Modelling Heat Transfer in t h e Conditioning Process Used in P l y w o o d Manufacturing Dino A. Gigliotti BSc, University of Northern British Columbia, 2002 Thesis Submitted In Partial Fulfillment Of The Requirements For The Degree Of Master Of Science in Mathematical, Computer and Physical Sciences (Physics) University of Northern British Columbia March 2008 © Dino A. 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Conformement a la loi canadienne sur la protection de la vie privee, quelques formulaires secondaires ont ete enleves de cette these. While these forms may be included in the document page count, their removal does not represent any loss of content from the thesis. Bien que ces formulaires aient inclus dans la pagination, il n'y aura aucun contenu manquant. Canada Abstract As part of an industrial process in the forestry industry, pre-cut segments of debarked logs are conditioned by placing the wood segments in a warm water shower. Simplistic models which assume a constant moisture content have previously been used to predict conditioning time. However, as is often the case in reality, the complexity of the problem is such that more accurate modelling is desired. Theoretical model equations are presented, and the problem is then simplified through some basic assumptions to pose a set of coupled differential equations for thermal energy and moisture transport in one spatial dimension. A finite difference numerical method is used to solve the initial boundary value problem which is then compared to internal time-dependent experimental temperature data of a log segment collected of over a full conditioning cycle. Limitations of the model are identified and discussed. 11 Contents Abstract ii Contents iv List of Tables vii List of F i g u r e s viii Acknowledgment xii 1 2 Introduction 1 1.1 Wood Structure 2 1.2 Defining Moisture Content 4 1.3 Conditioning Process 5 1.4 Approach 7 Theory 10 2.1 Introduction 10 2.2 Diffusive Approach 10 2.3 Continuum Mechanics Approach 12 2.3.1 Whitaker Model of General Porous Media 13 2.3.2 Coupling the Phases 17 iv 3 W o o d Theory 43 3.0.3 45 3.1 4 5 Constants and Wood Independent Coefficients Mechanistic Model 45 3.1.1 46 Structure Dependent Parameter Development 3.2 Capillary Pressure 49 3.3 Bulk Moisture Transfer and Permeability 54 3.3.1 Specific Liquid Permeability 55 3.3.2 Pit Aspiration 56 3.3.3 Relative Liquid Permeability 58 3.4 Conductive Thermal Energy Transfer 59 3.5 Diffusive Moisture Transfer 62 3.5.1 Discontinuity in Moisture Transfer 65 3.6 Liquid, Gas, Capillary Pressure Gradients 66 3.7 One Dimensional Radial Transport Equations 70 Boundary Conditions 74 4.1 Introduction 74 4.2 Energy and Mass Balance at the Surface 75 4.3 Inner Boundary Conditions 76 4.4 Outer Boundary (Inert Gas/Vapour Mixture) 77 4.4.1 Thermal Energy Transfer 77 4.4.2 Mass Transfer 80 Numerical M e t h o d 83 5.1 Discretization of Thermal Energy and Moisture Equations 84 5.2 Discretization of Boundary Conditions 85 5.3 Numerical Stability 87 6 7 Experimental D a t a 91 6.1 Experimental Setup 91 6.1.1 Thermal Data Collection 92 6.1.2 Moisture Content Data Collection 95 6.1.3 Experimental Errors 98 Results 104 7.1 Temperature Data Analysis 104 7.1.1 Predicted Versus Real Conditioning Cycle Times 104 7.1.2 Post Conditioning Cooling 105 7.1.3 Comparison of Model to Experimental Time Dependent Thermal Data 7.2 108 Experimental Moisture Content Data 109 7.2.1 Average Moisture Content During Conditioning . . . . . . . . . 109 7.2.2 Analysis of Moisture Content with Respect to Sapwood and Heartwood 7.2.3 7.3 7.4 8 110 Long Term Moisture Content Changes in Cylindrical Log Segments 110 Model Deficiencies 114 7.3.1 Frozen Wood 114 7.3.2 Longitudinal Transfer 117 Sensitivity Analysis 117 Summary and Conclusion 122 A Sensitivity Analysis 131 B Model t o Experimental Results Overlays 146 List of Tables 1.1 Optimal peeling temperatures 7 3.1 Thermal Conductivities, [23] 62 5.1 Initial conditions and parameters used in stability analysis 88 6.1 Comparison moisture content measurement methods 97 7.1 Experimental versus predicted conditioning times. Values given here correspond to the core (r=0) 7.2 106 Average moisture contents of cylindrical log segments before and after conditioning 109 7.3 Sapwood and Heartwood Moisture Content 110 7.4 Parameters used in sensitivity analysis. (T0 = 1.0°C, and T ^ = 40.0°C) 118 vn List of Figures 1.1 Inside fully loaded conditioning tunnel at North Central Plywood. . . 2.1 Liquid-vapour/inert gas interface. SOURCE: [28] 18 3.1 Simplification of wood structure used in the mechanistic model 47 3.2 Dimensions of tracheid taper by applying the mechanistic model. SOURCE: [24] 3.3 48 (A) Force imbalance at water-air interface in small cylindrical tube. (B) Water-air interface at equilibrium in small cylindrical tube 3.4 6 51 (A) Constant capillary pressure phase. (B) Critical saturation reached Scr, water begins to recede into the taper of the lumen, reaching the saturation dependent phase of the capillary pressure. (C) Irreducible saturation reached Si, water has recede into the funicular or discontinuous state, [24] 52 3.5 Principal radii, r\, and r 2 , shrinking as meniscus recedes 54 3.6 (A) Liquid from neighboring pit has receded past the pit aperture. (B) Meniscus has receded past the pit aperture. (C) Meniscus has receded to small openings in the membrane. (D,E) Capillary pressure has pulled the torus into the pit aperture. (F) Capillary pressure has pulled torus into disk shape, resulting in a completely aspirated pit. SOURCE: [23] viii 57 3.7 Cross section of a unit tracheid segment used in developing the thermal conductivity circuit illustrated in Figure 3.8 59 3.8 Resistor network equivalence of thermal conductivity 60 4.1 Boundary fluxes during heating 76 4.2 Illustration of the void space, shown in grey, created between the logs when stacked in the conditioning tunnel. The large circles represent the ends of the logs, and the small inner circle represents the hydraulic diameter D0 78 5.1 Fictitious point example 85 5.2 Temperature vs. Time plots for varying temporal step sizes 89 6.1 (A) Finished apparatus ready to accept another row of log segments. (B) Finished internal thermal sensor placement in log segment 93 6.2 Approximate radial placement of thermal sensors 94 6.3 (A) Drilling log segment to prepare for insertion of temperature sensors. (B) Cross section of thermal sensor insertion technique 95 7.1 Temperature of spruce sample after conditioning cycle is complete. . . . 107 7.2 Temperature of spruce sample after conditioning cycle is complete. . . . 108 7.3 Spruce, sample 1, long term moisture content change Ill 7.4 Spruce, sample 2. long term moisture content change 112 7.5 Spruce, sample 3, long term moisture content change 112 7.6 Pine, sample 1, long term moisture content change 113 7.7 Blue stain pine (beetle kill), sample 1, long term moisture content change. 113 7.8 Example of model overestimating temperature for frozen wood 7.9 Deviation from the average core heat-up time versus normalized parameter values 116 120 B.l Apr. 2004, Douglas Fir, 2=10cm, R =15.5cm, T 0 =9.5°C, T 0O =48.8°C, M o =0.76 147 B.2 Apr. 2004, Douglas Fir, z=30cm, i?=15.5cm, T 0 =7.1°C, T 00 =48.8°C, M 0 =0.76 148 B.3 Apr. 2004, Douglas Fir, z=128cm, #=15.5cm, TJ)=7.70C, 7^=48.8°C, Af0=0.76 B.4 June 2004, BK Pine, z=10cm, ft=15.0em, TQ=23.3°C, 149 T^bl.VG, MQ=0.37 150 B.5 June 2004, BK Pine, z=30cm, fl= 15.0cm, T 0 =21.0°C, Too=51.90C, M 0 =0.37 151 B.6 June 2004, BK Pine, z=128cm, i?=15.0cm, T 0 =21.9°C, T 00 =51.9°C ) M o =0.37 152 B.7 July 2004, BK Pine, z=10cm, i?-14.5cm, T 0 =20.9°C, T 0O =45.2°C, M 0 =0.50 153 B.8 July 2004, BK Pine, z=30cm, JR=14.5cm, r 0 =19.0°C, T 00 =45.2°C, M 0 =0.50 154 B.9 July 2004, BK Pine, z=128cm, 7?=14.5cm, T 0 =19.0 o C, T(X3=45.2°C, M 0 =0.50 B.10 Dec. 2004, Spruce, z=10cm, i2=17.0cm, T 0 =6.8°C, T 00 =43.3°C, M 0 =0.64 B . l l Dec. 155 156 2004, Spruce, z=30cm, i?=17.0cm, T 0 =2.0 o C, T 00 =43.3°C, M 0 =0.64 157 B.12 Dec. 2004, Spruce, «=128cm, i?-17.0cm, T 0 =0.1°C, T 0 0 -43.3°C, Af0=0.64 158 B.13 Jan. 2005, Pine, 2= 10cm, iJ=18.5cm, T 0 =4.5°C, T oc =53°C, Af0=0.49 . 159 B.14 Jan. 2005, Pine, z=30cm, R=18.5cm, T 0 =2.5°C, T 0O =53°C, M o =0.49 . 160 B.15 Jan. 2005, Pine, 2=128cm, R=18.5cm, T0=2.2°C, T00=53°C, M0=0.49 161 B.16 Feb. 2005, Pine, z=10cm, R=l8.5cm, 71,=1.10C, 7^=45°C, M Q = 0 . 4 9 . 162 B.17 Feb. 2005, Pine, z=30cm,fl=18.5cm,T0=1.4°C, T0C=45°C, M0=0.49 . 163 B.18 Feb. 2005, Pine, *=128cm,fl=18.5cm,T0=0.1°C, roo=45°C, M0=0.49 164 B.19 Mar. 3, 2005, Spruce, ^=10cm, i2=23.5cm, T0=2A°C, 7^=44.7°C, M0=1.49 165 B.20 Mar. 3, 2005, Spruce, z=30cm,fi=23.5cm,T0=2.8°C, T0O=44.7°C, M0=1.49 •. 166 B.21 Mar. 3, 2005, Spruce, z=128cm, i?=23.5cm, T0=2.8°C, T00=44.7°C, M0=1.49 167 B.22 Mar. 7, 2005, Spruce, 2=10cm, JR=23.5cm, T 0 = 10.5°C, T0O=51.5oC, M0=1.49 168 B.23 Mar. 7, 2005, Spruce, z=30cm, i?=23.5cm, T0=8.8°C, T0O=51.5°C, Af0=1.49 169 B.24 Mar. 7, 2005, Spruce, £=128cm, i2=23.5cm, r0=8.8°C, roo=51.5°C) M0=1.49 170 B.25 Mar. 22, 2005, Spruce, 2=10cm, /fc=23.5cm, T0=11.2°C, TOC=52.0°C, M0=1.49 171 B.26 Mar. 22, 2005, Spruce, z=30cm,fl=23.5cm,T0=9.2°C, Too=52.0°C, M0=1.49 172 B.27 Mar. 22, 2005, Spruce, 2=128cm, i?=23.5cm, TQ= 5.9°C, TOC=52.0°C, Af0=1.49 173 Acknowledgment I would like to thank the National Science and Engineering Research Council of Canada and North Central Plywoods (NCP) for there financial support in the form of an Industrial Postgraduate Scholarship. I would also like to thank, Garry Vinje, Nicholas Finch and Terry Breen of North Central Plywoods for their assistance in organizing data collection. I would like to extend a special thanks to Garry Vinje for his work in initiating the collaboration with NCP, and taking the time to design and facilitate construction of part of the data acquisition apparatus. I would like to thank my supervisors, Patrick Montgomery and Patrick Mann, for their enthusiasm and insightful comments during weekly morning meetings. I would also like to extend a special thanks to Dr. Montgomery for his role in initiating the collaboration with NCP, and for organizing an opportunity to present my thesis work at an international conference. In addition, I would like to extend a special thanks to committee member Ian Hartley for his assistance in the data collection process and for the use of his laboratory and equipment. Finally, I would like to thank my parents Quinto Gigliotti and Josie Gigliotti, and my brothers Dario Gigliotti and Davide Gigliotti, for supporting and encouraging me in my academic endeavors, and for being there whenever I need them. xii Chapter 1 Introduction The problem studied herein arises from an industrial application in the manufacturing of plywood. Of direct concern to industry partner, North Central Plywoods (NCP), is finding a method to predict the amount of time required to warm pre-cut and debarked cylindrical wooden log segments to a desired production temperature. This process is known as conditioning and can be achieved in a number of ways, for example, by submersing the segments in a hot water bath or by exposing them to hot steam. Currently at NCP, this is achieved by placing the wood segments in a shower of warm water for a pre-determined amount of time before moving the segments on to the next stage in production. Over-conditioning results in wasted time and energy, while under-conditioning leads to a lower quality final product. The specific problem is therefore to model the heating of a cylinder of wood through partial contact of its outer boundary with very humid air and/or liquid water. A similar problem was studied in the past by Steinhagen [26], [25], whose approach was to simplify the process to a radial heat conduction problem, at r or or where k is the thermal conductivity coefficient of the wood, c is the specific heat, 1 CHAPTER 1. INTRODUCTION 2 p is the density, r is the radius, t is the time, and T is the temperature within the wood segment. Steinhagen reported a relatively good fit between theory and experiment, stating that all computed results were within 10% of the experimental values. The extremely humid conditions encountered during the conditioning process suggests the logs may undergo changes in moisture content, and therefore a more general theoretical description should be considered. The sapwood moisture content of some softwoods can be 200% or more when in a green state. This means that, by weight, there is twice as much water in the sample than solid wood matrix. It therefore stands to reason that when heating the surface, not only will the thermal conductivity of the wood have a strong dependence on the moisture content, but the temperature gradients should give rise to pressure gradients, which will result in the movement of moisture in the form of vapour or liquid within the solid wood matrix. One of the goals of this research is to analyze not only thermal energy transfer during conditioning but also moisture transfer, and therefore the model proposed by Steinhagen will be insufficient in describing this coupled process. 1.1 Wood Structure A description of moisture transport in wood requires an understanding of the structure of wood itself. Like most biological material, wood can be classified as a porous solid [4], [16], [17], [23], and [24]. A porous material consists of a solid matrix which surrounds void space. The cell structure of porous solids falls under two main subcategories: open cell and closed cell. The void spaces in an open cell matrix are interconnected allowing fluid to flow through the porous matrix. The rate at which fluid is able to flow through a material is measured as permeability. The void spaces in closed cell porous materials are not interconnected and do not allow any fluid flow CHAPTER 1. INTRODUCTION 3 through the porous matrix, and are therefore referred to as impermeable. Wood is open celled and made up of long, hollow, straw-like cells called tracheitis, which are interconnected by small valve like openings known as pits, [9], [22], and [23]. The cell structure allows water and nutrients to move throughout the tree, although in softwoods only the outer 20-50% of the trunk is used for this process. This outer portion is known as sapwood and is the only truly live portion of the trunk. The inner portion of the trunk, known as heartwood, is left only to provide structural stability to the tree. Heartwood initially starts as sapwood but as new layers of wood grow at the surface older layers of sapwood turn to heartwood. In most softwoods, heartwood tends to take on a darker colour allowing for a visual distinction to be made. Physically, the only difference between sapwood and heartwood is the green moisture content. Sapwood tends to contain a much higher moisture content then the heartwood at the time of harvest, [9], [22], and [23]. Throughout the growing season new layers of wood are added to the surface; which can be seen as concentric circles expanding outwards in a circular pattern from the center. The layers are known as growth rings and are broken up into two distinct parts: early-wood and late-wood [23]. The wood added at the beginning of the growing season is known as early-wood and the wood added in the later part of the season is known as late-wood. Physically, early-wood and late-wood differ in the thickness of the cell walls, early-wood has slightly thinner cell walls, and late-wood has slightly thicker cell walls, resulting in a difference in density of the solid matrix. Wood is hygroscopic, meaning that its porous matrix forms a weak chemical bond with polar molecules such as water. The amount of bound water that can be absorbed by the solid wood matrix is limited, and depends on species, temperature, and humidity. Being chemically bonded to the cells' surface, hygroscopically bound water is the slowest to move through the solid matrix, and therefore tends only to be of CHAPTER 1. 4 INTRODUCTION interest during kiln drying of lumber. The fiber saturation point is defined by Siau [23] as the point at which physical properties, such as swelling, shrinkage, mechanical strength, heat of wetting, or electrical resistance begin to change with respect to moisture content. It is more loosely described as the point at which no more free water exists within the porous matrix and the cell structure is saturated with bound water. Fiber saturation point is often abbreviated in the literature as FSP, and therefore the moisture content at the fiber saturation point is denoted as 1.2 MFSP- Defining Moisture Content Moisture content of a porous material refers to the concentration of the saturating fluid, including hygroscopieally bound water within the porous matrix. Moisture content is defined as the following mass fraction, M(Moisture Content) = m g r e e n ~ m ° v e n dry , m (1.2) 'Oven dry where Trigj-een is the mass of the sample at the time of harvest, and moven dry is the mass of the sample after being dried in an oven for a sufficient amount of time to remove all moisture [23]. This results in a dimensionless quantity that ranges from 0 to well above 2 in the case of materials with large void volumes. Moisture content is often given as a percentage, understood in the framework of equation 1.2. Another dimensionless quantity often used when dealing with non-hygroscopic materials or in cases where the moisture content remains above the hygroscopic range, is called the saturation. Saturation is defined in a manner similar to moisture content because it is a mass fraction but differs in that it describes the mass of free fluid within some volume element of the material divided by the mass of maximum free fluid that can fit within the porous structure of the material, and therefore takes on CHAPTER 1. INTRODUCTION 5 a value between 0 and 1 ([24]). For wood, saturation is defined as, o "^green ^"saturated ^bound Abound ^ o v e n dry /-• „\ ^ o v e n dry Moisture content is more widely used since it is difficult to distinguish between hygroscopic and free fluid mass when making a measurement. Alternatively, saturation tends to be used in theoretical calculations, and therefore it is often necessary to map between both quantities. Saturation can be calculated from moisture content in the following way ([24]), Where A M is a function of the void volume fraction given by, Where the subscripts a and (5 represent the solid and liquid phases respectively, and p is the density. Since the solid phase of the saturating liquid is not considered in this study, the solid phase denoted as a will refer to the cell wall material of the wood only. It is also useful to note that what remains on the right side of equation (1.5) after dividing by the void volume is the bulk specific gravity. This is a useful quantity since the bulk specific gravity is a more widely used experimental value than the cell wall density pa, and is generally easier to determine. 1.3 Conditioning Process The plywood manufacturing process involves laminating together thin sheets of wood called veneer. Veneer is approximately 3 mm thick and is cut from 2 m log segments using a lathe. In order for the lathe to operate efficiently the fibers and knots must be softened through heating, which allows the lathe to work at faster speeds, require less maintenence, and consume less power. In addition, softening results in less breakage, CHAPTER 1. INTRODUCTION 6 less wasted wood, and a smoother, higher quality veneer which requires less postprocessing such as sanding. The conditioning process at North Central Plywoods (NCP) is acheived by pouring water at approximately 40-60°C over the log segments. By using heated water instead of heated air, the internal temperature of the timber segments can be raised without drying the wood. At NCP the conditioning chests consist of long cement tunnels approximately 30 m long by 2 m wide by 3 m high. The log segments are stacked horizontally and sprinklers are spaced approximately 2 m apart along the center of the ceiling. Figure 1.1 shows the orientation of logsegments in a typical conditioning tunnel. Currently, conditioning times range from 7 to 72 hours depending on species of wood. A complete conditioning cycle occurs Figure 1.1: Inside fully loaded conditioning tunnel at North Central Plywood. when the temperature at the center of the log segment reaches the optimal cutting temperature, which are listed in Table 1.1. CHAPTER 1. INTRODUCTION Species Douglas Fir Pine Blue Stain Pine1'2 Spruce 7 Temp. Acceptable Range 43 °C 32 - 60 °C 43 °C 21 - 71 °C 43 °C 21 - 71 °C 21 °C 10 - 38 °C Table 1.1: Optimal peeling temperatures. 1.4 Approach It is hoped that the model developed in this thesis will allow mill operators to predict average conditioning times for batches of logs, and be used as a research tool for the mill. Initially the problem seems to be one of non-isothermal saturation of a porous media, however under closer inspection it is revealed to be one of non-isothermal drying of porous media. Under normal circumstances, after harvesting, wood will begin to lose moisture through surface evaporation. The rate at w7hich this occurs depends on whether or not the bark has been removed, the species of wood, and ambient conditions such as temperature and humidity [23]. Fortunately modelling non-isothermal drying of porous media is well studied as it applies to kiln drying, ([13], [24], [14], [17], [16], [19], and [24]). Kiln drying is the process used to reduce the moisture content of dimensional lumber to approximately 15% at an accelerated rate. This is done to stabilize the lumber and keep it from warping significantly when in use. The majority of studies modelling thermal energy and moisture transfer in porous media have been established from a set of equations derived by Whitaker [28]. Whitaker completed a thorough characterization of the transport of energy and mass in general porous media from first principles. The set of governing equations developed by Approximation of optimal peeling temperature. 2 Also referred to as Beetle Kill Pine or BK Pine. CHAPTER 1. INTRODUCTION 8 Whitaker was later applied to the drying of dimensional lumber by Spolek [24], and Plumb [14]. Although similar, a number of differences exist between the drying model produced by Spolek and what is required in this study. For instance, Spolek [24] examined rectangular lumber while this thesis focuses on large cylindrical log segments. Additionally, in the Spolek study, drying to very low moisture content, well below the fiber saturation point, was of interest, whereas the current study deals almost exclusively at, or above, the fiber saturation point. The approach taken in this study will therefore be to apply the theory from an existing model such as Spolek [24], and Whitaker [28]. This is done in chapters 2 and 3. In order to ensure the accuracy of the model, results will be compared to timedependent experimental data. It was decided that the data would be collected from within the operating mill environment compared to a laboratory for two reasons. First, it would give mill operators a clear picture of conditions in the conditioning tunnels such as average water temperature over a real conditioning cycle. Until now this information w7as not available to NCP, and could be valuable on its own in optimizing the conditioning process. Second, it wrould allow a direct comparison of model data to data collected from actual conditioning log segments. Ideally, both data from carefully controlled laboratory experiments and data taken directly from the conditioning tunnel would be available to test the model. Unfortunately this was not possible due mainly to time constraints, and it was decided that data from on-site would provide more usable information then laboratory data. Further discussion on the experimental data collection is given in chapter 6. Another focus of this study is to investigate the heating and mass transfer boundaries at the surface of the logs during conditioning. The porous nature of wood means that both thermal energy and mass (moisture) will be transferred across the surface. Chapter 4 is devoted to examining the quantification of these boundary conditions. CHAPTER 1. INTRODUCTION 9 In addition, chapter 5 contains a description of the finite difference numerical technique developed to solve the equations for thermal energy and moisture transport. The results are portrayed along with experimental data in chapter 6 and 7. Chapter 2 Theory 2.1 Introduction As described in the introduction, this thesis considers the transfer of both thermal energy and moisture in wood during the conditioning process. Fortunately, the coupled process of heating and moisture transfer in wood has been well studied as it applies to the high temperature drying of rectangular lumber used in construction ([4], [6], [12], [24], [18], [19], [21], [24], and [29], among others). These studies have slightly different approaches to modelling the problem of wood drying, but all are based on either a diffusive approach or one developed from continuum mechanics. The objective of this chapter is to briefly describe the standard approaches from the literature. 