['exp no' int2str(i) ': ''Td=' num2str(Td(i)) ', ' 'K=' num2str(Kp(i)) ', ' 'f=' num2str(f(i)) ',' 'Tz=' num2str(Tz(i)) ', ' 'Tp1=' num2str(Tp1(i)) ', ' 'Tp2=' num2str(Tp2(i)) ', ' 'Ts=' num2str(Ts)]) legend('m out',['real ' '%Fit= ' num2str(fit(i))]) 125 subplot(3,1,2) plot(ze.sa,ze.u,'b.',ze.sa,ze.y,'k') title(['exp no' int2str(i) ': ' 'Estimation Data']) legend('input data','output data') subplot(3,1,3) plot(zv.sa,zv.u,'b.',zv.sa,zv.y,'k') title(['exp no' int2str(i) ': ' 'Validation Data']) legend('input data','output data') end % write2excel('C:\MATLAB7\work\Ali\Identifyingonedelaymodel\fitresult',0,'A1',' P2DZ,Arma1'); % write2excel('C:\MATLAB7\work\Ali\Identifyingonedelaymodel\fitresult',0,'A2',f it'); % write2excel('C:\MATLAB7\work\Ali\Identifyingonedelaymodel\fitresult',0,'B1',' Kp'); % write2excel('C:\MATLAB7\work\Ali\Identifyingonedelaymodel\fitresult',0,'B2',K p'); % write2excel('C:\MATLAB7\work\Ali\Identifyingonedelaymodel\fitresult',0,'C1',' Td'); % write2excel('C:\MATLAB7\work\Ali\Identifyingonedelaymodel\fitresult',0,'C2',T d'); % write2excel('C:\MATLAB7\work\Ali\Identifyingonedelaymodel\fitresult',0,'D1',' Tz'); % write2excel('C:\MATLAB7\work\Ali\Identifyingonedelaymodel\fitresult',0,'D2',T z'); % write2excel('C:\MATLAB7\work\Ali\Identifyingonedelaymodel\fitresult',0,'E1',' Tp1'); % write2excel('C:\MATLAB7\work\Ali\Identifyingonedelaymodel\fitresult',0,'E2',T p1'); % write2excel('C:\MATLAB7\work\Ali\Identifyingonedelaymodel\fitresult',0,'F1',' Tp2'); % write2excel('C:\MATLAB7\work\Ali\Identifyingonedelaymodel\fitresult',0,'F2',T p2'); % write2excel('C:\MATLAB7\work\Ali\Identifyingonedelaymodel\fitresult',0,'G1',' tu1'); % write2excel('C:\MATLAB7\work\Ali\Identifyingonedelaymodel\fitresult',0,'G2',t u1'); % write2excel('C:\MATLAB7\work\Ali\Identifyingonedelaymodel\fitresult',0,'H1',' tu2'); 126 % write2excel('C:\MATLAB7\work\Ali\Identifyingonedelaymodel\fitresult',0,'H2',t u2'); % write2excel('C:\MATLAB7\work\Ali\Identifyingonedelaymodel\fitresult',0,'I1',' f'); % write2excel('C:\MATLAB7\work\Ali\Identifyingonedelaymodel\fitresult',0,'I2',f '); Program “write2excel.m” function write2excel(fileloc,promptforsave,varargin) % write2excel(fileloc,promtforsave,range1,data1,range2,data2,...) % % % Uses ActiveX commands to write data_n into range_n in an existing Excel % spreadsheet. Inputs (excluding fileloc and promptforsave) must be paired. % As of 10/04 update, you may provide the target range (upper left % cell to lower right cell) OR just the upper right cell. If % the range is specified, the function will verify that the % corresponding data block is the correct size, and give an % error if not. (This may be useful for error checking, for instance.) % If only the upper left cell is provided, write2excel will % compute the target range. % % (Please use caution, as you can now overwrite data pretty easily.) % % Additionally, you may now specify cells by address (eg., 'H3') OR row, column % (eg, '[3,8]'). % % FILELOC: Enter a string representing the location of an Excel file. % Example: 'c:\brett\my archives\test1.xls' % PROMPTFORSAVE: binary variable. 