"Applied Science, Faculty of"@en .
"Chemical and Biological Engineering, Department of"@en .
"DSpace"@en .
"UBCV"@en .
"Alaqqad, Mohammed O. M."@en .
"2011-07-14T19:41:23Z"@en .
"2011"@en .
"Doctor of Philosophy - PhD"@en .
"University of British Columbia"@en .
"During Kraft pulping, cellulose fibres are liberated from a lignin-matrix, found in a wood chip, by reaction at elevated temperature and pressure using an alkaline solution. Poor lignin removal is a deleterious effect that leads to downstream operational difficulties and decreased product quality. A number of research groups speculate that this is caused by the uneven distribution of the alkaline solution through the wood chip bed during reaction. As a result, the goal of this thesis is to characterize the ease by which fluid flows, and disperses, through wood chip beds. One of the open remaining scientific questions is understanding the effect of bed compressibility on the resulting flow patterns.\n\nIn the first portion of this work we present a methodology to characterize the permeability of a compressible bed of wood chips under mechanical load. We show that under the limiting condition of when the mechanical load is large in comparison to hydraulic pressure the equations of motion can be linearized and solved to produce an expression approximating the variation in porosity along the length of the bed. We show how this may be used, in conjunction with multiple linear regression, to estimate permeability of the bed. The usefulness of these estimates was then tested by predicting the pressure drop versus flow relationship for conditions outside the range of the linearized solution. Good agreement was obtained.\n\nIn the second portion of this work we present a methodology to characterize the axial dispersion of a solute during steady-flow through a compressible bed of wood chips under mechanical load. We use a non-invasive imaging technique, namely electrical resistance tomography (ERT), to visualize the uniaxial displacement of a salt solution. Here we demonstrate that under two limiting cases the porosity of the porous bed varies slowly in the flow-direction and to the lowest order can be considered a constant. This simplified the optimization routine we used to match the experimental data to the numerical results of the advection-diffusion equation. Using this, a methodology to estimate the axial dispersion is given by a minimization scheme."@en .
"https://circle.library.ubc.ca/rest/handle/2429/35983?expand=metadata"@en .
"Characterizing the Permeability and Dispersion of Flows through Compressible Wood-Chip Beds by Mohammed Alaqqad B.Sc. (Chemical Engineering), The American University of Sharjah, 2007 A THESIS SUBMITTED IN PARTIAL FULFILLMENT 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) July 2011 c\rMohammed Alaqqad, 2011 Abstract During Kraft pulping, cellulose fibres are liberated from a lignin-matrix, found in a wood chip, by reaction at elevated temperature and pressure using an alkaline solution. Poor lignin removal is a deleterious effect that leads to downstream op- erational difficulties and decreased product quality. A number of research groups speculate that this is caused by the uneven distribution of the alkaline solution through the wood chip bed during reaction. As a result, the goal of this thesis is to characterize the ease by which fluid flows, and disperses, through wood chip beds. One of the open remaining scientific questions is understanding the effect of bed compressibility on the resulting flow patterns. In the first portion of this work we present a methodology to characterize the permeability of a compressible bed of wood chips under mechanical load. We show that under the limiting condition of when the mechanical load is large in comparison to hydraulic pressure the equations of motion can be linearized and solved to produce an expression approximating the variation in porosity along the length of the bed. We show how this may be used, in conjunction with multiple linear regression, to estimate permeability of the bed. The usefulness of these ii estimates was then tested by predicting the pressure drop versus flow relationship for conditions outside the range of the linearized solution. Good agreement was obtained. In the second portion of this work we present a methodology to characterize the axial dispersion of a solute during steady-flow through a compressible bed of wood chips under mechanical load. We use a non-invasive imaging technique, namely electrical resistance tomography (ERT), to visualize the uniaxial displace- ment of a salt solution. Here we demonstrate that under two limiting cases the porosity of the porous bed varies slowly in the flow-direction and to the lowest or- der can be considered a constant. This simplified the optimization routine we used to match the experimental data to the numerical results of the advection-diffusion equation. Using this, a methodology to estimate the axial dispersion is given by a minimization scheme. iii Preface A version of this thesis have been accepted for publication in The Canadian Jour- nal of Chemical Engineering as two articles; Alaqqad, M., C.P.J. Bennington and D.M. Martinez, \u00E2\u0080\u009DThe Permeability of Wood- Chip Beds: The Effect of Compressibility,\u00E2\u0080\u009D Can. J. Chem. Eng. (2011). Alaqqad, M., C.P.J. Bennington and D.M. Martinez, \u00E2\u0080\u009DAn Estimate of the Axial Dispersion During Flow through a Compressible Wood-Chip Bed,\u00E2\u0080\u009D Can. J. Chem. Eng. (2011). My contributions include: \u000F Planning the experiments and designing the experimental setup \u000F Conducting all laboratory work and developing the predictive models \u000F Performing experimental data analysis \u000F Being the principal author of the manuscripts iv Table of Contents Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xii Nomenclature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xviii Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxiii Dedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxiv 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.2 Flow Resistance through Porous Media . . . . . . . . . . . . . . . 10 v 2.3 Axial Dispersion through Porous Media . . . . . . . . . . . . . . 18 2.4 Electrical Resistance Tomography . . . . . . . . . . . . . . . . . 21 2.5 Research Objectives . . . . . . . . . . . . . . . . . . . . . . . . . 25 3 The Permeability of Wood-Chip Beds: The Effect of Compressibility 27 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 3.2 A Method to Determine r1 and r2 . . . . . . . . . . . . . . . . . . 28 3.3 Experimental Materials and Methods . . . . . . . . . . . . . . . . 36 3.4 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . 45 3.5 Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . 56 4 The Axial Dispersion in Wood-Chip Beds; ERT Visualization . . . . 57 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 4.2 The Methodology to Estimate De . . . . . . . . . . . . . . . . . . 58 4.2.1 The First Limiting Case: pc\u001D ph . . . . . . . . . . . . . 63 4.2.2 The second limiting case: pc = 0 and m\u001D ph . . . . . . . 65 4.3 Experimental Materials and Methods . . . . . . . . . . . . . . . . 67 4.4 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . 81 4.5 Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . 86 5 Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . . . 87 6 Recommendations for Future Work . . . . . . . . . . . . . . . . . . 89 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 vi Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 A Apparatus Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 B Wood Chips Screening Analysis . . . . . . . . . . . . . . . . . . . . 111 C Calculation of Sauter Mean Diameter of Wood Chips . . . . . . . . 115 D Porosity Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . 117 E Pneumatic Cylinder Desgin . . . . . . . . . . . . . . . . . . . . . . . 119 F Pressure Drop Across Screen Plate . . . . . . . . . . . . . . . . . . . 122 G Conductivity to Concentration Tomogram . . . . . . . . . . . . . . . 124 H Higher Order Estimates of e(x) . . . . . . . . . . . . . . . . . . . . . 126 I Summary of the Required Functions . . . . . . . . . . . . . . . . . . 128 J MATLAB Codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 J.1 Compressibility Function Regression Code . . . . . . . . . . . . . 129 J.2 Pressure Drop Regression Code . . . . . . . . . . . . . . . . . . . 133 J.3 Pressure Drop Prediction Code . . . . . . . . . . . . . . . . . . . 141 J.4 Axial Dispersion Minimization Code . . . . . . . . . . . . . . . . 145 K Experimental Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158 K.1 Pressure Drop Data . . . . . . . . . . . . . . . . . . . . . . . . . 158 vii K.2 Axial Dispersion Data . . . . . . . . . . . . . . . . . . . . . . . . 172 L Axial Dispersion Breakthrough Curves . . . . . . . . . . . . . . . . 173 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186 viii List of Tables Table 2.1 Literature values for the flow resistance parameters A and B as defined by Equation 2.5. . . . . . . . . . . . . . . . . . . . . . 13 Table 2.2 Literature values for the flow resistance parameters r1 and r2 as defined by Equation 2.9. The uncertainty in the estimates are given by the correlation coefficients r2, if available. \u00E2\u0080\u009D+\u00E2\u0080\u009D means retained on the plate. . . . . . . . . . . . . . . . . . . . . . . . 16 Table 2.3 Literature values for the compressibility parameters m, eg, and n as defined in Equation 2.8. These research groups indicate that m is a function of the degree of deliginification, i.e. the kappa number k , and advance the equationm(k)= (ao\u0000a1 lnk)\u0000n. 17 Table 3.1 Characteristics and dimensions with standard deviations of the wood chips before cooking. The total mixture was composed of 37.6%(wt/wt) of the 4 mm slot chips with the balance of mixture comprised of the 2 mm slot chips. Both dp as well as y for the total mixture are averages based upon the mass fraction of each species, see Appendices B and C. . . . . . . . 39 ix Table 3.2 Cooking conditions using the laboratory digester. . . . . . . . . 40 Table 4.1 Characteristics and dimensions with standard deviations of the wood chips before cooking. The total mixture was composed of 37.6%(wt/wt) of the 4 mm slot chips with the balance of mixture comprised of the 2 mm slot chips. Both dp as well as y for the total mixture are averages based upon the mass fraction of each species, see Appendices B and C. . . . . . . . 71 Table 4.2 Cooking conditions using the laboratory digester. . . . . . . . . 72 Table 4.3 A summary of the experimental conditions tested with esti- mates of the axial dispersion coefficients. . . . . . . . . . . . . 80 Table B.1 Statistical analysis of 2 mm thickness fraction in accepts. . . . . 113 Table B.2 Statistical analysis of 4 mm thickness fraction in accepts. . . . . 114 Table C.1 Average length, width, and thickness for mass fractions of ac- cepts. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 Table E.1 Calculation of compaction pressure. . . . . . . . . . . . . . . . 119 Table K.1 Pressure drop data of polypropylene spheres. . . . . . . . . . . 159 Table K.2 Pressure drop data of 100% accepts at k = 80 and pc = 0:0 kPa. 160 Table K.3 Pressure drop data of 100% accepts at k = 80 and pc = 5:6 kPa. 161 Table K.4 Pressure drop data of 100% accepts at k = 80 and pc = 9:8 kPa. 162 Table K.5 Pressure drop data of 100% accepts at k = 80 and pc = 14:0 kPa.163 Table K.6 Pressure drop data of 100% accepts at k = 53 and pc = 0:0 kPa. 164 x Table K.7 Pressure drop data of 100% accepts at k = 53 and pc = 5:6 kPa. 165 Table K.8 Pressure drop data of 100% accepts at k = 53 and pc = 9:8 kPa. 166 Table K.9 Pressure drop data of 100% accepts at k = 53 and pc = 14:0 kPa.167 Table K.10 Pressure drop data of 100% accepts at k = 25 and pc = 0:0 kPa. 168 Table K.11 Pressure drop data of 100% accepts at k = 25 and pc = 5:6 kPa. 169 Table K.12 Pressure drop data of 100% accepts at k = 25 and pc = 9:8 kPa. 170 Table K.13 Pressure drop data of 100% accepts at k = 25 and pc = 14:0 kPa.171 Table K.14 A summary of the experimental conditions tested with esti- mates of the axial dispersion coefficients with correlation coef- ficients r2 for 100% accepts wood chips. . . . . . . . . . . . . 172 xi List of Figures Figure 2.1 Schematic of a typical batch digester (Smook, 1992). . . . . . 8 Figure 2.2 Schematic of a typical one vessel Kamyr digester (Luis et al, 2000). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 Figure 2.3 Photograph of ITS P2000 ERT data acquisition system (ITS, 2006). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 Figure 2.4 Schematic diagram of electrode arrangement around the pro- cess vessel (ITS, 2006). . . . . . . . . . . . . . . . . . . . . . 23 Figure 2.5 Adjacent current injection and resultant voltage measurement of the remaining adjacent electrodes. 2.5a The first current injection. 2.5b The second current injection (ITS, 2006). . . . 24 Figure 2.6 ERT image reconstruction grid (ITS, 2006). . . . . . . . . . . 25 Figure 3.1 Schematic of the geometry considered. . . . . . . . . . . . . . 29 xii Figure 3.2 A comparison of the numerical (dashed line) and the asymp- totic solution (circles) at various d . The asymptotic solution is approximated using the first three terms in the series. This comparison is made with m = 100 kPa, n = 1:2, eg = 0:54, r1 = 3\u0002104 Pa:s=m2 and r2 = 1:9\u0002106 Pa:s2=m3. . . . . . . 34 Figure 3.3 Geometry of disk chipper chips. . . . . . . . . . . . . . . . . 37 Figure 3.4 3.4a Chip screening apparatus. 3.4b Chip size distribution. . . 38 Figure 3.5 Comparison of the accept chips before and after cooking to different k numbers. Scale in the picture is in cm. . . . . . . . 41 Figure 3.6 Schematic of the apparatus. . . . . . . . . . . . . . . . . . . . 43 Figure 3.7 Image of the apparatus. . . . . . . . . . . . . . . . . . . . . . 44 Figure 3.8 The resistance to flow of water through a incompressible bed of polystyrene spheres. The data is compared to the literature values (solid line) to assess the uncertainty in our measure- ment system. . . . . . . . . . . . . . . . . . . . . . . . . . . 46 Figure 3.9 The uniaxial compressibility of wood chips cooked to differ- ent k numbers. . . . . . . . . . . . . . . . . . . . . . . . . . 48 Figure 3.10 The measured pressure drop per unit length for a prescribed superficial velocity v at three different compaction pressures. The uncertainty in the estimate for ph=L is approximately 0.3 kPa=m. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 xiii Figure 3.11 The predicted pressured drop (dashed line), the experimen- tally measured values (circles), and the correlation without compressibility effects (solid line) for the case pc= 0. The un- certainty in the estimate for ph=L is approximately 0.3 kPa=m. The dashed line was calculated using the values of r1 and r2 determined above by numerical integration of the governing equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 Figure 3.12 3.12a Estimate of the permeability of Hemlock wood chips using the Kozeny-Carman expression given by Equation 2.2. The tortuosity and the specific surface of the chips were esti- mated through use of Equations 2.12 and 2.13. 3.12b The ratio of the permeabilities of a k 80 to k 25 wood chips compressed at different loads. . . . . . . . . . . . . . . . . . . . . . . . . 54 Figure 3.13 A comparison of the pressure drop correlations between our correlation and the literature correlations for porosity of 0.35. . 55 Figure 4.1 Schematic of the geometry considered. . . . . . . . . . . . . . 59 Figure 4.2 Geometry of disk chipper chips. . . . . . . . . . . . . . . . . 69 Figure 4.3 4.3a Chip screening apparatus. 4.3b Chip size distribution. . . 70 Figure 4.4 Comparison of the accept chips before and after cooking to different k numbers. Scale in the picture is in cm. . . . . . . . 73 Figure 4.5 Schematic of the apparatus. . . . . . . . . . . . . . . . . . . . 75 Figure 4.6 Image of the apparatus. . . . . . . . . . . . . . . . . . . . . . 76 xiv Figure 4.7 Contour images of a representative ERT measurement with k = 80, pc = 0 kPa, and e = 0:54. . . . . . . . . . . . . . . . 83 Figure 4.8 The evolution of the concentration of the brine solution as a function of time at different elevations in the column. P1-P7 are ERT planes 1-7. . . . . . . . . . . . . . . . . . . . . . . . 84 Figure 4.9 The estimated axial dispersion coefficient De with porosity e and inverse of permeability k. . . . . . . . . . . . . . . . . . . 85 Figure A.1 Schematic of the apparatus. . . . . . . . . . . . . . . . . . . . 98 Figure A.2 Schematic of the top screen plate. . . . . . . . . . . . . . . . 99 Figure A.3 Schematic of the top flange. . . . . . . . . . . . . . . . . . . 100 Figure A.4 Schematic of the top pipe section. . . . . . . . . . . . . . . . 101 Figure A.5 Schematic of the pipe with ERT ring. . . . . . . . . . . . . . 102 Figure A.6 Schematic of the ERT sensors. . . . . . . . . . . . . . . . . . 103 Figure A.7 Schematic of the pipe with ERT and DP ring. . . . . . . . . . 104 Figure A.8 Schematic of the bottom flange. . . . . . . . . . . . . . . . . 105 Figure A.9 Schematic of the ground ERT sensor. . . . . . . . . . . . . . . 106 Figure A.10 Schematic of the bottom screen plate. . . . . . . . . . . . . . 107 Figure A.11 Schematic of the bottom cap. . . . . . . . . . . . . . . . . . . 108 Figure A.12 Side view of the apparatus. . . . . . . . . . . . . . . . . . . . 109 Figure A.13 Top view of the apparatus. . . . . . . . . . . . . . . . . . . . 110 Figure B.1 Typical chip geometry and dimensions. . . . . . . . . . . . . 112 Figure B.2 2mm thickness fraction in accepts. . . . . . . . . . . . . . . . 113 xv Figure B.3 4mm thickness fraction in accepts. . . . . . . . . . . . . . . . 114 Figure E.1 Relationship between compaction pressure and air line pres- sure. Solid line is given by Equation E.2. . . . . . . . . . . . . 121 Figure F.1 Pressure drop across the screen plate. Dashed line is given by linear regression fit in MATLAB to give Equation F.1. . . . . . 123 Figure G.1 Concentration of Brine Solution. Dashed line is given by lin- ear regression fit in MATLAB to give Equation G.1. . . . . . . 125 Figure L.1 Axial dispersion breakthrough curves for k = 80 and pc = 0:0 kPa. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174 Figure L.2 Axial dispersion breakthrough curves for k = 80 and pc = 5:6 kPa. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 Figure L.3 Axial dispersion breakthrough curves for k = 80 and pc = 9:8 kPa. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176 Figure L.4 Axial dispersion breakthrough curves for k = 80 and pc = 14:0 kPa. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177 Figure L.5 Axial dispersion breakthrough curves for k = 53 and pc = 0:0 kPa. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178 Figure L.6 Axial dispersion breakthrough curves for k = 53 and pc = 5:6 kPa. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179 Figure L.7 Axial dispersion breakthrough curves for k = 53 and pc = 9:8 kPa. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180 xvi Figure L.8 Axial dispersion breakthrough curves for k = 53 and pc = 14:0 kPa. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181 Figure L.9 Axial dispersion breakthrough curves for k = 25 and pc = 0:0 kPa. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182 Figure L.10 Axial dispersion breakthrough curves for k = 25 and pc = 5:6 kPa. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183 Figure L.11 Axial dispersion breakthrough curves for k = 25 and pc = 9:8 kPa. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184 Figure L.12 Axial dispersion breakthrough curves for k = 25 and pc = 14:0 kPa. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185 xvii Nomenclature Symbols a radius of the circular pipe m A;B empirical constants a;b empirical constants Ac area of the compaction m2 As surface area of the particle m2 ao;1 empirical constants Apiston area of the piston m2 C concentration of the solute in the fluid M D molecular diffusivity m2=s dc diameter of the cylindrical particle m De effective diffusivity (dispersion) m2=s xviii dp equivalent diameter of the particle m d32 Sauter mean diameter of the particle m dp1;2;3;4 pressure drop across four sections of the wood-chip column Pa dpscreen pressure drop across the screen plate Pa F output force of the pneumatic cylinder N h height of the cylindrical particle m k permeability m2 L length of the wood-chip bed m l length of the parallelepipedal particle m Lo initial bed length before compressing the cooked wood chips m Lp bed length after compressing the cooked wood chips m m empirical constant of the stiffness of the network kPa n empirical constant p fluid pressure Pa pc compaction pressure (mechanical load) kPa ph hydraulic pressure (total pressure drop) Pa ps solid-stress Pa xix pair air line pressure kPa Pe Peclet number PF power factor (efficiency) of the pneumatic cylinder r arbitrary radius m r2 correlation coefficient r1 empirical constant of the flow resistance Pa:s=m2 r2 empirical constant of the flow resistance Pa:s2=m3 RePM Reynolds number for porous media sv specific surface area of the porous medium m2=m3 T temperature of the fluid \u000EC t arbitrary time s t thickness of the parallelepipedal particle m u local velocity m=s uo centreline velocity m=s v superficial velocity m=s Vp volume of the particle m3 w width of the parallelepipedal particle m xx x arbitrary vertical length m Y Yield of the cook Y distance over which the image is averaged in the y-coordinate m y;z coordinates in each individual ERT plane Z distance over which the image is averaged in the z-coordinate m Subscripts e experimental f final i initial Greek Letters b empirical constant d dimensionless group e porosity ec characteristic porosity eg gel porosity echip volume fraction of the cooked wood chips g dimensionless group xxi k kappa number m dynamic viscosity of the fluid Pa:s y sphericity shape factor r density of the fluid kg=m3 rob initial bulk packing density of the cooked wood chips kg=m 3 r pb bulk packing density of the cooked wood chips under compression kg=m 3 rchip basic density of the uncooked wood chips kg=m3 rcookchip basic density of the cooked wood chips kg=m 3 s electrical conductivity of NaCl brine solution mS=cm q shape factor constant usually assigned the value of 5:5 xxii Acknowledgments Many individuals have contributed to this research and their assistance is grate- fully acknowledged. I sincerely thank the following individuals for their help throughout the course of this project: My co-supervisors; Dr. Chad Bennington and Dr. Mark Martinez for their guidance, suggestions and valuable discussions throughout the course of this re- search. Without their insight both into my research and personal development, I don\u00E2\u0080\u0099t believe I could have ever achieved all that I have. FPInovations for providing the wood chips and their help in conducting the laboratory cooks. Members of Chemical and Biological Engineering Department, and Pulp and Paper Center for their help in many aspects of this research. The Natural Sciences and Engineering Research Council of Canada for finan- cial support of this work. Last but not least, my parents, my wife and my brothers for their love, encour- agement and support in realizing this dream. My deepest love and special thanks to my father who has always been my motivation, strength and greatest support. xxiii Dedication To My Great Parents; Omar Alaqqad, and Laila Alaqqad And My Lovely Wife; Najat And My Brothers; Emad, Iyad, and Alaa And My Friends, and Professors xxiv Chapter 1 Introduction The focus of the present work is to develop a methodology to characterize the per- meability and dispersion of a solute in a compressible porous medium. Although the goals of this work have relevance to a number of different scientific and indus- trial settings, such as the flow of groundwater through soil, this thesis, however, is motivated from one unit operation in the Kraft process in the pulp and paper industry, namely the chemical digester. Here, wood chips are cooked to remove lignin in large vessels typically 3\u00008 m in diameter and 25\u000060 m in height. The reaction solution is usually referred to as \u00E2\u0080\u009Dwhite liquor\u00E2\u0080\u009D, which is an aqueous so- lution of sodium hydroxide (NaOH) and sodium sulfide (Na2S) and is in contact with the wood chips at approximately 170\u000EC, 600 kPa, for approximately 4 hours. To ensure good thermal and mass transfer to the wood chips during reaction, pro- cess liquors are circulated through the digester after being heated in an external heat exchanger (Smook, 1992). 1 To achieve a high quality product, with small spatial variations in lignin con- tent, the process demands that each chip receives equivalent treatment. In other words, liquor flow should be maintained uniform and at an optimal level so that the reactants can be supplied to the chip and that the bi-products removed. Un- fortunately, this is not the case in most digesters as indications of non-uniformity, e.e. accelerated corrosion in the digester (Kiessling, 1995; Wensley, 1996), fluc- tuations in the exit kappa number, a measure of the extent of reaction (He et al, 1999), and temperature gradients (Pageau and Marcoccia, 2001) are evident. It is thought that spatial variations in the liquor flows result from chip softening as the reaction progresses. Chip softening results in a local compaction of the bed and the subsequent decrease in porosity limits penetration of the white liquor. This phenomenon is magnified as chip softening may cause a change in particle size distribution; this may also cause a local reduction in porosity (Lee, 2002). Char- acterizing this effect is one of the open remaining questions in this field and it is the primary scientific question addressed in this thesis. The motivation and the major findings of these two studies in this thesis are presented in three chapters. In Chapter 2, we present a concise summary of the literature review in order to set our results in the context of other work. Here a description of the Kraft pulping process as well as a number of studies relating the behavior of wood chips-liquor interaction is given. A brief description of our main experimental tool, Electrical Resistance Tomography (ERT), is presented here as well. In Chapters 3 and 4, we present the methodology and the results for the measurement of permeability and axial dispersion in a packed column of wood 2 chips under mechanical load. 3 Chapter 2 Literature Review 2.1 Background Chemical pulping of wood chips is a method of pulp manufacture in which wood chips are treated with aqueous alkaline or acidic solutions. The aim of chemical pulping is to remove enough lignin that binds the wood fibers together, to give them the required characteristics for their derived use, and to do so at the lowest possible cost. Accordingly, the liberated fibers can now be used to produce paper. Kraft pulping is today the predominant process for producing chemical pulp. In Canada, 80% of all the chemical pulp produced is by this process while 85% of the worldwide chemical pulp is produced via this process. The chemical pulping process can be accomplished with either batch or continuous digesters. Kraft pulping is a chemical process used to delignify wood chips by an aque- ous solution of NaOH and Na2S ,referred to as \u00E2\u0080\u009Dwhite liquor\u00E2\u0080\u009D, at a high temper- 4 ature reaching 170\u000EC and a high pressure reaching 600 kPa for approximately 4 hours to produce pulp. Kappa number k is used as a measure of the lignin con- tent and decreases as lignin is removed as reaction proceeds. It is noteworthy that cooked chips have similar dimensions as the original uncooked chips except that cooked chips has less lignin content (lower kappa number) depending on the degree of cooking. To ensure good thermal and mass transfer to the wood chips during reaction, process liquors are circulated through the digester after being heated in an external heat exchanger. Batch digesters range from 70 m3 to 340 m3 capacity, with a standard capacity of 170 m3 to 230 m3 for most modern mills. In batch cooking, the digester is filled with wood chips and enough white liquor is added to cover the chips. The con- tents are then heated to a predetermined schedule, usually by forced circulation of the cooking liquor through an external heat exchanger, as shown in Figure 2.1. Air and non-condensible gases generated during cooking are relieved through a pressure control valve at the top of the vessel. The maximum temperature, usually about 170\u000EC, is typically reached after 1.0-1.5 hours which allows for the cook- ing liquor to impregnate the chips. The cook is then maintained at the maximum temperature for up to 2 hours to complete the cooking reactions. After digestion, the contents are discharged into a blow tank where the softened chips are disinte- grated into fibers; the off-vapors are condensed in a heat exchanger where water is heated for pulp washing (Smook, 1992). Continuous digesters ranges from 6\u00009 m in diameter and 60\u000070 m in height with a typical capacity of 1000 tons per day of pulp (oven-dry basis). Most con- 5 tinuous digesters consist of three basic zones; an impregnation zone, one or more cooking zones, and a wash zone. A typical one-vessel Kamyr digester is shown in Figure 2.2. In continuous cooking, the wood chips are first carried through a steaming vessel where air and non-condensible gases are purged. The preheated chips and white liquor then enter the continuous digester where they move through the impregnation zone of an intermediate temperature (115\u0000 120\u000EC) to allow for uniform penetration of the chemical into the chips. As the chip mass moves through the cooking vessel, it enters the cooking zone where the mixture is heated to the cooking temperature, either by forced circulation of the cooking liquor through an external heat exchanger or by steam injection, and maintained at this temperature for up to 2 hours. Following completion of the cook, the chip mass enters the washing zone where hot spent liquor is extracted into a low-pressure tank where flash steam is generated for use in the steaming vessel. The pulp is usually quenched to below 100\u000EC with cool liquor to prevent mechanical damage to the fibers (Smook, 1992). Over the past decades some significant modifications to conventional batch and continuous cooking in Kraft pulping have been developed, and these improved methods and technologies are rapidly being incorporated into new installations and retrofitted into older mills. The new methodology involves additional equip- ment and processing steps, but doesn\u00E2\u0080\u0099t extend the cooking cycle (Smook, 1992). Current Kraft operating strategies rely on contacting the chips with liquors of different temperatures and compositions during cooking to optimize pulp quality and strength. This requires creation of uniform liquor flow throughout the chip 6 column. Local compaction play a key role in this process due to the presence of interstitial spaces between the chips producing non-uniformity in the liquor flow. Indications of non-uniformity include accelerated corrosion in the digester (Kiessling, 1995; Wensley, 1996), fluctuations in the exit kappa number (He et al, 1999) and circumferential temperature gradients in the digester (Pageau and Marcoccia, 2001). A number of researchers have measured the compressibility and flow resis- tance of wood-chip columns. He et al (1999) developed a 3-D coupled two- phase computer model of a continuoss digester simulating the flow distribution and delignification process. Harkonen (1987) developed a 2-D flow model of the digester using an Eulerian volume-averaged two-phase flow approach to account for interactions between the liquor and chip phases and measured pressure drop in support of his modeling. Other researchers, including Lindqvist (1994); Lammi (1996); Wang and Gullichsen (1999); Lee (2002) have experimentally measured pressure drop in wood-chip beds as a function of flow velocity and void fraction for different chip furnishes and size distributions. 7 Figure 2.1: Schematic of a typical batch digester (Smook, 1992). 8 Figure 2.2: Schematic of a typical one vessel Kamyr digester (Luis et al, 2000). 9 2.2 Flow Resistance through Porous Media Before we discuss the pertinent models relevant to flow through this class of ma- terial, it is instructive to first define the two parameters that will be used to charac- terize the resistance to flow in this work, i.e. the permeability k and the solid-stress ps. We begin this discussion with permeability. Traditionally permeability is de- fined using Darcy\u00E2\u0080\u0099s law dp dx = m k v (2.1) which relates the change in pressure drop per unit length dp=dx required to main- tain a superficial velocity v through a porous medium. The dynamic viscosity of the fluid is defined as m . There is no mechanistic justification of the form of Equation 2.1 except for empirical evidence demonstrating its utility for most media under creeping flow conditions (v! 0). There are various experimental methods for determining k; see for example the summary given by Scheidegger (1957) or the references given therein. For incompressible media, these methods simply involve the measurement of the su- perficial flow rate of the fluid for a given pressure drop and comparing these data to Equation 2.1. It is well known that permeability is a function of the poros- ity e and the most common expression outlining this dependency is given by the Kozeny-Carman relationship (Scheidegger, 1957), i.e. k = e3 qs2v(1\u0000 e)2 (2.2) 10 where sv is the specific surface area of the particles; and q is a constant usually ranges from 4:7\u0000 5:5. There are many other functional forms of permeability that have been developed for various ranges of volume fraction, particle geometry and network architecture. These models have been studied both theoretically and experimentally quite extensively (Jackson and James, 1986). Beyond this creep- ing flow limit, a number of authors have extended Darcy\u00E2\u0080\u0099s law and advanced an equation of the form dp dx = m k v+ bp ke 3 2 rv2 (2.3) where b is an empirical constant that must be determined from experiment, and r is the density of the fluid . This form of relationship is routinely found in a num- ber of works and is generally referred to as the either the Ergun or Forchheimer relationship. In Equation 2.3, a characteristic dimension should be used to repre- sent the particle size. Generally, the characteristic dimension is the diameter of a sphere having the same specific surface as the particle (Ergun, 1952; Comiti and Renaud, 1989; Niven, 2002), expressed as sv = As Vp = pd2p 1 6pd3p = 6 dp (2.4) where As is the surface area of the particle, Vp is the volume of the particle, and dp is the equivalent particle diameter. Substituting Equations 2.2 and 2.4 into Equation 2.3 results in the famous Ergun (1952) Equation, expressed as dp dx = A (1\u0000 e)2 e3 m y2d2p v+B (1\u0000 e) e3 r ydp v2 (2.5) 11 where A and B are empirical constants that must be determined experimentally, y is the sphericity shape factor. Sphericity is a measure of how spherical (round) an object is and it is defined as the ratio of the surface area of a sphere (with the same volume as the given particle) to the surface area of the particle, expressed as y = p1=3(6Vp)2=3 As (2.6) It is noteworthy that A is dominant for laminar flows (RePM \u0014 10), and B is dominant for turbulent flows (RePM \u0015 1000) where RePM is the modified Reynolds number for porous media, expressed as RePM = dprv m(1\u0000 e) (2.7) Values of A and B have been measured for several particles where it was found that they are functions of particle geometry and roughness (Ergun, 1952; Leva, 1959; MacDonald et al, 1979; Comiti and Renaud, 1989). A summary of the different studies, including the type and shape of the particles, is given in Table 2.1. We now continue with the discussion of the second parameter, the solid-stress relationship ps(e). The solid-stress is a property created by the connectivity and corresponding friction between the particles comprising the bed. If the frictional forces are sufficiently large, the bed may support an external load. If the load exceeds the compressive strength of the bed, the bed irreversibly densifies through a reduction of the void volume. The relationship between the applied pressure 12 Source Material Porosity Empirical Parameters e A B Ergun (1952) Spheres \u00E2\u0080\u0093 150 1:75 Leva (1959) Spheres \u00E2\u0080\u0093 200 1:75 MacDonald et al (1979) Spheres \u00E2\u0080\u0093 180 1:8\u00004:0 Comiti and Renaud (1989) Spheres, dp = 1:12 mm 0.36 140 1:68 Comiti and Renaud (1989) Spheres, dp = 4:99 mm 0.36 142 1:59 Comiti and Renaud (1989) Cylinders, h=dc = 5:49 0.39 166 3:20 Comiti and Renaud (1989) Plates, t=l = 0:102 0.46 216 12:2 Comiti and Renaud (1989) Plates, t=l = 0:209 0.35 161 6:69 Table 2.1: Literature values for the flow resistance parameters A and B as defined by Equation 2.5. and the void volume is defined as the solid-stress or compressibility function. This relationship is an implicit function of the particle geometry, packing and the orientation distributions (Toll, 1998). The solid-stress function usually obeys an empirical relationship of the form ps = f3(e) = m(eg\u0000 e)n (m;n\u0015 0) (2.8) where eg is the gel porosity, i.e. the porosity at which the network may first support an external load; andm and n are empirical constants that must be determined from experiment wherem is the stiffness of the network. For a more detailed discussion of this function the reader is encouraged to examine the earlier works of Wilder (1960), Wrist (1964), Han (1969) or the more recent works by Vomhoff (1998) and Pettersson et al (2008). For the particular case of flow through wood-chip beds, there are a limited 13 number of studies in which the permeability and compressibility functions are reported. As before, we shall report the values for permeability first and then move onto the compressibility function. Strictly speaking, the permeability for this material is never reported. Instead, we find that Equation 2.3 or or 2.5 is recast into the form dp dx = r1 f1(e)v+ r2 f2(e)v2 (2.9) f1(e) = (1\u0000 e)2 e3 (2.10) f2(e) = (1\u0000 e) e3 (2.11) through substitution of Equation 2.2 into Equation 2.3 and the parameters r1 and r2 are considered to be the flow resistance of the medium; they are determined by regression of a data set representing (vi;dpi=dx). Physically, r1 and r2 repre- sent the lumped effects of tortuosity, specific surface, and density and dynamic viscosity of the fluid, i.e. r1 = mqs2v = A m y2d2p (2.12) r2 = br p qsv = B r ydp (2.13) A summary of the different studies, including the type of wood chips and de- gree of deliginification, is given in Table 2.2. As shown the data set is highly variable and the uncertainty in the estimate, as determined by the correlation co- efficient, is large. It is interesting to note that for some estimates, the inertial 14 component of the Ergun relationship, i.e. the second term in this relationship, is reported as a negative value. This is physically impossible. Thus, for these negative cases, we may argue that r2 should not be different from zero due to the operation at near laminar regimes in such cases since RePM for flows through wood-chip beds usually lies within 50\u0014 RePM \u0014 2000. At this point, we attempt to gain insight into the problem by examining the literature for papermaking fibre suspensions. We do so as the wood chip is the precursor of the fibre suspension. For papermaking suspensions a fairly crude es- timate of the permeability is of the order k \u0018 10\u000012m2. We define this as a crude estimate as there is a vast literature with estimates of the permeability as a func- tion of porosity for various fibre and filler particle or refining treatments,see e.g. Robetrson and Mason (1949), Ingmanson and Whitney (1954) or Lindsay and his co-workers (Lindsay, 1990; Lindsay and Brady, 1993a,b). We anticipate that the permeability of the wood chip to be larger than this value. In addition we find that at low superficial velocities, Robetrson and Mason (1949) demonstrate a linear relationship between pressure drop and superficial velocity and that a Kozeny- Carman type relationship described the data quite well. At higher velocities, In- gmanson and Whitney (1954) indicate deviations from this type of behavior and inertial effects are evident. We now turn our attention to the characterization of the compressibility func- tion describing wood-chip beds. In this literature, water saturated wood chips were subjected to a static uniaxial load from which the displacement of the bed was measured. All researchers report compressibility behavior of the same form 15 Source Material Empirical Parameters Corr. Coeff. r1 r2 r2 (Pa:s=m2) Pa:s2=m3 Harkonen (1987) Scandinavian Pine 0:05\u0002105 3:9\u0002106 \u00E2\u0080\u0093 (distribution not specified) Lindqvist (1994) Scandinavian Pine 0:28\u0002105 \u00000:13\u0002106 0.36 + 6 mm slot 22.1% + 4 mm slot 44.2% + 2 mm slot 29.1% + 3 mm hole 4.6% Lindqvist (1994) Scandinavian Pine 0:51\u0002105 \u00000:35\u0002106 0.41 + 6 mm slot 23.2% + 4 mm slot 46.3% + 2 mm slot 30.5% Lindqvist (1994) Scandinavian Pine 0:06\u0002105 \u00000:01\u0002106 0.58 + 6 mm slot 60.3% + 4 mm slot 39.7% Lammi (1996) Scandinavian Birch 0:68\u0002105 2:47\u0002106 0.89 + 45 mm hole 0.6% + 8 mm slot 9.2% + 13 mm hole 51.2% + 7 mm hole 31.2% + 3 mm hole 6.4% + fines 1.1% Lammi (1996) Eucalyptus Camaldulensis 1:85\u0002105 \u00000:03\u0002106 0.91 + 45 mm hole 1.1% + 8 mm slot 4.8% + 13 mm hole 78.8% + 7 mm hole 12.6% + 3 mm hole 2.2% + fines 0.5% Wang and Gullichsen (1999) Scandinavian Pine 0:52\u0002105 1:50\u0002106 - + 7 mm hole 100% Wang and Gullichsen (1999) Scandinavian Pine 0:82\u0002105 \u00000:11\u0002106 - 4mm thick, 40mm length Lee (2002) White Spruce 0:11\u0002105 13:78\u0002106 0.76 + 3 mm hole 100% Lee (2002) White Spruce 0:66\u0002105 4:72\u0002106 0.82 + 7 mm hole 75% + 3 mm hole 25% Lee (2002) White Spruce 0:60\u0002105 4:57\u0002106 0.74 + 7 mm hole 87.5% + 3 mm hole 12.5% Lee (2002) White Spruce 0:40\u0002105 5:77\u0002106 0.91 + 7 mm hole 100% Table 2.2: Literature values for the flow resistance parameters r1 and r2 as defined by Equation 2.9. The uncertainty in the estimates are given by the correlation coefficients r2, if available. \u00E2\u0080\u009D+\u00E2\u0080\u009D means retained on the plate. 