2.2 Diffusive Approach Earlier models tend to be diffusion-based models where thermal energy and moisture transfer is modelled by directly applying Fick's law of mass transfer and Fourier's law of heat conduction, ^ = V-DmVM, 10 (2.1) CHAPTER 2. THEORY 11 and % = V • k^T. (2.2) In equations (2.1) and (2.2). kq is the heat conduction coefficient, and coupling between thermal energy and moisture transfer is achieved through the molecular diffusion Dm and thermal diffusion kq coefficients. This particular approach is applied by Horacek [8] and reports reasonable results although does not provide comparison to experimental data to justify the conclusion. By assuming constant temperature, equation (2.1) was applied on its own to wood drying by Baronas et al. [18], Liu et al. [12], and Cheng et al. [29] with varying levels of success. This simplification was made on the basis that moisture transfer within the porous matrix was much slower then heating. An expansion of the simple diffusion model, initially developed by Luikov [13] to describe energy and mass transport in the mixing of a binary gas, was subsequently applied to wood by Siau [23], fr = v • L« ( | ) V T - L^VM > (2-3) and - ^ = V • Lmq (-) VT - LmmVM , (2.4) where \i is chemical potential of the infiltrating fluid, and Lmq, Lqm, Lmm, and Lqq are referred to as kinetic coefficients. This approach is based on Osanger's theorem and the principles of irreversible thermodynamics by considering the increase of entropy of the system, resulting in what is essentially a generalization of equations (2.1) and (2.2) with more explicit representation of the coefficients. This model was applied to wood drying by Avramidis et al. [19], where a combination of theoretical and empirical expressions were used for the coefficients of proportionality. Theoretical moisture content and core temperatures were compared to experiment measurements and were reported to match reasonably well. CHAPTER 2. THEORY 12 Combined, equations (2.1) and (2.2) are considered to be a macroscopic approach to the problem, where the physical mechanisms such as capillary pressure and permeability are lumped into diffusion coefficients. The result is that the coefficients are defined as relatively complex empirical functions of temperature and moisture content and rely on many accurate experiments to be conducted to fine tune them rather than on a theoretical derivation. When dealing with cases involving different species, beetle infested wood, frozen wood and different drying conditions, the volume and accuracy of experimental data required to estimate the parameters is problematic. In addition, Whitaker [28] notes that the diffusive models of wood drying are somewhat accurate below FSP only but above FSP become inaccurate. 2.3 Continuum Mechanics Approach Aside from the absence of capillary, or bulk mass transfer, another problem in applying the previously discussed models is the lack of physical justification. More appealing, and seemingly more accurate, is to work from a continuum mechanics approach such as the ones described by Luikov [13], and Whitaker [28], to model thermal energy and moisture transfer in wood. A major difference in these approaches compared to the models previously discussed is that a differentiation between gas/vapour, liquids, and solid is made. In the review by Whitaker [28], previous models such as the diffusive models described earlier, are deemed to be unsatisfactory in accurately describing energy and mass transport in porous media below FSP and even less accurately above FSP. Whitaker argues that diffusive models are only accurate in late stages of drying, well after the fiber saturation point when capillary forces can be safely ignored. Although very similar to Luikov's approach, the model developed by Whitaker gives a much more rigorous derivation of the equations governing energy and mass transfer CHAPTER 2. THEORY 13 in porous media, resulting in more thorough and convincing arguments, and therefore offers the most reliable approach in this case. Whitaker's model was applied to the drying of dimensional (rectangular) lumber by Spolek [24], who reported a good match with experimental moisture and temperature profiles over time. In this study, what will be referred to as Spolek's model will be used as a basis for developing a set of equations to describe thermal energy and moisture transfer in cylindrical log segments during the conditioning process. A brief review of Whitaker's general theory of energy and mass transfer in porous media will be given in order to give the reader a clear understanding of the origins of the final governing equations utilized by Spolek, and to give meaning to the notations to be used. 2.3.1 Whitaker Model of General Porous Media Whitaker begins by considering the continuity of mass equation for a fluid of mass density p and velocity v, § j + V - p v = 0, (2.5) along with the ith species continuity equation, ^ + V • pm = ri. (2.6) In equation (2.6) r^ represents the mass rate of production of the i%th species (different elements of the gas phase) owing to chemical reaction. In addition, the thermal energy equation for enthalpy (h), Dh Pj^ ,_, = -V-q + $. . „. (2.7) In equation (2.7) q is the thermal energy, and <& is energy produced or absorbed internally. It is also assumed in equation (2.7) that compressive work and viscous dissipation are negligible in all phases. Whitaker uses (2.5) - (2.7) to pose a set of equations for each phase individually consisting of a continuity equation, species CHAPTER 2. 14 THEORY continuity equation and a thermal energy equation. The following subscripts are used to denote the phases: gas/vapour phase (7), liquid phase (/?), and the solid framework or dry body ( ) = kpV2Ta + $0. (2.13) Vapour/Inert Gas Phase The equations for describing the gas phase are slightly more complex since the gas is a mixture of water vapour and any number of other gases. This can be dealt with using the species continuity equation as well as the regular continuity equation along with a summation term in the thermal energy equation. The continuity equation for the entire vapour/inert gas mixture is, ^ + V • (p 7 v 7 ) = 0, (2.14) and the ith species continuity equation is written as ^ + V-(piVl)=0; (2.15) where the sink/source term $ is zero under the assumption that there will be no chemical reactions between the vapour and the gases in the mixture. The density p 7 and velocity v 7 for the vapour/inert gas phase are defined as the following summations, N Pl = Y,Pi ( 2 - 16 ) and, N i=l Now realizing that the bulk velocity of the species i is different from the diffusive velocity of species i through the mixture, the total velocity is written as the difference of bulk velocity and diffusive velocity, (2.18) CHAPTER! 16 THEORY It then follows that equation (2.14) can be expanded to include both the diffusive and bulk terms, ^ + V-(ftv7) = - V - ( p i m ) . (2.19) Whitaker expresses the diffusive flux as PiUi = -pyVv(^), (2.20) where V is the gas diffusion coefficient. The diffusive flux, (2.20), can now be substituted into equation (2.19) leading to dpi V • (Piv7) = V at Plvv{^ \pl (2.21) The thermal energy equation for a multicomponent vapour/inert gas phase, neglecting compressive work and viscous dissipation, is given by Whitaker as d_ di N Y,Pihi \ \ / N ) + V - ( ^Pi-Vihi ~ N N ) = -V-q7+$7+^PiUi-fi-/07^-^9—^- ^2'22) Whitaker assumes that the last two terms correspond to the diffusive body force rate of work and the time rate of change of the diffusive kinetic energy respectively and can be safely neglected, resulting in the simplified thermal energy equation for the vapour/inert gas phase, d_ N dt ( E , x P^) + V N • ( E P^h) = - V • q7 + $ 7 . (2.23) In (2.23) hi is the partial mass enthalpy and can be interpreted in a way similar to equation (2.8), hi = hi(T) = (cp)iT + constant. (2.24) Using the definition of /07/i7, N p1h1 = Y^,Pihh (2-25) CHAPTER 2. THEORY 17 along with equation (2.14), (2.18), and (2.25), the thermal energy equation (2.23) becomes, h ("a? + VT ' VhV = " V ' q7 ~ V ' ( ^ PiUihi) + *-r- (2-26) Finally, applying Fourier's law and equation (2.24) yields the thermal energy equation for the vapour/inert gas phase, PI^-P)I (j£ + v7 • VT7) - A;7V2r - V • ( J2 PIMA) + * r (2.27) i=i 2.3.2 Coupling the Phases The equations (2.11), (2.12), (2.13), (2.21), and (2.27) for the 3 phases must be coupled together through consideration of the interfaces. For wood there are three interfacial areas within the porous body; solid-liquid, solid-vapour/inert gas, and liquid-vapour/inert gas. The surface areas of the interfaces are denoted in the following way, Aa@ - solid-liquid interface area Aai - solid-vapour/inert gas interface area (2.28) Afj7 - liquid-vapour/inert gas interface area. By symmetry, it follows that, -A^ = -4/3(7, Aai = A7tT, A/37 = A ^ . (2.29) For the solid-liquid interface, Whitaker defines the following boundary conditions, V/3 = 0 qCT • n ^ + q/3 • n^^ = 0 Ta = Tp on where on Aal3 nff/3 = -n(i(J Aap. (2.30a) on Aa& (2.30b) (2.30c) CHAPTER 2. THEORY 18 As is standard, n denotes the outward normal vector to the surface of the interface. The first boundary condition, (2.30a), assumes that the velocity of the liquid at the interface is zero. The second boundary condition, (2.30b), means the thermal flux from the solid to the liquid and the liquid to the solid are equal and opposite. The boundary condition, (2.30c), means that the liquid and solid at the interface surface are in thermal equilibrium. The solid-vapour/gas interface is identical to the solid- Figure 2.1: Liquid-vapour/inert gas interface. SOURCE: [28] liquid interface, v7 = 0 qa • iVy 4- q 7 • TLya = 0 T^T^ on where on Ar 7 nCT7 = — njcr (2.31a) on Aai AaT (2.31b) (2.31c) The liquid-vapour/gas interface is more complicated because the interface A^ is a moving surface. The liquid-vapour boundary can be defined by first considering a material volume element Vm(t) such as in Figure 2.1. The volume element Vm(t) contains both liquid and gas phases separated by the singular surface A^ = A^p, with velocity denoted by w. The integral representation of the thermal energy equation (2.7) applies to any material volume despite possible discontinuities in ph or q, CHAPTER 2. 19 THEORY therefore the averaged thermal energy equation over the material element Vm(t) is D (ph)dV = - t>t JJjVm(t) q n e (- ) ea = e0(t)=ey(t) = l, (2.46) e* = y, T=v> 2 45 where, thereby relating the phase average and intrinsic phase average by, **(TaY = (Ta)- (2.47) The thermal energy equation for the solid phase can be volume averaged by integrating over Va and dividing by the total volume V, l v III, ^"°h°)iv - -v SSLv • "^+v III. *°iv ^ CHAPTER 2. 22 THEORY and therefore becomes, d •^(p*hir) = -(V-q/T) (2.49) + (Qa) What is known as the averaging theorem, can now be applied to equation (2.49). The averaging theorem can be understood by considering the arbitrary function ip. The volume average of the divergence of the arbitrary function ijj can be expanded in the following fashion by applying the averaging theorem, (V • V/3) = V • (fa) + ^ ff ip0 • n0adA + i // (fc • n, 9 7 dA (2.50) The gradient can be removed from the averaging brackets in the solid phase thermal energy equation by applying the averaging theorem, {pM = i v {q - - ^~v.,IA 1 qff • iLa0dA - ^ q<7 • naldA + {$„). (2.51) Similarly the averaging theorem can be applied to the volume averaged version of Fourier's law, and then substituted into the solid phase thermal energy equation, = -ka{S7Ta) = -k„ V(Ta) + i fl Tana[jdA + i ff , (2.52) T^dA leaving the volume averaged version of the solid phase thermal energy equation, eapACp)* m = V- V(ea(Tay) + i ff Tana0dA + i J-'Art — II qa • na(i&A ~y ft II J J TaiiG1dA Aa qVp{t) + ^fff QpdV. vV JJJVp(t) JJJVoit) (2-58) The general transport theorem and averaging theorem can now be applied to equation (2.58) resulting in, ~(pfih0) + V • {pphpVf}) + y = - V • (cfc) - ^ ff Piih(j(vfi - w) • n(ildA q / 3 • n^dA - ^ // = q/j • n0(7dA + <^>, (2.59) where the enthalpy can be expressed in terms of the temperature as in equation (2.8), hl3 = h0 + (cPMTe-T°0), (2.60) and h°p is the enthalpy at the reference temperature T@. After making this substitution into equation (2.59), the term, V • (Tpvp) emerges. In order to separate the V • (TpVp) CHAPTER 2. 24 THEORY term, Whitaker represents Tp and v^ in terms of average values and deviations from these values. The point functions T@ and v^ can therefore be represented in the following way, Tp = {Tp)p + 8T0 in the /3 phase, (2.61) Tfj = ST/3 = 0 in the a and 7 phases, (2.62) v^ = (v/j) + (5vy5 in the (3 phase, (2.63) V0 = Svg = 0 in the a and 7 phases. (2.64) {Tp)0 is the intrinsic phase average as defined earlier, and 6T@ is the thermal dispersion and 5-vp is the dispersion of the liquid phase velocity. We can now write (TpVp) in terms of the dispersion vector in the following way, <2>V/3) - <7»V/3> + (STpSvp). (2.65) The final form of the liquid phase thermal energy equation is now written as, ^P^p)p^^- dt + P,e(cP)^i3) • V(T0f + // = V ha V(e?(Taf) + ~ fj + Pp(cp)pV • (8T35v0) P,8{cP)pST0(vp - w) • iifrdA T9n0(TdA + i // T0nhdA where Fourier's law has been applied to express V • qp in terms of Tp, as was done in the case of the thermal energy equation for the solid phase. Volume Average Vapour/Inert Gas Phase The vapour/inert gas phase can now be examined, starting with equation (2.14), the continuity equation for the vapour/inert gas phase. In the same manner as the liquid CHAPTER 2. 25 THEORY phase continuity equation, the vapour/inert gas phase continuity equation can be volume averaged and after some algebra results in, ItL + V.favJ + ^JJ p 7 (v 7 - w) • nll3dA = 0. (2.67) Unlike the liquid phase equation the gas phase continuity equation cannot be further simplified since p1 depends on both temperature and composition of the gas phase. Similar to the liquid phase thermal energy equation, the term (p 7 v 7 ) can be separated be expressing the point functions p1 and v 7 in terms of the intrinsic phase averages and a deviation, p 7 = (p 7 ) 7 + <5p7 in the 7 phase, (2.68) p1 = 6p1 = 0 in the a and fi phases, (2.69) v 7 = (v 7 ) + 5v1 in the 7 phase, (2.70) v 7 = Sv1 = 0 in the a and j3 phases. (2-71) The intrinsic phase average will again be utilized here, which is defined for the 7 phase density as (p1) = e1{p1)\ (2.72) Using the above expressions, equation (2.67) can be written as, d { e ^V) + V • « p 7 » 7 ) ) + V • < + Whitaker reasons that deviations from the phase averages and intrinsic phase averages are much smaller then the phase averages and intrinsic phase averages themselves, therefore it is assumed that 5'ip^ = a ( £ f f i ) 7 ) + V • « f t } > 7 } ) . (2.77) The diffusion term on the right-hand side is approximated by Whitaker as, (p7V(p?;/p7)) = (p 7 > 7 (V((p i )/(p 7 D + SQi], (2.78) CHAPTER 2. 27 THEORY where 5f2j is the deviation term given as, Pi / <*Py 7 + ^ - \ ^ - . (2.79) >7> 7 \ 7 , The final form of the volume averaged vapour/inert gas phase species continuity equation is d{tl{ 1] df + V • « P ^ < v 7 » + ±ff P ? (v, - w) • n10dA = = V • [(p 7 ) 7 ©[V(( A )/(p 7 )'0 + m - (SpiSv-y)] • (2-80) The thermal energy equation for the vapour/inert gas phase can now be examined. Recalling the thermal energy equation for the vapour/inert gas phase, (2.23), and integrating each term over the volume V1 and dividing by V, the general transport theorem and averaging theorem can be applied in the usual manner resulting in the following, i IN \ N 1+ rr ( N \ ) dA ii • (X> *M v ILf ^ (" ^f \(Vi > •i^-- w)w) ••iLva v IV ~ ^ ~ dt \" ^X >' M + v\ ' = " V • (q 7 ) - — / / 7/3 q 7 • iLypdA - — / / q7-ivoL4+<$7). (2.81) Assuming the gas phase is ideal, the partial mass enthalpy, (2.24), can be represented in terms of the pure component capacity (cp)i instead of the partial mass heat capacity cPi giving, ^ = /i° + ( c P M T 7 - T 7 ° ) . (2.82) Since we are expressing the partial mass enthalpy then we are assuming that the gas phase is ideal. After substituting equation (2.82) into equation (2.81), as before, 28 CHAPTER 2. THEORY the terms containing products with p^v,; in them must be eliminated. Similar to the liquid phase, averages and deviations from the averages must be used to separate the products. Unlike the liquid phase, f\ cannot be treated as a constant. We can therefore use the following representation, T 7 - (T 7 ) 7 + ) -{J2p> dt = (X>>(a)^+ (it(cMPi^)) •v(T1r+ JT [% + ( c p M W - 7^)] f 1 / / fl(vi - w) • n^pdA) + 0 AT N i=l i=l a Substituting equation (2.87) into equation (2.81) and following the derivation of the CHAPTER 2. 29 THEORY liquid thermal energy equation, / N £<«>w.nr + £(<*)ifov<> )-V7 i=l U // Z ) ^( C p)^ T 7( V i - W ) ' n 7 / ^ ^ "M i=l 9 o7 ^ ( C p M f y i ^ ) + V ' S ( c p ) i ( 5 ( A : V , ) « ; at i=l v- fc, v( C7 (r 7 n + F - y // q7 * v ^ - y / / T-ylLy^dA + — T 7 n 7y gA4 A-fl 7/3 % • *h0dA + ($7>- (2.88) Total Thermal Energy Equation We can now make some reasonable assumptions to combine the thermal energy equations for each phase into one total thermal energy equation. A very reasonable assumption to make is that solid-liquid-gas system is in thermal equilibrium. This means that the intrinsic phase average temperatures are equal, (Tay = (T0f = {T,)\ (2.89) and since the spatial average temperature is defined as, (^^^(ny + e^r^ + e^y, (2.90) then it must be true that, (Tay = (Tpy = (T,y = (T) (2.91) CHAPTER 2. 30 THEORY Now if we add equations (2.53), (2.66), and (2.88) after imposing the above assumption, the following results, " d(T) , €*pA.Cp)a + €/3Pf}(Cp)/} + ^ I J ^ ( p J ) ( c p ) , at N vm- C + + P/3( P)/3{V ) + i5^(Cp)i,) + ($0) + <$7) + - ^ ( c ^ ^ T , ) , (2.93) i=l and V • (£) represents the liquid and gas phase dispersion, N (2.94) V • ( 0 = Pa(cP)0V • (ST05v0) + V • i=l ^ ( ^ M P ^ ) ^ ) . The following conditions are also utilized, n7(7 nCTj(5 = — naig, n.,7^ (2.95) n ^ = —n ^ , and, (2.96) 2> = TCT T 7 = 7> over over ,4^ = T^, A7/3 = T^, (2.97) TCT = T 7 over 4 (2.98) —T CHAPTER 2. THEORY 31 where it should also be rioted that the thermal conductivities within the respective phases are constant. Equation (2.92) can be simplified by substituting some more measurable quantities, starting with substituting the spatial average density which is a measurable quantity given by N (p) = e^y + e0(p0)0 + e7 J > ^ . (2.99) i=\ The spatial average density can then be used to define a mass fraction 'weighted average heat capacity as, Cp — . The first term on the left hand side of equation (2.92) now simplifies to ^.lUUj (p)cp-^-. Referring back to the solid-liquid and solid-vapour/inert gas boundary conditions it is apparent that two of the inter-phase flux terms on the right hand side of equation (2.92), terms 2 and 4, are zero. After some algebra the jump condition in equation (2.39) can be manipulated into the following form, JV Pphptvp - w) • n^ + Y^ Pihifa - w) • n 7/3 = - ( q ^ - q 7 ) • n ^ . (2.101) This can now be substitiited into the third term of equation (2.92), TV // JJAf* (q/3 - q 7 ) • n^dA = — \\ V JJABi p{ihp(v0 - w) • n3l + J2 pihi(vi - w) • n^dA, i=i (2.102) 32 CHAPTER 2. THEORY resulting in a simplified form of equation (2.92) (p)c, N d(T) Pp(cp)0(v0) at + ^2(cp)i(piVi) V(T)- i=l 1 V M v / 3 - w)((cp)/j5T/3 - h0) • n01+ •0y N + 5 1 ^(V': "" W)((cp)i5T7 - M • n10 ? + (CPM7 - T;)Mvi - w) • n7/5. (2.106) The species jump condition, equations (2.42), can now be substituted to simplify the right side of the above expression resulting in, A' v Pis( 0 - w)((q,)/3(5T/3 - h0) • n01 + ] P p;(v; - w)((cp)j<5T7 - ^ ) • n ^ = [h°0 - h\ + {cp)0{{T) - T°0) - (cM(T) ~ T;)]P0(V0 - w) • n01. (2.107) CHAPTER 2. THEORY 33 Whitaker identifies the enthalpy of vaporization per unit mass at the temperature (T) as, A ^ a p = [h} - hi + (cp)0((T) - T;) - {cM(T) - T;)\ (2.108) and the mass rate of vaporization per unit volume as, W = T7 / / ^ / / A•187 (2.109) M V /3 - W ) • * W M > reducing the total thermal energy eqiiation to an even simpler form d(T) Pp(cp)l3(Vp) + J](Cp)i(piVi dt V(7) + A/i,,,ap(m) 1 = V - V[(AveCT + fc^ + fc7e7)(7)] + [ka - kfi J- fj-J\y(jCtjX V J J A, aQ + (kg " M ^ J ^ Tpn^dA + (fc7 - Av) ~ J J X -yTL/yg-Gbfl. - v - ( e ) + ($). (2.110) At this point there are still some terms with Ta, Tg, and T 7 in them. Whitaker suggests a theorem developed by Gray [7], {i><,yVea = ly v {U'a - %IJa)n^dA + ^ V JJKe ff (6ipa - . 5T0n^dA + (fc7 - ka)— f Ahvap(m) 6TanyadA JJA„a II <5T7n7(TcL4 Ji-ycr (2.112) CHAPTER. 2. 34 THEORY Although equation (2.112) has been substantially simplified from its initial state, it still contains parts which are not readily obtained from experiment. The first term on the right hand side, or the conductive term, is still in a theoretical form and does not allow comparison to experiment at this point. Whitaker deals with the conductive term through a lengthy reasoning process, and conjectures that the deviations STa, STp, and bT1 can be expressed in the following functional form, (2.113) 5T0 = T(?^,V(T0)A. Further reasoning leads to the assumption that the deviations 8Ta, 5T0, and 5Tj can be approximated as linear functions. For instance, 5Tp = Cp-V{Tpf (2.114) therefore it would follow that, ~ If 5Tpn^dA = ^-ff Cy V ^ V ^ A (2.115) Whitaker also assumes that the variations in (Tp)13 will be small compared to the spatial variations of 6T@, therefore, i [[ ST0nlh4A = K0 • V<7»^, (2.116) where KQ is a second order tensor. It should then follow that solid and gas phase terms also follow that same line of reasoning, - f t V JJK& ^ [f 6Tana0dA = K(7-V{Tay, (2.117) 8TyihadA = K1-V(Tiy. (2.118) Equations (2.116), (2.117), and (2.118) can now be substituted into equation (2.112), and recalling that the phase average temperatures are equal, then the total thermal CHAPTER 2. THEORY 35 energy equation takes on the following form, N d 7 / \ ( ) v Pp{Cp)0 ( p) +53(cp)iNow by defining the following effective thermal conductivities, Keff = (eaka + e0k0 + e7fc7)?7 + (fcff - k&)Ka + (fc^ - /c7)A> + (k7 - ka)K17 (2.120) and, (2.121) (0 = -AD-V(T), where [/ is the unit tensor, and the dispersion V • (£) is interpreted in terms of a diffusion model as well, the total thermal energy equation simplifies even further to, d(T) C (P) P- dt N f (Cp)p(v0) M i + ^(CpJi^iVi V{TS) + Ahvap(m) »=i = V • (Keff • V ( T » + V • (tf D • V ( T » + {$). (2.122) The effective thermal conductivity can be taken even one step further to give JV PpiCphiVf}) + ^(Cp)i(piVi) V(T) + A/i mp - (2-128) The total thermal energy equation can now be written as, {p)Cp ^dt~ + to(cph(vp) + (P 7 ) 7 ((c p ) 7 ) 7 (v 7 )] • V(T)+Ahvap(m) = V - ( K e r / / - V ( T ) ) + ($), (2.129) leaving a simplified version of the original total thermal energy equation. Simplified Continuity Equations for Liquid and Vapour/Inert Gas Phases The mass rate of vaporization given by equation (2.109) can be used to simplify the liquid continuity equation to, % + V-(v /3 > + - ^ = 0. at pp (2.130) CHAPTER 2. 