1 (DEFAULT) = Prompt before saving % 0 = No prompt required % RANGE SPECIFIER(S): Enter the range(s) to read. You may use Excel % cell-references, as in: % 'B1:P5' % 'B1:B1' (or simply 'B1') % OR, alternatively, you may specify the row, column values, as % in: % '[8,3]:[12,4]' (to write from row 8, column % 3 to row 12, column 4); % '[8,3]' to write from row 8, column 3 % TO WHATEVER RANGE IS REQUIRED FOR THE DATA BLOCK. % % DATA: NOTE: To enter multiple strings, use cell arrays. Size compatibility is verified 127 % if a cell range is given, and is not if only the starting % position is specified. % % EXAMPLES: write2excel('c:\brett\my archives\test1.xls', 1, 'C1:E3',magic(3)); % write2excel('c:\brett\my archives\test1.xls', 0, 'C1:E3',{'string1','string 2', 'string3'}); % write2excel('c:\brett\my archives\test1.xls', 0, '[3,8]',magic(4)); % % Written by Brett Shoelson, Ph.D. % Last update: 1/04. % 10/04: Allow "dynamic specification" of cell ranges, and % allow specification of cells by row, column format. % % SEE ALSO: readfromexcel if nargin < 4 msgstr = sprintf('At a minimum, you must specify three input arguments.\nThe first is a string indicating the location of the excel file,\nthe second is a range to be written, and the third contains the data to write.'); error(msgstr); elseif ~iseven(nargin-2) msgstr = sprintf('Please enter input variables in pairs...\n''write range'',data,''write range'',data') error(msgstr) end tmp = varargin; sheetchanges = [];counter = 1; for ii = 1:length(tmp) if ischar(tmp{ii}) & (strcmp(tmp{ii},'sheet') | strcmp(tmp{ii},'sheetname')) sheetchanges(counter) = ii; counter = counter + 1; end end if ~isempty(sheetchanges) [sheetnames{1:length(sheetchanges)}] = deal(varargin{sheetchanges+1}); end [pathstr,name,ext] = fileparts(fileloc); if isempty(ext) fileloc = [fileloc,'.xls']; end if isempty(pathstr) fileloc = which(fileloc,'-all'); if size(fileloc,1) ~= 1 error('File was either not located, or multiple locations were found. Please reissue readfromexcel command, providing absolute path to the file of interest.'); end end % Ensure that range sizes and data are size-matched for ii = 1:2:nargin-2 128 if ismember(ii,sheetchanges) | ismember(ii,sheetchanges + 1) continue end % How are cells specified? if any(ismember(double(varargin{ii}),[65:90,97:122])) addrtype = 'letternumber'; else addrtype = 'rowcol'; end % Is range provided, or should it be auto-calculated? autorange = isempty(findstr(varargin{ii},':')); switch addrtype case 'letternumber' if autorange r1{ii} = varargin{ii}; [rx1,cx1] = an2nn(r1{ii}); rx2 = rx1 + size(varargin{ii+1},1)-1; cx2 = cx1 + size(varargin{ii+1},2)-1; r2{ii} = nn2an(rx2,cx2); else tmp = findstr(varargin{ii},':'); r1{ii} = varargin{ii}(1:tmp-1); r2{ii} = varargin{ii}(tmp+1:end); [rx1,cx1] = an2nn(r1{ii}); [rx2,cx2] = an2nn(r2{ii}); end case 'rowcol' if autorange r1{ii} = varargin{ii}; [t,r]=strtok(r1{ii},','); rx1 = str2num(t(2:end)); cx1 = str2num(r(2:end-1)); r1{ii} = nn2an(rx1,cx1); rx2 = rx1 + size(varargin{ii+1},1)-1; cx2 = cx1 + size(varargin{ii+1},2)-1; r2{ii} = nn2an(rx2,cx2); else tmp = findstr(varargin{ii},':'); r1{ii} = varargin{ii}(1:tmp-1); [t,r]=strtok(r1{ii},','); rx1 = str2num(t(2:end)); cx1 = str2num(r(2:end-1)); r2{ii} = varargin{ii}(tmp+1:end); [t,r]=strtok(r2{ii},','); rx2 = str2num(t(2:end)); cx2 = str2num(r(2:end-1)); r1{ii} = nn2an(rx1,cx1); r2{ii} = nn2an(rx2,cx2); end end if ~autorange % Validate size match for target range, data block sz = [rx2 - rx1 + 1, cx2 - cx1 + 1]; switch class(varargin{ii+1}) case {'double','cell'} sz2 = size(varargin{ii+1}); case 'char' sz2 = [size(varargin{ii+1},1),1]; 129 end if ~isequal(sz,sz2) error(sprintf('Mismatched