16 Source Material m(k) eg n a0 a1 (kPa) (kPa) Harkonen (1987) Scandinavian Pine 2:14\u000210\u00001 3:57\u000210\u00002 0.64 1.7 Lindqvist (1994) Scandinavian Pine 2:24\u000210\u00001 4:48\u000210\u00002 0.60 1.6 Lindqvist (1994) Scandinavian Pine 1:48\u000210\u00001 2:94\u000210\u00002 0.62 1.3 Lindqvist (1994) Scandinavian Pine 1:86\u000210\u00001 3:73\u000210\u00002 0.65 1.4 Lammi (1996) Scandinavian Birch 1:78\u000210\u00001 4:08\u000210\u00002 0.59 1.8 Lammi (1996) Eucalyptus Camald. 1:60\u000210\u00001 3:46\u000210\u00002 0.63 1.6 Wang and Gullichsen (1999) Scandinavian Pine 2:17\u000210\u00001 3:66\u000210\u00002 0.66 1.8 Wang and Gullichsen (1999) Scandinavian Pine 4:45\u000210\u00001 6:56\u000210\u00002 0.84 2.6 Table 2.3: Literature values for the compressibility parameters m, eg, and n as defined in Equation 2.8. These research groups indicate that m is a function of the degree of deliginification, i.e. the kappa number k , and advance the equation m(k) = (ao\u0000a1 lnk)\u0000n. as given by Equation 2.8. Here we find that the exponent n varies between 1.3 and 2.6 and that m varies according to the degree of deliginification of the chips (Harkonen, 1987; Lindqvist, 1994; Lammi, 1996; Wang and Gullichsen, 1999; Lee, 2002). There is some disagreement over the form of the gel point eg. Most authors in this literature indicate that this is simply the average sediment concen- tration of the wood chip pile; Lee (2002) indicates that this parameter varies also as a function of degree of deligninfication but with a function form slightly differ- ent than the body of work in the literature. A summary of the empirical models used to determine these parameters is given in Table 2.3. 17 2.3 Axial Dispersion through Porous Media Dispersion is a term commonly found in the literature but its definition, especially for flow in porous media, is less clear. Before we discuss the pertinent literature relevant to dispersion in wood-chip beds, it is instructive to define this term for the simplest case of the behavior of a diffusing species in uniform Poiseuille flow in a uniform circular capillary tube. In a series of classic papers, Taylor (Taylor, 1953, 1954a,b) demonstrates that a simple shear flow can surprisingly increase the effective diffusivity. Shear, in this case, acts to smear out the concentration distri- bution in the direction of the flow, enhancing the rate at which the solute spreads in that direction. To highlight this work, Taylor considered fully-developed Hagen- Pouiselle flow in a circular pipe of radius a, i.e. u(r) = uo \u0010 1\u0000 ( r a )2 \u0011 (2.14) where u and uo are the local and centreline velocities. If the concentration of the diffusing species is denoted by c and its molecular diffusivity by D, the evolution of the concentration distribution in the pipe is governed by a linear advection- diffusion equation of the form \u00C2\u00B6c \u00C2\u00B6 t +u(r)\u00C3\u0091c= \u00C3\u0091(D\u00C3\u0091c) (2.15) In this classic analysis, Taylor re-expressed the concentration and velocity fields as the sum of a cross-sectional average (indicated by an overbar) and a deviation 18 (indicated by a prime) u(r) = u\u00CC\u0084+u0(r) (2.16) c(r;x) = c\u00CC\u0084+ c0(r;x) (2.17) and assumed that both c0\u001C c\u00CC\u0084 and that the length scale of axial variation is much greater than a. By doing so, he shows that the advection-diffusion equation may be recast into the form \u00C2\u00B6 c\u00CC\u0084 \u00C2\u00B6 t + u\u00CC\u0084 \u00C2\u00B6 c\u00CC\u0084 \u00C2\u00B6x = De \u00C2\u00B6 2c\u00CC\u0084 \u00C2\u00B6x2 (2.18) where De is the effective diffusivity i.e. dispersion defined as De = D 1+ 1 192 \u0012 2au\u00CC\u0084 D \u00132! (2.19) Taylor\u00E2\u0080\u0099s analysis indicates that the effective diffusivity increases with the square of the average flowrate in the tube. Since this classic work, a vast body of literature has been developed on the subject of dispersion, especially in porous media. It seems to be a reasonably well- established fact that the extent of mixing between the solvent and solute can be characterized by an advection-diffusion-type equation in which an axial dispersion coefficient appears in place of the usual molecular diffusivity (Danckwerts, 1953; Brenner, 1962; Levenspiel and Bischoff, 1963). The physics of the enhancement of diffusion is governed by a stochastic process arising from the subtle interplay 19 of advective velocity gradients, molecular diffusion, boundary layer effects, and tortuosity (Khrapitchev and Callagha, 2003). There are various experimental methods for determining De. These methods simply involve the measurement of the concentration of a solute tracer as a func- tion of time and comparing these data to analytical or numerical solutions of an advection-diffusion model. Measurements and correlations of axial dispersion in packed beds using gaseous or liquid solvents at ambient conditions have been extensively studied and reviewed by many researchers (Chung and Wen, 1968; Edwards and Richardson, 1968; Kehinde et al, 1983; Gunn, 1987; Tsotsas and Schlunder, 1988). As an example of this, for a brine solution flowing through a packed bed of glass spheres, it is reported in the review by Delgado (2006), that the effective diffusivity is related to the porosity e of the bed according to De = aeb (2.20) where a and b are empirical constants determined by experiment. For packed beds of glass spheres, Rumer (1962) reports values of a and b to be 2:7\u0002 10\u00003m2=s and\u00001:11, respectively while Harleman and Rumer (1963) report values of 1:6\u0002 10\u00004m2=s and \u00001:18, respectively. For the particular case of flow through wood- chip beds in the pulp and paper industry, there are few, if any, studies in which the axial dispersion is reported (Michelsen and Foss, 1996). The only work that we could find directly related to this topic was that reported by Hradil et al (1993) for biomass conversion of wood chips into ethanol. The axial dispersion coefficient 20 in such beds was found to be in the range of De = 10\u00003\u000010\u00006 m2=s. At this point, we attempt to gain insight into estimates of the diffusivity by examining the related literature, i.e. for dispersion through wood pulp fibre mats. We do so as the wood chip is the precursor of the pulp fibre. In this case, we find that the axial dispersion coefficient is of the order De \u0018 10\u00006 m2=s. We consider this as a crude estimate as there is a vast literature with estimates of the axial dispersion coefficient as a function of porosity and velocity for various fibre and filler particle or washing treatments, see e.g. Poirier et al (1988), Arora et al (2006) or Potucek and Miklik (2010). 2.4 Electrical Resistance Tomography There are a number of experimental techniques currently available to visualize the motion of a fluid in a dense, opaque porous medium: X\u0000 or g ray attenuation, acoustics, or nuclear magnetic resonance. In this work we use electrical resistance tomography (ERT), a methodology which has been recently applied to a number of engineering studies (Williams and Beck, 1995; Lucas et al, 1999; Bolton et al, 2004; Vlaev and Bennington, 2004, 2005, 2006; ITS, 2006). Recently, electrical resistance tomography (ERT) has gained increasing popularity due to its appli- cations in the field of chemical and process engineering. ERT is a non-invasive technique that employs electrical measurements in order to obtain information about the contents of process vessels and pipelines such as concentration, tem- perature, velocity, and local void fraction in the measurement plane. Indeed, a number of researchers have used ERT successfully to image flow in packed beds 21 (Lucas et al, 1999; Bolton et al, 2004). Any tomography imaging system that is based on electrical fields is called electrical tomography or soft field systems due to the sensitivity of measurement in the region of interest. ERT has twomain advantages. First, measurements are conducted non-invasively without disturbing the system. Second, the region of interest is captured along the periphery of the measurement plane achieving an accurate representation of the cross sectional area of that region of interest. It is noteworthy that such imag- ing techniques have fast data processing capabilities and have low cost compared to other hard field imaging techniques such as computed axial tomography (CAT Scan). ERT images the spatial difference in conductivity through the region of in- terest using the tomographic sensors and data acquisition system. Accordingly, two-dimensional cross-sectional images of the region of interest are produced in real time by linear back projection algorithm used for image reconstruction. The ITS (Industrial Tomography Systems, Manchester, UK) P2000 system has eight sensor planes, each having 16 non-intrusive stainless steel electrodes lo- cated equidistantly around the vessel periphery (every 22:5\u000E), enabling the region of interest to be measured as it is illustrated in Figures 2.3 and 2.4. Basically, ITS P2000 system injects current between a pair of adjacent electrodes and measures the resultant voltage difference between the remaining adjacent electrode pairs ac- cording to a pre-defined measurement protocol (ITS, 2006), see Figures 2.5a and 2.5b. 22 Figure 2.3: Photograph of ITS P2000 ERT data acquisition system (ITS, 2006). Figure 2.4: Schematic diagram of electrode arrangement around the process vessel (ITS, 2006). 23 (a) (b) Figure 2.5: Adjacent current injection and resultant voltage measurement of the remaining adjacent electrodes. 2.5a The first current injection. 2.5b The second current injection (ITS, 2006). The accuracy and repeatability of the voltage measurements made using the ERT system are \u00060:5% and > 99:5%, respectively (ITS, 2006). Each plane is divided into a square 20\u000220 grid of pixels representing the interior cross-section of the vessel, for which some pixels lie on the outside of the vessel circumfer- ence; therefore, the reconstructed image is obtained from 316 of the 400 pixels, see Figure 2.6 . It is noted that ERT has an excellent temporal resolution but achieves a spatial resolution of only 5\u000010% of the vessel diameter (Williams and Beck, 1995). Image reconstruction in each plane could also be distorted due to the three-dimensional nature of the electric field, which is not accounted for in the reconstruction of the two dimensional planar images. Despite these limitations, ERT has been successfully used to study flow in a wide range of applications in- cluding liquor flow in scale-model continuous digesters by Vlaev and Bennington 24 Figure 2.6: ERT image reconstruction grid (ITS, 2006). (2004, 2005, 2006). 2.5 Research Objectives The literature shows that permeability and dispersion have been widely studied for a wide range of chemical engineering systems. This class of problem has re- ceived considerable attention in the pulp and paper literature as permeability and dispersion have been related to the efficiency of a number of different unit opera- tions, and are of great importance for flow through chip beds and consequently for digester design. However, very limited work has been published on permeability and dispersion through wood-chip columns and for most reported measurements compressibility effects have not been rigorously treated in these analyses (Harko- 25 nen, 1987; Hradil et al, 1993; Lindqvist, 1994; Lammi, 1996; He et al, 1999; Wang and Gullichsen, 1999; Lee, 2002). For the particular case of flow through wood-chip beds in the pulp and paper industry, there are few, if any, studies in which the axial dispersion is reported (Michelsen and Foss, 1996). The only work that we could find directly related to this topic was that reported by Hradil et al (1993) for biomass conversion of wood chips into ethanol. In other words, disper- sion should be characterized intensively in wood-chip beds while visualizing the flow in these beds. The objectives of this thesis are: \u000F To improve the understanding of flow resistance through compressible wood- chip beds, and check the applicability of the Ergun Equation to describe the flow resistance in these beds. \u000F To develop a procedure to estimate permeability of compressible wood-chip beds in which compressibility effects are treated rigourously. \u000F To visualize the flow and investigate the flow uniformity by using electrical resistance tomography (ERT) while using salt as a passive scalar. \u000F To characterize axial dispersion and mixing through compressible wood- chip beds. 26 Chapter 3 The Permeability of Wood-Chip Beds: The Effect of Compressibility 3.1 Introduction In this chapter, the resistance to flow of a Newtonian fluid through a compressible porous bed of wood-chips under axial load is investigated experimentally. The essential difficulty of characterizing the behavior of this material is that densifica- tion of the chip-bed occurs by both the application of an external mechanical force as well as by a hydraulic force created by the motion of the fluid. This sets itself apart from the traditional flow through incompressible porous medium studies as the additional complication of wood-chip bed deformation is introduced into the problem description. Compressible in this case indicates that the wood-chip bed densifies by the reduction of interparticle volume as opposed to particle defor- 27 mation. The reduction of interparticle volume may be associated with a change in particle orientation. We focus our efforts on the industrially-relevant case of understanding the flow in wood chips delignified to various kappa levels and esti- mate both the permeability and compressibility of these beds. This chapter is organized into three subsections. In x3.2, we present the methodology to estimate the permeability relationship. Here we present an al- ternative solution method to the governing equations, which have been posed pre- viously; this allows for a simpler means to estimate r1 and r2 through linear regres- sion. In x3.3, we demonstrate the utility of the approach through measurements of the pressure drop per unit length required to maintain the flow through a wood- chip bed under mechanical load. Finally, a comparison of the novel methodology and the experimental measurements are made in x3.4. 3.2 A Method to Determine r1 and r2 We begin this analysis by considering the one-dimensional flow of fluid through a compressible porous bed of length L and loaded mechanically by a compaction pressure pc, see Figure 3.1. The fluid is driven by a fluid pressure equal to a hy- draulic pressure ph, applied at the top of the bed; the fluid pressure at the bottom of the bed is considered to be zero. In a similar manner, we consider the solid- stress on the wood chips to be equal to pc at the top of the bed and pc+ ph at the bottom of the bed. For ease in the subsequent presentation, we define a charac- teristic porosity in the bed ec obtained by solving pc = f3(ec) i.e. e(L) where the solid-stress on the wood chips at the top of the bed is equal to pc. 28 Figure 3.1: Schematic of the geometry considered. Some of the earliest and most significant studies on compressible porous me- dia flows were conducted by Beavers and his coworkers (Beavers et al, 1981a,b), Burger and Concha (1998), Verhoff (1983), and Wakeman (1978). These are works with steady, one dimensional flow of non-compressible fluid through highly compressible material. Their analyses are not restricted to small deformations of flow within the Darcy regime (i.e. low velocity flow). Here it was observed that the gradient of solid-stress within the porous media is of equal and opposite mag- nitude to the fluid pressure gradient established in the flow dp dx = \u0000dps dx (3.1) 29 Equation 3.1 can be obtained by the mathematical addition of the Eulerian volume- averaged two-phase flow equations after neglecting the acceleration terms under the assumption of inviscid flow (Burger and Concha, 1998). With this, in com- bination with the Ergun equation and a compressibility relationship, formed a system of equations which could be integrated to compute the volumetric flowrate as a function of the solid-stress across finite porous materials. The boundary con- ditions, as introduced above, are p(0) = 0 ps(0) = pc+ ph (3.2) p(L) = ph ps(L) = pc (3.3) If we scale using p\u00CC\u0084= p ph p\u00CC\u0084s = ps\u0000 pc ph ; x\u00CC\u0084= x L e\u00CC\u0084 = e ec f\u00CC\u0084i = fi(e) fi(ec) (3.4) the governing equations reduce to d p\u00CC\u0084 dx\u00CC\u0084 = \u0000d p\u00CC\u0084s dx\u00CC\u0084 (3.5) d p\u00CC\u0084 dx\u00CC\u0084 = r\u00CC\u00841 f\u00CC\u00841(e\u00CC\u0084)+ r\u00CC\u00842 f\u00CC\u00842(e\u00CC\u0084) (3.6) p\u00CC\u0084s = 1 d \u0000 f\u00CC\u00843(e\u00CC\u0084)\u00001 \u0001 (3.7) 30 where r\u00CC\u00841 = r1 f1(ec)vL ph r\u00CC\u00842 = r2 f2(ec)v2L ph ; d = ph pc (3.8) Upon elimination of p\u00CC\u0084s and p\u00CC\u0084, the governing equation becomes \u0000 1 d f\u00CC\u0084 03(e\u00CC\u0084) de\u00CC\u0084 dx\u00CC\u0084 = r\u00CC\u00841 f\u00CC\u00841(e\u00CC\u0084)+ r\u00CC\u00842 f\u00CC\u00842(e\u00CC\u0084) (3.9) where the symbol 0 refers to differentiation with respect to e\u00CC\u0084 . The boundary con- ditions are assigned as the solid-stress is known at both the bottom and top of the column. In our notation these conditions read f\u00CC\u00843(e\u00CC\u0084(0)) = 1+d (3.10) f\u00CC\u00843(e\u00CC\u0084(1)) = 1 (3.11) As we have two conditions to satisfy a first-order ordinary differential equation, the extra condition will aid in determining the unknown resistances r\u00CC\u00841 and r\u00CC\u00842. Equation 3.9 can be integrated numerically using standard techniques. By doing so, the unknown resistances can be estimated using one of a number of different optimization schemes in conjunction with experimental data. However, for the special case where the hydraulic pressure is much less than the mechanical load, i.e. when 0< d \u001C 1, it is possible to solve this relationship using asymptotic methods to develop an analytical expression which can then be used to determine the unknown resistance to motion using a simpler multiple regression technique. 31 This is shown below. To begin, if we seek a solution of the form e\u00CC\u0084 = e\u00CC\u0084(0)+d e\u00CC\u0084(1)+ ::: (3.12) r\u00CC\u0084i = r\u00CC\u0084 (0) i +d r\u00CC\u0084 (1) i + ::: (3.13) Equation 3.9 is reduced to two linear differential equations i.e. \u0000 f\u00CC\u0084 03(e\u00CC\u0084(0)) de\u00CC\u0084(0) dx\u00CC\u0084 = 0 (3.14) \u0000 f\u00CC\u0084 03(e\u00CC\u0084(0)) de\u00CC\u0084(1) dx\u00CC\u0084 = r\u00CC\u0084(0)1 f\u00CC\u00841(e\u00CC\u0084 (0))+ r\u00CC\u0084(0)2 f\u00CC\u00842(e\u00CC\u0084 (0)) (3.15) with the boundary conditions e\u00CC\u0084(0)(0) = 1 e\u00CC\u0084(1)(0) = 1 f\u00CC\u0084 03(e\u00CC\u0084(0)(0)) (3.16) e\u00CC\u0084(0)(1) = 1 e\u00CC\u0084(1)(1) = 0 (3.17) With this, the solution of this set of equations is given by e\u00CC\u0084 = 1+d 1\u0000 x\u00CC\u0084 f\u00CC\u0084 03(1) +O(d 2) (3.18) subject to the constraint that r\u00CC\u0084(0)1 + r\u00CC\u0084 (0) 2 = 1 (3.19) 32 Equation 3.18 indicates that the first-order approximation of the porosity distribu- tion is linear, with the porosity decreasing with increasing distance away from the top of the wood-chip bed. This analysis can be easily extended to obtain higher- order approximations. This is shown in Appendix H where the O(d 2) term is reported. A comparison of the approximate analytical solution and the more accu- rate numerical solution are shown in Figure 3.2. Clearly this asymptotic method- ology yields a fairly reasonable approximation of e\u00CC\u0084(x\u00CC\u0084) over the region d < 0:3 Before continuing we must comment on the utility of this asymptotic solution. There is a possibility of degenerative behavior as f\u00CC\u0084 03(e\u00CC\u0084) approaches zero as the function e\u00CC\u0084 ! e\u00CC\u0084g. Under these conditions, the proposed series diverges. As a result, we must bound the useful range of solution to cases in which k f\u00CC\u0084 03(1) k\u0015 1. If we examine the form of this function, as given in Appendix I, we deduce that the useful range of this solution is given when e\u00CC\u0084g \u0015 n\u00001. Given a reasonable estimate of the variation of porosity along the length of the bed, we are now in a position to estimate the variation of the fluid pressure p\u00CC\u0084. Equation 3.6 may now be approximated using a similar series. If we seek a solution of the form p\u00CC\u0084= p\u00CC\u0084(0)+d p\u00CC\u0084(1)+O(d 2) (3.20) Equation 3.6 reduces to \u00C2\u00B6 p\u00CC\u0084(0) \u00C2\u00B6 x\u00CC\u0084 = 1 (3.21) \u00C2\u00B6 p\u00CC\u0084(1) \u00C2\u00B6 x\u00CC\u0084 = (1\u0000 x\u00CC\u0084) f\u00CC\u0084 03(1) \u0010 r\u00CC\u0084(0)1 f\u00CC\u0084 0 1(1)+ r\u00CC\u0084 (0) 2 f\u00CC\u0084 0 2(1) \u0011 + r\u00CC\u0084(1)1 + r\u00CC\u0084 (1) 2 (3.22) 33 0.7 0.8 0.9 1 1.1 0 0.2 0.4 0.6 0.8 1 \u00CE\u00B5/\u00CE\u00B5 c H ei gh t x /L \u00CE\u00B4 = 0.3 \u00CE\u00B4 = 0.2 \u00CE\u00B4 = 0.1 \u00CE\u00B4 = 0.05 Figure 3.2: A comparison of the numerical (dashed line) and the asymptotic solution (circles) at various d . The asymptotic solution is approxi- mated using the first three terms in the series. This comparison is made with m= 100 kPa, n= 1:2, eg = 0:54, r1 = 3\u0002104 Pa:s=m2 and r2 = 1:9\u0002106 Pa:s2=m3. When integrated, the fluid pressure is given by p\u00CC\u0084= x\u00CC\u0084+d (x\u00CC\u0084\u0000 x\u00CC\u00842) f\u00CC\u0084 03(1) \u0010 r\u00CC\u0084(0)1 f\u00CC\u0084 0 1(1)+ r\u00CC\u0084 (0) 2 f\u00CC\u0084 0 2(1) \u0011 +O(d 2) (3.23) with the constraint that r\u00CC\u0084(1)1 + r\u00CC\u0084 (1) 2 =\u0000 1 2 f\u00CC\u0084 03(1) \u0010 r\u00CC\u0084(0)1 f\u00CC\u0084 0 1(1)+ r\u00CC\u0084 (0) 2 f\u00CC\u0084 0 2(1) \u0011 (3.24) At this point we will now turn our attention to the overall problem considered in 34 this section, i.e. an estimate methodology for r\u00CC\u00841 and r\u00CC\u00842. Here, we will demonstrate how the two constraints, i.e. Equations 3.19 and 3.24, can be used in conjunction with linear regression of experimental data to estimate r1 and r2. To do so, we add Equations 3.19 and 3.24, after first multiplying Equation 3.24 by d , to yield \u0010 r\u00CC\u0084(0)1 +d r\u00CC\u0084 (1) 1 \u0011 + \u0010 r\u00CC\u0084(0)2 +d r\u00CC\u0084 (1) 2 \u0011 = 1\u0000 d 2 f\u00CC\u0084 03(1) \u0010 r\u00CC\u0084(0)1 f\u00CC\u0084 0 1(1)+ r\u00CC\u0084 (0) 2 f\u00CC\u0084 0 2(1) \u0011 (3.25) Upon substitution of Equation 3.13, the equation above simplifies to \u0012 1+d f\u00CC\u0084 01(1) 2 f\u00CC\u0084 03(1) \u0013 r\u00CC\u00841+ \u0012 1+d f\u00CC\u0084 02(1) 2 f\u00CC\u0084 03(1) \u0013 r\u00CC\u00842 = 1+O(d 2) (3.26) which can also be expressed as y= r1x1+ r2x2 (3.27) where x1 = \u0012 1+d 1 2 (3\u0000 ec) (1\u0000 ec) (eg\u0000 ec) ecn \u0013 (1\u0000 ec)2 e3c v (3.28) x2 = \u0012 1+d 1 2 (3\u00002ec) (1\u0000 ec) (eg\u0000 ec) ecn \u0013 (1\u0000 ec) e3c v2 (3.29) y = ph L (3.30) Equation 3.27 is in a form in which the unknowns r1 and r2 can be determined by multiple linear regression to a data set containing (vi; phi;Li). 