37 THEORY Virtually the same reasoning can be applied to the vapour/inert gas continuity equation. By equation (2.40) and (2.109) the mass rate of vaporization can also be written as, (m) = —— / / p 7 (v^ - w) • TLypdA (2.131) and therefore the vapour/inert gas phase continuity equation can be written as, 9{ ^f] + V • « p 7 » 7 » = . (2.132) The only remaining theoretical parts of the transport equations are, the deviation terms, SQ and (5/Oj5v7) in the vapour/inert gas species equation and the liquid and vapour/inert gas velocity fields (vp) and (t>7). Whitaker argues that the dispersion term (SpiSv-/) can be handled in the same way as the thermal dispersion terms by approximating it with a diffusion like expression, (6Pi6vJ • V((Pi)VW7). = -{PiynD (2.133) Similarly, Whitaker reasons that the 5Q term can be approximated relatively well by a diffusion expression such as the following, 5n = Dil-V((Pi)y(p^). (2.134) In this way the species continuity equation can be written in a more compacted form as, d ( £ T if > 7 ) + V • «A> 7 + V dt (P^S-v(g (2.136) and for the remaining gases, i = 2,3,... where phase changes do not occur, the continuity equation becomes, d(e 7 (p ; :) 7 ) + V - « p i » 7 » = Vdt \7 nW (2.137) w^g, • v In order to have a continuity equation representing the unsteady behavior of moisture, the liquid and vapour continuity equations can be combined into a total moisture continuity equation. This can be accomplished by adding equations (2.130) and (2.136) yielding, d ( e m + e 7 (pi) 7 ) + V.(^(v /3 ) + (p 1 )> 7 )) = V dt W^>.v(^l . (2.138) A further simplification can be made by defining the dimensionless quantity S, which was earlier given by equation (1.3). As mentioned earlier the saturation represents the mass fraction of moisture in the form of liquid and vapour with respect to the fully saturated mass. This can be calculated in the following way, S We + e 7 (Pi) 7 (2.139) is the volume of the void space in the averaging volume element. Substituting equation (2.139) into equation (2.138) gives, ,dS „ / . <** V . pi 7 v 7 ) PH v- (p7r (1) f(Ply (2.140) CHAPTER 2. THEORY 39 Liquid and Vapour/Inert Gas Velocity Fields In order to close the transport equations, expressions for the liquid and vapour/inert gas phase velocity fields are required. Whitaker starts with the equation of linear momentum for the liquid and vapour/inert gas, Pf^^T = P/*,7g + V • ? > , r (2-141) which must be solved for the liquid and vapour/inert gas velocities. Whitaker's solution of equation (2.141) is based heavily on an assumption of the fluid topology, the assumption being that the vapour/inert gas phase is continuous. What this means is that there exists a curve that can go through the porous material from one outer boundary to the opposite outer boundary in such a way to not cross a phase boundary. For instance, if an imaginary line is drawn through the liquid phase, it could be drawn from the surface to the core without crossing any phase boundaries. In such a case the liquid phase would be considered continuous. Additionally, the opposite follows: there does not exist a curve that can go through the porous material from one outer boundary to the opposite outer boundary in such a way to not cross a phase boundary. If so, then this phase would be considered discontinuous. In this study the majority of the wood is green, and therefore contains a high moisture content, which could therefore suggest quite strongly that the liquid phase is continuous and the gas phase is discontinuous. It is also very likely that the gas phase is never in a continuous phase due to either liquid blocking pit openings, or pit aspiration blocking pit openings to neighboring pits (these will be discussed further in chapter 3). The vapour/inert gas phase could therefore be described as trapped in a bubble within the lumens, the impact of which will be discussed in greater detail later on. It therefore does not make sense to follow Whitaker's approach in developing the liquid and vapour/inert gas velocity expressions since the approach is based on assumptions that are clearly not valid in CHAPTER! 40 THEORY this case. Instead a different approach is taken based on [24], where some relatively well accepted assumptions of convective fluid flow through wood are made in order to solve equation (2.141). The assumption that fluid flow within the cell structure of the wood is linear and that viscous forces are much greater then inertia! forces are often referred to in the literature [23], [24], and [14]. This means that the volumetric flow rate and the linear velocity are directly proportional to the applied pressure gradient. It must also be assumed that the fluid is incompressible and homogeneous, the porous media is homogeneous, there is no interaction between the fluid and cell wall, and permeability is independent of the length of the specimen in the flow direction. Using these assumptions the general solution of equation (2.141) is given as Darcy's Law, v = --(VP-/jg), (2.142) li where (j, is the fluid viscosity, K is the permeability, P is the pressure, and g is gravity. Darcy's Law applied to velocities for the liquid and vapour/inert gas phases give, respectively: v/j = KBHVPp-pps), (2.143) and A7 (VP7-p7g). (2.144) The phase average velocities are given by Spolek [24] as, = qV(P^-^g), (2.145) M/3 and, (v 7 ) = - ^ ( V ( F 7 V - p 7 g ) . (2.146) It is noted by Siau [23] that the assumptions made above are not entirely valid. For instance, when fluid moves from a large capillary to a small capillary, such as a fluid moving from a cell lumen to a pit opening, nonlinear effects from relatively high CHAPTER 2. 41 THEORY viscous drag may be present. Wood also has a relatively non-homogeneous structure, and the hygroscopic nature of the cell walls can exert forces on fluids made up of polar molecules such as water. Although this approach seems to require a number of somewhat unrealistic assumptions, it has been found to compare well with experiment [23], and will therefore be applied in this study. One conclusion that can be made at this point is that the literature seems to lack a satisfying theory of two-phase flow velocities in wood at high moisture contents such as in green wood. Further work in this area is required to develop a more convincing theory, but this is beyond the scope of this study. Simplified Transport Equations The above transformations are utilized to express the transport equations with respect to two dependent variables, temperature and saturation. Equations (2.145), and (2.146) can be substituted into the total thermal energy equation (2.129) and the total moisture continuity (2.140) equation, neglecting thermal energy sources and sinks {$), giving (P)CP d(T) dt Kf}pp{Cp)f: (V(F^-^g) + • ( v < p 7 r - 7>7g) • V(T> + Ahvap(m) P>7 = V • K?ff • V(T), (2.147) and, as - dt -PHW - ^ l + PuPi^ T O 1 (piY = V ^ D % • V n L Pa (PI)1 (2.148) {Pn Equations (2.147) and (2.148) contain several unknowns which must be determined prior to solving the equations. These unknowns can be obtained in two relatively different manners. They could be approximated by empirically developed expressions CHAPTER 2. THEORY 42 and constants, or they could be developed theoretically using a mechanistic model of wood structure. Acquiring such data experimentally would take a series of specially designed experiments and precise data collection, which may have to be repeated for different parameters such as species type. Additionally, to design experiments to collect this type of data would require that each coefficient be identified or transformed to a measurable value, which could become difficult for things such as the vapour/inert gas permeability. The experimental approach seems inefficient and is far beyond the scope of this research. Alternatively, theoretical approximations developed using the mechanistic model of wood structure will be taken from the literature to identify the unknown coefficients. In addition to the unknown coefficients, the phase pressures remain as unknown dependent variables. In order to solve this set of equations for moisture content and temperature, the phase pressures must be defined in terms of temperature and saturation. This will also be completed in the next chapter. Chapter 3 Wood Theory At this point a theoretical model of thermal energy and moisture transport has been given which contains the following unknowns: the liquid density - pp, gas density - p 7 , liquid heat capacity - (cp)p, gas heat capacity - (c p ) 7 , liquid permeability - Kp, gas permeability - K1, void fraction - , liquid viscosity - pp, enthalpy of vaporization Ahvap, mass rate of vaporization - (m), effective thermal conductivity tensor - K'l^p effective diffusivity tensor - D^L, the liquid phase average pressure - {Pp}13, and the gas phase average pressure - (P 7 ) 7 . The mechanistic model of wood structure will be utilized to develop theoretical expressions for the unknowns with the goal of decreasing dependence on experimentally determined parameters. This is important because it will result in a more robust model that can easily be adapted to different species and environmental parameters. North Central Plywoods utilizes at least three species of soft-woods in plywood production, and a transport model would require that an array of accurate experimental measurements be completed on each species in order to obtain the required coefficients. This is a cumbersome, time consuming task which in the end may still result in an inaccurate model due to experimental error. The mechanistic model of wood structure, which is well-accepted in the literature ([14], 43 CHAPTER 3. WOOD THEORY 44 [23], [24]), will be considered to gain an understanding in the development of the model parameters. Thermal energy and moisture flow through wood can each be categorized by two main groups: conduction and convection for thermal energy transfer, and bulk flow and diffusion for moisture transfer. These are reflected in the four main terms of equations (2.147) and (2.148). Conduction occurs as a result of neighboring molecules transferring energy to one another through molecular collisions, as energy transfers from regions of high to low temperature. Diffusive moisture transfer is analogous to conductive thermal energy transfer in that it occurs at a molecular level as a result of a concentration gradient. In the case of porous structures the concentration gradient is expressed as a moisture content gradient, therefore diffusive transfer moves form a region of higher moisture content to one with lower moisture content. Bulk fluid transfer, or molar transfer, differs from diffusive transfer in that instead of the fluid moving by way of molecular collisions, it moves in small parcels through the porous structure. Pressure gradients are created as a result of temperature gradients and capillary pressure gradients occur as a result of moisture content, or more specifically saturation gradients. This means that since temperature gradients are high at the surface and cooler at the core then a pressure gradient will exist which is higher at the surface and lower at the core resulting in convective liquid and vapour flow away from the surface towards the core. Alternatively, capillary flow, which moves in the direction of high to low saturation, will tend to push moisture from the more saturated areas inside of the wood to the surface, where surface evaporation results in overall moisture loss. This means that as the surface undergoes heating, the bulk moisture transfer mechanisms, capillary flow and convective flow, tend to oppose one another [4]From this reasoning alone it is difficult to deduce whether diffusive transfer of CHAPTER 3. WOOD THEORY 45 moisture, which tends to be a much slower process compared to convective transfer, will play a significant role in overall moisture transfer. As such, all of the processes will be examined in more detail in the following sections. 3.0.3 Constants and Wood Independent Coefficients Some of the unknowns found in equations (2.147) and (2.148) are constants and coefficients which are independent of wood species, such as enthalpy of vaporization Ahvap, liquid density - pp, liquid heat capacity - (cp)p, and liquid viscosity - \i$. The enthalpy of vaporization for liquid water has a dependence on temperature given by the expression [24], Ahvap = 2.525 x 106 - 2.824 x 103T, (3.1) where T is in degrees Celcius. The liquid viscosity - \i$ also contains a relatively significant temperature dependence, [24], which is given by the following, — = 0.2184((T- 8.435) + ((T - 8.435)2 + 8078)*) - 1 2 . 0 0 . H V (3.2) ' The liquid density - pp, and liquid heat capacity - {cp)p both contain some temperature dependence, but in this case will be considered negligible and therefore approximated as constant [2], ( C 3.1 p0 = 9 9 5 . 7 ^ . (3.3) ^ = 4118-°7^F (3.4) Mechanistic Model The mechanistic model simply refers to any theories arising from the geometrical description of wood cell structure [23], [24]. More specifically when talking about permeability, we refer to the Comstock model of permeability, named after G. L. CHAPTER 3. WOOD THEORY 46 Cornstock who first used a geometric model to describe the bulk flow of fluid through wood ([3]). The Cornstock model predicts the tangential and longitudinal permeabilities of softwoods fairly accurately, and requires that a few assumptions about the wood structure be made. These are: • Wood structure is homogeneous, • flow is through the cell cavity-pit system and the pits offer the only significant resistance to flow, and • the cells are square in cross section. When looked at in detail, wood structure is not homogeneous and the cells vary in size from early wood to late wood, as well as along the length of the tree. Along with the cavity-pit system, rays which run transverse to the cavity-pit system can also allow for the transport of fluid. Additionally the cells (tracheids) are not totally square, but instead have a more biological rounded shape. Although most of the conditions stated above contradict reality to varying extent, they have been shown in the literature to offer a good approximation ([14], [23], and [24]), and will therefore be taken as valid assumptions in this study. The mechanistic model can best be described by Figure 3.1, where the square tube structures represent the wood cell structures called tracheids. 3.1.1 Structure Dependent Parameter Development The mechanistic model provides a simplified structure to develop the model parameters which rely explicitly on geometric properties. One of the first parameters that can be defined is the void volume fraction, 0, also referred to as the porosity. The void volume fraction is the fraction of empty space within the tracheid, referred to as 47 CHAPTER 3. WOOD THEORY Traehetd Radial Transfer Figure 3.1: Simplification of wood structure used in the mechanistic model. CHAPTER 3. WOOD THEORY 48 IT W w w' J Figure 3.2: Dimensions of tracheid taper by applying the mechanistic model. SOURCE: [24] the cell lumen, to the total volume of the wood sample, 6 = 1 J J (3.5) VhT L denotes the total lumen length, and T denotes the total tracheid length. The numerator in equation (3.5) represents the lumen half volume, which can be calculated using the dimensions given in Figure 3.2, ',,,,2 ,Y 2 , VW ViL = (^-yW+ 2 (3.6) By similar triangles, IV y •• (3.7) = WV' which means, \L = •4 , V)W+ 9 2W yw3 (3.8) CHAPTER 3. WOOD 49 THEORY Now the denominator, which represents the tracheid half volume can be defined as, V =^ - ^ , (3.9) Using (3.8) and (3.9) in (3.5) gives, 2 2 Equation (3.10) can be further simplified by defining two parameters: the tracheid overlap fraction, a = |r, (3.11) a - £. (3.12) and the open space cell width fraction, Combining equation (3.10), (3.11), and (3.12), the void fraction can be written as, Tracheid dimensions such as the ones outlined in Figure 3.2 are readily available in the literature for various species. Throughout the remainder of this chapter it will be seen that very few experimental parameters, such as a and a, will be required to characterize different species in the model. 3.2 Capillary Pressure Using the mechanistic model, capillary forces can be examined more closely and a suitable expression for capillary pressure has been developed [24], [14], [28]. Capillary pressure can best be described as the pressure difference arising at the contact interface of two immiscible fluids as a result of the free energy surface at the interface and the CHAPTER 3. WOOD THEORY 50 attraction between the fluid and the solid container wall. The surface energy develops from the interaction of molecular cohesive and adhesive forces in a substance, cohesive being the attraction of molecules of a substance to itself, such as the intermolecular forces in a liquid, and adhesion being the attraction of the molecules of a substance to a different substance such as another fluid or a solid surface. The fluid's affinity for the solid surface depends on the chemical composition of the fluid. The fluid with the greater attraction to the solid is referred to as the wetting fluid and the other, the non-wetting fluid. This can best be described by considering an example such as a glass tube containing a water-air interface. In this case the water is the wetting fluid because it has a greater affinity to the glass then the air does and the water molecules at the edge of the interface will be pulled towards the sides of the glass tube. The free energy surface develops as a result of force imbalances at the surface of the wetting fluid. The water molecules below the surface of the interface have other water molecules surrounding them to balance any attractive forces, but at the surface this is not the case. At the surface there is only the much weaker water-air bonds to balance the intermolecular cohesive forces resulting in an imbalance, and the development of a free energy surface. As in nature, the free energy at the surface will seek to minimize itself causing the surface molecules to develop stronger bonds between other surface molecules manifesting itself into what is commonly referred to as the interfacial tension. In the case of a liquid surrounded by its own vapour, the interfacial tension is referred to as surface tension. At equilibrium the combination of force imbalances at the edge and middle of the interface result in the interface geometry taking on a concave shape into the wetting liquid which reflects the lowest energy configuration of the surface molecules. This is depicted in Figure 3.18. This compression of the wetting fluid leads CHAPTER 3. WOOD THEORY 51 (B) (A) AIR • AIR ® a « © • a • « i I I I I I I I i i i i i i i M ^ B M ^ft M ^ P H ^ P M ^ P M ^ P H ^ P M I I I I I X I •4 ^ P M ^ P •< ^ P M ^P M ^ P M ^ P M ^ P M « ® • _ I GLASS SURFACE WATER WATER Figure 3.3: (A) Force imbalance at water-air interface in small cylindrical tube. (B) Water-air interface at equilibrium in small cylindrical tube. to a pressure differential between the two fluids referred to as the capillary pressure, •< c — -* non-wetting * wetting- (3.14) In the case of a small cylindrical tube, or capillary, the capillary pressure Pc, is given by Laplace's Equation for capillary pressure, [1], A P = P 7 - P / 3 = Pc = a ( - + - J = — , \ ?i r-iJ r (3.15) where r\ and r-i are the two principal radii of curvature, and a is the surface tension which is proportional to the free energy at the interface [1]. For a cylindrical capillary, the radius of curvature is the harmonic average of the two radii of curvature, r-y and r-2- In the case of wood, the radius of curvature r is actually the lumen diameter, W, [24], shown in Figure 3.2, aW ~2~' (3.16) resulting in, 4(7 Pr = dW' (3.17) CHAPTER 3. WOOD 2 2 Vkpourtfrtert Gas ) 52 THEORY (A) Liquid ViMtsr L z I (B) (C) Figure 3.4: (A) Constant capillary pressure phase. (B) Critical saturation reached Scr, water begins to recede into the taper of the lumen, reaching the saturation dependent phase of the capillary pressure. (C) Irreducible saturation reached Si, water has recede into the funicular or discontinuous state, [24] As mentioned in Chapter 1, the capillary pressure is dependent both on temperature and saturation. The surface tension of water can be expressed as a function of temperature by the empirical expression, [24], a = 0.07564 - 0.000144T. (3.18) Since surface tension is temperature dependent, it follows from equation (3.17), that the capillary pressure is also temperature dependent. The saturation dependence is slightly less straightforward, and is a result of the CHAPTER 3. WOOD 53 THEORY tracheid tapering, as depicted in Figure 3.4. As the wood sample dries or absorbs liquid, the meniscus at the air-water interface either fills and moves toward the mid-point of the lumen or recedes towards the tapered end of the lumen. While the meniscus moves along the un-tapered section there is no effect on the capillary pressure since the lumen radius and surface tension remain constant. This is illustrated in Figure 3.4A. As the meniscus recedes into the tapered section the principal radius begins to change and therefore the capillary pressure is affected. This point is referred to as the critical saturation So- and is illustrated in Figure 3.4B. As the meniscus recedes further into the taper it gets to a point where the liquid phase in the individual lumens becomes disconnected from the liquid phase of its neighboring lumen. This point, illustrated in Figure 3.4C, is important because the liquid phase has transitioned from a continuous to discontinuous state which causes the bulk flow of liquid through the wood structure to cease. The saturation in this state is referred to as the irreducible saturation Si. The critical saturation and the irreducible saturation can be calculated from the structural geometry given in Figure 3.2. This is outlined by Spolek [24] where the critical saturation is given as, s -=i (T „ w (3 19) - 1 — [2 — a)a and the irreducible saturation is one quarter of the critical saturation, Si = ^ f • (3.20) As the meniscus moves along the tapered section of the tracheid, the rectangular tube-like shape causes one of the principal radii to change as illustrated in Figure 3.5. This reduction in the principal radii is the cause for the saturation dependence on the capillary pressure. Spolek [24] gives an in depth derivation of the saturation dependent capillary pressure resulting in the following expression for capillary pressure through CHAPTER 3. WOOD 54 THEORY Figure 3.5: Principal radii, r 1 ; and r2, shrinking as meniscus recedes. both saturation ranges, 4(7 Scr < S < 1 aW Pc = 2CT aW 1 + (¥ (3.21) s < sc In order to offer some verification of the theoretical expression given here, Spolek [24] offers an argument comparing the dimensionless capillary pressure of a sand bed to a dimensionless version of the theoretical expression given above. The resulting comparison shows good agreement with experiment and therefore equation (3.21) will be used in this study. 3.3 Bulk Moisture Transfer and Permeability All bulk movement of fluid within a porous medium is regulated by how permeable the particular material is to a particular fluid, where permeability can be defined as CHAPTER 3. WOOD THEORY 55 the ability with which a particular fluid can move through a porous body. Although the cell structure of wood would suggest a closed cell structure, on closer inspection the pit openings connecting the neighboring lumens allow the flow of fluid between neighboring cells. An interesting aspect of wood which separates it from other porous media is the phenomenon of pit aspiration which occurs during the evacuation of liquid from the cell lumens. Pit aspiration, which will be discussed in greater detail later, is the blocking of border pits as a result of capillary forces and chemical bonding which occur at the pit openings. Another aspect influencing the permeability is whether or not the fluid within the open-cell structure is continuous or discontinuous. As mentioned earlier, a fluid in an open-cell porous structure can exist in either the continuous or discontinuous state, or sometimes referred to as the funicular or pendular states respectively. This is important because once a phase becomes discontinuous then bulk flow of that phase would cease; however, not all flow stops when the pendular state is reached, as liquid can diffuse into the cell wall and emerge in the liquid phase of the neighboring lumen. 