range/data size for input pair %d. Specified range is %d x %d, data block is %d x %d.',(ii+1)/2,sz(1),sz(2),sz2(1),sz2(2))); end end end Excel = actxserver('Excel.Application'); Excel.Visible = 0; w = Excel.Workbooks; try excelarchive = invoke(w, 'open', fileloc); catch invoke(Excel, 'quit'); release(w); delete(Excel); error(sprintf('Sorry...unable to open file %s',fileloc)); end Sheets = Excel.ActiveWorkBook.Sheets; archive = Excel.Activesheet; initval = get(archive,'Index'); archive.Unprotect; % Read appropriate ranges into output variables chgcount = 1; for ii = 1:2:nargin-2 if ismember(ii,sheetchanges) try sheet = get(Sheets,'Item',sheetnames{chgcount}); invoke(sheet,'Activate'); archive = Excel.Activesheet; chgcount = chgcount + 1; continue catch invoke(Excel, 'quit'); release(w); delete(Excel); error(sprintf('\nUnable to find/open sheet %s.',sheetnames{chgcount})); end elseif ismember(ii,sheetchanges + 1) continue end archiverange = get(archive, 'Range', r1{ii}, r2{ii}); set(archiverange, 'value', varargin{ii+1}); release(archiverange); end sheet = get(Sheets,'Item',initval); 130 invoke(sheet,'Activate'); if ~promptforsave invoke(excelarchive,'save'); end invoke(Excel, 'quit'); release(excelarchive); release(w); delete(Excel); %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%% %SUBFUNCTIONS %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%% function k=iseven(x) k = x/2==floor(x/2); return function [r, c] = an2nn(cr) % convert alpha, number format to number, number format t = find(isletter(cr)); t2 = abs(upper(cr(t))) - 64; if(length(t2) == 2), t2(1) = t2(1) * 26; end c = sum(t2); r = str2num(cr(max(t) + 1:length(cr))); return function cr = nn2an(r, c) % convert number, number format to alpha, number format %t = [floor(c/27) + 64 floor((c - 1)/26) - 2 + rem(c - 1, 26) + 65]; t = [floor((c - 1)/26) + 64 rem(c - 1, 26) + 65]; if(t(1)<65), t(1) = []; end cr = [char(t) num2str(r)]; Program “testexcel.m”; this program converts the raw data in excel file to M-file format for processing %This file attains mat File of Farhad's Excel data. %It saves input consistency, output consistency, input signal, and %sampling time accordingly %There exist 41 excel files so that j goes from 1 to 41 clc clear close all k=1; [timee,cine,coute,setp] = readfromexcel('leo_4018pma','A:A','B:B','C:C','D:D');%Finding input % %Modifying cell to vector for i=1:length(cine) cin(i)=cine{i}; cout(i)=coute{i}; time(i)=timee{i}; end 131 %%%removing last component of cin cin=cin(1:end-1); cout=cout(1:end-1); time=time(1:end-1); cinnan=cin; coutnan=cout; %ploting cin and interploaed plot(time,cin,'c') hold on plot(time,cout) %saving results ts=time(2)-time(1); save('lab1','cin','cout','ts') Program “readfromexcel.m” function varargout = readfromexcel(fileloc,varargin) % VARARGOUT = READFROMEXCEL(FILELOC,VARARGIN) % % Uses ActiveX commands to read range(s) from an existing Excel % spreadsheet. % % FILELOC: Enter a string representing the (absolute or relative) % location of an Excel file. (Extension may be % omitted, and will be assumed to be .xls.) % Examples: 'c:\brett\my archives\test1.xls' % 'test1.xls' % 'myarchive' % % SHEETNAME: (Optional): Any occurrence in the variable argument list of % the strings 'sheetname' or 'sheet' prompts the function to change % active sheets to the value in the following variable. That specifier % must be a