35 3.3 Experimental Materials and Methods Air dried (to approximately 90% solid content) Hemlock wood chips were used in this study. These chips were made by a disk chipper which produces a chip shaped as a parallelepiped, see Figures 3.3a and 3.3b . The chips were classified using a Wennberg ChipClassTM classifier (Weyerhaeuser), shown in Figures 3.4a and 3.4b . The accept chips were further classified according to thickness using 2 and 4 mm slotted screens. For each chip fraction, the average length, width and thickness were measured by hand by randomly selecting a sample consisting of 250 individual chips, see Appendix B. Only accept chips fraction with 2\u00004 mm thickness was used in this study to avoid any non-uniformity of the oversized and undersized chip fractions, see Table 3.1. In addition both the equivalent particle mean diameter dp, defined as dp = 6 Vp As (3.31) as well as the sphericity y , are also given in the table to help characterize the wood chip further. 36 (a) (b) Figure 3.3: Geometry of disk chipper chips. 37 (a) oversize overthick accepts pins fines0 10 20 30 40 50 60 70 80 90 Chip size distribution Ai r d ry w ei gh t p er ce nt ag e (% ) (b) Figure 3.4: 3.4a Chip screening apparatus. 3.4b Chip size distribution. 38 Screen Thickness Width Length dp y Mass Fraction (mm) (mm) (mm) (mm) (%) 4 mm slot 4:4(\u00060:7) 19:1(\u00068:5) 22:9(\u00063:6) 9.2 0.60 37.6 2 mm slot 2:6(\u00060:6) 14:7(\u00066:5) 21:2(\u00062:0) 6.3 0.53 62.4 Total Mixture 7.1 0.56 100.0 Table 3.1: Characteristics and dimensions with standard deviations of the wood chips before cooking. The total mixture was composed of 37.6%(wt/wt) of the 4 mm slot chips with the balance of mixture com- prised of the 2 mm slot chips. Both dp as well as y for the total mixture are averages based upon the mass fraction of each species, see Appen- dices B and C. Cooks of accept chips were done to k numbers of 25, 53 and 80, which rep- resents the variation of kappa number during industrial cooking, using a 28 L Werverk laboratory digester (FPInnovation, Vancouver). Cooks were made using conventional kraft pulping conditions, see Table 3.2. After the cook, we washed the cooked wood chips gently, while they are in the cooking basket of the digester, by running water for about 15 mins to remove any residual alkali, and then we transferred the washed wood chips into storing buckets. The washing and trans- ferring processes were done very carefully in order to protect the dimensions of the cooked wood chips. Also before doing any experiments in our lab, we soaked and washed the cooked wood chips, while they are in our bed, with water until the effluent was clear. Our observations indicate that with lower kappa number chips, the longer the washing time that was required. Figures 3.5a-3.5d show a comparison of the wood chips before and after cooking to different k numbers where the cooked wood chips preserved their dimensions. 39 Wood chips species Hemlock (Accepts 2\u00004 mm) Wood chips charge (oven dry) 3:0 O:D:kg Liquor to Wood ratio 4:5 : 1 Effective Alkali charge (EA) 17% as Na2O Sulfidity 26:95% Heating rate 1:11\u000EC=min Maximum cooking temperature 170\u000EC Time to temperature 140 minutes Time at temperature To desired k number Table 3.2: Cooking conditions using the laboratory digester. 40 (a) k = 180 (b) k = 80 (c) k = 53 (d) k = 25 Figure 3.5: Comparison of the accept chips before and after cooking to dif- ferent k numbers. Scale in the picture is in cm. 41 A schematic diagram of the test apparatus is shown in Figures 3.6 and 3.7. The column has an inside diameter of 15 cm, a height of 45 cm and was made from 1.25 cm thick Plexiglas. The column has a perforated plate (having an open area of 40% using 3 mm diameter holes) equipped by a piston in order to simulate a compaction pressure (0-14 kPa) and create uniform flow through the wood chips. The compressive pressure was controlled by a pneumatic air cylinder (Ameri- can Cylinder R\rCo., Inc., Model 1500DVS-1\u00E2\u0080\u009D, Peotone, Illinois). The pressure drop across the column was divided into four equal sections (11.25 cm) and mea- sured by four differential pressure transducers (Omega R\rEngineering Inc., model PX2300-1DI, Laval, Quebec). For measuring the hydraulic pressure at the top of the Column, a gage pressure transducer (Omega R\rEngineering Inc., model PX01C1-015GI, Laval, Quebec) was used. For measuring the flow, a magnetic flow-meter (Rosemount R\rMeasurement, model 8712, Irvine, CA) was used. In addition, a centrifugal pump (Cole-Parmer R\rInc., Model K-07085-00, Concord, Ontario) was used to circulate the flow in the chip column. The column diameter and height were chosen with consideration of industrial and previous researchers\u00E2\u0080\u0099 designs. In addition, this column\u00E2\u0080\u0099s diameter helped in creating uniform force by the piston as well as reducing the quantity of cooked wood chips used since we didn\u00E2\u0080\u0099t have a limitless supply. Moreover, it is assumed that there is no channeling or other wall effects throughout the tests. According to Winterberg and Tsotsas (2000), wall effects can generally be neglected if the diameter of the packed bed is greater than ten times the diameter of the packing particles. Cooked wood chips at a prescribed k were added carefully, to avoid chip 42 Figure 3.6: Schematic of the apparatus. fracture as will as to avoid compaction in the chip column, into the test column to the initial height of 0.45 m. The packing was performed randomly. To start an experiment, water (T = 23\u000EC, m = 0:925\u000210\u00003 Pa:s, r = 997:3 kg=m3) from the 25 L tank was pumped upward through the column to remove any trapped air and to fill the column with fluid. Then, flow was shifted through the flow- meter to the top of the column and was allowed to stabilize and reach steady state before taking pressure drop data. During an experiment, the chip packed bed was 43 Figure 3.7: Image of the apparatus. compressed by the piston to give three compaction loads ranging from 0 to 14 kPa. The level of the compressive pressure was chosen to achieve porosities in the range of 0:2\u00000:5. This is the typical range found in industry and we determined this through literature values as well as in a pre-trial. The applied load was kept constant during an experiment while the flow rate was changed in increments up to 10 L=min. These ranges covered those encountered in operating digesters. Data was taken with increasing flow rate to avoid any hysteresis effects. 44 3.4 Results and Discussion In this section the results will be presented in four subsections. In the first part, a brief description of the uncertainty of our measurement will be addressed by mea- suring the resistance to flow in our apparatus with a packed-bed of monodisperse spheres. By doing so, we are able to compare our apparatus to other estimates given in the literature. We then report the values of the empirical constants for the compressibility function ps. Following this we turn our attention to the main results, that being on the estimates of r1 and r2 for wood chips at different k using the methodology reported in x3.2. Finally, we attempt to validate these estimates by attempting to predict the pressure drop for a given flow rate, with knowledge of r1 and r2 under the conditions of no applied mechanical load. To begin, in the first set of results, polypropylene spheres (dp = 1\u0002 10\u00002 m, r = 905 kg=m3) were used to verify our procedure. The results are shown in Fig- ure 3.8 alongside the estimates given by the Ergun Equation (Equations 2.5). For the range of data considered, we find that A = 210 and B = 2:33 with the corre- lation coefficient estimated to be r2 = 0:99 using Equation 2.5. The uncertainty in the experimental estimate stems from the range of pressures used in this case; we are at lower limit of the working range of the pressure transducers. In addition there is some variation in the estimates of A and B in the literature, see Table 2.1, due to, for example differences in the roughness of the particles, and as a result we consider our results shown comparable to those found in the literature. At this point we turn our attention to characterizing the behavior of wood 45 0 5 10 15 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 v (mm/s) P h /L (k Pa /m ) Measured Regression Ergun Equation Figure 3.8: The resistance to flow of water through a incompressible bed of polystyrene spheres. The data is compared to the literature values (solid line) to assess the uncertainty in our measurement system. chip beds under uniaxial compression. Here we do so by measuring the change in bed height as a function of compression pressure pc for beds that are water saturated. The experimentally measured data are shown in Figure 3.9a as circles with the dashed line representing the form of the function f3(e) as defined in Equation 2.8. The estimated values of eg is 0.54; n was found to be 1.2 over the range of conditions tested. The constant m was found to vary as a function of k where m was found to be 115;78, and 67kPa for k80;k53, and k25, respectively. This is shown in Figure 3.9b. The dashed line is the functional form for m(k) as advanced by Harkonen (1987). With the values of m, n, and eg determined we are 46 now able to estimate the values of r1 and r2 through the methodology proposed in x3.2. The results of this analysis are presented in Figures 3.10a - 3.10c. The dashed line represents the regression of Equation 3.27 performed using the entire data set. For the range of data considered, we find that r1 = 3\u0002104 Pa:s=m2 and r2 = 1:9\u0002106 Pa:s2=m3 with the correlation coefficient estimated to be r2 = 0:99. These results are similar to those reported by Harkonen (1987). We also report A and B values since they are independent of fluid properties and particle diameter using Equations 2.12 and 2.13. For the wood-chips tests, we find that A = 475 and B = 7:49 with the same correlation coefficient estimated to be r2 = 0:99. It should be noted that we are able to do the regression over the entire data set as the shape of the chips does not change significantly over the range of k . 47 0.2 0.3 0.4 0.5 0.6 0.7 0 5 10 15 Porosity \u00CE\u00B5 P s (kP a) \u00CE\u00BA = 25 \u00CE\u00BA = 53 \u00CE\u00BA = 80 (a) 20 30 40 50 60 70 80 50 60 70 80 90 100 110 120 m (k Pa ) \u00CE\u00BA (b) Figure 3.9: The uniaxial compressibility of wood chips cooked to different k numbers. 48 0 2 4 6 8 10 0 5 10 15 20 25 30 v (mm/s) P h /L (k Pa /m ) P c = 5.6 P c = 9.8 P c = 14.0 (a) k = 80 0 2 4 6 8 10 0 5 10 15 20 25 30 v (mm/s) P h /L (k Pa /m ) P c = 5.6 P c = 9.8 P c = 14.0 (b) k = 53 0 2 4 6 8 10 0 5 10 15 20 25 30 v (mm/s) P h /L (k Pa /m ) P c = 5.6 P c = 9.8 P c = 14.0 (c) k = 25 Figure 3.10: The measured pressure drop per unit length for a prescribed superficial velocity v at three different compaction pressures. The uncertainty in the estimate for ph=L is approximately 0.3 kPa=m. 49 To test the usefulness of these estimates for r1 and r2, we consider the case of flow through the packed bed without mechanical loading, i.e. pc = 0. Here, with the values of r1 and r2 we should be able to predict the pressure drop per unit length for a prescribed superficial velocity. For this case, the asymptotic method outlined in x3.2 does not apply as 1=d ! 0. What is required here is a numerical solution of the governing equations. The system of equations, in dimensional form, that are required to be solved is dps dx = \u0000(r1 f1(e)v+ r2 f2(e)v2) (ps(L) = 0) (3.32) e(x) = eg\u0000 ( ps(x)m ) 1 n (3.33) ph = ps(0) (3.34) Hence, with a prescribed superficial velocity v, Equation 3.32 can be integrated to find ps(x) and used in conjunction with Equation 3.34 to estimate the hydraulic pressure. This was performed in MATLAB using a Runge-Kutta scheme. The predictions for wood chips at three different k are shown in Figures 3.11a - 3.11c; clearly a reasonable prediction was obtained. 50 0 2 4 6 8 10 0 0.5 1 1.5 v (mm/s) P h /L (k Pa /m ) Measured Predicted No Compressibility Effects (a) k = 80 0 2 4 6 8 10 0 0.5 1 1.5 v (mm/s) P h /L (k Pa /m ) Measured Predicted No Compressibility Effects (b) k = 53 0 2 4 6 8 10 0 0.5 1 1.5 v (mm/s) P h /L (k Pa /m ) Measured Predicted No Compressibility Effects (c) k = 25 Figure 3.11: The predicted pressured drop (dashed line), the experimentally measured values (circles), and the correlation without compressibil- ity effects (solid line) for the case pc = 0. The uncertainty in the estimate for ph=L is approximately 0.3 kPa=m. The dashed line was calculated using the values of r1 and r2 determined above by numer- ical integration of the governing equations. 51 With the values of r1 and r2 estimated using the methodology given above, we are now able to estimate the permeability of the wood chip bed tested in this work. To do so we rearrange Equations 2.12 and 2.13 to find that qs2v = r1 m (3.35) b = r2 r r m r1 (3.36) from which we can estimate the permeability using Equation 2.2. This is shown graphically in Figure 3.12a from which we observe that the order of magnitude of this function over the porosity range tested is approximately k \u0018 10\u00008 m2. What we find significant with these findings is that our results are dependent on poros- ity which in turn is formed as a combination of degree of delignification, i.e. k number, and compaction pressure. We anticipated the possibility that the degree of delignification may affect permeability through changes in the specific surface; the chip may have a larger surface area through removal of the lignin. We argue that since this is not the case, the specific surface must be related solely to the geometry and orientation distribution of the chips in the compressible bed. We continue the discussion of the results to attempt to highlight the effect of k on the permeability. The degree of delignification affects permeability through the porosity; wood chips with different degree of delignification will compress to different porosities under the same external mechanical load. To characterize this, we report the relative permeability, i.e. the ratio of the permeabilities of a k80 bed to that of a k25, compressed with equivalent mechanical loads, see Figure 52 3.12b. Under the industrially relevant loads, the relative permeability increases with increasing loads. This results as the k25 chip-bed is more compressible, and compacts to a lower porosity, in comparison to the k80 chip bed, at equivalent mechanical loading. Finally, we compare our correlation to the literature for porosity of 0:35. The results are compared to Harkonen (1987) correlation since it is widely used in model digesters, as well as Lee (2002) correlation since Lee (2002) used sim- ilar chip size distribution to ours, i.e. 100 % accepts. We should note that Harkonen (1987) performed his experiments at a temperature T = 170\u000EC (m = 0:068\u0002 10\u00003 Pa:s, r = 885 kg=m3) using cooking liquor while Lee (2002) and our tests were made at T = 23\u000EC (m = 0:925\u000210\u00003 Pa:s, r = 997:3 kg=m3) us- ing water. We corrected our r1 and r2 values for Harkonen (1987) conditions to get r1 = 2\u0002 103 Pa:s=m2 and r2 = 1:7\u0002 106 Pa:s2=m3 with the same correlation coefficient estimated to be r2 = 0:99. This is shown graphically in Figure 3.13 from which we observe that our correlation has lower flow resistances than the literature correlations for similar chip size distribution and under all conditions. This can be justified by the fact that we accounted for compressibility effects in our correlation. As a result, the literature correlations over estimate the pressure drop. 53 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.5 1 1.5 2 2.5 x 10\u00E2\u0088\u00928 Porosity \u00CE\u00B5 Pe rm ea bi lity k (\u00CE\u00B5) (m 2 ) (a) 0 5 10 15 1 1.5 2 2.5 3 3.5 p c (kPa) R el at iv e Pe rm ea bi lity (b) The relative permeability of hemlock wood chips cooked to two different k . The rel- ative permeability is defined as the ratio of the permeability of a k = 80 wood chip bed to that of a wood chip bed cooked to k = 25. Figure 3.12: 3.12a Estimate of the permeability of Hemlock wood chips us- ing the Kozeny-Carman expression given by Equation 2.2. The tor- tuosity and the specific surface of the chips were estimated through use of Equations 2.12 and 2.13. 3.12b The ratio of the permeabilities of a k 80 to k 25 wood chips compressed at different loads. 54 0 2 4 6 8 10 0 2 4 6 8 10 12 v (mm/s) P h /L (k Pa /m ) Lee (2002) Harkonen (1987) Our Correlation at T=23\u00C2\u00B0C Our Correlation at T=170\u00C2\u00B0C Figure 3.13: A comparison of the pressure drop correlations between our correlation and the literature correlations for porosity of 0.35. 55 3.5 Summary and Conclusions In this work we present a methodology to characterize the resistance to fluid mo- tion through a porous compressible bed of wood chips. Here we advance the argument that under the limiting conditions of when the mechanical load is large in comparison to hydraulic pressure the equations of motion can be solved asymp- totically to produce an expression approximating the variation in porosity along the length of the bed. With this, a methodology to estimate the resistance is given by multiple linear regression. In the experimental portion of the work we mea- sured the resistance to flow of a bed of hemlock wood chips, cooked to three different k , and compacted to three different pc. With our methodology, we find that r1 = 3\u0002 104 Pa:s=m2 and r2 = 1:9\u0002 106 Pa:s2=m3. The usefulness of these estimates were tested for the extreme case with no applied mechanical load. A reasonable prediction was obtained, thus verifying the utility of the methodology and the estimated resistances. After accounting for compressibility effects, we find that our correlation has lower flow resistances than the literature correlations. 56 Chapter 4 The Axial Dispersion in Wood-Chip Beds; ERT Visualization 4.1 Introduction In this chapter, we characterize the dispersion of a solute flowing through a com- pressible porous bed of wood-chips under axial mechanical load. Here we exam- ine the flow of brine solution through a compressible wood chip bed. We focus our efforts on the industrially-relevant case of understanding the flow in wood chips delignified to various extents, i.e. decreasing kappa numbers, and visualize the displacement of the brine solution using electrical resistance tomography. In x4.2, we present a methodology to estimate the axial dispersion coefficient. Here we present the governing equations and examine two special cases in which the porosity of the bed varies slowly in the axial direction. Under these limiting cases, 57 the difficulty in estimating the diffusivity from experimental data is reduced sig- nificantly. In x4.3, we present the details of using electrical resistance tomography (ERT) to measure the evolution of the concentration profile of the displacement of a brine solution, with a different solution, from the wood chip bed, as well as the numerical methodology for estimating the axial dispersion. Here we use electrical resistance tomography (ERT), a non-invasive device, to reconstruct a tomographic image of the evolution of the concentration profile both spatially and temporally. In x4.4, we present qualitatively the images of the evolution of the concentration profiles as well as quantitative estimates of the axial dispersion coefficient. 