3.3.1 Specific Liquid Permeability As described by Comstock [3] and later revisited by Siau [23], the intrinsic (sometimes referred to as specific) permeability is a property of wood which depends only on the porous structure of the wood, and therefore differs slightly from species to species. The specific permeability is also independent of liquid viscosity and is therefore equal for the liquid and gaseous phases of a particular species. The anisotropic nature of wood causes the permeability to be directional but since this study is focused on the radial dimension, only the transverse permeability is of interest. Working on the premise that specific permeability is dependent on pore structure, and that pore CHAPTER 3. WOOD THEORY 56 structure differs slightly between earlywood and latewood, suggests a radial dependence to the specific permeability. Unfortunately, a scan of the literature could not uncover any radially dependent expressions of the specific permeability. In addition, no theoretical formulation of the specific permeability as a function of the geometric wood parameters, a and a, could be found. Instead, an experimental value for the average specific permeability taken directly from the literature ([17], [16], [14], [24]) will be utilized in this study, k0 = 1.0 x l(T 1 5 m 3 . 3.3.2 (3.22) Pit Aspiration Pit aspiration is a phenomenon! which occurs in wood as liquid escapes from the lumens during drying. Bordered pits are those connecting cell lumens in neighboring tracheids, and can become sealed to stop bulk gas flow through the wood structure. On closer inspection, the bordered pit aperture contains a circular disk suspended from a permeable membrane. The disk is known as the torus and contains small capillaries which allow fluid to flow through it. The membrane which is called the margo also contains small openings which allow fluid to move through it. The openings in the membrane are significantly smaller in diameter then the pit aperture but relatively large compared to the very small capillary openings in the torus. Siau [23] gives an explanation of how pit aspiration occurs as a result of capillary tension. Figure 3.6 illustrates the mechanism of this process. As liquid recedes from the tapered end of the lumen, liquid in the neighboring lumen is left to form a meniscus across the pit aperture. Further evaporation from the pit aperture during drying causes the meniscus to recede further into the pit aperture, first into the annular shape shown in Figure 3.6B, and then to the membrane divider. The smaller openings in the membrane result in a much higher surface tension at the liquid surface then in Figure CHAPTER 3. WOOD THEORY 57 Figure 3.6: (A) Liquid from neighboring pit has receded past the pit aperture. (B) Meniscus has receded past the pit aperture. (C) Meniscus has receded to small openings in the membrane. (D,E) Capillary pressure has pulled the torus into the pit aperture. (F) Capillary pressure has pulled torus into disk shape, resulting in a completely aspirated pit. SOURCE: [23] 3.6A and 3.6B thereby stopping evaporation at the surface. The smaller radius of the menisci causes the pressure difference between the liquid and air phases to increase thereby sucking the torus tight against the aperture wall as illustrated in Figure 3.6D and 3.6E. Whether the capillary pressure is enough to cause deflection of the torus into the aspirated position depends on a few things, assuming the liquid is water. Firstly the pit membrane openings must be small enough to produce the required pressure difference to deflect the torus. Secondly, the pit membrane must have relatively low rigidity: typically the early-wood of low density softwoods possess this property. The CHAPTER 3. WOOD THEORY 58 result of pit aspiration is that bulk flow of the vapour/inert gas phase does not occur at atmospheric pressure, thereby making the wood impervious to vapour/inert gas flow, or, AT7 = 0. (3.23) The effect of this on the transport equations will be seen in the next section. 3.3.3 Relative Liquid Permeability The relative permeability depends on the physical properties of the saturating fluid, differing between liquids and gases. Although slightly different models are presented in the literature they all agree that the relative permeability for a liquid is equal to unity when 100% saturated, and is equal to zero at the fiber saturation point. To recall, the critical saturation occurs when the meniscus of the liquid phase within the lumens is at or past the transition from the tapered part of the lumen to the uniform section of the lumen. Spolek [24] presents a relative permeability model based on the tapered lumen geometry, 1 Scr '(£)'-- 1 Si (3-25) \ <-5saturated J where Ssaturated is the maximum saturation of the porous medium. The latter is used to avoid a singularity that may exist in the model in the moisture content range between the fiber saturation point and the irreducible saturation 5j. Spolek [24] avoids this singularity in a slightly different way then Perre and Couture, and will be discussed CHAPTER 3. WOOD 59 THEORY _ || Ry Rjj where R is the thermal resistivity, and the subscripts represent the phases. The _L and || subscripts denote the crosswalls and sidewalls respectively since the thermal conductivity of the cell walls is anisotropic. The thermal conductivity coefficient is related to the resistivity by the following expression. R = hi' (3.29) where k is the thermal conductivity, A is the cross sectional area, and L is the length of the conductive path. The total thermal resistivity can be found by combining CHAPTER 3. WOOD THEORY 61 equations (3.28), and (3.29), Rtotal = ~1— + {aki{l^S)+akfjS + ka{1_a)) > ( 3 - 30 ) where ka, kp, andfc7are the thermal conductivities of the solid cell wall, liquid, and vapour/inert gas phases respectively, a is the open space cell width fraction, and S is the saturation. This approach assumes the conductive path is one dimensional, but as explained by Siau [23], this assumption is not totally valid and could lead to considerable error, since the much smaller thermal conductivity of the air vapour/inert gas phase compared to the cell wall or liquid phase causes non-uniformity in the thermal energy flux. This non-uniformity effectively reduces the conductive surface area of the crosswalls, leading to fringing effects. Siau [23] considers an empirical correction factor Z, to account for these effects, replacing (3.30) by, Rtotai = - ^ — + yak^ _ S) + ak(iS+K{l_a)j • ( 3 - 31 ) The correction factor accounts for the fraction of crosswalls surface area that is not fully conductive and is given by the following expression, ^ l - i l - " - 4 8 ' 1 - " ' - ^ . y a (3.32) ka J The details of the correction factor Z are given by Siau [23] and Spolek [24] and for brevity will not be discussed here. As a result the total thermal conductivity, which is the inverse of the total thermal resistivity, becomes, X ( a _ 1 ) - T T7 (lT -— F ^S)—+^akpS T r - l+- TfcT -^Y kq " 1.48(1 — a)k a + afc7+ f \afc a(l — a) J ' (3-33) Siau [23] approximates the longitudinal thermal conductivity for most softwoods on the order of, KL = 2.5V- ( 3 - 34 ) CHAPTER 3. WOOD THEORY Material Cell Wall substance _L Cell Wall substance || Water (STP) Air (STP) 62 A;(W/mK) 0.44 0.88 0.59 0.024 Table 3.1: Thermal Conductivities, [23]. Where kqL is the longitudinal thermal conductivity and kqj- is the tranverse thermal conductivity. Although this gives a good idea of the directional differences of the thermal conductivity in wood, the radial direction is being considered in this study therefore details of the longitudinal thermal conductivity are only important for comparative reasons. Additionally, Spolek [24] has shown that the temperature dependence of the thermal conductivity coefficient is negligible especially at saturations of 0.5 and less, and deviates a maximum of 14% from 0°C to 200°C at saturation near or at one. The thermal conductivity will therefore be expressed through use of equation (3.33). The thermal conductivities kqi, kqg, and ka are listed in table 3.1. The tracheid wall thickness ratio a can be calculated using equation (3.12) and the species dependent tracheid dimensions found in the literature [9]. 3.5 Diffusive Moisture Transfer Moisture diffusion in wood occurs through a combination of two different mechanisms, vapour diffusion, and bound water diffusion. A third mechanism, Knudsen diffusion, has also been identified in the literature [23], and occurs when the pore diameter is on the same order of magnitude as the mean free path of the molecules in the fluid. The border pits in wood have a diameter on the order of magnitude of the mean free CHAPTER 3. WOOD 63 THEORY path of the vapour/inert gas molecules but due to the assumption that the border pits become aspirated after the liquid phase has receded, Knudsen diffusion is not significant. Additionally, the mean free path of the liquid molecules is much larger then the border pit diameters, therefore Knudsen diffusion can safely be ignored for liquid flow. Bound water diffusion occurs as a result of water molecules traversing through the cell wall material driven by a concentration gradient. Vapour diffusion occurs across the lumens as bound water evaporates on one side of the lumen wall and condenses on the opposite side, which occurs throughout the wood structure as a condensation-evaporation cycle. Moisture diffuses into the cell wall, evaporates on the neighboring cell lumen wall, diffuses through the cell lumen, and then condenses on the opposite cell wall to start the cycle again. Any energy released through latent heat is re-absorbed at the same rate as vapour simultaneously condenses on the opposite side of the lumen. The diffusion coefficient itself is therefore a composition of both bound water diffusion and vapour diffusion. The diffusive term on the right-hand-side of equation (2.148) describes the diffusion of moisture in the form of vapour through the lumens. The quantity describing vapour diffusion is the vapour-inert gas fraction M ^ . and, being a dimensionless fraction describing moisture content, Spolek [24] reasons that the following approximation can be made, ^ V ( « 1 ) ^ V M . (3,5, Where Dm is describing a total diffusion coefficient composed of both bound water and vapour diffusion. An assumption is made here that moisture diffuses through the wood isotropically and therefore the effective diffusivity tensor reduces to a one-dimensional coefficient. This substitution also avoids a discontinuity that is encountered in wood drying models, the details of which are further discussed in the next section. The composition of the total diffusion coefficient is handled by Siau [23] in an CHAPTER 3. WOOD 64 THEORY analogous way to the total thermal conductivity coefficient. The difference with the total diffusion coefficient is that, opposite to the thermal conductivity, the conductivity of bound water in the cell walls is orders of magnitude lower then the vapour in the lumen, therefore it can be safely assumed that bound water diffusion from the sidewalls is negligible. The total diffusion coefficient is a serial path through the cell crosswalls and the lumen, Rm = R*,L + Ry, (3.36) where R is the diffusive resistivity. The individual diffusive resistivities are given by Siau [23] as, ^ = ( l - ^ l - O , *, = ^ A (3,7) (3-38) and since the diffusive conductivity is the reciprocal of the diffusive conductivity, Dm = ^ - , (3.39) the total diffusion coefficient is given as D = _J DbD ^ (3 40) The individual diffusivities have been given by Siau [23] as the following empirical equations, for the vapour diffusion through the lumens, and Db = 7.0 x 10~6e 38500-290M R T ^ (3.42) for bound water diffusion, where P is the total pressure, T is the temperature, and fty.i is the gas constant for water vapour. CHAPTER 3.5.1 3. WOOD THEORY 65 Discontinuity in Moisture Transfer It has been noted in the literature a number of times by, Siau [23], Plumb [14], Spolek [24], Couture [4], Goyeneche [6], and Krabbenhoft [11], that bulk flow of the liquid phase dominates overall fluid flow in the funicular state where the saturation ranges between the irreducible saturation and the maximum saturation Si < S < 1. It is also widely agreed upon in the literature that the cell wall is saturated with bound water above the fiber saturation point, therefore bound water concentration gradients in the cell wall only exist in the region below the fiber saturation point. The problem arises in the moisture content region between the fiber saturation point and the irreducible saturation, where the bulk flow of liquid is halted due to its transition into the pendular state, and diffusion of bound water has not yet began. Even in the event that the pit openings do not become totally aspirated, thereby allowing the vapour/inert gas phase to become continuous through the porous matrix, heating from the surface will cause a pressure gradient to push moisture away from the surface towards the core. As a result the model will predict no net change in the moisture content of the wood for the saturation level between the irreducible saturation and the fiber saturation point, contrary to experiment, which always shows the moisture content continuously changing throughout the moisture content range. The literature contains a few slightly different ways of addressing the discontinuity described here which are taken from slightly different permeability models. Perre [16], and [17] and Couture [4] avoid this discontinuity by considering the permeability model described by equation (3.25), which assumes that bulk liquid transport does not stop after the irreducible saturation is reached. Physically this seems unrealistic when looking at the mechanistic model, because once the meniscus recedes to the irreducible saturation no bordered pits exist to offer a connection between neighboring tracheids, but mathematically, it offers a solution to the problem of the discontinuity. CHAPTER 3. WOOD 66 THEORY Goyeneche [6] offers a more physically acceptable theory which assumes a thin liquid film is left behind to connect the remaining liquid phase even past the irreducible saturation and therefore bulk liquid flow can continue until the fiber saturation point is reached. The interested reader is directed to the work of Couture [4], Goyeneche [6], and Krabbenhoft [11] for a much more detailed discussion of the problem and possible solutions. Spolek [24] addresses the problem by assuming bound water diffusion occurs throughout the moisture content range thereby eliminating the discontinuity but introducing a small error into the solution. The error occurs at moisture contents above the irreducible saturation where diffusive transport of vapour and bound water is thought to be physically nonexistent but is still predicted by the model. Spolek reasons, in the case of drying, that the error produced is negligible in the final results and therefore acceptable. All the methods described above have been found to match experiment with varying degrees of success, although none seem to give a satisfactory physical argument to distinguish one as the definitive answer. Fortunately, moisture content ranges in logs used for plywood making are on the higher range of moisture content (near green) throughout their lifetime in the mill, and therefore any inaccuracy of the model in this saturation region is not as important in modeling the conditioning process as in the kiln drying models where the entire saturation range must be modeled accurately to produce usable results. 3.6 Liquid, Gas, Capillary Pressure Gradients In Chapter 2, the phase velocities were eliminated from the transport equations leaving only the phase pressures. In this section the phase pressures are expressed in terms of the dependent variables, temperature and moisture content, based on the work of Spolek [24]. CHAPTER 3. WOOD 67 THEORY Assuming the inert gas phase pressure, (P2) 7 , is dependent on both temperature and saturation, then the chain rule can be used on the inert gas pressure gradient giving, V7 = ^ | r ^ V { 7 ) + ^ I ^ V S . (3.43) The vapour/inert gas phase pressure can be quantified by making a few assumptions, first, the inert gas phase is assumed to behave as an ideal gas. This means the inert gas partial pressure can be calculated using the ideal gas law and the average temperature, (Pa>7 = n^RyAT) {3M) In equation (3.44), n7i2 is the number of moles of gas, i?^^ is the gas constant, and V1 is the inert gas volume. Due to the discontinuous nature of the inert gas phase, no expansion into neighboring cells is possible, therefore it is assumed that the moles of gas n remains constant and (P2) depends only on the local temperature and saturation. The temperature dependence of the inert gas phase is explicit from equation (3.44), and the saturation S appears through the gas phase volume Vy, Vy = (l- S), (3.45) Dividing the inert gas pressure by the initial inert gas pressure gives, (P2V = < ^ , o > 7 ^ f | j | , (3-46) where the subscript 2, indicates the inert gas phase and, 0 indicates initial conditions. To quantify the vapour phase similar reasoning is used and it is assumed that the vapour pressure depends on both the temperature and the saturation. Once again employing the chain rule, the vapour pressure can be expressed as a function of the temperature and saturation, VW1 = ^^V<7) + ^|^VS. (3.47) CHAPTER 3. WOOD 68 THEORY Whitaker [28] relates the vapour pressure within the lumens to the local temperature by combining the Kelvin Equation and the Clausius-Clapeyron Equation giving, 4<7 (Fi)7 = (Pio)7e . Ahygp I 1 1 ~\ aW »t3R-lMT)'+ «7,i ^ M ~mJJ. (3.48) The first term of the exponent represents the effects of the depressed vapour pressure at the curved meniscus, since the Clapeyron equation on its own only applies to saturated vapour pressure over a liquid with a flat liquid-vapour surface. Spolek [24] shows that the curvature effect of the meniscus does not significantly affect the vapour pressure in any way and can therefore be ignored, resulting in a simplification of equation (3.48), (ptf = (F 1;0 ) 7 e *?.i <^> < W . (3.49) It should be noted that the saturation dependence in the vapour pressure occurs as a result of the surface tension changing as the meniscus moves up or down the tapered part of the tracheid. Since the surface tension dependence has been eliminated from the vapour pressure expression then equation (3.47) can be reduced to, V(F)7 = ^ ~ ^ V < F , . (3.50) The total vapour/inert gas pressure gradient can then be calculated from the sum of the vapour partial pressure gradient (3.50) and the inert gas partial pressure (3.46) giving, Y7/PY7 l9^1 , dW\r>iTS , W H O r , r n Now adopting the notation of Spolek [24] the coefficients in (3.51) are written as, 7 d(p1r +d(p2) 7 - =— i#^ and, 7 ~ dS ' ( 3 - 52 > CHAPTER 3. WOOD 69 THEORY permitting (3.51) to be given as V(P 7 ) 7 = 6V7V(T) + CVyVS. (3.54) Equations (3.52) and (3.53) can be expressed by substituting equations, (3.46) and (3.49). The result is, Cr, = (P0,i)'W^e * ^ ^ + T p ( 1 _ g ) , (3-05) therefore C r T l =={-< 4 g = + < ^ , 2 (3.56) cSl = (Po*r-£z^T0' (3 57) R4T ' T and, - therefore, Cs, {P2V = T ^ n • (i-sy (3-58) An expression for the liquid pressure gradient is now required. The liquid pressure gradient can be defined by relating it to the vapour/inert gas phase pressure by way of capillary pressure. The capillary pressure is the pressure differential between the liquid phase and the vapour/inert gas phase, (Pc) = {P,y - (Ppf, (3.59) V(P /3 ) /3 = V ( P 7 r - V ( P c ) . (3.60) it then follows that, Capillary pressure is dependent on the surface tension and the curvature of the meniscus between the two fluids, and it follows that the capillary pressure gradient is dependent on the temperature and saturation, therefore V(PC> = ^ V < 7 ) + ^ V S . (3.61) CHAPTER. 3. WOOD 70 THEORY For convenience the notation of Spolek [24] will be used again, to label the coefficients in the following way, Or = ^ , (3-62) Cs = ^ , (3-63) and, allows equation (3.61) to be written more compactly as, V{PC) = CTV(T) + CSVS. (3.64) The liquid phase pressure gradient can also be written in the more compact form, V(P0f = {CTl - CT)V(T) + {Cs, - CS)VS. (3.65) Equations (3.62) and (3.63) can now be solved by substituting the capillary pressure given by equation (3.21), resulting in, 0.576CT CT= < -0.288 o r ^ Q /. xil (¥ aW (3.66) b < Ocr, and. 0 Cs={ , aW ' Ni m ' Srr < S ' "^ (3.67) rr " Leaving the liquid pressure gradient (3.65) as a function of known quantities. 3.7 One Dimensional Radial Transport Equations The coefficients have now been defined for the radial direction and therefore it is now possible to state the one-dimensional radial form of the transport equations (2.147) and (2.148), but a few simplifications can be made first. In Section 3.3 it was reasoned that the gas phase permeability is zero, and therefore the vapour/inert gas CHAPTER 3. WOOD THEORY 71 permeability terms vanish from the transport equations. Gravity causes hydrostatic pressure on the liquid phase, drawing liquid through the wood perpendicular to the earth's surface, but Spolek [24] reasons that hydrostatic pressure caused by gravity is negligible compared to convective and capillary pressure gradients. This reasoning is echoed throughout the literature and therefore the gravity related terms in the transport equations can be ignored. The absorption and release of energy as a result of evaporation and condensation within the wood structure is relatively small and therefore (m) is also reasoned to be a negligible term in the transport equations. As discussed previously, the evaporation-condensation cycle which occurs during bound water diffusion does not result in any net energy change because the energy absorbed in the evaporation from the lumen wall is released when condensing on the opposite cell wall. The result is a slightly simplified set of transport equations, « < ^ P)3{CP)0KI3 + (CT-CTy)V{T) + (Cs-Csy)VS •V<7} = V - W ( T ) , (3.68) and. ,es „ [( |V(T) + c c (£( »- *>) V(5) : V-.D m -V(M). (3.69) The measurable quantity representing fluid concentration is moisture content M, therefore the saturation S in the moisture transport equation should be substituted for the moisture content M, using equation (1.4). The gradient is vs = v M - Mfs:sp AM AM (VM-Viiy, (3.70) since AM is constant. Therefore AM V(M), (3.71) d(M) dt ' (3.72) and 1 AM 72 CHAPTER 3. WOOD THEORY By defining a dimensionless temperature Q, the temperature can be scaled to the same order of magnitude as the moisture content. Q may be substituted into equation (3.68) using the following transformation, v g = ^v. (3.74) V(T) = ATV(Q), (3.75) 9(1) = dt (3.76) Therefore and hence. ATd(Q) dl The transport equations, (3.68), and (3.69) then become, (P)CP-QT + pfahKp ({Qr _ CT,)ATV(Q) + (CS - C ^ l v W ) V(Q> = V-fc,V-(Q), (3.77) and, d(M) + V AM dt K,i M/3 (CT - CTl)\ ATV(Q) + f ^ ( C s - C, 7 ) M/3 rV(M) AM V-Dm-V(M). (3.78) At this point the volume average bracket notation will be dropped for convenience but will still be implied in the dependent variables. The radial form of the transport equations are then restated in a more visually appealing form, dt up M-c^xr^HC-c*,)^™ 09. - -— t — dr r dr Q dr ' (3.79) CHAPTER 3. WOOD THEORY 73 and, cp dM AM dt 1 d Kp ( T r dr fip c -cTlW§Hcs-cs^™ 1 d „ dM r or (3.80) In chapter 4, boundary and initial conditions will be posed to create an initial boundary value problem for equations (3.79) and (3.80) Chapter 4 Boundary Conditions 4.1 Introduction One of the research questions posed in the introduction is to examine the boundary conditions and their effect on conditioning times. This question is complicated by the fact that the log's surface is exposed to both liquid water as well as moist air, or multi-phase boundary flow. In the case of moist air, it can be assumed that convective currents will form along the log's surface. This is evident by the relatively sharp rise in temperature at the log's surface after the conditioning cycle starts, which is likely to suggest convective flow, since the much slower process of conductive heating alone could not produce such a sharp temperature increase. The forced spray coining from the sprinklers may contribute to convective flow forming at the surface, but the exact cause of the convective currents is less important then the effect on the boundary conditions. In addition to the convective flow of hot, moist air, liquid driven by gravity should make its way over the log's surface. The result would be a much different boundary, composed of liquid flowing axially over the log's surface. The difficulty arises in distinguishing which parts of the log is covered in moist air and 74 CHAPTER 4. BOUNDARY 75 CONDITIONS which is covered in liquid. In addition, it is not clear whether or not thermal energy and/or moisture transfer is dependent on phase, and if so to what extent. A careful search of the literature could not uncover studies with similar multi-phase boundary conditions, and attempting to describe this type of boundary from first principles is a relatively difficult exercise, which lies outside the scope of this project. In order to produce usable boundary conditions some simplifying assumptions must first be made. The boundary is reduced to a single phase by assuming the relatively small volume of water sprayed compared to the exposed surface area of the logs is enough to ignore the liquid phase at the surface entirely. This assumption becomes increasingly accurate for logs below the first couple layers of logs at the top of the stack. 