string argument matching exactly the name of an existing % sheet in the opened file. If this argument is omitted, the function % defaults to reading from the first sheet in the file. % % RANGE SPECIFIER(S): Enter the range(s) to read. The values stored in these % ranges will be returned in consecutive output arguments. % Example: 'B1:B5' % 'B1:B1' (or simply 'B1') % 'B1:P4' % 'B:B' or 'B' (Entire second column) % '2:2' or '2' (Entire second row) % 'ALL' (Entire sheet) % (Additional ranges: Comma separated ranges in the same form as above; % contents of archive will be returned in output arguments % 2...n) % % NOTE: Specifying range as 'ALL' returns entire used portion of sheet; 132 % Specifying range as 'B:B' or '2:2' returns % appropriate row of UsedRange. (Data are selected in % block form as for 'ALL', then the selected row/column % is returned. % % OUTPUT: If specified range is 1 cell, variable returned is of the same % class as cell contents. If the range spans more than 1 cell, the variables will be cell arrays. % % EXAMPLES: a = readfromexcel('c:\brett\my archives\test1.xls','C1:C5'); % reads from the currently active sheet % [a,b] = readfromexcel('c:\brett\my archives\test1.xls','sheet','sheet2','C1:C5','C1:P3'); % reads from sheet2 % [a,b,c] = readfromexcel('myarchive','C3:D5','sheet','mysheet','E4','sheet','sheet2','B3 '); % reads a from currently active sheet, switches to sheet % 'mysheet' to read b, then to sheet 'sheet2' to read c. % % Written by Brett Shoelson, Ph.D. % shoelson@helix.nih.gov % Update History: 1/04. Version 1. % 2/2/04. Now allows multiple specifications of sheet name % (at the suggestion of R. Venkat), and support of % relative paths (thanks to Urs Schwarz). Also, inclusion of the % extension '.xls' is now superfluous. % 7/21/04. Implements try/catch structure for reading of % ranges to avoid errors that leave open activex % connections. (Response to Chris Paterson's CSSM % query). % 7/29/04. Accomodates reading of entire sheet, or of % entire row/column. (Response to email queries by Kinan Rai % and CSSM query by Xiong.) % % SEE ALSO: write2excel if nargin < 2 msgstr = sprintf('\nAt a minimum, you must specify three input arguments.\nThe first is a string indicating the location of the excel file,\nand the second is a range to be read.'); error(msgstr); end sheetchanges = [strmatch('sheet',varargin,'exact');strmatch('sheetname',varargin,'exact')]; if ~isempty(sheetchanges) [sheetnames{1:length(sheetchanges)}] = deal(varargin{sheetchanges+1}); end [pathstr,name,ext] = fileparts(fileloc); if isempty(ext) fileloc = [fileloc,'.xls']; end 133 if isempty(pathstr) fileloc = which(fileloc,'-all'); if size(fileloc,1) ~= 1 error('File was either not located, or multiple locations were found. Please reissue readfromexcel command, providing absolute path to the file of interest.'); end end Excel = actxserver('Excel.Application'); Excel.Visible = 0; w = Excel.Workbooks; try excelarchive = invoke(w, 'open', fileloc); catch invoke(Excel, 'quit'); release(w); delete(Excel); error(sprintf('Sorry...unable to open file %s',fileloc)); end Sheets = Excel.ActiveWorkBook.Sheets; archive = Excel.Activesheet; initval = get(archive,'Index'); % Read appropriate ranges into output variables chgcount = 1; argcount = 1; for ii = 1:nargin-1 readinfo = []; if ismember(ii,sheetchanges) try sheet = get(Sheets,'Item',sheetnames{chgcount}); invoke(sheet,'Activate'); archive = Excel.Activesheet; chgcount = chgcount + 1; continue catch invoke(Excel, 'quit'); release(w); delete(Excel); error(sprintf('\nUnable to find/open sheet %s.',sheetnames{chgcount})); end elseif ismember(ii,sheetchanges + 1) continue end %Parse range rangespec = 0; if strcmp(lower(varargin{ii}),'all') %Range of the form 'ALL' rangespec = 1; else tmp = findstr(varargin{ii},':'); if isempty(tmp) %Range of the form 'A' or '2' or 'A2' r1 = varargin{ii}; 134 r2 = r1; if ~any(ismember(r1,num2str([1:9]))) %Range of the form 'A' rangespec = 2; elseif all(ismember(r1,num2str([1:9]))) %Range of the form '2' rangespec = 3; else %Range of the form 'A2' rangespec = 4; end else % Range of the form 'A2:B3', '2:2', or 'A:A' r1 = varargin{ii}(1:tmp-1); if all(ismember(r1,num2str([1:9]))) %Range of the form '2:2' r2 = r1; rangespec = 5; elseif ~any(ismember(r1,num2str([1:9]))) %Range of the form 'A:A' r2 = r1; rangespec = 6; else %Range of the form 'A1:B2' r2 = varargin{ii}(tmp+1:end); rangespec = 7; end end end try switch rangespec case 1 readinfo = get(archive,'UsedRange'); case {2,6} readinfo = get(archive,'UsedRange'); [r,c] = an2nn(r1); r1 = nn2an(readinfo.row,c); [m,n] = size(readinfo.value); r2 = nn2an(readinfo.row+m,c); readinfo = get(archive, 'Range', r1, r2); case {3,5} readinfo = get(archive,'UsedRange'); [m,n] = size(readinfo.value); r2 = nn2an(r1,readinfo.row+n); r1 = nn2an(r1,readinfo.column); readinfo = get(archive, 'Range', r1, r2); case {4,7} readinfo = get(archive, 'Range', r1, r2); otherwise readinfo.value = {}; fprintf('Error parsing input argument %d.',ii+1); end catch fprintf('Error reading range specified by input argument %d.',ii+1); invoke(Excel, 'quit'); release(excelarchive); release(w); delete(Excel); return end varargout{argcount} = readinfo.value; argcount = argcount + 1; 135 end % Reset to initial active sheet sheet = get(Sheets,'Item',initval); invoke(sheet,'Activate'); try release(readinfo); end invoke(excelarchive,'close'); %This closes without saving, so changing the active sheet is temporary %invoke(excelarchive,'save'); %Note: Instead of invoke(excelarchive,'close'), I use the save option after % switching back to the initially active sheet. This stops Excel from showing % "previously saved versions" when the file is % next opened. invoke(Excel, 'quit'); release(excelarchive); release(w); delete(Excel); return %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%% %SUBFUNCTIONS %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%% function cr = nn2an(r, c) % convert number, number format to alpha, number format %t = [floor(c/27) + 64 floor((c - 1)/26) - 2 + rem(c - 1, 26) + 65]; t = [floor((c - 1)/26) + 64 rem(c - 1, 26) + 65]; if(t(1)<65), t(1) = []; end cr = [char(t) num2str(r)]; function [r, c] = an2nn(cr) % convert alpha, number format to number, number format t = find(isletter(cr)); t2 = abs(upper(cr(t))) - 64; if(length(t2) == 2), t2(1) = t2(1) * 26; end c = sum(t2); r = str2num(cr(max(t) + 1:length(cr))); 136 APPENDIX C: Program for cavern volume determination in CFD #include "udf.h" DEFINE_ON_DEMAND(cavern_calc_s) { Domain *d; real strain, volume, vol_tot; Thread *t; cell_t c; d = Get_Domain(1); thread_loop_c(t,d) { begin_c_loop(c,t) { volume = C_VOLUME(c,t); strain = C_STRAIN_RATE_MAG(c,t); if (strain >= 0.984) vol_tot += volume; } end_c_loop(c,t) } printf("The cavern volume