4.2 The Methodology to Estimate De In this section, we discuss the methodology required to image the evolution of the concentration profile in the wood-chip bed as well as the mathematical model used to estimate dispersion. This section is presented in three parts. In the first part, we present the model equations to describe our experimental conditions. The model presentation is relatively straightforward as the equations have been derived pre- viously. Our contribution in this section is the scaling of the equations of motion, rendering them dimensionless, so that it is possible to see two limiting cases in which the porosity is nearly constant in the compressible bed. We do so as this directs our experimental efforts to measure the axial dispersion for cases which are relatively easy to analyze mathematically. Here we show two limiting cases which are experimentally achievable. In x4.2.1 and x4.2.2, we present asymp- totic solutions for these limiting cases and demonstrate that to leading order the 58 Figure 4.1: Schematic of the geometry considered. porosity is constant. We extend the solution and also present in these sections, the first-order approximations for porosity. We begin this presentation of the governing equation by defining our geome- try. We consider the one-dimensional flow of fluid through a compressible porous bed of length L and loaded mechanically by a compaction pressure pc, see Figure 4.1. The fluid is driven by a fluid pressure equal to a hydraulic pressure ph, ap- plied at the top of the bed; the fluid pressure at the bottom of the bed is considered to be zero. In a similar manner, we consider the solid-stress on the wood chips to be equal to pc at the top of the bed and pc+ ph at the bottom of the bed. In this experiment we consider the displacement of a fluid in the bed, with an initial concentration Ci; at t > 0, a second fluid is introduced through the top at a con- centration C f . In this case, a fluid with a lighter density is displacing a fluid with a heavier density. 59 As the wood chip bed is compressible, that is, the porosity varies in the di- rection of the flow, we must consider both the evolution of the concentration dis- tribution, as well as the spatial variations in the porosity and velocity fields for a complete description of this experiment. The literature in this area is vast and the reader should consult the classic works by Beavers and his coworkers (Beavers et al, 1981a,b), Burger and Concha (1998), Verhoff (1983), and Wakeman (1978) for a full description of the derivation of the equations describing the flow field or of the work by Poirier et al (1988) for the a description of the evolution of the concentration field. Using the notation defined in Chapter 3, the equations of motion for this geometry are dp dx =\u0000dps dx (4.1) dp dx = f1(e)v+ f2(e)v2 (4.2) ps = f3(e) (4.3) \u00C2\u00B6C \u00C2\u00B6 t = \u00C2\u00B6 \u00C2\u00B6x \u0012 De \u00C2\u00B6C \u00C2\u00B6x \u0013 \u0000 v e \u00C2\u00B6C \u00C2\u00B6x (4.4) where p and ps are the fluid pressure and the solid stress of the fluid and solid phases, respectively; e is the porosity of the compressible wood-chip bed; v is the superficial velocity of the fluid; C is the concentration of a solute in the fluid; f1(e) and f2(e) are empirically determined coefficients related to the resistance of flow of the fluid; and f3(e) is an empirically determined function that relates the applied solid stress to the local porosity. It is traditionally referred to as the 60 compressibility relationship. These empirical relationships have been recently reported in Chapter 3 for wood-chip beds and are given as f1(e) = r1 (1\u0000 e)2 e3 (4.5) f2(e) = r2 (1\u0000 e) e3 (4.6) f3(e) = m(eg\u0000 e)n (4.7) where r1, r2, m, n, and eg are experimentally determined constants. The boundary conditions, as introduced above, are p(0) = 0 ps(0) = pc+ ph (4.8) p(L) = ph ps(L) = pc (4.9) For ease in the subsequent presentation, we define a characteristic porosity in the bed ec obtained by solving pc+ ph = f3(ec) i.e. e(0) where the solid-stress on the wood chips at the bottom of the bed is equal to pc+ ph. If we scale the equations using p\u00CC\u0084= p ph ; p\u00CC\u0084s = ps\u0000 pc ph ; x\u00CC\u0084= x L ; e\u00CC\u0084 = e ec f\u00CC\u0084i = fi(e) fi(ec) ; C\u00CC\u0084 = C\u0000Ci C f \u0000Ci ; t\u00CC\u0084 = tv Lec ; D\u00CC\u0084e(e\u00CC\u0084) = De(e) De(ec) = e\u00CC\u0084b 61 and form the following dimensionless groups Pe= vL ecDe(ec) ; r\u00CC\u00841 = r1 f1(ec)vL ph ; r\u00CC\u00842 = r2 f2(ec)v2L ph ; d = ph ph+ pc the governing equations reduce to d p\u00CC\u0084 dx\u00CC\u0084 = \u0000d p\u00CC\u0084s dx\u00CC\u0084 (4.10) d p\u00CC\u0084 dx\u00CC\u0084 = r\u00CC\u00841 f\u00CC\u00841(e\u00CC\u0084)+ r\u00CC\u00842 f\u00CC\u00842(e\u00CC\u0084) (4.11) p\u00CC\u0084s = 1 d \u0000 f\u00CC\u00843(e\u00CC\u0084)\u00001 \u0001 +1 (4.12) \u00C2\u00B6C\u00CC\u0084 \u00C2\u00B6 t\u00CC\u0084 = 1 Pe \u00C2\u00B6 \u00C2\u00B6 x\u00CC\u0084 \u0012 D\u00CC\u0084e(e\u00CC\u0084) \u00C2\u00B6C\u00CC\u0084 \u00C2\u00B6 x\u00CC\u0084 \u0013 \u0000 1 e\u00CC\u0084 \u00C2\u00B6C\u00CC\u0084 \u00C2\u00B6 x\u00CC\u0084 (4.13) where Pe is the Peclet number. Upon elimination of ps and p, the governing equation for porosity becomes \u0000 1 d f\u00CC\u0084 03(e\u00CC\u0084) de\u00CC\u0084 dx\u00CC\u0084 = r\u00CC\u00841 f\u00CC\u00841(e\u00CC\u0084)+ r\u00CC\u00842 f\u00CC\u00842(e\u00CC\u0084) (4.14) Here the symbol 0 refers to differentiation with respect to e\u00CC\u0084 . The boundary con- ditions for Equation 4.14 are assigned as the solid-stress, and hence porosity, is known at both the bottom and top of the column. In our notation these conditions read p\u00CC\u0084s(0) = 1! e\u00CC\u0084(0) = 1 (4.15) p\u00CC\u0084s(1) = 0! e\u00CC\u0084(1) = e\u00CC\u0084g\u0000 (e\u00CC\u0084g\u00001)(1\u0000d )1=n (4.16) 62 For the advection-diffusion equation the initial and boundary conditions are those prescribed by Danckwerts (1953) t\u00CC\u0084 = 0 : C\u00CC\u0084(x\u00CC\u0084;0) = 0 (4.17) x\u00CC\u0084= 0 : \u00C2\u00B6C\u00CC\u0084 \u00C2\u00B6 x\u00CC\u0084 = 0 (4.18) x\u00CC\u0084= 1 : e\u00CC\u0084D\u00CC\u0084(e\u00CC\u0084) Pe \u00C2\u00B6C\u00CC\u0084 \u00C2\u00B6 x\u00CC\u0084 = C\u00CC\u0084\u00001 (4.19) The objective in presenting these relationships is to develop a methodology for determining De, which is contained in the Peclet number Pe, which is robust and accurate. The complication in experimentally determining this variable is that this material is compressible and e varies as a function of x. Clearly, the certainty of our estimate for De would increase if we could estimate the molecular diffusivity, though measurement of the evolution of the concentration of a solute, under con- ditions where the porosity is nearly constant. In the following two subsections, we will demonstrate two limiting cases, which are experimentally achievable, in which the porosity varies slowly in the direction of flow. We will then conduct our experiments under these conditions. 4.2.1 The First Limiting Case: pc\u001D ph Before we proceed it is interesting to note that there is a one-way coupling be- tween Equations 4.14 and 4.13. Here the porosity is independent of the concen- tration field; the concentration field however is affected by the porosity. Because of this one-way coupling we can operate on Equation 4.14 independently of Equa- 63 tion 4.13. In this subsection, we will demonstrate that under this limiting case, the porosity varies slowly in x. Under the conditions that the mechanical load is much greater than the hydraulic pressure, we find that 0< d \u001C 1. If we seek a solution of the form e\u00CC\u0084 = e\u00CC\u0084(0)+d e\u00CC\u0084(1)+ ::: (4.20) Equation 4.14 is reduced to two linear differential equations i.e. \u0000 f\u00CC\u0084 03(e\u00CC\u0084(0)) de\u00CC\u0084(0) dx\u00CC\u0084 = 0 (4.21) \u0000 f\u00CC\u0084 03(e\u00CC\u0084(0)) de\u00CC\u0084(1) dx\u00CC\u0084 = r\u00CC\u00841 f\u00CC\u00841(e\u00CC\u0084(0))+ r\u00CC\u00842 f\u00CC\u00842(e\u00CC\u0084(0)) (4.22) with the boundary conditions e\u00CC\u0084(0)(0) = 1 e\u00CC\u0084(1)(0) = 0 (4.23) e\u00CC\u0084(0)(1) = 1 e\u00CC\u0084(1)(1) = (e\u00CC\u0084g\u00001) n (4.24) With this, the solution of this set of equations is given by e\u00CC\u0084 = 1+d (e\u00CC\u0084g\u00001)x\u00CC\u0084 n +O(d 2) (4.25) subject to the constraint that r\u00CC\u00841+ r\u00CC\u00842 = 1+O(d 2) (4.26) 64 We interpret Equation 4.25 to state that to the lowest-order, the porosity is constant and approximately equal to the characteristic porosity, i.e. e \u0019 ec. The porosity gradient which must exist is small; its magnitude is proportional to the ratio of the hydraulic pressure to the compressive pressure. As we are able to con- duct experiments with ph=pc! 0, we consider e to be a constant for the purpose of estimating Pe. 4.2.2 The second limiting case: pc = 0 and m\u001D ph Physically we interpret this case as one in which there is no applied mechanical load and that the applied hydraulic pressure is much less than the stiffness of the porous bed. In simpler terms, we consider the case of very slow flow through an unloaded bed. This case is quite subtle and to help highlight why the gradients are small we examine the boundary conditions given by Equation 4.16. As d = 1 under these conditions, this boundary condition reduces to e\u00CC\u0084 = e\u00CC\u0084g which can be expressed as e\u00CC\u0084 = eg ec = 1 1\u0000 1eg \u0000 ph m \u00011=n \u0019 1+ 1eg \u0010 ph m \u00111=n + ::: (4.27) If m\u001D ph, we find that porosity at the upper and lower bounds of the bed, to the lowest order of approximation, are the same and equal to 1, see 4.15 to verify this argument. This indicates a regime in which the porosity gradients are small and achievable experimentally for the material we are using. To formalize this argument and to rigourously deduce the form of the porosity 65 gradient, we define g = 1 eg \u0010 ph m \u00111=n (4.28) such that 0< g \u001C 1 and seek a solution of the form e\u00CC\u0084 = 1+ ge\u00CC\u0084(1)+ ::: (4.29) With this Equation 4.14 is reduced to de\u00CC\u0084(1) dx\u00CC\u0084 = 1 (4.30) where f\u00CC\u0084 03(1) =\u0000 n e\u00CC\u0084g\u00001 =\u0000 n g (4.31) subject to r\u00CC\u00841+ r\u00CC\u00842 = n+O(g) (4.32) and e\u00CC\u0084(1)(0) = 0 e\u00CC\u0084(1)(1) = 1 (4.33) 66 With this, the solution of this set of equations is given by e\u00CC\u0084 = 1+ g x\u00CC\u0084+O(g2) (4.34) For both limiting cases, we find that to the lowest order of approximation, the porosity gradients are essentially constant and as a result, the linear advective- diffusion equation can be simplified, considerably to read \u00C2\u00B6C\u00CC\u0084 \u00C2\u00B6 t\u00CC\u0084 = 1 Pe \u00C2\u00B6 2C\u00CC\u0084 \u00C2\u00B6 x\u00CC\u00842 \u0000 \u00C2\u00B6C\u00CC\u0084 \u00C2\u00B6 x\u00CC\u0084 (4.35) Under these conditions, we will estimate Pe by numerically integrating Equation 4.35 and comparing this to experimental data. The methodology for collecting the experimental data is given in the following section. 4.3 Experimental Materials and Methods Air dried (to approximately 90% solid content) Hemlock wood chips were used in this study. These chips were made by a disk chipper which produces a chip shaped as a parallelepiped, see Figures 4.2a and 4.2b . The chips were classified using a Wennberg ChipClassTM classifier (Weyerhaeuser), shown in Figures 4.3a and 4.3b . The accept chips were further classified according to thickness using 2 and 4 mm slotted screens. For each chip fraction, the average length, width and thickness were measured by hand by randomly selecting a sample consisting of 250 individual chips, see Appendix B. Only accept chips fraction with 2\u00004 mm thickness was used in this study to avoid any non-uniformity of the oversized and 67 undersized chip fractions, see Table 4.1. In addition both the equivalent particle mean diameter dp, defined as dp = 6 Vp As (4.36) as well as the sphericity y , are also given in the table to help characterize the wood chip further. 68 (a) (b) Figure 4.2: Geometry of disk chipper chips. 69 (a) oversize overthick accepts pins fines0 10 20 30 40 50 60 70 80 90 Chip size distribution Ai r d ry w ei gh t p er ce nt ag e (% ) (b) Figure 4.3: 4.3a Chip screening apparatus. 4.3b Chip size distribution. 70 Screen Thickness Width Length dp y Mass Fraction (mm) (mm) (mm) (mm) (%) 4 mm slot 4:4(\u00060:7) 19:1(\u00068:5) 22:9(\u00063:6) 9.2 0.60 37.6 2 mm slot 2:6(\u00060:6) 14:7(\u00066:5) 21:2(\u00062:0) 6.3 0.53 62.4 Total Mixture 7.1 0.56 100.0 Table 4.1: Characteristics and dimensions with standard deviations of the wood chips before cooking. The total mixture was composed of 37.6%(wt/wt) of the 4 mm slot chips with the balance of mixture com- prised of the 2 mm slot chips. Both dp as well as y for the total mixture are averages based upon the mass fraction of each species, see Appen- dices B and C. Cooks of accept chips were done to k numbers of 25, 53 and 80, which rep- resents the variation of kappa number during industrial cooking, using a 28 L Werverk laboratory digester (FPInnovation, Vancouver). Cooks were made using conventional kraft pulping conditions, see Table 4.2. After the cook, we washed the cooked wood chips gently, while they are in the cooking basket of the digester, by running water for about 15 mins to remove any residual alkali, and then we transferred the washed wood chips into storing buckets. The washing and trans- ferring processes were done very carefully in order to protect the dimensions of the cooked wood chips. Also before doing any experiments in our lab, we soaked and washed the cooked wood chips, while they are in our bed, with water until the effluent was clear. Our observations indicate that with lower kappa number chips, the longer the washing time that was required. Figures 4.4a-4.4d show a comparison of the wood chips before and after cooking to different k numbers where the cooked wood chips preserved their dimensions. 71 Wood chips species Hemlock (Accepts 2\u00004 mm) Wood chips charge (oven dry) 3:0 O:D:kg Liquor to Wood ratio 4:5 : 1 Effective Alkali charge (EA) 17% as Na2O Sulfidity 26:95% Heating rate 1:11\u000EC=min Maximum cooking temperature 170\u000EC Time to temperature 140 minutes Time at temperature To desired k number Table 4.2: Cooking conditions using the laboratory digester. 72 (a) k = 180 (b) k = 80 (c) k = 53 (d) k = 25 Figure 4.4: Comparison of the accept chips before and after cooking to dif- ferent k numbers. Scale in the picture is in cm. 73 There are a number of experimental techniques currently available to visualize the motion of a fluid in a dense, opaque porous medium: X\u0000 or g ray attenuation, acoustics, or nuclear magnetic resonance. In this work we use electrical resistance tomography (ERT), a methodology which has been recently applied to a number of engineering studies (Williams and Beck, 1995; Lucas et al, 1999; Bolton et al, 2004; Vlaev and Bennington, 2004, 2005, 2006; ITS, 2006). A schematic diagram of the test apparatus is shown in Figures 4.5 and 4.6. The column has an inside diameter of 15 cm, a height of 45 cm and was made from 1.25 cm thick Plexiglas. The column has a perforated plate (having an open area of 40% using 3 mm diameter holes) equipped by a piston in order to simulate a compaction pressure (0-14 kPa) and create uniform flow through the wood chips. The compressive pressure was controlled by a pneumatic air cylinder (Ameri- can Cylinder R\rCo., Inc., Model 1500DVS-1\u00E2\u0080\u009D, Peotone, Illinois). The pressure drop across the column was divided into four equal sections (11.25 cm) and mea- sured by four differential pressure transducers (Omega R\rEngineering Inc., model PX2300-1DI, Laval, Quebec). For measuring the hydraulic pressure at the top of the Column, a gage pressure transducer (Omega R\rEngineering Inc., model PX01C1-015GI, Laval, Quebec) was used. For measuring the flow, a magnetic flow-meter (Rosemount R\rMeasurement, model 8712, Irvine, CA) was used. In addition, a centrifugal pump (Cole-Parmer R\rInc., Model K-07085-00, Concord, Ontario) was used to circulate the flow in the chip column. The column diameter and height were chosen with consideration of industrial and previous researchers\u00E2\u0080\u0099 designs. In addition, this column\u00E2\u0080\u0099s diameter helped in 74 Figure 4.5: Schematic of the apparatus. creating uniform force by the piston as well as reducing the quantity of cooked wood chips used since we didn\u00E2\u0080\u0099t have a limitless supply. Moreover, it is as- sumed that there is no channeling or other wall effects, which was confirmed using ERT, for the experimental conditions tested. According to Winterberg and Tsotsas (2000), wall effects can generally be neglected if the diameter of the packed bed is greater than ten times the diameter of the packing particles. With ERT, differences in electrical conductivity s of the fluid are mapped 75 Figure 4.6: Image of the apparatus. by conductivity sensors located on the periphery of the apparatus. With the tomo- graph used, an ITS P2000 (Industrial Tomography Systems, Manchester, UK), im- ages are scanned along 8 different parallel planes, approximately 5.625 cm thick, to give an axial extent of approximately 45 cm. Each axial plane is made up of 20\u0002 20 pixels, 58 mm2 formed from 16 conductivity probes separated by 22:5\u000E along the circumference of the apparatus representing the interior cross-section of the vessel, for which some pixels lie on the outside of the vessel circumference. Before we process the conductivity signals, we need to convert the conductivity 76 tomogram produced by ERT to a concentration tomogram since it is a 1:1 map as shown in Appendix G. We then processed the concentration signals to create two different reconstructed images. To help in this we define Ce(x;y;z; t), where (y;z) are the coordinates in each individual ERT plane, to represent the experimen- tal concentration of of the solute. In the first series of reconstructed images we transform the measurements in each plane to create a map hCe(x;y; t)i from which we can examine the curvature in the advancing front. Specifically,hCe(x;y; t)i is calculated using hCe(x;y; t)i= 1Z Z Z 0 Ce(x;y;z; t)dz (4.37) where Z is the distance over which the image is averaged in the z-coordinate. In the second reconstructed images, we average the values at each plane to create a data point, i.e. hhCe(x; t)ii= 1ZY Z Y 0 Z Z 0 Ce(x;y;z; t)dydz (4.38) to estimate the extent of axial dispersion where Y is the distance over which the image is averaged in the y-coordinate. To start an experiment, cooked wood chips at a prescribed k were added care- fully, to avoid chip fracture as will as to avoid compaction in the chip column, into the test column to the initial height of 0.45 m. The packing was performed randomly. Then the chip packed bed was compressed by the piston to a known load, somewhere in the range from 0 to 14 kPa. The level of the compressive pres- 77 sure was chosen to achieve porosities in the range of 0:2\u00000:5. This is the typical range found in industry and we determined this through literature values as well as in a pre-trial. A 5:36\u000210\u00002 M NaCl salt solution (T = 23\u000EC, 4 mS=cm) was introduced by prescribing the inlet flowrate of 5 L=min and measuring the inlet pressure ph. At t = 0, the valve was switched to the second tank, instead of the first tank, and a 2:52\u0002 10\u00002 M NaCl solution (T = 23\u000EC, 2 mS=cm) was intro- duced at the top of the column, at the same pressure and flowrate to displace the higher conductivity brine solution in the wood chip column. In this case a lower density solution displaces a higher density one to avoid any gravity currents. Ta- ble 4.3 shows a listing of the experimental conditions tested. Both limiting cases were tested using Hemlock wood chips cooked to three different k . In total 12 different cases were tested and the conditions used are summarized in Table 4.3. These ranges covered those encountered in operating industrial digesters. Only the two limiting cases were tested. As shown cases under limiting conditions 1 is given when d ! 1; for the second limiting case g ! 0. To estimate the axial dispersion coefficient we formally define the case in which we want to minimize the function S(De), which represents the L2 norm of the difference between the experimental measurements and the solution of Equa- tion 4.35. Formally, we state that we seek minS(De) = kC\u00CC\u0084(x\u00CC\u0084; t\u00CC\u0084)\u0000 C\u00CC\u0084e(x\u00CC\u0084; t\u00CC\u0084) \u000B\u000Bk (4.39) 78 subject to \u00C2\u00B6C\u00CC\u0084 \u00C2\u00B6 t\u00CC\u0084 = 1 Pe \u00C2\u00B6 2C\u00CC\u0084 \u00C2\u00B6 x\u00CC\u00842 \u0000 \u00C2\u00B6C\u00CC\u0084 \u00C2\u00B6 x\u00CC\u0084 (4.40) C\u00CC\u0084(0; t\u00CC\u0084) = C\u00CC\u0084e(0; t\u00CC\u0084) (4.41) C\u00CC\u0084(1; t\u00CC\u0084) = C\u00CC\u0084e(1; t\u00CC\u0084) (4.42) C\u00CC\u0084(x\u00CC\u0084;0) = C\u00CC\u0084e(x\u00CC\u0084;0) (4.43) It should be noted that we are not using the (Danckwerts, 1953) conditions as indicated previously. Here we are using experimental data to represent both the boundary and initial conditions. By doing so we estimate De using the data from the six remaining ERT planes. This minimization problem was conducted using a MATLAB built-in function for unconstrained optimization, i.e. fminbnd. In ad- dition Equation 4.40 was integrated using another built-in function in MATLAB, pdepe. 