4.2 Energy and Mass Balance at t h e Surface The energy and mass boundary conditions can be formulated by stipulating that energy and mass are conserved across the surface. In the case of heating, the thermal energy flux given by equation (4.7), or mass flux given by equation (4.16), crossing the inner surface of the boundary must balance the thermal energy and moisture flux crossing the outer surface of the boundary. That is, Thinner surface TTl outer surface \^,'~) Houter surface V*'^) and Qinner surface The thermal energy and moisture fluxes within the wood are known to be a combination of the transport mechanisms discussed earlier, as illustrated in Fig\ire 4.1. The thermal energy and moisture flux balances at the surface can be written as Robin boundary conditions, [14], [24] as, QQ —k0——h Ahvapm = q, at r — R, (4.3) 76 CHAPTER 4. BOUNDARY CONDITIONS and, dr dr LH-i fj,/3 AM dr (4.4) The thermal energy and moisture fluxes from the free stream q, and rh must now be examined. i outside n sjrface m n inside i Figure 4.1: Boundary fluxes during heating. 4.3 Inner Boundary Conditions The inner boundary, which lies at the center of the log can easily be obtained by assuming the solution is symmetric. Therefore the derivative of the temperature and moisture content must be zero. CHAPTER 4. BOUNDARY CONDITIONS 77 and, ^ = 0, or 4.4 at r = 0. (4.6) Outer Boundary (Inert Gas/Vapour Mixture) Applying convective boundary layer flow theory, fluid flow is stationary at the point of contact with the surface and equal to the convective flow stream velocity at some distance from the surface. This is generally known as the no-slip condition. Another result of the no-slip condition is that the temperature is equal to the surface temperature at the point of contact to the surface and equal to the convective free stream temperature some distance from the surface. The cylindrical shape of the logs results in void spaces in between the stacked logs, as depicted in Figure 4.2. As a result, the majority of the log's surface is exposed to longitudinal air flow. By approximating the longitudinal voids as cylinders, the geometry of the boundary layer can be simplified to a more well-studied case of cylindrical duct flow ([2], [10]). The transfer coefficients for thermal energy and mass between a fluid stream and the walls of a cylindrical duct will be applied to the heating and transfer of moist air to the surface of a log in the next section. 4.4.1 Thermal Energy Transfer Applying the theory discussed above, the thermal energy flux at the surface is given by Bejan [2] as, q = hg(T8-T00), (4.7) where T^ is the stream temperature, Ts is the surface temperature, and hq is the thermal conductivity of the boundary layer, which is also referred to as the surface thermal energy transfer coefficient. This expression is derived from Fourier's law CHAPTER 4. BOUNDARY CONDITIONS 78 Figure 4.2: Illustration of the void space, shown in grey, created between the logs when stacked in the conditioning tunnel. The large circles represent the ends of the logs, and the small inner circle represents the hydraulic diameter D0. and implies that the surface temperature depends only on the temperature difference between the free stream and the surface, and the thermal energy transfer coefficient of the boundary layer. Bejan [2] and Incropera [10] give the surface thermal energy transfer coefficient for internal convection of turbulent flow through a cylindrical duct as, where hq is the surface thermal energy transfer coefficient, Nu is the Nusselt number, k is the thermal conductivity of the bathing fluid, moist air in this case, and D0 is the hydraulic diameter. The hydraulic diameter is an effective diameter describing the cross section of the flow. In this case, the void cross section is assumed to be circular, and therefore the hydraulic diameter is simply the diameter of the circular region. The roughness of the logs surface and disturbances from random spray patterns of hot water sprinklers strongly suggest that the resulting flow in the voids will be in CHAPTER 4. BOUNDARY 79 CONDITIONS the turbulent regime, but the flow regime in the voids can be better understood by calculating the Reynolds number. The Reynolds number. Re, is a dimensionless quantity describing the flow type, and in the case of internal flow through a duct, the Reynolds number is calculated as, Re = l -^, (4.9) where Uoo is the stream velocity, and v is the viscosity of the bathing liquid. The Reynolds number can be useful in identifying the flow regime, for example Reynolds numbers in the range Re > 2300 suggest turbulent flow whereas Re < 2300 suggests laminar flow. Since a free stream does not technically exist in internal flow, the free stream velocity U^ is more accurately described by the mean fluid velocity um, over the duct's width. Calculating the Reynolds number requires some detailed information of the flow such as the mean fluid velocity, um% but since no fluid velocity data are available for the conditioning tunnel it will have to be approximated. Using a conservative approximation for the mean fluid velocity of 1.5 —, and an approximate hydraulic diameter of 0.10m. a rough calculation can be done to approximate the Reynolds number in the voids between the log segments, umD0 (1.5 s )(0.10m) Re = _J2_£ ~ ^ dl > = 10000, v (1.5 x 1 0 - 5 ^ ) 4.10 which clearly lies in the turbulent range. Another dimensionless quantity, the Nusselt number is required in the calculation of the surface thermal energy transfer coefficient. The Dittus-Boelter equation [10], approximates the Nusselt number for turbulent flow through a cylindrical tube as, Nu = 0.023Re% Pr11, where n = 0.4 for heating (T^, > Ts), and n = 0.3 for cooling (T^ < Ts). (4.11) This correlation has been validated through experiment for the following conditions; 0.7 < 80 CHAPTER 4. BOUNDARY CONDITIONS Pr < 160, Re > 10000, and ~ > 10, where L is the tube length. The Pr number is another dimensionless quantity known as the Prandtl number and is defined as, Pr = - . a (4.12) where a is the thermal diffusivity of the bathing fluid. It is interesting to note that in the case where the temperature difference between the surface and the free stream is extreme it is recommended that the temperature dependence of the viscosity be taken into account giving, Nu = Q.027Re^Pri / \ °'14 ( ^- J , (4.13) where \is is the viscosity of the fluid at T ^ . Equations (4.8), and (4.11) can now be combined to form an expression for the surface thermal energy transfer coefficient, hq= (^-\ 0.023 RelPrn. (4.14) The above approximation assumes that the voids between log segments are not only close to cylindrical but also that the log segments surface is smooth. A rougher surface would result in a higher Nu and therefore higher hq, which means that equation (4.11) will probably underestimate the value of hq. Substituting equation (4.7), and (4.14) into equation (4.3) and substituting the the dimensionless temperature gives the Robin thermal energy flux boundary, _kdQ(R,t) 4.4.2 + Ah^ifh = f^\Q023RelPrnAT(Q(R, t) - l). (4.15) Mass Transfer Being a porous material, both energy and mass are exchanged with the environment at the log surface. Bejan [2] and Incropera [10] analyze the mass transfer boundaries analogously to the thermal energy transfer boundary, for instance the mass flux across CHAPTER 4. BOUNDARY CONDITIONS 81 the surface is given as a function of the concentration difference between the air stream Coo a n d the surface Cs, m^h^Cs-Coo), (4.16) where hm is the surface mass transfer coefficient. The moisture content of moist air cannot be defined as it is for the wood surface and therefore not usable in describing the mass concentration. Instead, the vapour pressure difference is considered to describe the mass concentration ([10], [24], [2]), giving the mass flux as m = /iro(F1-Fli00). (4.17) Although vapour pressure is defined for both the surface and free stream, the mass flux described by equation (4.17) does contain an explicit dependence on moisture content at the surface. This lack of coupling between the mass flux and the moisture content causes the initial boundary value problem to become ill-posed and therefore unusable in its current form. Attempts to resolve this have not been successful and therefore as a first approximation to the problem mass boundary will be reduced to a Dirichlet boundary, M(R,t) = M0. (4.18) This simplified approach introduces another problem since this type of boundary is somewhat unphysical. This can be understood by examining the physical significance of the Dirichlet boundary. The application of a Dirichlet boundary implies the convective boundary layer described at the beginning of the chapter is non-existent. Instead, the moisture content at the surface is artificially set to a constant value, and it is assumed that the external conditions are able to sustain the surface moisture content at a constant level regardless of the temperature and moisture content gradients inside the wood and the temperature and humidity outside of the wood. It also implies that the mass flux at the surface is dependent only on the internal conditions of the wood. CHAPTER 4. BOUNDARY CONDITIONS 82 An argument can be made that during the conditioning process, if the humidity stays at a relatively constant level, which is a plausible condition, and the boundary value for the moisture content is assumed to be the equilibrium moisture content (EMC), the Dirichlet condition becomes a relatively accurate representation of the boundary. Nevertheless, this still remains an undesirable representation of the mass boundary, but unfortunately further investigation is not within the scope of the project. Chapter 5 Numerical Method For this one-dimensional problem a simple explicit finite-difference scheme has been chosen. In particular, a first order forward difference is used for the temporal terms along with a second order central difference for the spatial terms giving a scheme which is first order in time and second order in space or a FTCS scheme ([5]). The mixed boundary conditions will be handled using a fictitious point approach at the outer surface boundaries, and cylindrical symmetry can be used to address the inside boundaries. The numerical code was developed in C + + . One of the drawbacks of finite difference methods is that a regularly spaced grid can be limiting when solving over the irregular topology of logs. In this study, the irregular boundary is handled by assuming, that the log does not have any significant deformations on the surface, and the segment of interest is short enough that the natural taper of the tree is not significant and it can therefore be approximated as a perfect cylinder. Additionally, by transforming the defining equations into cylindrical coordinates, a regular spatial grid can be formed over the log segment. Further details on finite difference schemes for solving differential equations can be found in the literature [5]. 83 CHAPTER 5.1 5. NUMERICAL 84 METHOD Discretization of Thermal Energy and Moisture Equations The FTCS scheme is applied to the system of equations given by (3.68) and (3.69). Starting with the thermal energy equation (3.68), Q?+1-Qi At n r.;_t_ i k1., i {pC^nAr (pcPy; * • ( > - < * ) ) ; 1 Ar Qi'+l Qi-1 2Ar Ml i+l M?-i\f<&-i-QLi 2Ar 2Ar (Kp A M V /i/3 cT-c ' 5 «-5V Ar TJ , (5.1) and the mass equation (3.69), At AM 1 ri,i.U,,x | d> r,Ar *+2 i+s 1 1 r 0 rj Ar *+i #/3 I - r,-_iiA. i 5 '^2 \ : Ar Ar Mr+1-Aff- ( CT — C T 7 *+^ tffl '' 2 (Cr — Cr7J AMAT 1 r r^Ar *+4 Ka r,; i 2 \ A*/3 (c T d Ar te^L( Ar Q\l-QU Ar (5.2) Where the subscript i is the spatial grid indices, the superscript n is the temporal grid indices, Ar is the spatial grid size, and At is the temporal grid size. CHAPTER 5.2 5. NUMERICAL METHOD 85 Discretization of B o u n d a r y Conditions The boundary conditions given in equations (4.15) and (4.18) are different, in that, the thermal energy boundary is a Robin condition and the moisture boundary is a Dirichlet condition. Unlike Dirichlet boundaries where the boundary point is given explicitly as a constant value, Robin type boundaries involve the gradient of the dependent variable at the boundary. The Robin boundary can therefore be dealt N : — i w P— +:E' k 1 -•X *-h-+ Figure 5.1: Fictitious point example. with numerically by applying the fictitious point method. This is accomplished by considering an imaginary point E' which exists outside the boundary illustrated in Figure 5.1, where the derivatives are specified, for instance, dx = g(x,t). (5.3) By applying a backwards difference, the second derivative at the boundary point P is given by, d2T dx2 2(TW -TP + hgP) + 0{h). h2 (5.4) For further details on the ficticious point approach the reader is referred to Yardley [5]. CHAPTER 5. NUMERICAL 86 METHOD The temperature boundary condition is given by rearranging equation (4.15) to give the derivative of the dimensionless temperature, dQ, q-mLvap A aT (iM)= -kq (5.5) ' and the Dirichlet moisture boundary is given by equation (4.18), (5.6) M(R, t) = M 0 . The discretized temperature boundaries are essentially of the same form as equations (5.1) and (5.2), the only difference being in the calculation of the i = 7 + 1 and i = I+\ terms. At the boundary the i = I + 1 and i — I + \ terms are ficticious points and can be expressed in terms of f(t) by applying a forward difference approximation. For instance, dQ, 1 ^ = '(*>• (5.7) Q1+1 = 2Arf{t) + QU, (5.8) *-<*<> = ^ therefore, and similarly for Q.i+i QnI+± = krf{t)+Qri_,. (5.9) The above discretization can now be applied to equation (5.1), keeping in mind that f H M = 0, to give QT ~ Qi Ar •i- , hn (pCXnAr v_KT ] l 2 ~~l pacPf}AT Qi 2Ar/(t) + Q ? _ 1 - Q ? Ar Ql-i Ar fa): CT, fit) fit). (5.10) The discretized boundary condition for M(R,t) is given by, M}1+1 = M0. (5.11) CHAPTER 5.3 5. NUMERICAL METHOD 87 Numerical Stability Choosing an appropriately sized spatial step is important because a grid that is too course could result in a loss of information and a grid that is too fine can lead to an inefficient numerical scheme. The choice of spatial step size is especially important in an explicit scheme because the spatial step size is coupled to the temporal step size and therefore affects the stability of the numerical scheme. This dependence can be analyzed by considering the Courant Number, Where At is the temporal step and Ar is the spatial step. Ideally, a Fourier stability analysis of the numerical scheme could be used to calculate the constraints on the Courant Number for which the numerical scheme is stable. In this case, the nonlinearities in the governing equations make a rigorous stability analysis difficult, and instead, an experimental approach will be utilized here by considering a range of Courant Numbers and visually analyzing the results for signs of instability. Being a predominately diffusion dominated model, it will be hypothesized that the numerical method will also behave as it would for the general one-dimensional diffusion equation, such as (1.1), for which the following restriction is necessary for stability of the scheme, D W? < \ where D is the thermal diffusion coefficient. (5 13) - It is thought that a spatial step of Ar = 5 mm will be small enough to produce enough spatial resolution to accurately represent the conditioning process, therefore applying the above condition, the time step requirement would be, At < 1.25 x 1 ( T 5 ^ . kq (5.14) CHAPTER 5. NUMERICAL Parameter r AT M0 K0 a a hq 88 METHOD Value 15.0 cm 60.0°C 0.60 5.00 x 10-16 m2 0.80 0.25 15.0 Y„ Reasoning average log diameter To = 5 ° C , I'max = 6 5 C average initial MC [24], [14], [4], [17], [16] average for softwoods, [24], [9] average for softwoods, [24] Table 5.1: Initial conditions and parameters used in stability analysis. The thermal diffusion coefficient, density, and heat capacity are calculated by equation (3.33), (2.99), and (2.100) respectively, along with mean temperatures, moisture content, and tracheid ratio values of T = 30 °C, M = 0.60, and a = 0.2 respectively, giving, /U o \ -*• -*• o o Ol O Ol O o o o ^S.. *"'^|V o CJ1 - ill3 - - o \ o O 1 1 o Ol Temperature (°C) o O N> Ol O ro o o o o -J ?^ ii o o en o o CD 4^ 3 H ro o o o o o O 3 3 3 - oi o oi o o o o 3g _i _i O . o o o o X\ '•ON. ^'O^ N \ \ o O 1 1 ^ o Ol Temperature (°C) ^ N L ~ ~ -• ^ - o Ol Temperature (°C) o o II o o o o II CD 00 >> o> o o o O O W O O v -fe. O V v V | 1 V \ V 1 \ O) O \* \* X *\ VV . %, s Temperature (°C) "••.JN. *"-.r^^^ o o o o _i _i b b M O Temperature (°C) o - Ol O Ol O o o o o 3g o 3 3 3 o o CD 4^ _, LU o o o Ij^ 4^ O O o o o o o >> II IO 00 CO to fcq O I 2 1 CHAPTER 5. NUMERICAL METHOD 90 instability include the large oscillatory behavior seen in Figures 5.2c, 5.2d and 5.2e. Although A ~ 1.20 x 106, seems to be more then sufficient to produce a stable scheme, to assure stability is achieved in all situations A = 0.40 x 106, has been chosen for all the calculations in the following chapters which corresponds to a temporal step size of At = 105. Chapter 6 Experimental Data 6.1 Experimental Setup In order to ensure the accuracy of the model, results will be compared to timedependent experimental data. Thermal data consists of measurements taken from the interior and exterior surface of the log segment along with some strategically placed thermal sensors in and around the conditioning tunnel to collect ambient temperatures. In addition to thermal data, it was first thought that time dependent moisture content data could be obtained as well. It was later realized that moisture content measurements could not efficiently be carried out within the working conditioning tunnel environment and subsequently left out. A suitable alternative was to take average moisture content measurements from the log segments before and after conditioning. There were 27 experiments, and the data are displayed in Appendix B, Figures B.l(a) to B.27(e) superimposed on numerical solutions given in Chapter 7. This chapter contains discussion of the experimental procedures and errors. 91 CHAPTER 6.1.1 6. EXPERIMENTAL DATA 92 Thermal D a t a Collection The conditioning tunnels of a working mill are a very hostile environment for data measurement equipment. The environment within the conditioning tunnel is very hot and humid with sharp temperature changes when warmed logs are replaced with a new batch of cold, and sometimes frozen logs. In addition to temperature and humidity, the log segments, sometimes weighing upwards of 400 kg, are loaded and unloaded by heavy duty machinery which moves approximately 5-12 segments at a time, depending on diameter. This can produce very high crushing and acceleration forces on the log segments and any measuring equipment on or around them. In order to facilitate data collection a steel cage was constructed of 15 cm square tubing to hold 6-8 segments at a time which could easily be handled by the equipment used to load and unload log segments. The steel cage served a few purposes; it would allow for the experiment to be setup away from the hazards of the operating conditioning tunnels and later be maneuvered into the conditioning tunnel as a single unit. It also served to help protect the sensors and data-logging equipment from being crushed by the handling equipment or other logs. By making the cage able to hold 8-10 log segments the test log could still be surrounded on all sides by other logs, permitting the test log to be exposed to the same exterior conditions as during a normal conditioning cycle. The cage and data logging equipment are shown in Figure 6.1. Due to cost constraints the number of thermal sensors was limited, and therefore it was decided to limit thermal data collection to two dimensions within the log segment. The chosen dimensions were the radial (r) and longitudinal (z) dimensions, the radial being the most significant in terms of thermal gradients. The z dimension was chosen because it was thought the warming process was essentially axially symmetric, and therefore z would show the only significant thermal gradients besides the radial dimension. CHAPTER 6. EXPERIMENTAL DATA 93 Figure 6.1: (A) Finished apparatus ready to accept another row of log segments. (B) Finished internal thermal sensor placement in log segment. The final thermal sensor configuration consisted of a row of five sensors located as follows: one at the core, one 5 cm from the core, one at approximately 9 — 11cm from the core, one in the heartwood just past the sapwood-heartwood interface, and one on the surface. The radial positions where chosen to reflect points of interest within the log during heating which is further illustrated in Figure 6.2. The core temperature is of interest because it reflects the minimum temperature in the log being the farthest point from the heat source. The 5 cm position is important since this is the maximum radial depth reached by the lathe during veneer cutting. The heartwood sensor is required to get an idea of any thermal discontinuities between the heartwood and sapwood. The surface sensor was deemed important in order to analyze the boundary conditions, and the 9 — 11 cm position was chosen to fill in the sometimes large distance between the 5 cm position and heartwood. In the longitudinal direction three sets of the thermocouple configurations described above were setup at specific intervals from the mid point of the log segment to the end. The chosen intervals were at 10 cm from the end, 30 cm from the end, and at the midpoint, approximately 128 cm from the end. The midpoint was recorded CHAPTER 6. EXPERIMENTAL 94 DATA 1Ocm0 unused core Thermal sensors M•M -Sapwood Heartwood Figure 6.2: Approximate radial placement of thermal sensors. because, logically it is the least affected by heating form the log ends. The 10 cm and 30 cm intervals were chosen to analyze the effects of heating from the log ends. These two intervals, although far apart, were used to give an approximation of the thermal energy gradient in the longitudinal direction. The assumption was made that heating of the log segment was more or less symmetrical about the midpoint and therefore sensors were only placed on one end of the log. To verify this assumption, thermal sensors were placed on opposite sides of the steel cage, which could be used to uncover any end-to-end temperature differences. Each radial row of internal sensors had a corresponding surface sensor which was attached to the surface of the log segment using staples. In addition to the three main radial rows of sensors, one sensor was placed on the end of the log segment at r = 0, z — 0 to record surface temperature. The internal sensors were inserted by first drilling a 9.5 mm hole into the log to a depth required to intersect a radius from the centre of the log segment at right angles, which is illustrated in Figure 6.3. This technique was thought to give the most accurate placement of the sensors within the log segment, while at the same time minimizing interference with surface flows over the log. The thermocouple wires CHAPTER 6. EXPERIMENTAL DATA 95 were routed around the log using staples into a shock and water proof enclosure, and in order to protect the thermocouple wires from the sharp edges of the staples thin cardboard strips were sandwiched between the wire and staple. The enclosure was then attached near the bottom of the steel cage in order to help protect it from damage. Figure 6.1 shows a log segment being prepared for conditioning. Figure 6.3: (A) Drilling log segment to prepare for insertion of temperature sensors. (B) Cross section of thermal sensor insertion technique. 