is %10.8f", vol_tot); } 137 APPENDIX D: Program for tracer analysis in CFD #include <udf.h> DEFINE_PROFILE(hwtracer_mf_profile, t, i) { real flow_time = CURRENT_TIME; face_t f; begin_f_loop(f,t) { if (flow_time > 3411) F_PROFILE(f, t, i) = 0.0847; else if (flow_time > 3284) F_PROFILE(f, t, i) = 0.1096; else if (flow_time > 3192) F_PROFILE(f, t, i) = 0.0847; else if (flow_time > 3070) F_PROFILE(f, t, i) = 0.1096; else if (flow_time > 2986) F_PROFILE(f, t, i) = 0.0847; else if (flow_time > 2531) F_PROFILE(f, t, i) = 0.1096; else if (flow_time > 2453) F_PROFILE(f, t, i) = 0.0847; else if (flow_time > 2208) F_PROFILE(f, t, i) = 0.1096; else if (flow_time > 2130) F_PROFILE(f, t, i) = 0.0847; else if (flow_time > 1992) F_PROFILE(f, t, i) = 0.1096; else if (flow_time > 1803) F_PROFILE(f, t, i) = 0.0847; else if (flow_time > 1566) F_PROFILE(f, t, i) = 0.1096; else F_PROFILE(f, t, i) = 0.0847; } end_f_loop(f,t) } 138 APPENDIX E: ERT data Pulp type: hardwood; Mass concentration: 2%; E/D = 0.6; Z/T = 1.0 Impeller speed (N) (rpm) ERT images 125 150 200 Conductivity scale P1 P2 P4 P5 P1 P1 P2 P2 P4 P4 P5 P5 low high P3 P3 P3 139 Pulp type: hardwood; Mass concentration: 2%; E/D = 0.6; Z/T = 1.0 Impeller speed (N) (rpm) ERT images 225 250 Conductivity scale P1 P2 P3 P5 P1 P2 P4 low high P4 P5 P3 140 Pulp type: hardwood; Mass concentration: 2%; E/D = 0.4; Z/T = 1.0 Impeller speed (N) (rpm) ERT images 125 150 200 Conductivity scale P1 P2 P4 P5 P1 P1 P2 P2 P4 P4 P5 P5 low high P3 P3 P3 141 Pulp type: hardwood; Mass concentration: 2%, E/D = 0.4; Z/T = 1.0 Impeller speed (N) (rpm) ERT images 225 250 Conductivity scale P1 P2 P3 P5 P1 P2 P4 low high P4 P5 P3 142 Pulp type: hardwood; Mass concentration: 2%; E/D = 0.4; Z/T = 0.8 Impeller speed (N) (rpm) ERT images 125 150 200 Conductivity scale P1 P2 P3 P4 P1 P1 P2 P2 P3 P3 P4 P4 low high 143 Pulp type: hardwood; Mass concentration: 2%; E/D = 0.4; Z/T = 0.8 Impeller speed (N) (rpm) ERT images 225 250 Conductivity scale P1 P2 P3 P4 P1 P2 P3 P4 low high 144 Pulp type: hardwood; Mass concentration: 3%; E/D = 0.4; Z/T = 0.8 Impeller speed (N) (rpm) ERT images 425 450 475 Conductivity scale P1 P2 P3 P4 P1 P1 P2 P2 P3 P3 P4 P4 low high 145 Pulp type: hardwood; Mass concentration: 3%; E/D = 0.4; Z/T = 0.8 Impeller speed (N) (rpm) ERT images 500 525 Conductivity scale P1 P2 P3 P4 P1 P2 P3 P4 low high 146 Pulp type: hardwood; Mass concentration: 4%; E/D = 0.4; Z/T = 0.8 Impeller speed (N) (rpm) ERT images 550 575 600 Conductivity scale P1 P2 P3 P4 P1 P1 P2 P2 P3 P3 P4 P4 low high 147 Pulp type: hardwood; Mass concentration: 4%; E/D = 0.4; Z/T = 0.8 Impeller speed (N) (rpm) ERT images 650 675 Conductivity scale P1 P2 P3 P4 P1 P2 P3 P4 low high 148 Pulp type: softwood; Mass concentration: 3%; E/D = 0.4; Z/T = 0.8 Impeller speed (N) (rpm) ERT images 425 450 475 Conductivity scale P1 P2 P3 P4 P1 P1 P2 P2 P3 P3 P4 P4 low high 149 Pulp type: softwood; Mass concentration: 3%; E/D = 0.4; Z/T = 0.8 Impeller speed (N) (rpm) ERT images 500 525 Conductivity scale P1 P2 P3 P4 P1 P2 P3 P4 low high
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Mixing dynamics of pulp suspensions in cylindrical vessels Hui, Kwok Wai Leo 2011
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Title | Mixing dynamics of pulp suspensions in cylindrical vessels |
Creator |
Hui, Kwok Wai Leo |
Publisher | University of British Columbia |
Date Issued | 2011 |