79 Limiting Case k m pc ph L ec d g De Pe r2 (kPa) (kPa) (kPa) (m) (m2=s) pc\u001D ph 80 115 5.6 0.163 0.386 0.462 0.028 0.008 9:09\u000210\u00003 0.43 0.997 80 115 9.8 0.234 0.351 0.408 0.023 0.011 2:01\u000210\u00002 0.20 0.999 80 115 14.0 0.391 0.330 0.370 0.027 0.016 3:78\u000210\u00002 0.11 0.999 53 78 5.6 0.286 0.343 0.414 0.049 0.017 9:18\u000210\u00003 0.42 0.997 53 78 9.8 0.545 0.309 0.354 0.053 0.030 3:31\u000210\u00002 0.12 0.999 53 78 14.0 1.736 0.277 0.280 0.110 0.078 9:64\u000210\u00002 0.05 0.999 25 67 5.6 0.267 0.336 0.379 0.046 0.019 3:40\u000210\u00002 0.12 0.999 25 67 9.8 0.663 0.291 0.283 0.063 0.040 1:02\u000210\u00001 0.05 1.000 25 67 14.0 2.409 0.267 0.225 0.147 0.116 2:16\u000210\u00001 0.03 0.999 m\u001D ph 80 115 0.0 0.135 0.450 0.538 1.000 0.007 3:72\u000210\u00003 1.06 0.989 53 78 0.0 0.118 0.450 0.537 1.000 0.008 3:17\u000210\u00003 1.24 0.992 25 67 0.0 0.107 0.450 0.536 1.000 0.009 3:25\u000210\u00003 1.21 0.992 Table 4.3: A summary of the experimental conditions tested with estimates of the axial dispersion coefficients. 80 4.4 Results and Discussion We begin the presentation of the results by presenting a case representative of the phenomenon of the brine solution being displaced from the column. To aid in the clarity, we present a contour plot of hCe(x;y; t)i at 6 different times, see Figures 4.7a - 4.7f. As shown, the column is initial one concentration and as time progresses, the more dense salt solution is swept vertically out through the bottom of the column. From these figures we see the evolution of the concentration and physically the dispersion coefficient represents the gradient of the concentration at the interface between the two fluids. In all cases measured, a larger concentration gradient existed in the vertical direction rather than in the horizontal direction; this indicates that we nearly achieved a one-dimensional displacement and wall effects are negligible. In all cases measured, channeling was not observed. The evolution of the axial concentration profiles are shown for a number of cases in Figures 4.8a and 4.8b. Two dramatically different cases are shown in which in Figure 4.8b the porosity is much lower than in Figure 4.8a. What is evident in these images is that the concentration gradients for the conditions with the lower porosity are smaller than for those measured at the higher porosity. For the lower porosity case, the results appear seemingly like the idealized results of a well-stirred tank. The dashed lines in these figures represent the best fit for De as estimated using the optimization. We speculate that the results stem from the fact that there is greater tortuosity, created by more dead end channels, in the highly compressed bed. This may represent one source of the heterogeneity which is 81 found industrially. Estimates ofDe for each case tested are shown in Table 4.3, along with the cor- relation coefficients. This is also shown in Figure 4.9a as a function of porosity for different kappa numbers. The trend in this data is similar to that indicated by Equation 2.20 and we find a = 7:2\u0002 10\u00004 m2=s and b = \u00003:8. The estimates of De using Equation 2.20, with our experimentally determined a and b, are suitable for all industrial situations since the tested conditions covered those encountered in operating industrial digesters. We also report in Figure 4.9b that the axial dis- persion is related to the inverse of the permeability, as measured in Chapter 3 where axial dispersion scales out with the inverse of the permeability. 82 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 Horizontal Position D im en si on le ss H ei gh t 0 0.2 0.4 0.6 0.8 1 (a) t\u00CC\u0084 = 0 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 Horizontal Position D im en si on le ss H ei gh t 0 0.2 0.4 0.6 0.8 1 (b) t\u00CC\u0084 = 0:29 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 Horizontal Position D im en si on le ss H ei gh t 0 0.2 0.4 0.6 0.8 1 (c) t\u00CC\u0084 = 0:59 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 Horizontal Position D im en si on le ss H ei gh t 0 0.2 0.4 0.6 0.8 1 (d) t\u00CC\u0084 = 0:88 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 Horizontal Position D im en si on le ss H ei gh t 0 0.2 0.4 0.6 0.8 1 (e) t\u00CC\u0084 = 1:17 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 Horizontal Position D im en si on le ss H ei gh t 0 0.2 0.4 0.6 0.8 1 (f) t\u00CC\u0084 = 1:76 Figure 4.7: Contour images of a representative ERT measurement with k = 80, pc = 0 kPa, and e = 0:54. 83 0 0.5 1 1.5 2 2.5 3 0 0.2 0.4 0.6 0.8 Dimensionless Time D im en si on le ss C on ce nt ra tio n Measured Model P1 P3 P2 P4 P5 P7 P6 (a) k = 80, pc = 0 kPa, e = 0:54 0 0.5 1 1.5 2 2.5 3 0 0.2 0.4 0.6 0.8 Dimensionless Time D im en si on le ss C on ce nt ra tio n Measured Model P6 P7 P5 P4 (b) k = 25, pc = 14:0 kPa, e = 0:25 Figure 4.8: The evolution of the concentration of the brine solution as a function of time at different elevations in the column. P1-P7 are ERT planes 1-7. 84 0.2 0.3 0.4 0.5 0.6 0.7 0 0.05 0.1 0.15 0.2 0.25 Porosity \u00CE\u00B5 D e (m 2 /s ) \u00CE\u00BA = 25 \u00CE\u00BA = 53 \u00CE\u00BA = 80 (a) 0 0.5 1 1.5 2 x 109 0 0.05 0.1 0.15 0.2 0.25 1/k (1/m2) D e (m 2 /s ) (b) Figure 4.9: The estimated axial dispersion coefficientDe with porosity e and inverse of permeability k. 85 4.5 Summary and Conclusions In this work we present a methodology to characterize axial dispersion through a porous compressible bed of wood chips while visualizing the flow using ERT. Here we demonstrate that under two limiting cases the porosity of the porous bed varies slowly in the flow-direction and to the lowest order can be considered a constant. This simplified the optimization routine we used to match the experi- mental data to the numerical results of the advection-diffusion equation. Using this, a methodology to estimate the axial dispersion is given by a minimization scheme. We find that for Hemlock chips, cooked to k in the range 25-80, the axial dispersion followed an equation of the form De = aeb where a and b were determined to be 7:2\u000210\u00004 m2=s and b=\u00003:8, respectively. 86 Chapter 5 Summary and Conclusions In the first portion of this work we present a methodology to characterize the re- sistance to fluid motion through a porous compressible bed of wood chips. Here we advance the argument that under the limiting conditions of when the mechan- ical load is large in comparison to hydraulic pressure the equations of motion can be solved asymptotically to produce an expression approximating the variation in porosity along the length of the bed. With this, a methodology to estimate the re- sistance is given by multiple linear regression. In the experimental portion of the work we measured the resistance to flow of a bed of hemlock wood chips, cooked to three different k , and compacted to three different pc. With our methodology, we find that r1 = 3\u0002104 Pa:s=m2 and r2 = 1:9\u0002106 Pa:s2=m3. The usefulness of these estimates was tested for the extreme case with no applied mechanical load. A reasonable prediction was obtained, thus verifying the utility of the methodol- ogy and the estimated resistances. After accounting for compressibility effects, 87 we find that our correlation has lower flow resistances than the literature correla- tions. In the second portion of this work we present a methodology to characterize axial dispersion through a porous compressible bed of wood chips while visual- izing the flow using ERT. Here we demonstrate that under two limiting cases the porosity of the porous bed varies slowly in the flow-direction and to the lowest order can be considered a constant. This simplified the optimization routine we used to match the experimental data to the numerical results of the advection- diffusion equation. Using this, a methodology to estimate the axial dispersion is given by a minimization scheme. We find that for Hemlock chips, cooked to k in the range 25-80 and compacted to pc in the range 0\u000014 kPa, the axial dispersion followed an equation of the form De = aeb where a and b were determined to be 7:2\u000210\u00004 m2=s and \u00003:8, respectively. 88 Chapter 6 Recommendations for Future Work The following recommendations are suggested for future work in flow uniformity on digester operations: \u000F Introduction of radial flow in the wood-chip column. This will allow for ad- vancing 2-D/3-D models for estimating radial permeability and radial dis- persion. \u000F Usage of high spatial resolution imaging techniques such as computed axial tomography (CAT-Scan) or nuclear magnetic resonance imaging (NMRI). 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Wrist, P.E., \u00E2\u0080\u009DThe present state of our knowledge on the fundamentals of wet press- ing,\u00E2\u0080\u009D Pulp and Paper Magazine of Canada 65(7), T284-T296 (1964). 96 Appendix A Apparatus Design 97 Figure A.1: Schematic of the apparatus. 98 Figure A.2: Schematic of the top screen plate. 99 Figure A.3: Schematic of the top flange. 100 Figure A.4: Schematic of the top pipe section. 101 Figure A.5: Schematic of the pipe with ERT ring. 102 Figure A.6: Schematic of the ERT sensors. 103 Figure A.7: Schematic of the pipe with ERT and DP ring. 104 Figure A.8: Schematic of the bottom flange. 105 Figure A.9: Schematic of the ground ERT sensor. 106 Figure A.10: Schematic of the bottom screen plate. 107 Figure A.11: Schematic of the bottom cap. 108 Figure A.12: Side view of the apparatus. 109 Figure A.13: Top view of the apparatus. 110 Appendix B Wood Chips Screening Analysis Wood chips are produced by cutting the wood perpendicular to the grain direction with wedged knives. The average chip length, width, and thickness can vary due to the chipper setting depending on the structure of the wood. Common chip dimensions are 12 to 25 mm long, 12 to 25 mm wide, and 2 to 10 mm thick. Wood chips require screening before cooking to eliminate oversize pieces that cause heterogeneous delignification, and fines that easily overcook and increase flow resistance in liquor circulation (Gullichsen et al, 1992). Figure B.1 shows a common wood chip geometry and dimensions. Different thickness fractions were analyzed by randomly selecting 250 pieces of wood chips from the pile to determine their length, width, and thickness. The statistical analysis of different thickness fractions are tabulated in Table B.1 and B.2 111 Figure B.1: Typical chip geometry and dimensions. 112 Statistics Length Width Thickness (mm) (mm) (mm) Sample Size 250 250 250 Mean 21.21 14.69 2.76 Median 20.99 13.80 2.77 Mode 20.74 11.35 2.00 Standard Deviation 1.99 6.47 0.57 Sample Variance 3.97 41.89 0.32 Standard Error 0.13 0.41 0.04 Minimum 16.14 2.87 1.29 Maximum 31.89 39.73 4.08 Range 15.75 36.86 2.79 Sum 5302.88 3673.72 690.40 Confidence Level (95%) 0.25 0.80 0.07 Table B.1: Statistical analysis of 2 mm thickness fraction in accepts. Figure B.2: 2mm thickness fraction in accepts. 113 Statistics Length Width Thickness (mm) (mm) (mm) Sample Size 250 250 250 Mean 22.91 19.14 4.36 Median 22.62 17.52 4.22 Mode 22.04 17.95 4.03 Standard Deviation 3.64 8.48 0.71 Sample Variance 13.26 71.92 0.50 Standard Error 0.23 0.54 0.04 Minimum 2.17 4.95 2.21 Maximum 50.63 57.78 6.38 Range 48.46 52.83 4.17 Sum 5726.92 4785.52 1090.25 Confidence Level (95%) 0.45 1.05 0.09 Table B.2: Statistical analysis of 4 mm thickness fraction in accepts. Figure B.3: 4mm thickness fraction in accepts. 114 Appendix C Calculation of Sauter Mean Diameter of Wood Chips The method of Sauter mean diameter based on weight fraction was used to calcu- late the Sauter mean diameter d32 of wood chips. Screen Thickness Width Length Mass Fraction (mm) (mm) (mm) (xi) 4 mm slot 4:4(\u00060:7) 19:1(\u00068:5) 22:9(\u00063:6) 0.376 2 mm slot 2:6(\u00060:6) 14:7(\u00066:5) 21:2(\u00062:0) 0.624 Table C.1: Average length, width, and thickness for mass fractions of ac- cepts. To determine d32 of wood chips (parallelepipedal shape), the specific surface 115 area Sv is used Sv = As Vp = 2\u0002 (w\u0002 t+w\u0002 l+ t\u0002 l) l\u0002w\u0002 t (C.1) where As is the mean surface area of the particles,Vp is the volume of the particles, l is the mean length of the particles, w is the mean width of the particles, and t is the mean thickness of the particles. For non-spherical particles, the equivalent particle diameter dp can be written as (Comiti and Renaud, 1989): dp = 6 Sv (C.2) For a wide range of size distribution, the Sauter mean diameter of particle can be given by (Coulson et al, 1978): d32 = 1 \u00C3\u00A5 i xi dp;i (C.3) 116 Appendix D Porosity Measurement In the two-phase test column and at no flow, cooked chips occupy the volume fraction echip and the remaining interstitial space is filled with liquid. This liquid occupies the characteristic void fraction (porosity) ec . As a result ec+ echip = 1 (D.1) The basic density of uncooked wood chips rchip was determined using a water displacement method (CPPA standard A.8P) to be 398.27 (oven dry kg)=(m3 green volume). To account for the decreased density after cooking due to the loss of lignin, the chip density was multiplied by the cook yieldY to determine the cooked wood chips density rcookchip rcookchip = Yrchip (D.2) After adding the cooked wood chips to the test column, they have an initial bulk 117 packing density of rob (volume occupied by the oven-dry mass of cooked wood chips added). To account for compression of the cooked chip column, a bulk packing density of r pb was used which varies according to Equation D.3 r pb = r o b ( Lo Lp ) (D.3) where Lo is the initial bed length before compressing the cooked wood chips, and Lp is the bed length after compressing the cooked wood chips. The volume fraction of cooked wood chips becomes echip = r pb rcookchip (D.4) By substituting Equation D.4 into Equation D.1 and rearranging to get ec = 1\u0000 r pb rcookchip (D.5) or ec = 1\u0000 rob ( Lo Lp ) Yrchip (D.6) 118 Appendix E Pneumatic Cylinder Desgin Below shows the conversion from air pressure to actual force on the chip column based on the area of the piston. Power Factor (PF) 0:89 Piston Diameter (m) 2:70\u000210\u00002 Piston Area Apiston (m2) 5:72\u000210\u00004 Compaction Plate Diameter (m) 0:1524 Compaction Area Ac (m2) 1:820\u000210\u00002 Air Line Pressure Air Line Pressure Cylinder Output Force Compaction Pressure (bar) (kPa) (N) (kPa) 2.0 200 101.9 5.6 3.5 350 178.4 9.8 5.0 500 254.8 14.0 Table E.1: Calculation of compaction pressure. 119 The pneumatic cylinder output force F can be given by F = pair\u0002PF\u0002Apiston (E.1) where pair is the air line pressure, PF is the power factor (efficiency) of the pneu- matic cylinder, and Apiston is the piston area. The compaction pressure can be given then by pc = F Ac = 0:028 pair (E.2) where Ac is the compaction area. Figure E.1 shows the relationship between com- paction pressure and air line pressure. 120 0 100 200 300 400 500 600 0 5 10 15 20 p air (kPa) p c (kP a) p c (kPa) = 0.028 p air (kPa) Figure E.1: Relationship between compaction pressure and air line pressure. Solid line is given by Equation E.2. 121 Appendix F Pressure Drop Across Screen Plate Before the test run, we need to determine pressure drop across the screen plate by measuring the screen plate resistance at empty column with different flow rates. The pressure drop across the screen plate as a function of superficial velocity is shown in Figure F.1. By simple curve fitting of the data using linear regression in MATLAB we determine the Equation of the fitting line to calculate pressure drop across the screen plate as a function of superficial velocity dpscreen (kPa) = 0:0025v(mm=s) (F.1) During the test, the differential pressure transducer measures the pressure drops including; screen plate, hydrostatic water, and bed. In order to obtain the pressure drop across the bed only, we need to subtract pressure drop across screen plate, and hydrostatic pressure drop of water from the measured pressure drop. 122 0 2 4 6 8 10 12 0 0.005 0.01 0.015 0.02 0.025 0.03 v (mm/s) dp sc re e n (kP a) dp screen (kPa) = 0.0025 v (mm/s) Figure F.1: Pressure drop across the screen plate. Dashed line is given by linear regression fit in MATLAB to give Equation F.1. 123 Appendix G Conductivity to Concentration Tomogram Before the test run, we need to convert the conductivity tomogram produced by ERT to a concentration tomogram by measuring the conductivity of NaCl brine solution with different brine concentrations. The brine concentration as a function of brine conductivity is shown in Figure G.1. By simple curve fitting of the data using linear regression in MATLAB we determine the Equation of the fitting line to calculate brine concentration as a function of brine conductivity C (M) = 0:0142s (mS=cm)\u00000:0032 (G.1) 124 0 2 4 6 8 10 12 0 0.05 0.1 0.15 \u00CF\u0083 (mS/cm) C (M ) C (M) = 0.0142 \u00CF\u0083 (mS/cm) \u00E2\u0088\u0092 0.0032 Figure G.1: Concentration of Brine Solution. Dashed line is given by linear regression fit in MATLAB to give Equation G.1. 125 Appendix H Higher Order Estimates of e(x) We can extend the accuracy of the analysis in Chapter 3 by extending the number of terms of the form of the solution. If e\u00CC\u0084 = e\u00CC\u0084(0)+d e\u00CC\u0084(1)+d 2e\u00CC\u0084(2)::: (H.1) r\u00CC\u0084i = r\u00CC\u0084 (0) i +d r\u00CC\u0084 (1) i +d 2r\u00CC\u0084(2)i ::: (H.2) the O(d 2) terms must satisfy 8><>: \u0000e\u00CC\u0084 (1) f\u00CC\u0084 003 (1) de\u00CC\u0084(1) dx\u00CC\u0084 \u0000 f\u00CC\u0084 03(1) de\u00CC\u0084(2) dx\u00CC\u0084 = e\u00CC\u0084(1) \u0010 r\u00CC\u0084(0)1 f\u00CC\u0084 0 1(1)+ r\u00CC\u0084 (0) 2 f\u00CC\u0084 0 2(1) \u0011 + r\u00CC\u0084(1)1 + r\u00CC\u0084 (1) 2 (H.3) 126 subject to e\u00CC\u0084(2)(0) =\u00001 2 f\u00CC\u0084 003 (1)\u0000 f\u00CC\u0084 03(1) \u00013 e\u00CC\u0084(2)(1) = 0 (H.4) This relationship can be solved to yield e\u00CC\u0084(2) =\u00001 2 (x\u00CC\u0084\u0000 x\u00CC\u00842)\u0000 f\u00CC\u0084 03(1) \u00012 \u0010r\u00CC\u0084(0)1 f\u00CC\u0084 01(1)+ r\u00CC\u0084(0)2 f\u00CC\u0084 02(1)\u0011\u0000 f\u00CC\u0084 003 (1)2\u0000 f\u00CC\u0084 03(1)\u00013 (1\u00002x\u00CC\u0084+ x\u00CC\u00842) (H.5) 127 Appendix I Summary of the Required Functions Here is a summary of the required functions in Chapters 3 and 4: f\u00CC\u00841(e\u00CC\u0084) = \u0012 1\u0000 ece\u00CC\u0084 1\u0000 ec \u00132 1 e\u00CC\u00843 (I.1) f\u00CC\u00842(e\u00CC\u0084) = \u0012 1\u0000 ece\u00CC\u0084 1\u0000 ec \u0013 1 e\u00CC\u00843 (I.2) f\u00CC\u00843(e\u00CC\u0084) = \u0012 e\u00CC\u0084g\u0000 e\u00CC\u0084 e\u00CC\u0084g\u00001 \u0013n (I.3) f\u00CC\u0084 01(1) = ec\u00003 1\u0000 ec (I.4) f\u00CC\u0084 02(1) = 2ec\u00003 1\u0000 ec (I.5) f\u00CC\u0084 03(1) =\u0000 n e\u00CC\u0084g\u00001 (I.6) f\u00CC\u0084 003 (1) = n(n\u00001) (e\u00CC\u0084g\u00001)2 (I.7) 128 Appendix J MATLAB Codes J.1 Compressibility Function Regression Code The code written in MATLAB 7.5 for the compressibility function regression is included here. The main file is: \u000F Ps_regression.m 129 Program J.1 Main Program: Ps regression.m %*************************************************** % Author: Mohammed Alaqqad (2011) % Description: This is a program that correlates the % compressibility as a function of void fraction. % The method used for the correlation is least % square method (Linear regression). % In the form Ps=m(voidg-void)\u00CB\u0086n %*************************************************** % Clear History clear all % kappa(voidc,Pc(kPa),k1(80)-k2(53)-k3(25)) kappa = zeros(4,2,3); kappa(:,:,1) = [0.538 0.0; 0.462 5.6; 0.416 9.8; 0.372 14.0]; kappa(:,:,2) = [0.537 0.0; 0.414 5.6; 0.354 9.8; 0.280 14.0]; kappa(:,:,3) = [0.536 0.0; 0.397 5.6; 0.316 9.8; 0.250 14.0]; [npx npy npe] = size(kappa); for i = 1:npe for j=2:npx xdata(j-1) = log(kappa(1,1,i)... - kappa(j,1,i)); ydata(j-1) = log(kappa(j,2,i)*1000); end p = polyfit(xdata,ydata,1); fitP(i,:) = [p(1),exp(p(2))/1000,kappa(1,1,i)]; 130 EpsI(i,:) = linspace(kappa(1,1,i),... kappa(end,1,i),100); psI(i,:)= exp(p(2))*(kappa(1,1,i)... - EpsI(i,:)).\u00CB\u0086p(1); end hold on box on fitP plot(kappa(:,1,3), kappa(:,2,3),\u00E2\u0080\u0099sk\u00E2\u0080\u0099,kappa(:,1,2),... kappa(:,2,2),\u00E2\u0080\u0099vk\u00E2\u0080\u0099,kappa(:,1,1), kappa(:,2,1),\u00E2\u0080\u0099ok\u00E2\u0080\u0099) xlabel(\u00E2\u0080\u0099Porosity \epsilon\u00E2\u0080\u0099,\u00E2\u0080\u0099Fontsize\u00E2\u0080\u0099,16) ylabel(\u00E2\u0080\u0099P_s (kPa)\u00E2\u0080\u0099,\u00E2\u0080\u0099Fontsize\u00E2\u0080\u0099,16) legend(\u00E2\u0080\u0099\kappa = 25\u00E2\u0080\u0099,\u00E2\u0080\u0099\kappa = 53\u00E2\u0080\u0099,\u00E2\u0080\u0099\kappa = 80\u00E2\u0080\u0099) plot(EpsI(1,:), psI(1,:)/1000,\u00E2\u0080\u0099--k\u00E2\u0080\u0099,EpsI(2,:),... psI(2,:)/1000,\u00E2\u0080\u0099--k\u00E2\u0080\u0099,EpsI(3,:), psI(3,:)/1000,\u00E2\u0080\u0099--k\u00E2\u0080\u0099); axis([0.2 0.7 0 18]) % use this command in all figures to change all % legends and axis etc to this % font size set(gca,\u00E2\u0080\u0099fontsize\u00E2\u0080\u0099,16) mdata = fitP(:,2) kdata = [80 53 25]\u00E2\u0080\u0099 p = polyfit(log(kdata),mdata,1) kI = linspace(kdata(1),kdata(end),50); %mI = polyval(p,log(kI)); mI = (0.0774-0.0129.