6.1.2 Moisture Content D a t a Collection Moisture content data collection is a slightly more complex task which can be understood in the following discussion. Although there are a number of moisture content measuring techniques, each has its own limitations which made it unrealistic to collect time dependent moisture content gradients within the log segments during conditioning. Some of these methods include: resistivity/conductivity measurements, capacitance measurements, computed tomography, nuclear magnetic resonance, and the mass difference method. In order to collect time dependent moisture content, the method would have to meet certain requirements such as: it must be reasonably CHAPTER 6. EXPERIMENTAL DATA 96 accurate (on order of thermal-couple accuracy), nondestructive, portable, automatable, robust, and relatively inexpensive. The resistivity/conductivity and capacitance methods are relatively similar in that the measurements can be correlated to moisture contents if some parameters of the wood are known such as species. Computed tomography uses X-rays to build a 3D picture of the different internal structures of the wood. Since wet wood fibers and liquid water can be made to appear different than drier wood fiber under X-ray, it becomes possible to extrapolate the moisture content in the given specimen. This method is more suited to studying wood structure but has also been adapted to studying moisture content ([20]). Nuclear magnetic resonance (NMR). also known as magnetic resonance imaging (MRI), works by using a strong magnetic field to align the spins of hydrogen atoms in hydrogen containing materials. An RF pulse is then applied at a specific frequency known as the Larmor frequency putting some of the hydrogen atoms into an excited state. Once the RF pulse has passed the excited hydrogen atoms revert to a low energy state thereby emitting electromagnetic energy at a particular frequency which is then measured by sensitive measuring equipment. The frequency and time it takes for the hydrogen molecule to drop back to its original state can then be used to differentiate between and obtain the concentration of the chemical species the hydrogen atoms are bound to. In the case of measuring moisture content in wood, bound water produces a different resonance frequency than liquid water or wood fibers, therefore the moisture content in the form of bound or free water can be found quite accurately in this manner. Computed tomography and nuclear magnetic resonance, although technically very different, are similar in the way they are applied to the specimen as well as being expensive and not portable. The last and most widely used method is the mass difference method. This method requires a sample of the specimen to be taken, weighed, and then dried to what is known as oven dry conditions. Oven dry means all free and bound water has CHAPTER 6. EXPERIMENTAL Method Resistivity Capacitance CT 1 NMR 2 MD 3 Accuracy low-medium low-medium high high medium DATA Portability high high very low very low medium-high 97 Automatable yes yes no no no Durability medium medium low low high Cost medium medium very high very high low Table 6.1: Comparison moisture content measurement methods. been removed from the sample and only wood fiber is left. By doing this, the mass of water that was in the sample can be found and divided by the oven dry mass, giving the fraction of water to wood fiber in the sample. This is the standard measure of moisture content in the literature ([23]). The major downfall of this method is that it is not automatable, and can be destructive, but on the positive side is relatively accurate, and inexpensive. The attributes of each method are given in Table 6.1. For this study it was decided that mass difference would be used to collect moisture content. This choice was made primarily based on cost and low experimental error. Methods such as CT and NMR were out of the question because of their very high cost and lack of portability. Although resistivity/conductivity, and capacitance are automatable they would require data-logger hardware in addition to the thermal datalogger hardware. The addition of further data-logging equipment put the cost of this method far beyond the scope of this project. Additionally, the resistivity/conductivity and capacitance sensors would require careful calibration and lack the rugged nature of thermal sensors and therefore would not be suitable for the conditioning tunnel environment. It was therefore decided that measuring time dependent moisture content gradients was not feasible and only before and after conditioning moisture contents would be available for the log segment being studied. 1 NMR-Nuclear magnetic resonance. CT-Computed Tomography. 3 MD-Mass Difference. 2 CHAPTER 6. EXPERIMENTAL DATA 98 Mass difference measurements were conducted using a coring tool which is used to extract an approximately 4 mm diameter cylinder of wood from the surface to the core of the log. The sample is then weighed on site, taken back to the lab and dried for 24-48 hours in an oven, then weighed again. Using the two moisture content measurements, the moisture content of the specimen was then calculated. Some difficulties were encountered using this method such as resolution of the data. The core samples represented an average moisture content from the surface to the core of the segment, therefore no radial moisture content gradients were available. Additionally, since only a few samples were taken from each log segment, resolution in the longitudinal direction was limited to two or three points and therefore did not produce meaningful moisture content gradient data in the longitudinal direction either. In an attempt to achieve a more meaningful measurement of moisture content, later moisture content measurements were enhanced to show the difference between sapwood and heartwood. This was done by separating the extracted core into sapwood and heartwood before taking mass measurements. This was difficult in healthy pine and spruce since the sapwood and heartwood are relatively similar in colour, resulting in additional measurement error, but ultimately a much more accurate picture of moisture content in the log segment was obtained. 6.1.3 Experimental Errors Temperature The thermal sensors used in the study are narrow band T-type thermocouples. Thermocouples were chosen for their cost effectiveness and durability. Although the T-type thermocouple temperature range is the most well suited for this study, thermocouples tend to suffer from a lack of resolution and therefore accuracy in narrow temperature CHAPTER 6. EXPERIMENTAL DATA 99 ranges. The T-type thermocouples and data-loggers used in this study are rated from the manufacturer with the following accuracy for the temperature range —35°C to 200°C: Terror = ± 1 % of span + resolution, (6.1) Terror = ± 2.35°C + 1.7°C, (6.2) Terror = ±4.05°C. (6.3) which is, therefore, In an effort to increase resolution and accuracy of the surface temperature measurements a set of thermistors was also employed in the study. The thermistors used were housed in stainless steel tips for faster response time and have the following accuracy for the temperature range —35°C to 95°C, Terror = ± 0.2°C°C + 0.5°C, (6.4) Terror (6-5) therefore, = i 0.7°C. Although much more accurate then the thermocouples, thermistors lack durability and therefore suffered extensive damage after only a fewT runs and were discontinued for the remaining data runs. Aside from the limitations of the thermocouples, random error is also introduced into the measurements as a result of the two phase flow within the conditioning tunnel. For example, although the majority of the logs surface is in contact with humid air, there is the possibility of liquid water, which could be at a slightly different temperature then the humid air, coming into contact with the logs surface. Although precautions were taken to seal the thermocouple insertion points, there is still a possibility of liquid either entering the hole where the thermocouple is placed, or coming CHAPTER 6. EXPERIMENTAL DATA 100 into contact with one of the surface sensors. In addition, measurements could be affected by damage to the thermocouple wiring and electronic hardware. Although precautions were taken to protect the equipment, and necessary repairs were done after each data run, the relatively harsh environment encountered within the conditioning chambers made it nearly impossible to avoid such damage. This type of damage is apparent in some of the latter data runs where unreasonable temperature measurements are seen in some cases, and the only possible reason is a malfunctioning data-logger. Another significant source of error is the estimation of the thermal initial and boundary conditions. The model requires the initial temperature and maximum temperature to be specified as constant model parameters in order to estimate initial and boundary conditions. Unfortunately the initial temperature of the log is not uniform and cannot be represented by a constant value, therefore an average initial temperature is calculated for each of the three radial sets of experimental data. The maximum temperature should theoretically be a constant value for a given data set since the maximum temperature is essentially that of the water/air temperature in the conditioning tunnel. Unfortunately, due to various uncontrollable factors in the mill, the water temperature is not constant, and making this assumption can introduce a fairly large source of error. To address this, the variance in the maximum temperature can be calculated and used to estimate error in the experimental data. The variance is calculated as, where Tj is the ith sample, T is the average of the entire sample, and n is the sample size. In this case the sample is the set of temperature data points which lie in the steady-state portion of the temperature vs. time plots. After a sufficient time t the entire log segment should reach some final temperature equal to the maximum CHAPTER 6. EXPERIMENTAL DATA 101 temperature, this state is known as the steady-state, where the temperature no longer depends on time. The average, T is computed by considering the steady-state portion of all the surface data points, f E£°i (TJ) + E S ft;) + E S 8 (Ti) ^ "10 + ^30 + ^128 where the subscripts 10, 30, and 128 represent the three surface data points. The surface data points are used since they are the first to reach the steady-state temperature thereby providing the largest sample size, and being on the surface they should most closely represent the maximum temperature. Then for each set of radial data points the standard deviation from the average maximum temperature seen by the log segment can be computed and used to represent the measurement error in the experimental data, Terror = ± l / ^ V ^ ~ ^ . (6.8) n—1 The time accuracy of the loggers is rated at ± 2 seconds per day, which is ± 0.000023 % over one day and can therefore be safely neglected for the purposes of this study. Position The spatial accuracy of the measurements is heavily reliant on the experimenter. Error in the placement of the internal thermocouples is dependent on both the angle and position of the drill when boring the holes, as well as non-uniformity on the dimensions of the log segment. It is standard practice to calculate precision of a measuring tool to be half of the smallest increment, which in this case is ±0.5mm. Although, in this case, the precision of the measuring tape is dwarfed by the precision associated with the angle of the drill when drilling the hole to insert the thermocouple. Since the holes where essentially drilled freehand, meaning there was no sort of brace used to ensure the holes went in exactly perpendicular to the radius of the log segment, CHAPTER 6. EXPERIMENTAL DATA 102 then it makes sense to associate an error in the drill angle. This angular error was estimated at approximately ±2.5°. The angular error translates to a linear uncertainty of approximately ±10 m,m therefore, rerror = ± 1 0 771771, (6.9) (6.10) The spatial error associated with the radial measurement is estimated to be relatively high due to the method used to drill and place the internal thermocouples, and although relatively error prone, is thought to be the most effective method for placing the internal thermocouples. Moisture Content Error in moisture content measurements is likely to come from a few sources which are listed below from greatest to least importance: • moisture content gradients, • experimenter error (separating sapwood and heartwood), • time delays between extracting sample and measurement, and • accuracy of mass measurement. The log may contain initial moisture content gradients in the r, z, and 6 directions. Since it is feasible to only take a limited number of samples from the wood for moisture content measurement, it is impossible to get an accurate average measurement of moisture content throughout the log segment. For instance three samples are taken before conditioning in approximately the same area as the temperature sensors are embedded. After conditioning, because the samples require material to be removed, CHAPTER 6. EXPERIMENTAL DATA 103 the same area cannot be tested again, and instead a location near to the first is chosen. This, in effect, changes both the time and position variables, thereby giving rise to error in moisture content measurements. Initial moisture gradients in the wood may be caused by factors such as non-uniform exposure to a heat source such as the sun, and the inhomogeneous nature of wood structure. The separation of sapwood and heartwood moisture content measurement introduced error because of difficulties differentiating between sapwood and heartwood. Since sapwood moisture content, in most cases, is much higher then heartwood, including some of the sapwood in the heartwood measurement could result in a larger value then expected. Time delays between the end of the conditioning cycle and moisture content measurement is another source of error. Time delays occur since the test apparatus is not always removed from the conditioning tunnel immediately after the cycle is complete. The delay between conditioning cycle completion and measurement, ranges from minutes to hours. Since the drying of wood is a relatively slow process, this source of error was relatively insignificant compared to the first. The accuracy of the scale when weighing the samples could be considered a source of error, but in this case is thought to be insignificant compared to the previously discussed sources. It is difficult to quantitatively give the error associated with these sources and therefore it will be estimated as, Merror = ± 25%. (6.11) Chapter 7 Results 7.1 Temperature Data Analysis In this chapter experimental results from the previous chapter will be compared with theoretical results from the model. In addition, experimental moisture content data will be discussed, such as moisture content change during conditioning, moisture content differences between heartwood and sapwood, and moisture content changes of debarked logs over long periods. A sensitivity analysis is preformed on the numerical model to analyze the output variability of the model. This is of interest in understanding which input parameters affect the model most significantly, and therefore how error in the input parameters affect the model results. 7.1.1 Predicted Versus Real Conditioning Cycle Times As mentioned earlier, it is of great interest to the industry partner North Central Plywoods, to optimize the relatively energy hungry conditioning process by predicting conditioning cycle times for log batches. By comparing measured conditioning times to predicted conditioning times for individual logs the accuracy of the model can be 104 CHAPTER 7. RESULTS 105 judged. Table 7.1 gives a comparison of predicted conditioning times to measured conditioning times, by comparing the amount of time it takes for the center of the log to reach optimal cutting temperature. Optimal cutting temperature is given in the table as Topumai- The time predicted by the model is given in the table as tpred and the experimental temperatures are taken from the center of the log at three separate locations, z = 10 cm,,30 em, 128 cm from the end of the log, denoted in the table as texP\z--=wcm.,texp\z=3(scm, and texp\z^i2sCm respectively. The error in the total measured conditioning times was extracted from error in the temperature by considering the function, T = f(t) (7.1) t = f-l(T). (7.2) then taking the inverse, By including the error in the temperature T ± 8T, the temporal error can be found, t±St = /^(Ti^T), (7.3) ±5t = f-\T±5T)-t. (7.4) In addition, Table 7.1 also lists the amount of time the batch of logs actually spent conditioning denoted by tactuai, and the actual temperature inside the conditioning tunnel denoted as T^.. 7.1.2 Post Conditioning Cooling After a conditioning cycle is complete it takes approximately two hours to process one conditioning tunnel full of logs, and in this time, significant cooling can occur, depending on the ambient conditions. Although the information given in Table 7.1 is sufficient for making a comparison of model data to real data, it is impractical when directly applied to mill applications. In order for the logs at the back of the tunnel to ''actual (hrs) 81.2 9.16 9.95 72.1 64.8 114 19.3 33.3 94.2 tpred (hrs) 7.83 3.97 6.03 9.78 12.0 15.25 24.4 17.8 17.3 — — 23.0 ± 5.06 — — 23.0°C± 16.2* — — — Lexp] z=128cm (hrs) 7.60 ± 1.97 3.78 ± 1.00 6.01 ± 0.75 13.2 ± 0.97 14.3 ± 2.39 25.4 ± 2.92 -1.47°C± 4.39* ^exp\z=30cm (hrs) 7.57 ± 0.47 4.87 ± 0.56 6.60 ± 0.56 12.0 ± 1.50 14.5 ± 3.5 21.6 ± 3.58 (hrs) 5.87 ± 3.72 2.48 ± 1.22 4.44 ± 0.50 6.78 ± 9.69 9.18 ± 3.81 £exp|z=10cm 35.0 30.8 30.8 26.7 30.8 30.8 26.7 26.7 26.7 °c -* optimal * Did not reach optimal temperature during conditioning, instead final temperature is given. — Time values have unreasonably large errors. Diameter (an) 31.0 30.0 29.0 34.0 37.0 37.0 47.0 47.0 47.0 °C 48.8 51.9 45.2 43.3 53.1 45.3 44.7 51.5 52.0 Too Table 7.1: Experimental versus predicted conditioning times. Values given here correspond to the core (r=0). Fir BK Pine Pine Spruce Pine Pine Spruce Spruce Spruce Species CHAPTER 7. 107 RESULTS reach the lathe within the appropriate temperature range, they must be heated slightly above the optimal peeling temperature. In addition, the model should be able to take into account cooling as well as heating. An example of the cooling that occurs near the surface of the log after the conditioning cycle has stopped is illustrated in Figures 7.1-7.2. The temperature profiles seen here reflect the surface temperature of the • 60 V \l A "» O \ 2 40 ' \ 3 CO 03 ' *av -1 -\ a. £E 20 [ ^ *• .A A Optimal Temperature Range for Peeling A i A ** . - • . . i ' r=13.0cm, Experimental Surface, Experimental r=13.0cm, Theoretical Surface, Theoretical Ambient Temperature '-. '•- Unloadinc1 Period i 1 0 1 i 0 2 * ** ' - 1 -" A A * T A - * _ * _ * * • • • • i I • • • -r*- Mj T -rl hn J 4 6 Time (hours) Figure 7.1: Temperature of spruce sample after conditioning cycle is complete. log and the internal sensor closest to the surface. The two hour unloading window is shown as the vertical grey bar, where it can be seen that significant cooling occurs in both cases within this time span. For instance, in Figure 7.1 the surface temperature of the log dropped approximately 28°C and the temperature 2cm from the surface dropped approximately 19°C. CHAPTER 7. 108 RESULTS 60 r=21.0cm, Experimental Surface, Experimental r=21.0cm, Theoretical Unloading Period • O 4 0 t* \'".' \ 0 -". ' A A t 2> \ •:• • m V ® on "1 Q . 20 *> E 1*-* H ' [•*« J |***J Optimal Temperature Range for ! Peeling T""t — I- I • . T ' t **4 •" ..I 1 r 0 1 i 0 . 1 1 2 4 , , r 1 , > . 6 Time (hours) Figure 7,2: Temperature of spruce sample after conditioning cycle is complete. 7.1.3 Comparison of Model to Experimental Time Dependent Thermal D a t a To verify that the theoretical model is exhibiting physically accurate results, experimental temperature data have been plotted over model data. The model data were produced by inputting experimental parameters, specific to the particular data set, into the numerical solver to produce time dependent data. By including experimental measurement error a quantitative comparison of theoretical data to experimental data can be made thereby making it possible to draw conclusions on the accuracy of the model as well as identify model deficiencies. The fits being referred to are given in Appendix B, figures B.l(a) to B.27(e), where experimental temperature data are given by the square points and the error in the temperature measurements are given as vertical bars. The theoretical data produced by the model is given by the solid lines, and the dotted lines represent error in the positioning of the temperature sensors. CHAPTER 7. RESULTS 109 Species Conditioning Time Mmtial (%) Mfinal (%) BK Pine Spruce Pine Spruce Spruce Spruce 09h 56m 51s 72h 04m 13s 64h 47m 01s 19h 20m 28s 35h 15m 33s 94h 10m 45s 36.5 ± 9.13 63.5 ± 15.9 48.9 ± 12.2 117 ± 29.3 93.4 ± 23.4 102 ± 25.5 31.8 ± 7.95 63.6 ± 15.9 54.7 ± 13.8 93.4 ± 23.4 89.7 ± 22.4 75.5 ± 18.9 Table 7.2: Average moisture contents of cylindrical log segments before and after conditioning. The temperature error bars seem to vary significantly between data sets, in some cases becoming orders of magnitude larger then the measurements themselves. This seemingly unrealistic error likely reflects technical problems with measuring equipment discussed in Chapter 6, and instead of discarding these data sets they have been kept for continuity and completeness. 