Description | Cylindrical agitated chests are frequently used to facilitate manufacturing processes in pulp and paper industry and one of their main functions is to attenuate any process disturbances. However, owing to the inherited non-Newtonian nature of pulp suspensions, it is not easy to achieve complete mixing and with the improper chest design, these agitated chests do not always perform ideally or satisfactorily. The cavern formation in incomplete mixing may induce bypassing and dead zones, which significantly affect the chest performance. A study of cavern formation in a cylindrical agitated chest was thus carried out. Also, a dynamic model developed by Soltanzadeh et al. (2009) was used to quantify the mixing dynamics of the cylindrical chest. In addition, using computational fluid dynamics (CFD), the simulated results of the flow in the chest were compared with the experimental results to verify the applicability of the CFD model on the study of pulp suspension agitation. To investigate the cavern formation in a lab-scale cylindrical chest, electrical resistance tomography (ERT) and ultrasonic Doppler velocimetry (UDV) were applied to estimate the cavern shape and size. Both methods gave satisfactory results and as expected, the cavern size was found to increase with impeller speeds. The cavern shape was best described as a truncated right-circular cylinder. Based on this observation, a model considering the interaction between the cavern and chest walls was developed to calculate the cavern volume. With the dynamic model, a series of dynamic tests were carried out to characterize the mixing behavior of the lab-scale cylindrical chest. It was found that the proposed flow configuration with the outlet close to the cavern could minimize the bypassing which affects mixing quality. Also, ERT verified the presence of cavern and dead zone when the chest was not completely agitated in continuous-flow operation. Numerical simulations using CFD were compared with the experimental results under different operating conditions. Pulp suspensions are a mixture of water and wood fibres that can entangle each other to form flocs affecting the mixing flow. Owing to this complex rheology, it is not easy to model the agitation precisely in CFD using a homogeneous fluid model. The floc formation and air entrapment observed in experiments were difficult to be numerically taken into account in the simulations. Although the CFD model could not exactly predict the mixing situation of pulp suspensions, it still can be used to estimate the mixing flow patterns, e.g., flow directions, in the proposed chest designs. |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Date Available | 2011-05-02 |
Provider | Vancouver : University of British Columbia Library |
Rights | Attribution-NonCommercial-NoDerivatives 4.0 International |
DOI | 10.14288/1.0059243 |
URI | http://hdl.handle.net/2429/34187 |
Degree |
Doctor of Philosophy - PhD |
Program |
Chemical and Biological Engineering |
Affiliation |
Applied Science, Faculty of Chemical and Biological Engineering, Department of |
Degree Grantor | University of British Columbia |
GraduationDate | 2011-11 |
Campus |
UBCV |
Scholarly Level | Graduate |
Rights URI | http://creativecommons.org/licenses/by-nc-nd/4.0/ |
AggregatedSourceRepository | DSpace |
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