*log(kI)).\u00CB\u0086-1.2; figure(2) plot(kdata,mdata,\u00E2\u0080\u0099ok\u00E2\u0080\u0099,kI,mI,\u00E2\u0080\u0099--k\u00E2\u0080\u0099) ylabel(\u00E2\u0080\u0099m (kPa)\u00E2\u0080\u0099,\u00E2\u0080\u0099Fontsize\u00E2\u0080\u0099,16) xlabel(\u00E2\u0080\u0099\kappa\u00E2\u0080\u0099,\u00E2\u0080\u0099Fontsize\u00E2\u0080\u0099,16) axis([20 86 50 120]) set(gca,\u00E2\u0080\u0099fontsize\u00E2\u0080\u0099,16) figure(3) mI = (0.214-0.0357.*log(kI)).\u00CB\u0086-1.7; 131 plot(kI,mI,\u00E2\u0080\u0099-k\u00E2\u0080\u0099) ylabel(\u00E2\u0080\u0099m (kPa)\u00E2\u0080\u0099,\u00E2\u0080\u0099Fontsize\u00E2\u0080\u0099,16) xlabel(\u00E2\u0080\u0099\kappa\u00E2\u0080\u0099,\u00E2\u0080\u0099Fontsize\u00E2\u0080\u0099,16) axis([20 86 50 130]) set(gca,\u00E2\u0080\u0099fontsize\u00E2\u0080\u0099,16) 132 J.2 Pressure Drop Regression Code The code written in MATLAB 7.5 for the pressure drop regression is included here. The main file is: \u000F Ph_regression.m 133 Program J.2 Main Program: Ph regression.m %*************************************************** % Author: Mohammed Alaqqad (2011) % Description: This is a program that correlates the % pressure as a function of velocity and void % fraction. The method used for the correlation is % least square method (Multiple Linear regression). %*************************************************** %*************************************************** % USER NEEDS TO SPECIFY/INPUT THE NECESSARY % PARAMETERS AND FILENAME:- % name : The name of the data set (Pressure Drop) % datafile: The file name that the data store % (pressuredrop.txt) %*************************************************** % Clear History clear all name=\u00E2\u0080\u0099Pressure Drop\u00E2\u0080\u0099; datafile=\u00E2\u0080\u0099dp_regression.txt\u00E2\u0080\u0099; % Clear Matlab command window clc disp(\u00E2\u0080\u0099In progress...\u00E2\u0080\u0099) % Get data from text file % column 1 column 2 column 3 column 4 % column 5 column 6 % voidc U(mm/s) DP/L(Pa/m) Ph (Pa) % k Pc (kPa) % fid =fopen(filename, permission) fid = fopen(datafile,\u00E2\u0080\u0099r\u00E2\u0080\u0099); [x1 x2 x3 x4 x5 x6 x7]=textread... 134 (datafile,\u00E2\u0080\u0099%f%f%f%f%f%f%f\u00E2\u0080\u0099); status=fclose(fid); % Determine number of data points count=length(x1); voidc=x1; % Convert velocity from (mm/s) to (m/s) u=x2./1000; dp=x3; Ph=x4; % Convert Pc from (kPa) to (Pa) kappa=x5; Pc=x6.*1000; %ne=x7; ne=1.2; % viscosity of liquid (Pa.s) mu=0.000925; % density of liquid (kg/m3) ro=997.3; % equivalent particle diameter (m) d=0.00714; % sphericity () sp=0.556; delta=Ph./Pc; voidg=0.54; term1=(1+((delta./2).*((3-voidc)./(1-voidc)).*... ((voidg-voidc)./(voidc.*ne)))).*... (((1-voidc).\u00CB\u00862)./(voidc.\u00CB\u00863)).*u; term2=(1+((delta./2).*((3-2.*voidc)./(1-voidc))... .*((voidg-voidc)./(voidc.*ne)))).*((1-voidc)... ./(voidc.\u00CB\u00863)).*(u.\u00CB\u00862); % Create the x matrix from these vectors % using a for loop 135 % The dependent variable goes into a n*1 % column vector. for n=1:count x(n,1)=term1(n); x(n,2)=term2(n); y(n,1)=dp(n); end % Use the matrix division operation. a=(x\u00E2\u0080\u0099*x)\(x\u00E2\u0080\u0099*dp); A=(a(1)*(d\u00CB\u00862*sp\u00CB\u00862))/mu; B=(a(2)*(d*sp))/ro; %Harkonen correction for r1 and r2 mu_h=0.000068; ro_h=885; r1_h=(A*mu_h)/(d\u00CB\u00862*sp\u00CB\u00862); r2_h=(B*ro_h)/(d*sp); %z=inv(x\u00E2\u0080\u0099*x); % Calculate the values of the pressure drop % predicted by the equation. % Then calculate the difference between % the experimental and predicted values % dppred == dp predicted % res == residuals % diff == percent different dppred =x*a; res=(dp-dppred); diff=abs(res./dp)*100; % Statistical Analysis % sum-of-squares terms % sum of squares, error (SSE) SSE=0; % initialize 136 sumres=0; for n=1:count SSE=SSE+res(n)*res(n); sumres=sumres+res(n); end MeanErr=abs(sumres./count); % Sum of squares, regression (SSR) avedppred=0; SSR= 0; for n=1:count avedppred=avedppred+dppred(n); end avedppred=avedppred./count; for n=1:count SSR=SSR +(dppred(n)-avedppred).\u00CB\u00862; end % Sum of squares, total (SST) avedp=0; SST=0; for n=1:count avedp=avedp+dp(n); end avedp=avedp./count; for n=1:count SST=SST+(dp(n)-avedp).\u00CB\u0086 2; end SST; % R-squared R2=1-(SSE./SST); R22=SSR./SST; % Adjusted R-squared 137 R2adj=1-(count-1).*(1-R2)./(count-2); % Residual Mean Square (MSE) % K = total number of predictors or number of % independent variables % n = sample size n=count; K=2; MSE=SSE./(count-K-1); % Standard Error of the Estimate (SEE) SEE = sqrt(MSE); % 95% Confidence Interval [b,bint] = regress(dp,x); c1=bint(1,2)-a(1); c2=bint(2,2)-a(2); c12=(c1*(d\u00CB\u00862*sp\u00CB\u00862))/mu; c22=(c2*(d*sp))/ro; c12_h=(c12*mu_h)/(d\u00CB\u00862*sp\u00CB\u00862); c22_h=(c22*ro_h)/(d*sp); %Sa1=sqrt(z(1,1)*MSE); %Sa2=sqrt(z(2,2)*MSE); %talpha_554=1.964255187; %t1=talpha_554*Sa1; %t2=talpha_554*Sa2; %L1=a(1)-talpha_554*Sa1; %U1=a(1)+talpha_554*Sa1; %L2=a(2)-talpha_554*Sa2; %U2=a(2)+talpha_554*Sa2; % Create output report disp(\u00E2\u0080\u0099 \u00E2\u0080\u0099) disp(\u00E2\u0080\u0099 =======================================\u00E2\u0080\u0099) disp(\u00E2\u0080\u0099Author:Mohammed Alaqqad\u00E2\u0080\u0099) fprintf(\u00E2\u0080\u0099Run at:%s\n\u00E2\u0080\u0099,datestr(now)) fprintf(\u00E2\u0080\u0099Title: Analysis of %s\n\u00E2\u0080\u0099,name) disp(\u00E2\u0080\u0099Equation: y=a1*x1+a2*x2\u00E2\u0080\u0099) disp(\u00E2\u0080\u0099Method: Multiple Linear Regression\u00E2\u0080\u0099) 138 disp(\u00E2\u0080\u0099\u00E2\u0080\u0099) disp(\u00E2\u0080\u0099\u00E2\u0080\u0099) disp(\u00E2\u0080\u0099\u00E2\u0080\u0099) disp(\u00E2\u0080\u0099=========================================\u00E2\u0080\u0099) fprintf(\u00E2\u0080\u0099Total numbers of data points (n)=%f\n\u00E2\u0080\u0099, count) fprintf(\u00E2\u0080\u0099\n The Units for dP/L (Pa/m) and v (m/s)\u00E2\u0080\u0099) fprintf(\u00E2\u0080\u0099\n The parameters r1 and r2 are\n\u00E2\u0080\u0099) %fprintf(\u00E2\u0080\u0099a1=%9.2f\n\u00E2\u0080\u0099, b(1)) %fprintf(\u00E2\u0080\u0099a2=%9.2f\n\n\u00E2\u0080\u0099,b(2)) fprintf(\u00E2\u0080\u0099r1=%9.2f+-%9.2f\n\u00E2\u0080\u0099,a(1),c1) fprintf(\u00E2\u0080\u0099r2=%9.2f+-%9.2f\n\n\u00E2\u0080\u0099,a(2),c2) fprintf(\u00E2\u0080\u0099A=%9.2f+-%9.2f\n\u00E2\u0080\u0099,A,c12) fprintf(\u00E2\u0080\u0099B=%9.2f+-%9.2f\n\n\u00E2\u0080\u0099,B,c22) fprintf(\u00E2\u0080\u0099r1_h=%9.2f+-%9.2f\n\u00E2\u0080\u0099,r1_h,c12_h) fprintf(\u00E2\u0080\u0099r2_h=%9.2f+-%9.2f\n\n\u00E2\u0080\u0099,r2_h,c22_h) fprintf(\u00E2\u0080\u0099Void Velocity ExptdP PreddP Percent Diff Residuals\n\u00E2\u0080\u0099) fprintf(\u00E2\u0080\u0099 [mm/s] [Pa/m] [Pa/m] \n\u00E2\u0080\u0099) for n= 1 :count fprintf(\u00E2\u0080\u0099%4.3f %6.2f %5.2f %5.2f %5.1f %6.2f\n\u00E2\u0080\u0099,... voidc(n), u(n)*1000, dp(n) , dppred(n)... ,diff(n),res(n) ./1000) end disp(\u00E2\u0080\u0099\u00E2\u0080\u0099) disp(\u00E2\u0080\u0099\u00E2\u0080\u0099) disp(\u00E2\u0080\u0099STATISTICAL ANALYSIS\u00E2\u0080\u0099) disp(\u00E2\u0080\u0099--------------------\u00E2\u0080\u0099) fprintf(\u00E2\u0080\u0099Sum of Squares,Error(SSE)=%f\n\u00E2\u0080\u0099,SSE) fprintf(\u00E2\u0080\u0099Mean of Residuals Error=%f\n\u00E2\u0080\u0099,MeanErr) fprintf(\u00E2\u0080\u0099sum of Square,Total(SST)=%f\n\u00E2\u0080\u0099,SST) fprintf(\u00E2\u0080\u0099Sum of Squares,Regression(SSR)=%f\n\u00E2\u0080\u0099,SSR) fprintf(\u00E2\u0080\u0099Coefficient of Determination (R\u00CB\u00862)=%f\n\u00E2\u0080\u0099,R2) fprintf(\u00E2\u0080\u0099Adjusted R-squared=%f\n\u00E2\u0080\u0099,R2adj) fprintf(\u00E2\u0080\u0099Standard Error of Estimate(SEE)=%f\n\u00E2\u0080\u0099,SEE) % Save the data into text file R=[u dp dppred]; save results.txt R -ASCII -TABS 139 % Plot the data figure plot(u.*1000,dp./1000,\u00E2\u0080\u0099*r\u00E2\u0080\u0099,u.*1000,dppred./1000,\u00E2\u0080\u0099+k\u00E2\u0080\u0099) xlabel(\u00E2\u0080\u0099Velocity m/s\u00E2\u0080\u0099) ylabel(\u00E2\u0080\u0099Pressure drop Pa/m\u00E2\u0080\u0099) str =[\u00E2\u0080\u0099Experimental Data\u00E2\u0080\u0099]; str1 =[\u00E2\u0080\u0099Correlated Data\u00E2\u0080\u0099]; legend(str,str1,\u00E2\u0080\u0099Location\u00E2\u0080\u0099,\u00E2\u0080\u0099best\u00E2\u0080\u0099) %xmin=0; xmax=10; ymin=0; ymax=20; %axis([xmin xmax ymin ymax]) fprintf(\u00E2\u0080\u0099\n\n ******End of Analysis*******\n\n\u00E2\u0080\u0099) 140 J.3 Pressure Drop Prediction Code The codes written in MATLAB 7.5 for the pressure drop prediction are included here. The main files are: \u000F Main.m \u000F Valid_Pc0.m 141 Program J.3 Main Program: Main.m %*************************************************** % Author: Mohammed Alaqqad (2011) % Description: This is a program that predict the % pressure as a function of velocity and void % fraction. The method used for the prediction is % numerical solution using ode15s. %*************************************************** %*************************************************** % USER NEEDS TO SPECIFY/INPUT THE NECESSARY % PARAMETERS AND FILENAME:- % name : The name of the data set (Pressure Drop) % datafile: The file name that the data store % (valid.txt) %*************************************************** % Clear History clear all name=\u00E2\u0080\u0099Pressure Drop\u00E2\u0080\u0099; datafile=\u00E2\u0080\u0099valid.txt\u00E2\u0080\u0099; % Clear Matlab command window clc disp(\u00E2\u0080\u0099In progress...\u00E2\u0080\u0099) % Get data from text file % column 1 column 2 column 3 column 4 % v(mm/s) DP/L(Pa/m) m(kPa) n % fid =fopen(filename, permission) fid = fopen(datafile,\u00E2\u0080\u0099r\u00E2\u0080\u0099); [v dP m n]=textread(datafile,\u00E2\u0080\u0099%f%f%f%f\u00E2\u0080\u0099); v=v./1000; dp=dP.*1000; m=m.*1000; status=fclose(fid); 142 % find the data for the three different kappa numbers mexp = [115 78 67]*1000; %nexp = [1.3 1.2 1.2]; nexp = [1.2 1.2 1.2]; np = 100; for i=1:3 IDX = find(m == mexp(i)); RNG = [IDX(1) IDX(end)]; vfit = linspace(v(RNG(1)),v(RNG(end)),np); for j = 1:np dPdx(j)=Valid_Pc0([vfit(j),mexp(i),nexp(i)]); end figure(i) midP = round(mean(RNG)); r1=3e4; r2=3.3e6; voidc=0.54; term1=(((1-voidc)\u00CB\u00862)/(voidc\u00CB\u00863)).*vfit; term2=((1-voidc)/(voidc\u00CB\u00863)).*(vfit.\u00CB\u00862); % without compressibility effects dp_model=r1*term1+r2*term2; plot(v(RNG(1):RNG(2))*1000,dP(RNG(1):RNG(2))... /1000,\u00E2\u0080\u0099ok\u00E2\u0080\u0099,vfit*1000,dPdx/1000,\u00E2\u0080\u0099--k\u00E2\u0080\u0099,... vfit*1000,dp_model/1000,\u00E2\u0080\u0099-k\u00E2\u0080\u0099) xlabel(\u00E2\u0080\u0099v (mm/s)\u00E2\u0080\u0099,\u00E2\u0080\u0099fontsize\u00E2\u0080\u0099,16) ylabel(\u00E2\u0080\u0099P_h/L (kPa/m)\u00E2\u0080\u0099,\u00E2\u0080\u0099fontsize\u00E2\u0080\u0099,16) axis([0 10 0 1.5]) legend(\u00E2\u0080\u0099Measured\u00E2\u0080\u0099,\u00E2\u0080\u0099Predicted\u00E2\u0080\u0099,... \u00E2\u0080\u0099No Compressibility Effects\u00E2\u0080\u0099,\u00E2\u0080\u0099location\u00E2\u0080\u0099,\u00E2\u0080\u0099best\u00E2\u0080\u0099) % use this command in all figures to change all % legends and axis etc to this font size set(gca,\u00E2\u0080\u0099fontsize\u00E2\u0080\u0099,16) end 143 Program J.4 ODE Solution: Valid Pc0.m function phl=Valid_Pc0(D) global epsg v m n r1 r2 v = D(1); m = D(2); n = D(3); %v = 8/1000; epsg = 0.54; r1 = 3e4; r2 = 1.9e6; %r2 = 1.5e6; L = 0.45; [x,y] = ode15s(@porous,[L 0],0); phl = y(end)/L; function dy = porous(x,y) global v r1 r2 dy(1) = -(r1*f1(y)*v + r2*f2(y)*v\u00CB\u00862); function yout = f1(y) global epsg m n if y<0, y=0; end epsilon = epsg - (y/m)\u00CB\u0086(1/n); yout = (1-epsilon)\u00CB\u00862/(epsilon)\u00CB\u00863; function yout = f2(y) global epsg m n if y<0, y=0; end epsilon = epsg - (y/m)\u00CB\u0086(1/n); yout = (1-epsilon)/(epsilon)\u00CB\u00863; 144 J.4 Axial Dispersion Minimization Code The codes written in MATLAB 7.5 for the axial dispersion estimation are included here. The main files are: \u000F Main.m \u000F Importfile.m \u000F Inputdata.m \u000F Pdetest.m 145 Program J.5 Main Program: Main.m %*************************************************** % Author: Mohammed Alaqqad (2011) % Description: This is a program that estimates the % breakthrough curves for a convective-diffusion % equation of the form % % C_T = 1/Pe*C_XX - C_X [T,X] \in [0,1] % % T=0 C=0 % X=1 (C-1) - C_X = 0 % X=0 C_X = 0 % The method used for the estimation is minimization % between the experimental data and the numerical % solution using fminsearch and pdepe. %*************************************************** % clear history clear all % Clear Matlab command window clc disp(\u00E2\u0080\u0099In progress...\u00E2\u0080\u0099) global Xe Te Ce Ci frames_c x_planes % Get data from text file and modify Inputdata % Set initial values of Dax Dax=1.8e-4;%m2/s % Use the optimization function to find Dax Dr=fminsearch(@Pdetest,Dax) [rows columns]=size(Ce); count= rows; 146 count2=columns*rows; % Calculate the values of the concentration % predicted by the equation. % Then calculate the difference between the % experimental and predicted values . % Ce == C experimnetal % Ci == C predicted % res == residuals % diff == percent different res=(Ce-Ci); diff=abs(res./Ce).*100; % Statistical Analysis % sum-of-squares terms % sum of squares, error (SSE) SSE=0; % initialize sumres=0; for n=1:count SSE=SSE+sum(res(n,:).*res(n,:)); sumres=sumres+sum(res(n,:)); end MeanErr=abs(sumres./count2); % Sum of squares, regression (SSR) aveCi=0; SSR= 0; for n=1:count aveCi=aveCi+sum(Ci(n,:)); end aveCi=aveCi./count2; for n=1:count SSR=SSR +sum((Ci(n,:)-aveCi).\u00CB\u00862); end 147 % Sum of squares, total (SST) aveCe=0; SST=0; for n=1:count aveCe=aveCe+sum(Ce(n,:)); end aveCe=aveCe./count2; for n=1:count SST=SST+sum((Ce(n,:)-aveCe).\u00CB\u0086 2); end SST; % R-squared R2=1-(SSE./SST); R22=SSR./SST; % Adjusted R-squared R2adj=1-(count2-1).*(1-R2)./(count2-2); % Residual Mean Square (MSE) % K = total number of predictors or number of % independent variables % n = sample size n=count2; K=1; MSE=SSE./(n-K-1); % Standard Error of the Estimate (SEE) SEE = sqrt(MSE); % Create output report disp(\u00E2\u0080\u0099 \u00E2\u0080\u0099) disp(\u00E2\u0080\u0099 ===========================================\u00E2\u0080\u0099) disp(\u00E2\u0080\u0099Author:Mohammed Alaqqad\u00E2\u0080\u0099) fprintf(\u00E2\u0080\u0099Run at:%s\n\u00E2\u0080\u0099,datestr(now)) fprintf(\u00E2\u0080\u0099Title: Analysis of %s\n\u00E2\u0080\u0099,\u00E2\u0080\u0099Axial Diffusion\u00E2\u0080\u0099) disp(\u00E2\u0080\u0099Equation: 1D convective-disffusive model\u00E2\u0080\u0099) disp(\u00E2\u0080\u0099Method: Solve and minimize\u00E2\u0080\u0099) 148 disp(\u00E2\u0080\u0099\u00E2\u0080\u0099) disp(\u00E2\u0080\u0099\u00E2\u0080\u0099) disp(\u00E2\u0080\u0099\u00E2\u0080\u0099) disp(\u00E2\u0080\u0099============================================\u00E2\u0080\u0099) fprintf(\u00E2\u0080\u0099Total numbers of data points (n)=%f\n\u00E2\u0080\u0099... ,count2) fprintf(\u00E2\u0080\u0099\n The Units for D (m2/s)\u00E2\u0080\u0099) fprintf(\u00E2\u0080\u0099\n The parameters Dax is\n\u00E2\u0080\u0099) fprintf(\u00E2\u0080\u0099Dax=%9.6f\n\u00E2\u0080\u0099,Dr) disp(\u00E2\u0080\u0099\u00E2\u0080\u0099) disp(\u00E2\u0080\u0099\u00E2\u0080\u0099) disp(\u00E2\u0080\u0099STATISTICAL ANALYSIS\u00E2\u0080\u0099) disp(\u00E2\u0080\u0099--------------------\u00E2\u0080\u0099) fprintf(\u00E2\u0080\u0099Sum of Squares,Error(SSE)=%f\n\u00E2\u0080\u0099,SSE) fprintf(\u00E2\u0080\u0099Mean of Residuals Error=%f\n\u00E2\u0080\u0099,MeanErr) fprintf(\u00E2\u0080\u0099sum of Square,Total(SST)=%f\n\u00E2\u0080\u0099,SST) fprintf(\u00E2\u0080\u0099Sum of Squares,Regression(SSR)=%f\n\u00E2\u0080\u0099,SSR) fprintf(\u00E2\u0080\u0099Coefficient of Determination(R\u00CB\u00862)=%f\n\u00E2\u0080\u0099,R2) fprintf(\u00E2\u0080\u0099Adjusted R-squared=%f\n\u00E2\u0080\u0099,R2adj) fprintf(\u00E2\u0080\u0099Standard Error of Estimate(SEE)=%f\n\u00E2\u0080\u0099,SEE) figure1=figure; hold on box on plot(Te\u00E2\u0080\u0099,Ce(:,2),\u00E2\u0080\u0099--k\u00E2\u0080\u0099,Te\u00E2\u0080\u0099,Ci(:,2),\u00E2\u0080\u0099-k\u00E2\u0080\u0099) legend(\u00E2\u0080\u0099Measured\u00E2\u0080\u0099,\u00E2\u0080\u0099Regression\u00E2\u0080\u0099,\u00E2\u0080\u0099Location\u00E2\u0080\u0099,\u00E2\u0080\u0099best\u00E2\u0080\u0099) %title(\u00E2\u0080\u0099Solution at X=0 to 1\u00E2\u0080\u0099) plot(Te\u00E2\u0080\u0099,Ce,\u00E2\u0080\u0099--k\u00E2\u0080\u0099,Te\u00E2\u0080\u0099,Ci,\u00E2\u0080\u0099-k\u00E2\u0080\u0099) errorbar(Te(24),Ce(24,2),0.02,\u00E2\u0080\u0099k\u00E2\u0080\u0099) xlabel(\u00E2\u0080\u0099Dimensionless Time T\u00E2\u0080\u0099,\u00E2\u0080\u0099fontsize\u00E2\u0080\u0099,16) ylabel(\u00E2\u0080\u0099Dimensionless Concentration C\u00E2\u0080\u0099,\u00E2\u0080\u0099fontsize\u00E2\u0080\u0099,16) axis tight set(gca,\u00E2\u0080\u0099fontsize\u00E2\u0080\u0099,16) % Create textbox annotation(figure1,\u00E2\u0080\u0099textbox\u00E2\u0080\u0099,\u00E2\u0080\u0099String\u00E2\u0080\u0099,{\u00E2\u0080\u0099P1\u00E2\u0080\u0099},... \u00E2\u0080\u0099FontName\u00E2\u0080\u0099,\u00E2\u0080\u0099Times New Roman\u00E2\u0080\u0099,... \u00E2\u0080\u0099FitBoxToText\u00E2\u0080\u0099,\u00E2\u0080\u0099off\u00E2\u0080\u0099,... 149 \u00E2\u0080\u0099LineStyle\u00E2\u0080\u0099,\u00E2\u0080\u0099none\u00E2\u0080\u0099,... \u00E2\u0080\u0099Position\u00E2\u0080\u0099,[0.1798 0.67 0.03858 0.06616]); % Create textbox annotation(figure1,\u00E2\u0080\u0099textbox\u00E2\u0080\u0099,\u00E2\u0080\u0099String\u00E2\u0080\u0099,{\u00E2\u0080\u0099P2\u00E2\u0080\u0099},... \u00E2\u0080\u0099FontName\u00E2\u0080\u0099,\u00E2\u0080\u0099Times New Roman\u00E2\u0080\u0099,... \u00E2\u0080\u0099FitBoxToText\u00E2\u0080\u0099,\u00E2\u0080\u0099off\u00E2\u0080\u0099,... \u00E2\u0080\u0099LineStyle\u00E2\u0080\u0099,\u00E2\u0080\u0099none\u00E2\u0080\u0099,... \u00E2\u0080\u0099Position\u00E2\u0080\u0099,[0.1874 0.5613 0.03858 0.06616]); % Create textbox annotation(figure1,\u00E2\u0080\u0099textbox\u00E2\u0080\u0099,\u00E2\u0080\u0099String\u00E2\u0080\u0099,{\u00E2\u0080\u0099P3\u00E2\u0080\u0099},... \u00E2\u0080\u0099FontName\u00E2\u0080\u0099,\u00E2\u0080\u0099Times New Roman\u00E2\u0080\u0099,... \u00E2\u0080\u0099FitBoxToText\u00E2\u0080\u0099,\u00E2\u0080\u0099off\u00E2\u0080\u0099,... \u00E2\u0080\u0099LineStyle\u00E2\u0080\u0099,\u00E2\u0080\u0099none\u00E2\u0080\u0099,... \u00E2\u0080\u0099Position\u00E2\u0080\u0099,[0.1954 0.4744 0.03858 0.06616]); % Create textbox annotation(figure1,\u00E2\u0080\u0099textbox\u00E2\u0080\u0099,\u00E2\u0080\u0099String\u00E2\u0080\u0099,{\u00E2\u0080\u0099P4\u00E2\u0080\u0099},... \u00E2\u0080\u0099FontName\u00E2\u0080\u0099,\u00E2\u0080\u0099Times New Roman\u00E2\u0080\u0099,... \u00E2\u0080\u0099FitBoxToText\u00E2\u0080\u0099,\u00E2\u0080\u0099off\u00E2\u0080\u0099,... \u00E2\u0080\u0099LineStyle\u00E2\u0080\u0099,\u00E2\u0080\u0099none\u00E2\u0080\u0099,... \u00E2\u0080\u0099Position\u00E2\u0080\u0099,[0.2215 0.2671 0.03858 0.06616]); % Create textbox annotation(figure1,\u00E2\u0080\u0099textbox\u00E2\u0080\u0099,\u00E2\u0080\u0099String\u00E2\u0080\u0099,{\u00E2\u0080\u0099P5\u00E2\u0080\u0099},... \u00E2\u0080\u0099FontName\u00E2\u0080\u0099,\u00E2\u0080\u0099Times New Roman\u00E2\u0080\u0099,... \u00E2\u0080\u0099FitBoxToText\u00E2\u0080\u0099,\u00E2\u0080\u0099off\u00E2\u0080\u0099,... \u00E2\u0080\u0099LineStyle\u00E2\u0080\u0099,\u00E2\u0080\u0099none\u00E2\u0080\u0099,... \u00E2\u0080\u0099Position\u00E2\u0080\u0099,[0.2075 0.4046 0.03858 0.06616]); % Create textbox annotation(figure1,\u00E2\u0080\u0099textbox\u00E2\u0080\u0099,\u00E2\u0080\u0099String\u00E2\u0080\u0099,{\u00E2\u0080\u0099P6\u00E2\u0080\u0099},... \u00E2\u0080\u0099FontName\u00E2\u0080\u0099,\u00E2\u0080\u0099Times New Roman\u00E2\u0080\u0099,... \u00E2\u0080\u0099FitBoxToText\u00E2\u0080\u0099,\u00E2\u0080\u0099off\u00E2\u0080\u0099,... \u00E2\u0080\u0099LineStyle\u00E2\u0080\u0099,\u00E2\u0080\u0099none\u00E2\u0080\u0099,... \u00E2\u0080\u0099Position\u00E2\u0080\u0099,[0.2116 0.3217 0.03858 0.06616]); 150 % Create textbox annotation(figure1,\u00E2\u0080\u0099textbox\u00E2\u0080\u0099,\u00E2\u0080\u0099String\u00E2\u0080\u0099,{\u00E2\u0080\u0099P7\u00E2\u0080\u0099},... \u00E2\u0080\u0099FontName\u00E2\u0080\u0099,\u00E2\u0080\u0099Times New Roman\u00E2\u0080\u0099,... \u00E2\u0080\u0099FitBoxToText\u00E2\u0080\u0099,\u00E2\u0080\u0099off\u00E2\u0080\u0099,... \u00E2\u0080\u0099LineStyle\u00E2\u0080\u0099,\u00E2\u0080\u0099none\u00E2\u0080\u0099,... \u00E2\u0080\u0099Position\u00E2\u0080\u0099,[0.2436 0.2022 0.03858 0.06616]); 151 Program J.6 Import File: Importfile.m function Importfile(fileToRead1) %IMPORTFILE(FILETOREAD1) % Imports data from the specified file % FILETOREAD1: file to read % Import the file rawData1 = importdata(fileToRead1); % For some simple files (such as a CSV or JPEG % files), IMPORTDATA might return a simple array. % If so, generate a structure so that the output % matches that from the Import Wizard. [unused,name] = fileparts(fileToRead1); %#ok newData1.(genvarname(name)) = rawData1; % Create new variables in the base workspace from % those fields. vars = fieldnames(newData1); for i = 1:length(vars) assignin(\u00E2\u0080\u0099base\u00E2\u0080\u0099, vars{i}, newData1.