7.2 Experimental Moisture Content Data 7.2.1 Average Moisture Content During Conditioning Average moisture content measurements were collected, before and after the conditioning cycle. Since measuring moisture content by mass difference required material to be removed from the log segment, the same position could not be measured before and after conditioning, therefore evenly spaced spots around the log segment were chosen and average moisture content for the entire log was calculated before and after conditioning. The average change in moisture content of the log segments before and after conditioning can be seen in Table 7.2. For the most part Table 7.2 shows a decrease or no change in average moisture content of the log segments during CHAPTER 7. RESULTS Species Pine Spruce Spruce Spruce Spruce Spruce ••" sapwood 110 \'Q) 127 ± 31.8 246 ± 61.5 183 ± 45.8 162 ± 40.4 212 ± 53.0 122 ± 16.6 Mheartwood {%) 45.6 ± 11.4 50.8 ± 8.00 38.7 ± 9.70 41.8 ± 10.4 41.0 ± 10.3 54.0 ± 13.5 •M average \'0) 67.7 ± 16.9 11.7 ± 29.3 93.4 ± 23.4 89.7 ± 22.4 102 ± 25.5 75.5 ± 18.9 Table 7.3: Sapwood and Heartwood Moisture Content conditioning. Comparisons of experimental initial and final moisture content to theoretical values were not considered since the physically artificial nature of the Dirichlet moisture boundary would not yield any useful insight. 7.2.2 Analysis of Moisture Content with Respect to Sapwood and Heartwood Sapwood of softwood has a significantly higher moisture content than the heartwood in a green state. This is evident, and therefore further reinforced in some of the more detailed moisture content measurements where sapwood was separated from heartwood and measured separately. The results of these measurements are given in Table 7.3. Table 7.3 quite clearly illustrates that large differences in moisture content between sapwood and heartwood in green wood in some cases. 7.2.3 Long Term Moisture Content Changes in Cylindrical Log Segments As mentioned in earlier sections, de-barked log segments can sometimes be stockpiled outside where they are exposed to ambient conditions. In these relatively extreme conditions, significant changes in temperature and moisture content in the wood is CHAPTER 7. RESULTS 111 Sapwood MC Heartwood MC c o o 2h O CD O) 2 CD 3 i ! ^ - ^ TO—^rr 2tr ^ Time (days) Figure 7.3: Spruce, sample 1, long term moisture content change. likely to occur. In these cases assuming green moisture content as the initial moisture would be inaccurate and could produce significant errors in the final results. In order to examine the impact of this, a group of log segments was left outside in ambient conditions for approximately one month and Figures 7.3, 7.4, 7.5, 7.6, and 7.7 show moisture content versus time for a selection of log segments exposed to ambient conditions for this time period. The lower section of the plot gives the ambient relative humidity and temperature while the upper section gives average sapwood and heartwood moisture content for the particular log segment. As expected, all three spruce samples, which all had relatively high initial moisture contents, show an overall trend of decreasing moisture content over time, which agrees with the earlier discussed theory. This trend is apparent in all samples until approximately 20 days from the beginning of data collection when there is a definite increase in moisture content. This increase is most likely attributed to an CHAPTER 7. RESULTS 112 • • Sapwood MC Heartwood MC c •| 1.5 TO o 0 D5 2 > 0.5 "10 15" Time (days) Figure 7.4: Spruce, sample 2, long term moisture content change. Sapwood MC Heartwood MC |o0 . 8 co O 0.6 g)0.4 CD CD ^ 0.2 -100 f 50 H E X Relative Humidity Temperature 0 tr ^V Time (days) ^0 ^fe -10 Figure 7.5: Spruce, sample 3, long term moisture content change. CHAPTER 7. RESULTS 113 Sapwood MC Heartwood MC c 0.8 o o(0 H— O.b O S 0) 0 4 C3) m CD ^ 0.2 Time (days) Figure 7.6: Pine, sample 1, long term moisture content change. 0.5 Sapwood MC Heartwood MC Io 0.4 50 O 0.3 S>0.2 to i_ d) •^ 0.1 - 10 T5 Time (days) 20 25 |2 Figure 7.7: Blue stain pine (beetle kill), sample 1, long term moisture content change. CHAPTER 7. RESULTS 114 approximately 15 hour period of wet snowfall which occurred at approximately 18.5 days from the beginning of data collection. Within a 30 hour time period, following the snowfall, the ambient temperature was relatively mild which caused the snow to melt and hence liquid to be absorbed at the logs' surface. Also in agreement with theory is the moisture content recorded for the beetle kill pine, which is thought to have a relatively constant moisture content, which is close to the fiber saturation point in both the sapwood and heartwood. 7.3 Model Deficiencies There are a number of known deficiencies in the model which are exposed in the theoretical-experimental data overlay plots. This is partially due to the fact that the heating of log segments in a humid environment is a relatively complex problem to model, and the fact that a one-dimensional model is being used to model a three dimensional problem. As a result virtually any attempt at modeling such a problem will be an approximation and inevitably contain some deficiencies. Although, the approximations made when building the model tend to reveal such deficiencies, there magnitude is only truly seen wrhen theoretical and experimental data are directly compared, as in Appendix B. 7.3.1 Frozen Wood Some of the discrepancies seen in the plots shown in Appendix B are due to wood which is fully or partially frozen before entering the conditioning tunnel. What is referred to as frozen wood is actually wood that has been exposed to subzero (Celsius) temperature allowing the trapped liquid water to convert to the solid, or ice phase. Previously, it was thought that the bound, or hygroscopic water also underwent some CHAPTER 7. RESULTS 115 sort of phase change, but NMR experiments [20] suggest the bound water remains unchanged in subzero (Celsius) temperatures. During the winter months, as well as the early spring months, log segments enter the conditioning tunnel after having spent a significant amount of time in freezing temperatures and therefore contain significant amounts of frozen liquid water. Experimental proof of frozen log segments entering the conditioning tunnel can be seen in the following plots; B.2(a), B.12(a-c), B.14(a), B.14(b), B.15(a-c), B.17(a-c), B.18(a-d), B.21(a-d), B.27(a-d). The plots also serve to illustrate the limitations of the model to predict temperature changes in the case of frozen wood. In order for the frozen water in wood to transition to liquid, extra energy, in the form of latent heat, is required before temperature change occurs. For example, if we consider a time period from 0 to 10 hours of Figure 7.8 (taken from Figure B.18(a)), the temperature can be seen to remain at approximately zero degrees Celsius even though energy is being added to the system through heating from the surface. As a result, it seems reasonable that the model will tend to overestimate the temperature in the case of frozen wood. The plots mentioned above, where frozen wood is initially present, seem to support this hypothesis since each case of frozen wood shows a tendency for the model to overestimate the measured temperature at that point. The magnitude of error associated with this deficiency is dependent on the initial temperature of the frozen portion, and the rate at which energy is flowing into the frozen portion, and therefore varies for each case. Further discussion on phase change was not in the scope of this study but is given in the literature by Ozisik [15], Voller [27], and Steinhagen [26], and [25]. CHAPTER 7. RESULTS 116 r=0.0cm, theoretical r=0.0cm, z=30.0cm, experimental 5r = +1.5 40 O o CD i_ "2 20 CD Q. E CD J 0 I I l_l_l 5 I I L_J 10 I I I I I I I I 15 I I 20 I I I I I 25 I L_l L_l 30 Time (hrs) Figure 7.8: Example of model overestimating temperature for frozen wood. CHAPTER 7.3.2 7. RESULTS 117 Longitudinal Transfer Another model deficiency can be attributed to thermal energy transfer in the longitudinal direction. The model presented here assumes that thermal energy transfer only occurs in the radial direction and the 9 direction is also neglected. This assumption is valid in the case of a sufficiently long cylinder, but begins to break down when the diameter of the cylinder approaches the length of the cylinder. All the wood cylinders used in plywood manufacture are cut to the same length of approximately 2m, and the diameters vary depending on the age and section of the tree the segment was cut from. The diameter to length ratio can vary significantly from sample to sample and therefore error attributed to this deficiency can also change form sample to sample. It can be reasoned that any discrepancy attributed to longitudinal thermal energy transfer should be at a minimum at the mid-point of the log segment and at a maximum near the end. This makes sense because any temperature gradient that forms in the longitudinal direction is a result of heating from the end surfaces. Just as expected a qualitative analysis of the plots given in Appendix B seems to show more discrepancy between model and experiment for data points near the end (z = 10.0 cm) of the log segment as compared to the midpoint [z — 128.0 cm). 7.4 Sensitivity Analysis The model considered here relies on six experimental values to define a particular log: the radius - r, initial moisture content - M 0 , intrinsic liquid phase permeability - Kp, tracheid wall thickness ratio - o, tracheid overlap ratio - a, and the surface thermal energy transfer coefficient - hq. Although there are other experimental parameters involved in the model, these six tend to contain the most uncertainty, whether due to experimental measurement error or variances from log to log. By conducting a CHAPTER 7. RESULTS 118 Parameter r - Radius M 0 - Initial Moisture Content A'/3 - Liquid Intrinsic Permeability a - Tracheid Wall Thickness Ratio a - Tracheid Overlap Ratio hq - Surface Thermal Energy Transfer Coefficient Range of Values 10-20 cm 0.350-1.60 1.00-10.0 (xl0~~ 16 m 2 ) 0.70-0.90 0.25-0.50 9.0-20.0 ^ ^ Table 7.4: Parameters used in sensitivity analysis. (T0 = 1.0°C, and T^, = 40.0°C) sensitivity analysis the impact of the uncertainties in the parameter values can be quantified. The sensitivity analysis was accomplished by considering a meaningful range of the aforementioned parameters and using the model to calculate the time required for the core to reach steady state temperature (^- = 0). A range of three values was considered for each of the six parameters therefore giving 729 unique sets of parameters. This large number of numerical solutions was easily handled by breaking the 729 individual runs into 9 separate sets and running the numerical code in parallel on a cluster of 9 processors, which reduced the time to complete the computation from hours to minutes. Table 7.4 gives the range of values utilized in the sensitivity analysis. The range of radii and moisture content was chosen to reflect the range of values seen in the mill. The range of values for Kg, a, and a are taken from the literature, [4], [14], [16], [17], and [24], for the species utilized in the mill. The surface thermal energy transfer coefficient, hq, relies on a few individual experimental parameters itself, such as mean fluid velocity - um. and the Irydraulic diameter - D0. To reduce the number of experimental parameters in the sensitivity analysis, a range of thermal energy transfer coefficients are considered which correspond to a particular range of mean fluid velocities (1.0-10.0 ^ ) and hydraulic diameters (1.0-10.0 cm). The resulting analysis produced the results shown in Appendix A. The first six CHAPTER 7. RESULTS 119 columns of the table contain the unique set of parameters being tested, the last column contains the deviation from the average time taken for the core to reach constant temperature, and the last column contains the same deviation normalized between one and negative one. The deviation from the average time is found by calculating the average of all 729 times and subtracting the time for the particular set of parameters. In order to give a more qualitative view of the data produced in the sensitivity analysis an averaging scheme has been devised. As described earlier, for each of the six parameters a range of three values has been chosen, and therefore each of the three values has one third of total unique sets of parameters associated with it. The matrix of values is given as, ' SA \ ?'i r2 r3 Mi M2 M3 Kx K2 K3 a-i a2 a3 Oil Oi2 «3 h\ h2 h3 » j where the subscripts have been dropped to simplify the notation. For each value in the SA matrix there exists 243 unique sets of parameters, for instance for r\, '''i, / Mi \ I K\\ I (li \ I Qi \ M2 K2 a2 a2 \ M3 J \K3J \a3 J \<*3 h2 / \h3J The conditioning time for each of the 243 unique sets of parameters can then be CHAPTER 7. RESULTS 120 Radius (r) in 10 in m or Initial Moisture Content (MQ) 0 0.2 0.4 0.6 0.8 Tracheid Wall Thickness Ratio (a) o O o Tracheid Overlap Ratio (a) §> > Surface Heat Transfer Coefficient (hq) Oh 0.2 0.5 0.4 0.6 0.8 Liquid Permeability (Kg) 0 -0.5 0.2 0.4 0.6 0.8 1 Normalized Parameter Values Figure 7.9: Deviation from the average core heat-up time versus normalized parameter values. CHAPTER 7. RESULTS 121 calculated giving a vector of conditioning times / t l \ w h n,m \t,j Where I = 243 and represents the number of unique sets of parameters, n and m represent the indices of the parameter matrix SA, and t is the conditioning time. By calculating the 1-norm of t n>m and dividing by / , the average conditioning time for a particular value in the parameter matrix can be calculated. ZTI (7.5) The result is, for each of the radius values, ri,r2, and r 3 , an associated average conditioning time can be calculated titi, t^2, and t l j 3 respectively. These can then be plotted as the deviation from the average conditioning time versus the normalized parameter value, and the results are given in Figure 7.9. Chapter 8 Summary and Conclusion The modelling of any industrial process can be plagued by oversimplification which causes the problem to be either trivial or too far removed from reality to be of much use. In this case, the modelling of log conditioning is a complicated process involving a consideration of the vapour and liquid phases of moisture moving within a the porous wooden cellular network. Through volume averaging, a system of differential equations, derived from a lumber drying model, were applied and found to model the experimental results relatively well within experimental error. As mentioned before, of interest to industry partner North Central Plywoods, is the models' ability to predict conditioning times. This can be explored in Table 7.1, where predicted versus experimental heat up times are compared. From this table it can deduced that 40% of the predicted heat up times at r — 0.0cm and z — 10.0cm are in agreement with the model, within experimental error, 60% at z = 30.0cm,, and 80% at z = 128.0cm. This is also excluding data which contains frozen wood. This result is relatively encouraging for the z = 128.0cm position especially, and reinforces the earlier discussed problem of using a 1-dimensional model to describe a 3-dimensional problem. 122 CHAPTER 8. SUMMARY AND CONCLUSION 123 The accuracy of the model can be explored by analyzing the plots in Appendix B. Excluding data sets where the error was unreasonably large (>100%), nearly 60% of the temperature data sets match theory within error. This does not seem like a particularly encouraging result, but the model deficiencies described in Chapter 7 account for a substantial amount of this discrepancy. If the data sets containing frozen wood are neglected, the percentage of data which match theory within error increases to 65%. In addition, taking only the data sets from the midpoint of the log (z = 128.0cm), where heating from the log ends is much less of a factor, and frozen samples are ignored, the number of data sets matching theory within error rises to 78%. This result shows that the deficiencies described in Chapter 7 play a relatively significant role in the accuracy of the model, and therefore correction of these deficiencies should give improvement in the model output quality. It is also noted in Chapter 7 that an important aspect of modelling the conditioning cycle is to take into consideration the cooling of the logs when the conditioning cycle is complete. It was shown that in this time significant cooling can occur in the period of time between ending the conditioning cycle and moving the log segments to the lathe. Part of the proposed study was to examine the thermal energy and moisture fluxes at the boundary of the log. This was done in chapter 5 where a mixed Robin-Dirichlet boundary condition was proposed. To examine the impact of the boundary conditions on the model results the experimental-theoretical temperature data overlays in Appendix B can once again be considered. A visual inspection of the thermal data collected from the surface suggests the model data tend to overestimate the surface temperature. This is apparent in nearly all the data overlay plots and is thought to be attributed to the model's boundary conditions. The coupled nature of thermal energy and moisture transfer means the discrepancy at the boundaries could be due CHAPTER 8. SUMMARY AND CONCLUSION 124 to the thermal energy or moisture flux boundary. Unfortunately further investigation of the boundary flux would require further study of the fluid dynamics inside the conditioning tunnel, which is beyond the scope of this study. If data collected from the surface sensor (r = R) are neglected, as well as frozen wood, at z = 10cm, 76% of the data sets match theory within experimental error, 59% at z = 30cm, and 80% at z = 128.0cm, which is a marked improvement. Although modelling the heating of logs is the primary interest of the study, it has been hypothesized that moisture transfer plays a significant role in the process and is therefore examined experimentally. Table 7.2 shows that a relatively low amount of moisture content change occurs during the conditioning. It is also apparent that, contrary to previous beliefs, the average moisture content decreases during conditioning, with the exception of one sample in table 7.2 which instead showrs an increase in moisture content during conditioning. Increasing moisture content is seen again in Figures 7.3, 7.4, 7.5, 7.6, and 7.7, when the logs are in direct contact with liquid water at the surface as a result of melting snow. It therefore appears that wood will tend to lose moisture content until reaching its equilibrium moisture content unless directly exposed to water, at which time moisture content can increase. It can therefore be concluded that the sample in line 3 of Table 7.2 may have had its surface in contact with water for a substantial amount of time during conditioning. This also tends to support the need for further understanding of the fluid dynamics of the conditioning process in order to derive physically accurate moisture flux boundaries. The sensitivity analysis gives an overall appreciation on how the model results are affected by initial conditions and experimental parameters, and the results are summarized in Figure 7.9. The log radius tends to produce the largest impact on predicted conditioning times. This result is relatively intuitive since it is saying that a large diameter log takes longer to heat then a small diameter one. Of interest is the CHAPTER 8. SUMMARY AND CONCLUSION 125 range of the deviation from the average conditioning time. This means that a log with radius 20cm will take, on average, 40 hours longer to heat from an initial temperature of VC to 40°C at the center, than one of 10cm radius. Also of interest is the sensitivity to the moisture content range, which gives a good representation of the moisture content dependence on the heating time. A deviation of 20 hours on average between the green moisture content of beetle kill pine (M 0 ~ 0.35), to the most moisture laden green spruce sample (Mo ~ 1.60) was seen in the sensitivity analysis. This suggests the model has a relatively significant moisture content dependence on the heating time, which stresses the importance of accurate initial moisture content data. The species dependent parameters a,a, and Kp seem to have the least significant effect on predicted conditioning times. It can therefore be concluded that the model has relatively weak species dependence with the exception of the tracheid wall thickness a which shows a deviation of 4 hours on average. The model also seems to have a relatively weak surface thermal energy transfer coefficient hq dependence. In addition, the dependence seems to be relatively nonlinear compared to the others, which can be seen in figure 7.9. This means that lower values, which correspond to higher mean fluid velocities, and smaller hydraulic diameters, have greater impact than lower fluid velocities and larger hydraulic diameters. The model presented here is thought to be sufficient in predicting conditioning times for the industrial process of log conditioning when supplied with accurate initial conditions. Nevertheless, it is thought that model accuracy could be improved through further research. Generalizing the model to the longitudinal (z) spatial dimension is thought to be the next logical step, thereby allowing heating from the log ends to be taken into account. It is also thought that augmenting the model to include phase change would result in a substantial improvement in the models ability to predict conditioning times of initially frozen logs. The addition of thawing would require CHAPTER 8. SUMMARY AND CONCLUSION 126 a differentiation between the solid wood matrix (a) and the ice phase in the model, as well as a significant change to the numerical scheme to account for the thawing front or moving boundary. In addition, a number of assumptions were made about the air flow surrounding the logs in the conditioning tunnel. Investigation of the flow field surrounding the logs through experiment or computational fluid dynamics simulation would be extremely beneficial in understand the thermal energy and moisture boundary conditions better. Further refinements to the model include; differentiation of the sapwood and heartwood portion of the log, experimental investigation of timedependent moisture content inside the wood, and experimental investigation of the moisture flux at the surface. Bibliography [1] Jacob Bear. Dynamics of Fluids in Porous Media. Dover Publications, Inc., New York, 1988. [2] Adrian Bejan. Heat Transfer. John Wiley and Sons, Inc., 1993. [3] G. L. Comstock. Directional permeability of softwoods. Wood and Fiber Science, 14:283-289, 1970. [4] F. Couture. Relative permeability relations a key factor for a drying model. Transport in Porous Media, 23:303-335, 1996. [5] J. Blackledge G. Evans and P. Yardley. Numerical Methods for Partial Differential Equations. Springer, 2000. [6] M. Goyeneche. A film-flow model to describe free water transport during drying of a hygroscopic capillary porous medium. Transport in Porous Media, 48:125158, 2002. [7] W. G. Gray. A derivation of the equations for multi-phase transport. Chemical Engineering Science, 30:229-233, 1974. [8] P. Hofacek. Modelling of coupled moisture and heat transfer during wood drying. tfh International IUFRO Wood Drying Conference, 2003. 