(vars{i})); end 152 Program J.7 Input and Modify Data: Inputdata.m % Importing the data file where the first column % contains the time data with rows equal to the % number of frames. % The second column till the end contains the % concentration data where each column corresponds % to one frame that contains the eight images % sequentially in 2528 rows. % Import data Importfile(\u00E2\u0080\u0099test.txt\u00E2\u0080\u0099); % return rows and columns [Rows Columns]=size(test); % number of frames (starts from second column) frames=Columns-1; % array of frames for f=1:frames fr(f)=f; end % array fo time data for f=1:frames t(f)=test(f,1); end % extract only the concentration data for c=2:frames+1 data(:,c-1)=test(:,c); end % averaging data over z and y to get Cee(x,f) x_planes=7; start_plane=1; 153 end_plane=(x_planes-1)+start_plane; adj=start_plane-1; for f=1:frames for x=start_plane:end_plane x2=x-adj; for i=1+316*(x-1):316*x; j=i-316*(x-1); m(j,x2,f)=data(i,f); end Cee(x2,f)=mean(m(:,x2,f)); end end % normalizing the data after finding Co (initial) % and Cf (final) for each pixel. % average frames before fo and after ff fo=20; ff=250; for x=start_plane:end_plane x=x-adj; for f=1:fo dataCo(x,f)=Cee(x,f); end for f=ff:frames dataCf(x,f-(ff-1))=Cee(x,f); end Co(x,1)=mean(dataCo(x,:)); Cf(x,1)=mean(dataCf(x,:)); end for f=1:frames Cen(:,f)=(Cee(:,f)-Co)./(Cf-Co); end % cropping the data 154 fco=44; fcf=125; frames_c=fcf-fco+1; fr_c=[1:1:frames_c]; for f=1:(fcf-fco+1) Cec(:,f)=Cen(:,((f-1)+fco)); end % flip the data for x=start_plane:end_plane x=x-adj; Cer(x,:)=Cec(((x_planes+1)-x),:); end % transpose of data Ce=Cec\u00E2\u0080\u0099; 155 Program J.8 PDE Solution: Pdetest.m function res=Pdetest(D) global Pe Xm Tm Xe Te Ce Ci frames_c x_planes % Input: % Xf = grid size % Tf = maximum time % np = number of grid points %D=1.8e-4;\ void=0.538; v = 0.00474/void; %m/s L=0.450; %m Pe=(v*L)/D; %Pe=10; ti=0; %s xi=0; %m tf=L/v; %s tow, L/(Q/A), A*L/Q, V/Q xf=L; %m Ti=ti*v/L; Xi=xi/L; Tf=(tf*v/L)*3; Xf=xf/L; np=40; m = 0; X = linspace(Xi,Xf,np); T = linspace(Ti,Tf,np); Tm=T\u00E2\u0080\u0099; Xm=X; 156 Te=(linspace(Ti,Tf,frames_c))\u00E2\u0080\u0099; Xe=linspace(Xi,Xf,x_planes); sol = pdepe(m,@pdex1pde,@pdex1ic,@pdex1bc,X,T); % Extract the first solution component as C(T,X) C = sol(:,:,1); % Interpolate the solution to match our Xe and Te Ci=interp2(Xm,Tm,C,Xe,Te); size(Ci); D % Residuals res=sum(sum(((Ci-Ce).\u00CB\u00862))) % -------------------------------------------------- function [c,f,s] = pdex1pde(X,T,C,DCDX) global Pe c = 1; f = (1/Pe)*DCDX; s = -DCDX; % -------------------------------------------------- function C0 = pdex1ic(X) C0 = 0; % -------------------------------------------------- function [pl,ql,pr,qr] = pdex1bc(Xl,Cl,Xr,Cr,T) global Te Ce Cleft=interp1(Te,Ce(:,1),T); Cright=interp1(Te,Ce(:,end),T); pl = Cl-Cleft; ql=0; pr= Cr-Cright; qr = 0; 157 Appendix K Experimental Data K.1 Pressure Drop Data 158 Condition polypropylene spheres Volume of column (m3) 8:209\u000210\u00003 Mass of polypropylene spheres (kg) 4.4284 Bulk density (kg=m3) 539.48 Polypropylene spheres density (kg=m3) 905.00 Porosity at no flow (fraction) 0.404 Flow rate ph L v ph=L (L=min) (Pa) (m) (mm=s) (Pa=m) 0.00 0.00 0.450 0.00 0.00 1.80 9.51 0.450 1.65 21.14 3.25 19.95 0.450 2.97 44.33 4.67 35.43 0.450 4.27 78.74 5.76 54.65 0.450 5.26 121.44 7.22 74.50 0.450 6.60 165.57 8.57 95.65 0.450 7.83 212.56 10.49 134.21 0.450 9.59 298.25 12.02 160.21 0.450 10.98 356.01 13.79 204.20 0.450 12.60 453.78 14.30 230.61 0.450 13.07 512.46 Table K.1: Pressure drop data of polypropylene spheres. 159 Condition 100% accepts Kappa number 80 Compaction Pressure (kPa) 0.0 Volume of column (m3) 8:209\u000210\u00003 Mass of O.D. cooked chips (kg) 0.8133 Initial cooked bulk density (kg=m3) 99.08 Yield (%) 53.86 Basic chip density (kg=m3) 398.27 Cooked chip density (kg=m3) 214.51 Porosity at no flow (fraction) 0.538 Flow rate ph dp1 dp2 dp3 dp4 L v ph=L (L=min) (Pa) (Pa) (Pa) (Pa) (Pa) (m) (mm=s) (Pa=m) 0.00 0.000 0.000 0.000 0.000 0.000 0.450 0.00 0.00 1.82 30.843 7.679 7.532 7.963 7.669 0.450 1.66 68.54 3.30 69.305 17.378 17.169 17.589 17.170 0.450 3.02 154.01 4.54 112.251 28.103 27.668 28.675 27.805 0.450 4.15 249.45 5.15 135.381 33.902 33.344 34.625 33.510 0.450 4.71 300.85 5.97 169.312 42.353 41.666 43.334 41.960 0.450 5.45 376.25 7.26 232.569 57.975 57.290 59.336 57.968 0.450 6.63 516.82 8.71 315.079 77.825 77.998 79.455 79.802 0.450 7.96 700.18 10.50 424.909 104.776 105.256 106.958 107.918 0.450 9.59 944.24 Table K.2: Pressure drop data of 100% accepts at k = 80 and pc = 0:0 kPa. 160 Condition 100% accepts Kappa number 80 Compaction Pressure (kPa) 5.6 Volume of column (m3) 8:209\u000210\u00003 Mass of O.D. cooked chips (kg) 0.8133 Initial cooked bulk density (kg=m3) 99.08 Yield (%) 53.86 Basic chip density (kg=m3) 398.27 Cooked chip density (kg=m3) 214.51 Porosity at no flow (fraction) 0.462 Flow rate ph dp1 dp2 dp3 dp4 L v ph=L (L=min) (Pa) (Pa) (Pa) (Pa) (Pa) (m) (mm=s) (Pa=m) 0.00 0.000 0.000 0.000 0.000 0.000 0.386 0.00 0.00 1.79 33.334 4.188 9.818 9.473 9.854 0.386 1.63 86.36 3.36 79.962 10.047 23.325 23.063 23.527 0.386 3.07 207.15 4.61 129.646 16.290 37.756 37.521 38.080 0.386 4.21 335.87 5.26 162.766 20.451 47.270 47.284 47.760 0.386 4.81 421.67 5.91 194.347 24.419 56.351 56.558 57.020 0.386 5.40 503.49 7.47 282.346 35.476 81.510 82.620 82.741 0.386 6.82 731.47 8.82 370.377 46.537 106.696 108.723 108.420 0.386 8.06 959.53 10.65 505.173 63.474 145.728 147.644 148.327 0.386 9.73 1308.74 Table K.3: Pressure drop data of 100% accepts at k = 80 and pc = 5:6 kPa. 161 Condition 100% accepts Kappa number 80 Compaction Pressure (kPa) 9.8 Volume of column (m3) 8:209\u000210\u00003 Mass of O.D. cooked chips (kg) 0.8133 Initial cooked bulk density (kg=m3) 99.08 Yield (%) 53.86 Basic chip density (kg=m3) 398.27 Cooked chip density (kg=m3) 214.51 Porosity at no flow (fraction) 0.416 Flow rate ph dp1 dp2 dp3 dp4 L v ph=L (L=min) (Pa) (Pa) (Pa) (Pa) (Pa) (m) (mm=s) (Pa=m) 0.00 0.000 0.000 0.000 0.000 0.000 0.356 0.00 0.00 1.87 47.226 2.402 15.168 15.032 14.624 0.355 1.71 133.03 3.36 114.631 5.701 36.464 36.372 36.094 0.355 3.07 322.90 4.62 190.075 8.861 60.410 60.407 60.398 0.354 4.22 536.94 5.26 234.383 10.928 74.496 74.489 74.470 0.354 4.81 662.10 6.01 294.590 13.643 93.362 93.534 94.051 0.354 5.49 832.18 7.37 417.687 18.128 132.453 132.893 134.213 0.353 6.73 1183.25 8.72 550.894 22.336 174.874 175.661 178.023 0.352 7.97 1565.04 10.46 754.426 28.468 239.825 241.122 245.012 0.351 9.55 2149.36 Table K.4: Pressure drop data of 100% accepts at k = 80 and pc = 9:8 kPa. 162 Condition 100% accepts Kappa number 80 Compaction Pressure (kPa) 14.0 Volume of column (m3) 8:209\u000210\u00003 Mass of O.D. cooked chips (kg) 0.8133 Initial cooked bulk density (kg=m3) 99.08 Yield (%) 53.86 Basic chip density (kg=m3) 398.27 Cooked chip density (kg=m3) 214.51 Porosity at no flow (fraction) 0.372 Flow rate ph dp1 dp2 dp3 dp4 L v ph=L (L=min) (Pa) (Pa) (Pa) (Pa) (Pa) (m) (mm=s) (Pa=m) 0.00 0.000 \u00E2\u0080\u0093 0.000 0.000 0.000 0.331 0.00 0.00 1.86 79.442 \u00E2\u0080\u0093 25.556 26.884 27.003 0.331 1.70 240.01 3.54 205.104 \u00E2\u0080\u0093 65.942 69.446 69.716 0.331 3.23 619.65 4.65 317.405 \u00E2\u0080\u0093 102.064 107.453 107.888 0.331 4.25 958.93 5.26 390.914 \u00E2\u0080\u0093 125.658 132.383 132.873 0.331 4.81 1181.01 6.11 497.348 \u00E2\u0080\u0093 158.833 168.950 169.564 0.330 5.58 1507.11 7.47 684.651 \u00E2\u0080\u0093 218.651 232.577 233.423 0.330 6.83 2074.70 8.83 898.464 \u00E2\u0080\u0093 286.933 305.212 306.320 0.330 8.07 2722.62 10.69 1236.401 \u00E2\u0080\u0093 392.184 421.398 422.818 0.329 9.77 3758.06 Table K.5: Pressure drop data of 100% accepts at k = 80 and pc = 14:0 kPa. 163 Condition 100% accepts Kappa number 53 Compaction Pressure (kPa) 0.0 Volume of column (m3) 8:209\u000210\u00003 Mass of O.D. cooked chips (kg) 0.7614 Initial cooked bulk density (kg=m3) 92.75 Yield (%) 50.35 Basic chip density (kg=m3) 398.27 Cooked chip density (kg=m3) 200.53 Porosity at no flow (fraction) 0.537 Flow rate ph dp1 dp2 dp3 dp4 L v ph=L (L=min) (Pa) (Pa) (Pa) (Pa) (Pa) (m) (mm=s) (Pa=m) 0.00 0.000 0.000 0.000 0.000 0.000 0.450 0.00 0.00 1.89 24.177 6.228 6.522 5.699 5.728 0.450 1.73 53.73 3.41 57.128 14.311 14.965 13.741 14.110 0.450 3.12 126.95 4.41 89.274 21.882 22.960 21.718 22.715 0.450 4.03 198.39 5.21 118.457 28.820 30.240 28.955 30.442 0.450 4.76 263.24 6.01 152.776 36.758 38.590 37.590 39.838 0.450 5.49 339.50 7.65 219.116 53.251 54.789 54.465 56.611 0.450 6.99 486.92 8.99 303.817 72.296 75.184 75.839 80.498 0.450 8.21 675.15 10.85 365.225 90.484 90.707 91.621 92.412 0.450 9.91 811.61 Table K.6: Pressure drop data of 100% accepts at k = 53 and pc = 0:0 kPa. 164 Condition 100% accepts Kappa number 53 Compaction Pressure (kPa) 5.6 Volume of column (m3) 8:209\u000210\u00003 Mass of O.D. cooked chips (kg) 0.7614 Initial cooked bulk density (kg=m3) 92.75 Yield (%) 50.35 Basic chip density (kg=m3) 398.27 Cooked chip density (kg=m3) 200.53 Porosity at no flow (fraction) 0.414 Flow rate ph dp1 dp2 dp3 dp4 L v ph=L (L=min) (Pa) (Pa) (Pa) (Pa) (Pa) (m) (mm=s) (Pa=m) 0.00 0.000 0.000 0.000 0.000 0.000 0.355 0.00 0.00 1.81 50.663 2.355 16.087 16.103 16.118 0.354 1.65 143.12 3.35 121.512 5.304 38.657 38.736 38.815 0.353 3.06 344.23 4.23 187.812 7.538 59.555 60.091 60.627 0.352 3.87 533.56 5.26 286.276 9.716 90.606 92.187 93.767 0.350 4.80 817.93 6.05 372.320 10.562 118.113 120.586 123.059 0.348 5.53 1069.89 7.49 502.892 13.042 160.338 163.283 166.229 0.347 6.85 1449.26 8.80 614.494 12.795 197.684 200.566 203.449 0.345 8.04 1781.14 10.68 788.688 12.383 256.939 258.769 260.598 0.343 9.76 2299.38 Table K.7: Pressure drop data of 100% accepts at k = 53 and pc = 5:6 kPa. 165 Condition 100% accepts Kappa number 53 Compaction Pressure (kPa) 9.8 Volume of column (m3) 8:209\u000210\u00003 Mass of O.D. cooked chips (kg) 0.7614 Initial cooked bulk density (kg=m3) 92.75 Yield (%) 50.35 Basic chip density (kg=m3) 398.27 Cooked chip density (kg=m3) 200.53 Porosity at no flow (fraction) 0.354 Flow rate ph dp1 dp2 dp3 dp4 L v ph=L (L=min) (Pa) (Pa) (Pa) (Pa) (Pa) (m) (mm=s) (Pa=m) 0.00 0.000 \u00E2\u0080\u0093 0.000 0.000 0.000 0.322 0.00 0.00 1.75 103.937 \u00E2\u0080\u0093 32.645 35.907 35.384 0.322 1.60 322.79 2.71 190.417 \u00E2\u0080\u0093 59.833 65.882 64.702 0.321 2.48 593.20 3.46 272.618 \u00E2\u0080\u0093 84.123 94.857 93.638 0.320 3.16 851.93 4.60 435.773 \u00E2\u0080\u0093 133.658 151.794 150.322 0.320 4.21 1361.79 5.12 545.348 \u00E2\u0080\u0093 162.033 191.809 191.507 0.318 4.67 1714.93 6.11 726.022 \u00E2\u0080\u0093 215.351 255.895 254.776 0.317 5.58 2290.29 7.56 1053.386 \u00E2\u0080\u0093 306.981 373.804 372.601 0.315 6.91 3344.08 8.85 1358.608 \u00E2\u0080\u0093 391.578 484.856 482.174 0.313 8.09 4340.60 10.76 1907.267 \u00E2\u0080\u0093 529.375 688.874 689.018 0.310 9.83 6152.47 Table K.8: Pressure drop data of 100% accepts at k = 53 and pc = 9:8 kPa. 166 Condition 100% accepts Kappa number 53 Compaction Pressure (kPa) 14.0 Volume of column (m3) 8:209\u000210\u00003 Mass of O.D. cooked chips (kg) 0.7614 Initial cooked bulk density (kg=m3) 92.75 Yield (%) 50.35 Basic chip density (kg=m3) 398.27 Cooked chip density (kg=m3) 200.53 Porosity at no flow (fraction) 0.280 Flow rate ph dp1 dp2 dp3 dp4 L v ph=L (L=min) (Pa) (Pa) (Pa) (Pa) (Pa) (m) (mm=s) (Pa=m) 0.00 0.000 \u00E2\u0080\u0093 0.000 0.000 0.000 0.289 0.00 0.00 1.86 313.704 \u00E2\u0080\u0093 69.484 122.095 122.125 0.289 1.70 1085.48 2.62 568.513 \u00E2\u0080\u0093 124.458 221.919 222.137 0.288 2.39 1974.00 3.49 887.160 \u00E2\u0080\u0093 194.259 346.229 346.671 0.288 3.19 3080.42 4.72 1430.260 \u00E2\u0080\u0093 309.338 560.042 560.880 0.287 4.31 4983.48 5.22 1735.799 \u00E2\u0080\u0093 370.680 682.024 683.095 0.286 4.77 6069.23 6.02 2341.460 \u00E2\u0080\u0093 493.594 923.155 924.711 0.285 5.50 8215.65 7.57 3292.519 \u00E2\u0080\u0093 675.716 1307.274 1309.529 0.283 6.91 11634.34 8.84 4104.871 \u00E2\u0080\u0093 819.172 1641.445 1644.254 0.281 8.07 14608.08 10.61 5184.375 \u00E2\u0080\u0093 1004.668 2088.250 2091.456 0.279 9.69 18581.99 Table K.9: Pressure drop data of 100% accepts at k = 53 and pc = 14:0 kPa. 167 Condition 100% accepts Kappa number 25 Compaction Pressure (kPa) 0.0 Volume of column (m3) 8:209\u000210\u00003 Mass of O.D. cooked chips (kg) 0.7143 Initial cooked bulk density (kg=m3) 87.02 Yield (%) 47.14 Basic chip density (kg=m3) 398.27 Cooked chip density (kg=m3) 187.74 Porosity at no flow (fraction) 0.536 Flow rate ph dp1 dp2 dp3 dp4 L v ph=L (L=min) (Pa) (Pa) (Pa) (Pa) (Pa) (m) (mm=s) (Pa=m) 0.00 0.000 0.000 0.000 0.000 0.000 0.450 0.00 0.00 1.69 21.018 5.750 5.151 5.105 5.012 0.450 1.55 46.71 3.31 55.310 14.258 13.737 13.697 13.617 0.450 3.02 122.91 4.59 90.150 22.422 22.562 22.573 22.594 0.450 4.20 200.33 5.09 107.360 26.710 26.867 26.879 26.903 0.450 4.65 238.58 6.11 143.125 34.929 35.960 36.039 36.197 0.450 5.59 318.06 7.37 194.336 47.222 48.869 48.996 49.249 0.450 6.73 431.86 8.73 253.038 61.156 63.700 63.895 64.287 0.450 7.97 562.31 10.52 338.593 84.778 84.621 84.609 84.585 0.450 9.62 752.43 Table K.10: Pressure drop data of 100% accepts at k = 25 and pc = 0:0 kPa. 168 Condition 100% accepts Kappa number 25 Compaction Pressure (kPa) 5.6 Volume of column (m3) 8:209\u000210\u00003 Mass of O.D. cooked chips (kg) 0.7143 Initial cooked bulk density (kg=m3) 87.02 Yield (%) 47.14 Basic chip density (kg=m3) 398.27 Cooked chip density (kg=m3) 187.74 Porosity at no flow (fraction) 0.397 Flow rate ph dp1 dp2 dp3 dp4 L v ph=L (L=min) (Pa) (Pa) (Pa) (Pa) (Pa) (m) (mm=s) (Pa=m) 0.00 0.000 0.000 0.000 0.000 0.000 0.346 0.00 0.00 1.79 49.0687 1.089 15.868 15.962 16.149 0.345 1.64 142.23 3.38 124.868 2.405 40.528 40.748 41.188 0.344 3.09 362.99 4.47 194.833 3.182 63.446 63.774 64.431 0.343 4.08 568.03 5.20 267.061 4.343 87.113 87.458 88.147 0.343 4.76 778.60 5.95 353.443 3.665 116.151 116.482 117.144 0.341 5.43 1036.49 7.52 548.924 4.067 181.118 181.494 182.246 0.340 6.87 1614.48 8.63 732.868 1.093 243.226 243.750 244.798 0.338 7.88 2168.25 10.30 1073.822 \u00E2\u0080\u0093 356.665 357.622 359.535 0.336 9.42 3195.90 Table K.11: Pressure drop data of 100% accepts at k = 25 and pc = 5:6 kPa. 169 Condition 100% accepts Kappa number 25 Compaction Pressure (kPa) 9.8 Volume of column (m3) 8:209\u000210\u00003 Mass of O.D. cooked chips (kg) 0.7143 Initial cooked bulk density (kg=m3) 87.02 Yield (%) 47.14 Basic chip density (kg=m3) 398.27 Cooked chip density (kg=m3) 187.74 Porosity at no flow (fraction) 0.316 Flow rate ph dp1 dp2 dp3 dp4 L v ph=L (L=min) (Pa) (Pa) (Pa) (Pa) (Pa) (m) (mm=s) (Pa=m) 0.00 0.000 \u00E2\u0080\u0093 0.000 0.000 0.000 0.305 0.00 0.00 1.75 132.347 \u00E2\u0080\u0093 34.928 48.628 48.791 0.305 1.60 433.92 2.51 217.522 \u00E2\u0080\u0093 56.885 80.181 80.456 0.304 2.29 715.53 3.32 336.120 \u00E2\u0080\u0093 87.061 124.321 124.737 0.303 3.03 1109.31 4.59 557.653 \u00E2\u0080\u0093 144.440 206.263 206.951 0.303 4.19 1840.44 5.10 662.764 \u00E2\u0080\u0093 170.033 245.954 246.778 0.302 4.66 2194.58 6.02 879.482 \u00E2\u0080\u0093 223.443 327.472 328.567 0.301 5.50 2921.87 7.27 1258.020 \u00E2\u0080\u0093 313.275 471.594 473.150 0.299 6.65 4207.42 8.60 1779.281 \u00E2\u0080\u0093 434.018 671.528 673.736 0.297 7.86 5990.85 9.66 2458.271 \u00E2\u0080\u0093 573.885 940.743 943.643 0.293 8.83 8390.00 Table K.12: Pressure drop data of 100% accepts at k = 25 and pc = 9:8 kPa. 170 Condition 100% accepts Kappa number 25 Compaction Pressure (kPa) 14.0 Volume of column (m3) 8:209\u000210\u00003 Mass of O.D. cooked chips (kg) 0.7143 Initial cooked bulk density (kg=m3) 87.02 Yield (%) 47.14 Basic chip density (kg=m3) 398.27 Cooked chip density (kg=m3) 187.74 Porosity at no flow (fraction) 0.250 Flow rate ph dp1 dp2 dp3 dp4 L v ph=L (L=min) (Pa) (Pa) (Pa) (Pa) (Pa) (m) (mm=s) (Pa=m) 0.00 0.000 \u00E2\u0080\u0093 0.000 0.000 0.000 0.278 0.00 0.00 1.87 460.557 \u00E2\u0080\u0093 88.411 185.755 186.391 0.278 1.71 1656.68 2.49 707.437 \u00E2\u0080\u0093 133.707 286.384 287.346 0.277 2.28 2553.92 3.37 1140.454 \u00E2\u0080\u0093 215.504 461.723 463.227 0.277 3.08 4117.16 4.57 1850.151 \u00E2\u0080\u0093 344.055 751.869 754.228 0.276 4.17 6703.45 5.20 2408.650 \u00E2\u0080\u0093 440.570 982.591 985.488 0.275 4.75 8758.73 5.98 3129.978 \u00E2\u0080\u0093 553.488 1286.446 1290.044 0.273 5.47 11465.12 7.26 4484.264 \u00E2\u0080\u0093 764.191 1858.255 1861.818 0.271 6.63 16547.10 8.61 6524.210 \u00E2\u0080\u0093 1069.980 2725.389 2728.841 0.269 7.86 24253.57 Table K.13: Pressure drop data of 100% accepts at k = 25 and pc = 14:0 kPa. 171 K.2 Axial Dispersion Data Limiting Case k m pc ph L ec d g De Pe r2 (kPa) (kPa) (kPa) (m) (m2=s) pc\u001D ph 80 115 5.6 0.163 0.386 0.462 0.028 0.008 9:09\u000210\u00003 0.43 0.997 80 115 9.8 0.234 0.351 0.408 0.023 0.011 2:01\u000210\u00002 0.20 0.999 80 115 14.0 0.391 0.330 0.370 0.027 0.016 3:78\u000210\u00002 0.11 0.999 53 78 5.6 0.286 0.343 0.414 0.049 0.017 9:18\u000210\u00003 0.42 0.997 53 78 9.8 0.545 0.309 0.354 0.053 0.030 3:31\u000210\u00002 0.12 0.999 53 78 14.0 1.736 0.277 0.280 0.110 0.078 9:64\u000210\u00002 0.05 0.999 25 67 5.6 0.267 0.336 0.379 0.046 0.019 3:40\u000210\u00002 0.12 0.999 25 67 9.8 0.663 0.291 0.283 0.063 0.040 1:02\u000210\u00001 0.05 1.000 25 67 14.0 2.409 0.267 0.225 0.147 0.116 2:16\u000210\u00001 0.03 0.999 m\u001D ph 80 115 0.0 0.135 0.450 0.538 1.000 0.007 3:72\u000210\u00003 1.06 0.989 53 78 0.0 0.118 0.450 0.537 1.000 0.008 3:17\u000210\u00003 1.24 0.992 25 67 0.0 0.107 0.450 0.536 1.000 0.009 3:25\u000210\u00003 1.21 0.992 Table K.14: A summary of the experimental conditions tested with estimates of the axial dispersion coefficients with correlation coefficients r2 for 100% accepts wood chips. 172 Appendix L Axial Dispersion Breakthrough Curves 173 0 0.5 1 1.5 2 2.5 3 0 0.2 0.4 0.6 0.8 Dimensionless Time D im en si on le ss C on ce nt ra tio n Measured Model P1 P3 P2 P4 P5 P7 P6 Figure L.1: Axial dispersion breakthrough curves for k = 80 and pc = 0:0 kPa. 174 0 0.5 1 1.5 2 2.5 3 0 0.2 0.4 0.6 0.8 Dimensionless Time D im en si on le ss C on ce nt ra tio n Measured Model P2 P3 P4 P5 P6 P7 Figure L.2: Axial dispersion breakthrough curves for k = 80 and pc = 5:6 kPa. 175 0 0.5 1 1.5 2 2.5 3 0 0.2 0.4 0.6 0.8 Dimensionless Time D im en si on le ss C on ce nt ra tio n Measured Model P3 P4 P7 P5 P6 Figure L.3: Axial dispersion breakthrough curves for k = 80 and pc = 9:8 kPa. 176 0 0.5 1 1.5 2 2.5 3 0 0.2 0.4 0.6 0.8 Dimensionless Time D im en si on le ss C on ce nt ra tio n Measured Model P6 P7 P5 P4 P3 Figure L.4: Axial dispersion breakthrough curves for k = 80 and pc = 14:0 kPa. 177 0 0.5 1 1.5 2 2.5 3 0 0.2 0.4 0.6 0.8 Dimensionless Time D im en si on le ss C on ce nt ra tio n Measured Model P1 P2 P3 P4 P6 P5 P7 Figure L.5: Axial dispersion breakthrough curves for k = 53 and pc = 0:0 kPa. 178 0 0.5 1 1.5 2 2.5 3 0 0.2 0.4 0.6 0.8 Dimensionless Time D im en si on le ss C on ce nt ra tio n Measured Model P6 P4 P5 P3 P2 P7 Figure L.6: Axial dispersion breakthrough curves for k = 53 and pc = 5:6 kPa. 179 0 0.5 1 1.5 2 2.5 3 0 0.2 0.4 0.6 0.8 Dimensionless Time D im en si on le ss C on ce nt ra tio n Measured Model P7 P4 P3 P5 P6 Figure L.7: Axial dispersion breakthrough curves for k = 53 and pc = 9:8 kPa. 180 0 0.5 1 1.5 2 2.5 3 0 0.2 0.4 0.6 0.8 Dimensionless Time D im en si on le ss C on ce nt ra tio n Measured Model P6 P5 P7 P4 Figure L.8: Axial dispersion breakthrough curves for k = 53 and pc = 14:0 kPa. 181 0 0.5 1 1.5 2 2.5 3 0 0.2 0.4 0.6 0.8 Dimensionless Time D im en si on le ss C on ce nt ra tio n Measured Model P7 P8 P6 P5 P4 P3 P2 P1 Figure L.9: Axial dispersion breakthrough curves for k = 25 and pc = 0:0 kPa. 182 0 0.5 1 1.5 2 2.5 3 0 0.2 0.4 0.6 0.8 Dimensionless Time D im en si on le ss C on ce nt ra tio n Measured Model P5 P7 P6 P4 P3 Figure L.10: Axial dispersion breakthrough curves for k = 25 and pc = 5:6 kPa. 183 0 0.5 1 1.5 2 2.5 3 0 0.2 0.4 0.6 0.8 Dimensionless Time D im en si on le ss C on ce nt ra tio n Measured Model P7 P6 P5 P4 Figure L.11: Axial dispersion breakthrough curves for k = 25 and pc = 9:8 kPa. 184 0 0.5 1 1.5 2 2.5 3 0 0.2 0.4 0.6 0.8 Dimensionless Time D im en si on le ss C on ce nt ra tio n Measured Model P6 P7 P5 P4 Figure L.12: Axial dispersion breakthrough curves for k = 25 and pc = 14:0 kPa. 185 Bibliography Comiti, J. and M. Renaud, \u00E2\u0080\u009DA New Model for Determining Mean Structure Pa- rameters of Fixed Beds from Pressure Drop Measurements: Application to Beds Packed with Parallelepipedal Particles,\u00E2\u0080\u009D Chem. Eng. Sci. 44(7), 1539- 1545 (1989). Coulson, J.M., J.F. Richardson, J.R. Backhurst and J.H. Harker,Chemical Engi- neering, A. Wheaton & Co. Ltd Great Britain, Vol. 2, 3rd Edition, 8-12 (1987). Gullichsen, J., H. Kolehmainen and H. Sundqvist, \u00E2\u0080\u009DOn the nonuniformity of the kraft cook,\u00E2\u0080\u009D Paperi Ja Puu 74(6), 486-490 (1992). 186"@en .
"Thesis/Dissertation"@en .
"2011-11"@en .
"10.14288/1.0059100"@en .
"eng"@en .
"Chemical and Biological Engineering"@en .
"Vancouver : University of British Columbia Library"@en .
"University of British Columbia"@en .
"Attribution-NonCommercial-NoDerivatives 4.0 International"@en .
"http://creativecommons.org/licenses/by-nc-nd/4.0/"@en .
"Graduate"@en .
"Characterizing the permeability and dispersion of flows through compressible wood-chip beds"@en .
"Text"@en .
"http://hdl.handle.net/2429/35983"@en .