127 BIBLIOGRAPHY 128 [9] R. Bruce Hoadley. Identifying Wood: Accurate Results With Simple Tools. Tauton Press, 1990. [10] Frank P. Incropera and David P. De Witt. Introduction to Heat Transfer. John Wiley and Sons, Inc., 1990. [11] K. Krabbenhoft. Double porosity models for the description of water infiltration in wood. Wood Science and Technology, 38:641-659, 2004. [12] J. Y. Lin and W. T. Simpson. Two-stage moisture diffusion in wood with constant transport coefficients. Drying Technology, 17:257 269, 1999. [13] A. V. Luikov. Heat and Mass Transfer in Capillary-Porous Bodies. Pergamon Press, Oxford, 1966. [14] G. A. Spolek O. A. Plumb and B. A. Olmstead. Heat and mass transfer in wood during drying. International Journal of Heat and Mass Transfer, 128:1669-1678, 1985. [15] M. Necati Ozisik. Heat Conduction, Second Edition. John Wiley and Sons, Inc., 1993. [16] P. Perre and I. W. Turner. A 3d version of transpore: A comprehensive heat and mass transfer computational model for simulating the drying of porous media. International Journal of Heat and Mass Transfer, 42:4501 4521, 1999. [17] P. Perre and I. W. Turner. Transpore: A generic heat and mass transfer computational model from understanding and visualising the drying of porous media. Drying Technology, 42:4501 4521, 1999. BIBLIOGRAPHY 129 [18] F. Ivanauskas R. Baronas and M. Sapagovas. Numerical investigation of moisture movement in wood under isothermal conditions. Mathematical Modelling and Analysis, 6:167-177, 2001. [19] S. G. Hatzikiriakos S. Avramidis and J. F. Siau. An irreversible thermodynamics model for unsteady-state nonisothermal moisture diffusion in wood. Wood Science and Technology, 28:349-358, 1994. [20] I. D. Hartley S. Gadzdinski and H. Peemoeller. Freezing in trembling aspen by h-nmr relaxometry. Molecular Physics Reports, 29:144 149, 2000. [21] T. A. G. Langrish S. Pang and R. B. Keey. Moisture movement in softwood timber at elevated temperatures. Drying Technology, 12:897-914, 1994. [22] J. F. Siau. Transport Processes in Wood. Springer-Verlag, New New York, 1984. [23] J. F. Siau. Wood: Influence of Moisture on Physical Properties. Virginia Polytechnic Institute and State University, 1995. [24] G. A. Spolek. A Model of Simultaneous Convenctive, Diffusive, and Capillary Heat and Mass Transport in Drying Wood. Washington State University, 1981. [25] H. P. Steinhagen. Computerized finite-difference method to calculate transient heat conduction with thawing logs. Wood and Fiber Science, 18(3):460-467, 1986. [26] H. P. Steinhagen and H. W. Lee. Enthalpy method to compute readial heating and thawing of logs. Wood and Fiber Science, 18(3):460-467, 1986. [27] V. Voller and M. Cross. Accurate solutions of moving boundary problems using the enthalpy method. 24(3):545-556, 1981. International Journal of Heat and Mass Transfer, BIBLIOGRAPHY 130 [28] S. Whitaker. Simultaneous heat, mass, and momentum transfer in porous media: A theory of drying. Advances in Heat Transfer. 13:119 203, 1977. [29] E. T. Choong Y. Chen and D. M. Wetzel. A numerical analysis technique to evaluate the moisture-depenent diffusion coefficient on moisture movement during drying. Wood and Fiber Science, 28:338 345, 1996. Appendix A Sensitivity Analysis CO I CO I ft, ft. li-l 1! 1 llll 10 i !a "i •^ > o 5 JB -S I i 3 .a I I TJ JS J& ^ S /* 2 .B c 13 .a .a JB •& ^ s r- f „ r- •*- f^ ^ 'H i*- i*» ^ r- *"- f" ry ••* ^1 h* r- r- ^ r-- »- i ^ t s ' ^ '"- »•- *- f<- i-- i- ^S3^g cii w CM p. r~ (-- r*» h* »- M «>l (\| i**. ifM.i »i io ai »i oi oi t« "^ ui «i in in «) O' «i si io u m jf tii o o o o si ' j O' f i i i i i o o o ft is c» o m n o o o ci ci * i'. 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C 3 o c 3 0 0 o a c 3 C J Q E 3 a o c 3 £ 3 Q a a a a o o o o a o s s c 3 0 c i a a e j C ! C 3 C i o i-H <1 too ..—1 APPENDIX A. SENSITIVITY Initial Log Radius, (m) Moisture Content (fraction) 0.100 0.100 0.10Q 0.100 0.100 0.100 0.100 0.100 0.100 0.100 0.100 0.100 0.100 0.100 0.100 0.10O 0.100 0.100 0.100 0.100 0.100 0.1 QO 0.10O 0.100 0.10O 0.100 0.100 0.100 0.100 0.100 0.100 0.100 0.100 0.100 0.100 0.100 0.1 QQ 0.100 0.1QO 0.100 0.100 0.100 0.100 0.100 0.100 0.100 0.100 0.100 0.1 DO 0.100 0.100 0.100 0.100 0.100 0.350 0.350 O.350 0.350 0.350 O.350 O.350 0.350 0.350 O.350 O.350 0.350 0.350 0.350 0.350 0.350 0.350 0.350 O.350 0.350 0.350 0.350 O.350 O.350 0.30 0.350 0.350 o.eoo o.eoo o.eoo o.eoo o.aoo G.8G0 O5O0 o.goo o.goo o.wo 0.900 0.900 0.900 O.S3G 0.900 O.9O0 O.9O0 O.9O0 Q.9O0 o.goo o.goo o.eoo o.»o 0.900 0.900 0.900 o.goo O.930 o.goo o.goo o.goo 0.800 o.eoo o.roo 0.800 0.800 0.800 0.800 0.800 0.800 0.700 0.7O0 0.7O0 0.7DO O.7O0 0.70G o.eoo 0.800 O.800 0.800 0.800 0.800 0.800 O.800 0.800 0.800 0.800 0.800 0.800 o.eoo 133 TrsshiaBd Wall Surface Heat Inside to Trachied Transfer Outside Overlap Ratio Coefficient Thickness (W*n«K) Ratio 0.700 Q.7O0 O.7O0 O.7O0 0.70Q O.7D0 0.700 O.7O0 O.7O0 0.700 0.700 0.700 0.700 0.700 G.7GG Q.7O0 0.700 0.700 0.7DO O.7O0 0.800 ANALYSIS 0.250 0.25O 0.25O 0.250 0.250 0.2m 0.25O 0.28O 0250 0.380 0.3SO 0.380 0.38O 0.38S 0.38O 0.38O 0.380 0.38O 0.5OO 0.500 0.5OO 0.5OO 0.500 0.500 0.500 0.5OO 0.500 0550 0560 0550 0.25O 0.2EO 0.25O 0550 G.250 0.25O 0.38O 0.38O 0.38O 0.38O 0.38O 0.380 0.38O 0.380 0.380 0.5OQ 0.500 0.500 0.BOO 0.E0O 0.50O 0.5OQ 0.5CO 0.500 ' • • 9.00 9.00 9.00 20.0 20.G 20.0 40.9 40.O 40.O 9.00 9.00 900 20.Q 20.O 20.O 40.0 40.0 4Q.0 9.00 9.03 9.QQ 20.0 20.0 20.O 40.O 40.O 40.0 9.Q0 9.00 9.00 20.Q 20.0 20.O 40.0 40.O 40.O 9.00 9.00 9.00 20.O 2Q.Q 20.O 40.O 40.O 40.0 9.00 9.00 9.00 20.O 20.0 20.O 40.0 40.0 40.0 Figure A.2 Liquid Permeability trtfj 1.001-16 5.00E-16 1.OOE-15 1.0OE-16 5.0QE-16 1.00E-15 1.00E-16 5.00E-16 1.00E-15 1.00E-16 5.00E-16 1 0OE-15 1.00E-16 5.00E-16 1.00E-15 1.00E-16 5.00E-16 1 .OOE-15 1.00E-16 5.OOE-16 1.OOE-15 1.OOE-16 5.OOE-16 1.OOE-15 1.00E-16 5.OOE-16 1.OOE-15 1.OOE-16 5.OOE-16 1.00E-15 1.OOE-16 5.OOE-16 1.00E-15 1.00E-16 5.Q0E-16 1.OOE-15 1.0QE-16 5.ODE-16 1.OOE-15 1.00E-16 5.00E-18 1.OOE-15 1.00E-16 5.OOE-16 1.OOE-15 1.OOE-16 5.00E-16 1.00E-15 1.OOE-16 5.0QE-16 1.OOE-15 1.OOE-18 5.00E-16 1.OOE-15 Delation from Deviation Airerage from Average Time Time. • traction • i hours) -0 577 -0 577 -0.5T -0.6G5 -0 602 -0.GO2 -0.5U •0.512 -0.S12 -0.577 -0.577 -0577 -O.S02 -osoa •o.eos -o.su -0.612 -0.611 -0 577 -0.077 •OJV** -0SQ2 -oecz -Q6G2 -0 «S1? -0 015 -0.61? -0.523 -0 5OS -O^OC-0.577 •ftset -0.556 -0.59S -0 587 -0 530 0.D3& •0.5E1 -0.515 -Q.530 -0 57c -0.57C -0CO1 -CM' -0.592 -0.527 -0.527 -0.527 -0.581 0.9J1 -0581 -0 602 -0 602 -0.GQ2 -22.5 -22 3 -22 5 -23 2 -23.2 23 2 23.6 -ss.e -S36 -22 S -22.3 -223 -23.2 -2S.2 -23.2 -236 -23.6 -23.6 -223 -223 223 •23.2 -"3.2 25.2 -23 6 •PS a -23.6 -2C.2 -1S.6 -!'?.4 -22.3 -218 -21 5 -23.1 -22.6 •22.5 -205 -20.1 -is.a -22.4 •22.2 -22.0 -23 2 -23.0 •22.8 -2C.3 -20,3 •20 3 -22.4 224 -22.4 -23.3 -23.2 -23.2 CO CO 1 CO OH lL* si ||]I 1- iiil llf t p cvj o . 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SENSITIVITY Initial Moisture Content (fraction) Log Radius I'm) 0.150 0.1 SO 0.150 0.1 SO 0.150 0.150 0.150 0.150 D.150 0.150 0.150 D.150 0.150 0.150 D.1S0 D.150 0.150 0.150 0.150 0.150 0.150 0.150 0.150 0.150 0.150 0.150 0.150 D.150 D.150 0.150 0.150 0.150 0.150 0.150 0.150 0.150 0.150 Q.150 0.150 0.150 0.150 0.150 0.150 0.150 0.1SO D.150 0.150 0.150 0.150 0.150 D.150 D.150 0.150 0.150 0.800 0.800 0.800 Q.SOO 0.800 0.800 O.SOO o.soo 0.800 0.800 o.eoo o.soo • 0.800 0.800 0.800 O.800 0.800 0.800 0.800 Q.800 0.800 0.800 O.SOO O.800 0.800 O.SOO 0.800 1.600 1.600 1.600 1.600 1.600 1.600 1.600 1.S00 1.600 1.600 1.600 1.S00 1.600 1.600 1.600 1.600 1.S00 1.S00 1.600 1.600 1.600 1.600 1.600 1.600 1.800 1.S00 1.600 ANALYSIS 139 Trschied Wsll Surface Heat Inside to Trachied Transfer Outside Overlap Ratio Coefficient Thickness (W/msK) Ratio o.soo O.9O0 0.900 0.9QO O.9O0 O.9O0 0.900 0.9QD 0.900 0.900 0.900 0.900 0.900 O.SOO O.SOO 0.900 0.900 G.9GG O.9C0 0.900 0.900 O.SOO o.goo 0.900 0.900 0.90Q o.aoo Q.70Q O.7Q0 0.700 Q.7O0 O.7O0 O.7O0 O.7O0 O.7O0 O.TOD 0.700 O.7O0 Q.7O0 O.7O0 0.700 O.7O0 0.700 Q.7O0 O.7O0 0.700 0.700 0.700 0.70D Q.7D0 0.700 0.700 0.700 0.700 0.250 0.250 o.sso 0.25Q 0.25O 0550 0.25O 0.25G • mm 0.38O 0.38O 0.380 0.38Q 0.380 0.38O 0.38O 0.3SO 0.380 0.500 0.500 0.5QO 0.5OO O.BOQ 0.5OO 0.500 0.500 0.5O0 0.25O 0.26O 0.25O 0.25O 0.250 0.25O 0.SSO 0.25O 0.25G 0.38O 0.38O 0.38O 0.38O 0.380 0.380 0.38O 0.380 0.380 0.500 0.5QQ 0.5OO 0.5QO 0.50O O.EOO 0.5O0 0.5OO 0.5CX) ' ' • • - ' 9.00 9.00 9.00 20.O 20.O 20.Q 40.O 40.O 40.O S.OQ 9.00 9.00 20.O 20.O 20.0 40.0 40.0 40.0 9.00 9.00 9.00 20.O 20.O 20.0 40.0 40.0 40.O 9.00 9.00 9.00 20.Q 20.0 20.0 40.0 40.O 40.O 9.00 9.00 9.00 20.O 20.O 20.O 40.0 4Q.0 40.O 9.00 9.00 9.00 20.O 20.O 20.0 40.0 40.0 40.O Figure A.8 Liquid Permeability 1W) 1.00E-18 5.0OE-18 1.00E-15 1.00E-16 5.0OE-18 1.00E-15 1.00E-16 5.0QE-16 1.0OE-1S 1.00E-16 S.OQE-16 1.0OE-15 1.00E-16 5.00E-16 1.0OE-15 1.00E-16 5.00E-16 1.0OE-15 1.0OE-16 5.0OE-16 1.00E-15 1.0OE-16 5.00E-16 1.0OE-15 1.0QE-16 5.0QE-16 1.00E-15 1.0GE-16 5.DOE-16 1.00E-15 1.0GE-16 5.QQE-16 1.0OE-15 1.00E-16 5.00E-16 1.0OE-16 1.0OE-18 5.QQE-16 1.Q0E-15 1.0OE-16 5.0QE-16 1.0OE-15 1.00E-16 5.0GE-18 1.0OE-16 1.QQE-16 5.0OE-16 1.00E-15 1.00E-16 5.0OE-16 1.00E-15 1.00E-16 5.0OE-16 1.00E-15 O r a t i o n from Deviation Average Fsom Average Tims Time ihouisi • fraction; -0 084 -0 069 -0 057 -0.164 -0 151 -0140 0.195 -0.184 -0 173 -0.0BE -0 083 -Q.07S -0.165 -0.1 SS -0.199 •0196 -0.195 -0.191 -0.086 -ooae 0.086 -0.165 -0165 -0 135 -0196 -0.196 -0.196 0 381 0393 0.399 0.225 0 243 0 245 0.164 0.181 0.184 0 380 0.397 0.399 0224 0241 0.244 0.1S3 0180 0 134 0.3M 0 358 0.398 0.219 0.2i2 0.240 0153 0.171 0179 -3-2 -2.7 -2.2 -6 3 -0 6 -5 4 "5 •'I -S.7 -3,3 -3.2 -3 0 -S.4 -6 3 -61 -7 6 -7 5 •I A -35 -3 3 -33 -6 4 -6 4 64 -7 6 -7 6 -7 6 14 7 15.3 154 8 7 94 94 6 3. f.O 71 146 155 154 3D 93 94 S3 59 /I 14.4 150 15.3 84 83 93 £1 66 69 APPENDIX A. SENSITIVITY Initial Lag Radius (m) 0.1 SO 0.1 SO o.i m 0.1 SO 0.150 D.150 0.150 0.1 SO 0.150 D.1S0 D.150 0.1 SO 0.150 0.1 SO D.150 0.1 SO 0.1 IQ 0.150 0.150 0.150 0.15O 0.150 D.150 0.150 0.1 SO 0.1 §0 D.150 0.150 0.1 SO 0.150 0.150 0.150 0.150 0.150 0.150 0.150 C.150 0.150 0.158 0.150 0.150 0.1 SO 0.1 SO 0.1 SO 0.1 SO 0,150 0.150 0.150 D.150 0.1 SB 0.150 0.15S 0.150 0.150 Moisture Content (fraction) 1.S00 1.600 1.600 1.600 1.S00 1.600 1.600 1.600 1.500 1.S00 1.600 1.G00 1.600 1.800 1.600 1.600 1.600 1.600 1.600 1.800 1.600 1.600 1.600 1.S00 1.600 1.600 1.600 1.600 1.600 1.600 1.600 1.600 1.600 1.600 1.600 1.600 1.600 1.600 1.600 1.600 1.600 1.S00 1.600 1.600 1.600 1.600 1.600 1.600 1.600 1.600 1.600 1.600 1.600 1.600 ANALYSIS 140 Trachisd Wall Surface Hast Inside to Transfer Trachied Outside Overlap Ratia Coefficient Thickness (Wrn'K) Ratio o.aoo O.8O0 O.8O0 O.8O0 O.8O0 0.8QO O.SD0 Q.8O0 0.800 0.800 0.800 O.8O0 O.8O0 O.8O0 G.8G0 0.800 O.8O0 O.8O0 O.8O0 O.8O0 0.800 0.8O0 o.aoo O.8O0 Q.8Q0 0.800 0.8GQ Q.90G 0.900 O.W0 O.QOO G.90Q O.9O0 O.9O0 0.900 0.9QO O.9O0 G.9Q0 O.9O0 0.900 O.SOG 0.900 O.9O0 0.9O0 O.S30 Q.SO0 o.goo 0.900 O.9O0 0.900 o.soo 0.900 O.9O0 0.900 0.2SO 0.250 0.26O 0.25O 0.2SO 0.250 0.25O 0.25O 02m 0.38O 0.380 0.38O 0.38O 0.380 0.38O 0.380 0.38O 0.38O 0.5OO 0.5O0 0.50O 0.500 0.5CO 0.5O0 0.S0O 0.5OO 0.5OO 0.SSO 0250 Q25Q 0.2EO 0250 02m 0550 0.25O 0.290 0.38O 0.38O 0.3W 0.38O 0.38O 0.38O 0.33O 0.36O Q.3SQ 0.500 0.500 0.500 0.50O 0.5CO 0.500 0.500 0.5OO 0.5OQ ' . ' ' 9.GG 9.00 9.00 20.0 20.G 20.Q 40.O 40.0 40.0 9.00 9.00 9.00 20.0 20.O 20.O 40.O 40.0 40.O 9.00 9.00 S.00 20.0 20.0 20.O 40.O 40.0 40.0 9.00 9.00 9.00 20.0 20.O 20.0 40.O 40.0 40.0 9.00 9.00 9.00 20.0 20.O 20.O 40.O 40.O 40.0 9.00 9.CQ 9.GO 20.O 20.0 20.O 40.O 40.O 40.0 Figure A.9 Liquid Permeability M1) 1.00E-16 5.00E-16 1.0OE-15 1.00E-16 5.0OE-16 1.0OE-15 1.0OE-16 5.00E-16 1.00E-15 1.00E-16 s.ooE-ie 1.0OE-15 1.00E-16 5.0QE-16 1.00E-15 1.00E-16 5.0OE-16 1.0OE-15 1.0OE-16 5.0OE-16 1.0OE-1S 1.00E-16 5.0OE-1S 1.0OE-16 1.00E-16 5.00E-16 1.00E-15 1.0OE-16 5.ODE-16 1.0OE-15 1.00E-16 5.0OE-18 1.0OE-15 1.00E-16 5.00E-16 1.00E-15 1.00E-16 5.00E-18 1.00E-15 1.00E-16 5.0OE-16 1.0OE-1S 1.0OE-16 5.0OE-16 1.0QE-15 1.00E-16 5.0OE-16 1.0OE-15 1.00E-16 5.0OE-16 1.00E-15 1.00E-16 5.0OE-16 1.0OE-15 Deviation Peiiatoon from From Average Average Tims Tims • fraction'' 'hoursi 0.24? 0.2BO 0.262 0.06? C.103 0.105 0.033 0 041 0044 0.245 0259 0.352 0.058 0.102 0.105 0.027 0.04Q 0.044 0.241 0.252 0268 0.085 0.03* 0.101 00Z4 0 033 0.039 0077 0.087 0.090 -0.082 -0 072 -0.069 -0.144 -0.1 X -0.13* O.Q7S 0.086 OG69 -0.083 -0 074 -0.070 -0.145 -o.iae -0.133 0.074 0.081 0.085 -0.084 -0.075 -0.074 -0.146 -0.141 -0.137 d5 100 101 34 4.0 41 1 1 1.5 1 7 0.4 10.0 101 3.4 3.$ 4.1 1.0 1.5 17 93 97 100 3 2 3.S 33 09 1 3 1.5 30 34 35 -3.2 -2G -2 7 •5.6 -5.2 -5.1 2.3 3.3 34 -3.2 -2 6 -2.7 -5.6 -55 -5.1 *.'.? 31 33 -3.3 3.0 -2.9 -5 5 -5 4 -5.3 APPENDIX A. SENSITIVITY ANALYSIS Traohisd Wail Inside to initial Log Radius Mo stuns Content (Fraction) 0.20O 0.20O 0.20Q 0.200 C.20O 0.200 0.20O 0.200 0.200 G.200 0.2QQ 0.20O 0.200 0.200 0.2QO 0.200 0.200 0.200 0.20O 0.20O 0.200 0.20O 0.200 0.20O 0.200 0.200 0.20O 0.20O 0.200 0.200 0.200 0.20O 0.20O 0.200 0.200 0.2QO 0.20Q 0.20O 0.2Q0 0.20O 0.200 0.200 0.200 0.20O 0.20O 0.200 0.200 0.20O 0.200 0.200 0.2QQ 0.20O 0.20O 0.200 0.350 O.350 O.S50 0.350 0.350 0.350 0.350 0.350 O.350 0.350 0.360 0.350 0.350 0.350 Q.350 0.350 0.350 0.350 0.350 O.350 0.350 0.350 O.350 0.350 0.350 0.350 0.350 O.350 0.350 O.350 0.350 O.350 0.350 0.350 0.350 O.350 0.350 O.350 0.350 0.350 0.350 0.350 0.350 0.350 0.350 O.350 0.350 O.350 0.350 0.350 0.350 0.350 0.350 0.350 Outside Thcknass Ratio 0.7GO Q.7Q0 0.700 O.7O0 O.7O0 0.7QQ 0.700 Q.70Q 0.700 Q.7G0 G.7QQ 0.7DO 0.7DO 0.700 O.7B0 Q.7O0 0.700 O.?Q0 0.7QO 0.780 O.7O0 0.700 0.700 0.700 0.700 0.7GO 0.700 0.80Q 0.800 0.800 0.630 0.8O0 0.800 o.aoo Q.SO0 o.eso 0.80Q o.aoo 0.800 0.8SO o.aoo Q.80G o.aao '> 0.800 Q.SOO 0.8O0 Q.8GG S.K30 Q.8Q0 O.gOO o.eoo o.aoo G.8QG 0.830 Surface Heat Liquid Transfer Permeability Coefficient (nf) (W/maK) Trachied Overlap Ratio 0.250 0250 0.25O 0.25O 0.250 0.250 0.25O 0.2SO 0.2SO 0.380 0.38O 0.38O 0.3SO 0.380 0.3&3 0.38O 0.38O 0.38O O.SQu O.EOO 0.500 0.500 0.500 0.5O0 0.500 0.500 0.5OQ 0.25Q 0.2SO 0.25G 0.250 0550 0.26O 0.250 0550 0250 0.380 0.380 0.380 0.38O 0.38O 0.38O 0.38O 0.380 0.393 0.5O0 0.500 0.5OO 0.5OO 0.5OO 0.500 0.5O0 O.SOO 0.500 141 i S.OO 9.00 9.00 20.O 2Q.Q 20.0 40.0 40.O 40.O 9.00 9.0Q 9.00 20.O 20.O 20.0 40.0 40.0 40.O 9.00 9.00 9.00 20.O 20.0 20.0 40.O 40.O 40.0 9.00 9.00 9.00 20.0 20.0 20.0 40.O 40.0 40.0 9.00 8.00 9.00 20.0 20.0 2Q.0 4Q.0 40.0 40.0 9.00 9.00 9.QO 20.0 20.0 20.O 40.O 40.0 40.O Figure A. 10 : : 1.00E-1S S.0OE-18 1.00E-15 1.00E-1S 5.0OE-16 1.00E-15 1.00E-1S 5.0QE-16 1.0GE-1S 1.0QE-16 5.00E-16 1.00E-15 1.0QE-16 5.QQE-16 1.0OE-15 1.00E-16 5.QGE-16 1.00E-15 1.00E-16 5.0QE-16 1.00E-15 1.Q0E-16 5.00E-16 1.0OE-15 1.0GE-16 5.00E-16 1.00E-15 1.00E-16 5.00E-16 1.00E-15 1.GQE-16 5.00E-16 1.00E-15 1.0GE-16 5.00E-1S 1.0OE-15 1.00E-16 5.00E-16 1.00E-15 1.QQE-18 5.00E-18 1.00E-15 1.00E-16 5.00E-16 1.00E-15 1.0GE-16 5.00E-16 1.00E-15 1.0OE-16 5.00E-1S 1.0GE-15 1.0GE-18 5.00E-16 1.0OE-15 Deviation Derail :>n from Average From Average Tirra Time i hour 5,i ill acton' 0259 0.253 0.2fO O210 0.210 0.210 01*1 0.191 0.191 02^9 0.2EB 0.259 0.21 J 0 21Q 0.210 0.191 0191 0.1W1 0 2t9 0.239 0.559 G.210 0.210 0.210 0191 0.191 0131 0 421 0431 0421 0.272 0 372 0 272 0 353 0.353 0.353 0.421 0.421 :M2I 0.372 0.372 0.272 0.353 0353 0.353 0.421 0.421 0 421 0.372 0.3*2 03~2 0 353 0.153 0.353 100 1Q.0 100 SI 81 8.1 74 74 74 100 10.0 100 S.1 31 ei 74 74 '4 10.0 100 1G0 a.i ai ai 74 74 74 135 165 165 144 14 4 144 135 136 136 16.2 162 155 144 14* 14.4 136 13 6 13.C 165 162 165 14 4 14.4 144 13 5 136 136 o o o o c s r a c s o r a o o o o o r a o o QQOOOCSCD U?l - t - * £2* —i ~ * £31 - 1 :ipocipppooppppppo3pppopppo ka Ka K* ba CD bo bo bo ba bo « L? * lr# La b* l» fo in LT W w ^ J U i Jh LA - * CT V U O x> -M O O O —t —i _k U3 b£ { ^ O O O - t _• - * U U > tn ui b u u u a '-* ~* i> 0> D ^ hi J t ' ^Q ^ O >3 O tt t i Id Hi Ol lii O O 6 C* U CM v V* & O Q O - i - i fc - i t 3 j5 (3 y 5 sfeis I (J1 U I U t O (O fcfcfe U U) - aas i J l ' i la lX^ j Ds O t t &£ l C ^ r Oass - A U>a - sii isis fee o o o o o o c o o o o o o o o o o o o o o o o o o o o c o mrnrnmmrniTirTimmiTimmmmnimmmmmmmmrnnimmrnmmrnmmrnmmrTirniTimmFnrnmmm »ii)iSi}ig)gta(»giiii^gsaa)0)!]ii)!ni)i3i«iui9)t!)i>igigiSi3iRiii3)«(!iai(tttii^ffiitig)9iBen!iiiffia)ui9i9!ii{iiii» c s Q o o r a o o o o o o o o r a a o o o o a o o o o o o o C T O Q o a o o Q o a o o o o o Q i O o a a Q Q Q o a Q o O O O O O O O O O O O O O O O O O O Q O O O O O O O O O O O O O O O Q O O O O Q O O O O O O O O O O O O O O , KJi -*> ~* oooobooooobbbobooobbobbooooboboboooobooob 888888888SSgggS18Siaggi8gii88888888eBlSiSiSSSi8g8g8ga8 o o o m o o o o o o a r a o r a o o o o o o o m o o o p o p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p O O O O O O O O O S3 O O O O O O O O O O O O O O O O O _. _ . _ . _ . _ . _ . _ . _ . _. _.. _ , _- —. . . . _. ._. _ . _ . _ . _ . _. _ . _ . i Q i O 0 O O O 0 0 £ 3 O S 3 O ! a i 0 C a O 0 O C a O O i 0 C 5 O O 0 C > C 3 C 3 O 0 O Q 0 O 0 0 e 5 O O O O O O C * S 3 O O ^ 0 O O i 0 0 o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O Q O O O O O O O O C 3 0 G O O O O O O O O C 3 0 0 0 0 0 0 0 0 0 0 C 3 0 S ^ O O O O O O O O O C 3 0 0 0 0 C 3 0 0 0 0 e 3 0 0 o o o o o o o o r a o o r a o o o o o c 3 0 0 o o r a o o o o o o o o o o o o o o o o o o o o r a o o o : r a r a o o r a o o CO CO j—I Q • X t ! 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O O O O O O O O O S 3 0 a i Q O O O O O C 3 C 5 0 0 0 0 0 0 0 S 3 Q O O O Q O O O O O O O Q O O O Q O O O O Q O Q O O O O O O O O Q i Q O Q O O O O O O O O O Q O O O Q O O Q Q O Q O Q O O Q O O O O O O O O O O Q Q O Q O O O O O O O O O O O Q Q O O O O O O Q Q O O O O O i Q O O O O Q O o o o o o o o o o o o o o ' o o o o o o r a o o o o o o a o o o o o o 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ^ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Ol S-H < APPENDIX A. SENSITIVITY Log Radius I'm) Initial Moisture Content (fraction) 0.200 0.2QG 0.200 0.2QQ Q.2QQ 0.200 0.200 0.20Q Q.20O 0.20O 0.200 0.20O 0.20O 0.20O 0.20O 0.200 0.200 0.20O 0.200 0.20O 0.20O 0.20O 0.200 0.20O 0.20O 0.200 0.20O 0.20O 0.200 0.20O 0.20O 0.20O 0.20O 0.200 0.20O 0.20O 0.20O 0.20O 0.200 0.20O 0.20O 0.200 0.20O 0.20O 0.200 0.200 0.20O 0.200 Q.200 0.200 0.20O 0.2QO D.200 0.200 1.800 1.600 1.600 1.600 1.600 1.600 1.600 1.600 1.600 1.600 1.600 1.600 1.600 1.600 1.600 1.600 1.S00 1.S00 1.S00 1.600 1.600 1.600 1.600 1.600 1.600 1.600 1.600 1.600 1.600 1.600 1.600 1.600 1.600 1.600 1.600 1.600 1.S00 1.600 1.600 1.600 1.600 1.600 1.600 1.600 1.600 1.600 1.600 1.600 1.600 1.S00 1.600 1.600 1.600 1.600 ANALYSIS 144 Trachied Wall Surface Hsat Inside to Traohisd Transfer Outside Overlap Ratio Coefficient Thcknass (W/m«K) Ratio 0.700 G.7O0 0.700 0.700 0.7GO 0.700 0.7DO Q.7QG O.TQO O.7D0 G.7O0 0.70G O.7O0 0.7TO O.TQO Q.7G0 0.700 O.TQO 0.700 0.700 0.700 0.700 O.TQO 0.7GO O.7D0 Q.70G 0.700 O.8O0 O.9O0 O.3O0 O.8O0 O.8O0 O.8O0 0.800 O.8O0 O.8O0 O.8O0 O.8O0 0.800 O.8O0 O.8O0 0.900 Q.8O0 Q.8G0 0.800 0.800 O.8O0 0.3GQ O.8O0 O.8O0 O.8O0 O.SOO 0.800 O.8O0 G.250 Q250 0.250 0.25O 0.26O 0.2SO Q.2SO 0.250 0.25Q 0.38O 0.38O 0.3SO 0.330 0.38O 0.380 0.380 Q.3SQ 0.3SO 0.500 Q.50G 0.500 0.500 0.5OD 0.5QQ D.500 O.SOO 0.500 0.25Q 02S0 025O 02m 02EO 02SO 0.250 0.250 0.250 0.38O 0.38O 0.38O 0.38O 0.38Q 0.380 0.380 0.380 0.3S0 0.500 0.500 0.5O0 0.5OQ 0.500 0.500 0.5O3 0.5OO 0.5OO . , ' . : • 9.00 9.00 9.00 20.0 20.0 2Q.0 40.0 40.0 40.0 9.00 9.00 9.00 20.0 20.0 20.0 40.0 40.O 40.0 9.00 3.00 9.00 20.O 20.0 20.O 40.0 40.O 40.O 9.00 9.00 9.00 20.0 20.O 20.O 40.0 40.0 40.0 9.00 9.00 9.00 20.0 20.O 20.0 40.O 40.0 40.0 9.QG 9.00 9.00 20.O 20.O 20.0 40.0 40.O 40.O Figure A. 13 Liquid Permeability 1.Q0E-16 5.Q0E-16 1.00E-15 1.00E-16 5.00E-16 1.00E-15 1.00E-16 5.00E-16 1.00E-15 1.00E-16 5.Q0E-16 1.0OE-15 1.00E-16 5.00E-16 1.00E-15 1.00E-16 5.0OE-16 1.00E-15 1.00E-16 5.00E-16 1.00E-15 1.0QE-16 5.00E-16 1.00E-15 1.00E-16 5.0QE-16 1.0OE-16 1.0OE-16 5.0QE-16 1.0OE-15 1.00E-16 5.0GE-1S 1.00E-15 1.00E-16 5.00E-16 1.0OE-15 1.00E-16 5.00E-16 1.00E-15 1.0QE-16 5.0OE-18 1.0OE-15 1.00E-16 S.OOE-16 1.00E-15 1.00E-16 5.0GE-16 1.00E-15 1.00E-16 5.00E-16 1.0OE-15 1.00E-16 S.0OE-16 1.0OE-15 Decalun troTi Deviation A m i age Time Mr acton', from Average T.me ihours) 1 327 1 3oS 1 361 1 123 1.114 1 159 1042 1073 1079 1.355 1 25S 1 360 1.121 1.152 1 1:3 1.040 515 52 4 525 433 44.5 44.7 402 414 41$ 51.1 52.3 524 432 44.4 44 S 401 413 41.5 50 7 51.7 525 42 9 438 443 39 8 40 7 412 419 42 9 43,0 54.0 34.9 351 30 8 318 32 0 418 42 8 43 0 339 348 351 30.8 31.7 31.9 416 42.3 42.7 33.7 343 343 30 5 315 316 1 en 10T 1.315 1.940 1.353 1 112 1 135 1 149 1.03? 1.054 1 «9 1 067 1 111 1.115 0.4c 1 0905 0.911 0303 0.824 0.823 1.064 1.109 1 114 0.8"3 0.903 0.910 0.^8 0.822 0.829 1078 1.095 1.107 0.374 0.890 0.901 0 793 0.809 0.82J APPENDIX A. SENSITIVITY Initial Log Radius 0.200 0.200 0.2DO 0.20Q 0.20O Q.2D0 Q.2QQ Q.2QQ 0.20O 0.20O 0.20O 0.20Q D.ZD0 0.2QO 0.2QO D.20Q 0.200 0.200 0.20O 0.20O 0.20Q 0.200 0.2QQ 0.2QQ Q.200 0.200 0.20O Moisture Content (traction) 1.600 1.600 1.600 1.600 1.600 1.800 1.600 1.600 1.600 1.600 1.600 1.800 1.600 1.600 1.600 1.600 1.600 1.600 1.600 1.600 1.600 1.600 1.600 1.600 1.600 1.600 1.600 145 ANALYSIS Trachied Wall Surface Haat In&ideto Trachied Transfer Outside Overlap Ratio Coefficient Thfcknass (W.rnsK) I Ratio O.SOO Q.3O0 0.900 O.9O0 o.soo 0.900 O.SOO 0.900 0.900 Q.90Q 9.90Q O.9O0 0.900 0.900 0.900 0.900 0.900 O.SOO O.SOO 0.900 0.900 O.SOO Q.9Q0 0.900 0.900 0.900 0.900 0.250 0.250 Q.29Q 0.2SO 0.25O 0.25O 0.2SO 0.25O 0.2SO 0.380 0.38O 0.38O 0.38O 0.38O 0.38O Q.3&G 0.380 0.38O 0.5QO Q.50Q 0.5OG 0.SOO 0.500 0.500 0.5QO 0.5O0 0.5O0 - 3.00 9.00 9.00 20.0 20.Q 20.O 40.0 40.0 40.0 9.00 9.00 9.00 20.O 20.0 20.0 40.O 40.0 40.0 9.00 9.00 9.00 20.O 20.O 20.0 40.0 40.0 40.O Figure A. 14 Liquid Permeability (rrf) 1.00E-16 5.00E-1S 1.0OE-15 1.0QE-16 S.OOE-16 1.QGE-15 1.00E-16 5.00E-16 1.00E-1S 1.00E-16 5.Q0E-1B 1.00E-15 1JJOE-16 5.00E-16 1.00E-1S 1.00E-1S 5.0QE-1S 1.0OE-15 1.0OE-16 5.00E-1B 1.00E-15 1.0OE-1S 5.Q0E-1S 1.0OE-15 1.0OE-16 S.Q0E-1S 1.00E-15 -• I J K 1 " _ r r« . s' an •*ll " 1 3 f , t 1 Ti- 1 -? 1 i-i.in. ll . : i jTL4 31Z _ '•}f> l n.".; r JIJ _* •_ • r_ 2 ••! •" i::« l»'b i .2 i - l l 144 •-'1 " - -1 • 'J ~ C'.'" •J " t i J 7"[ -J v&,i :?4 ^.'••u if tv •J jj~ " jCb •f. i •P3 "5 C Appendix B Model to Experimental Results Overlays 146 APPENDIX B. MODEL TO EXPERIMENTAL RESULTS OVERLAYS O 147 40 r=0.0cm, theoretical r=0.0cm, z=10.0cm, experimental 6r = ±1.5 r=4.0cm, theoretical r=4.0cm, z-10.0cm, experimental Sr = ±1.5 10 Time (hrs) 15 ; Time (hrs) (a) (b) : 50 O - .-^3£* y" 40 : O 40 ,-""' — / vs^ :/// 10 w * z^ ._» • / 3 „„ CO I- .- "i" / .<' • i £ / / " / / '/ / • i , , , i i 15 20 Time (hrs) (d) (c) O 40 CD Q. E 20 r=15.5cm, theoretical r=15.5cm, z=10.0cm, experimental Time (hrs) (e) Figure B.l: Apr. 2004, Douglas Fir, z=10cm, R =15.5cm, T0=9.5°C, T 3 30 15 r- • " " 15 20 25 10 30 15 Time (hrs) Time (hrs) (a) (b) 20 1^_J_; 1 • 1 o r=4.0cm, theoretical F4.0cm, z=30.0cm, experimental 5r = ±1.5 t .".-l-i>5^*===^ • .—.40 *h r=0.0cm, z=30.0cm, experimental 5r = ±1.5 1 / '-'* t," /• \^^ , . ' • ' ' ' // " / / '/ / / / - I/ / 1/ 1 J' .f 1/ . " , , , 1 ) r=13.5cm, theoretical r=13.5cm, z=30.0cm, experimental r=10.0cm, z=30.0cm, experimental 8r = ±1.5 15 ' 5r = i1.5 10 20 15 Time (hrs) Time (hrs) (c) (d) • \ • \ • \ • I • I- ^L 20 •±± 2 : a. f,/ E . /1 / n=13.5cm, theoretical r-13.5cm, z=128.0cm, experimental 5r = ±1.5 • " 5 10 15 20 25 30 Time (hrs) (d) (c) . *J•J |• 50 I — 40 L" I ' o a> 3 30 OS a Q.2U„ „ E " © H 10 r=15.5cm, theoretical r=15.5cm, z=128.0cm, experimental * 0 5 10 15 20 25 30 Time (hrs) (e) Figure B.3: Apr. 2004, Douglas Fir, z=128cm, #=15.5cm, T0=7.7°C, rl\ M0=0.76 =48.8°C, APPENDIX B. MODEL TO EXPERIMENTAL RESULTS OVERLAYS eo 60 55 T i ! I I , i " L^^~" ^, 35 - l 50 o ,„ o 40 . ro 3 0 "r : 55 !-| 50 r £ 3 150 " ! • \ \ * / £ 3 35 ra 3° - g.25 ziy CL25 I 2° '•- I 1 2 ° i- ' 1-15 r n n ~ m than a c « i 5 10 C fl 10 : 0: r=0.0cm, z=10.0cm, experimental 6r=±1.5 0; Ay LJ^ 1-15 • I --^iS^ o _„- j^> / 10 , UJji-i-H""!-^ r 15 20 fh af 1 r=5.0cm, z=10.0cm, experimental 5r = t 1 . 5 5 10 Time (hrs) Time (hrs) (a) (b) 15 20 60 ? 55 50 - ^ 4 5 r O o 40 '- £ 3 35 l±J^?A^'>4 ^-H^^r^T^T ^.• - 1 5 i. „ n n 10 r » '- n t-__. r=13.0cm, theoretical r=13.0cm, z=10.0cm, experimental 5r = ±1.5 r=9.0cm, z=10.0cm, experimental 5r = ±1.5 0 5 10 15 20 Time (hrs) Time (hrs) (c) (d) 60 55 r 50 I ^45 r, \ • o^40 £ 35 i ro 3° E § . 2 5 •*70 -q> • dU >-15^ 10 r=15.0cm, theoretical r=15.0cm, z=10.0cm, experimental 5r? Time (hrs) (e) Figure B.4: June 2004. BK Pine. £=10cm, #=15.0cm, Tn=23.3°C, Tn =51.9°C, Af0=0.37 APPENDIX B. MODEL TO EXPERIMENTAL RESULTS OVERLAYS 151 60 j. 55 h o^40 S> 35 ,,l,.,,!.. o '-- ^ ^ j j - H - ^ 50 r „45 ^-r*"' A~ ro 3° I 2° L t" 5 r=0.0cm, z=30.0cm, experimental 5r = ±1.5 10 15 zz 10 15 (a) (b) 55 „~-~^0z=srf*!^''1 ^-^^^. * * »•" ,45 * s^-' / / / *»i 50 j. ' o o_^40 £ 35 3 t o 30 • ' '-!//• , + . * " ' • * • i - i ' , " „; ;/" ^—~~ ......"r"~~ \ ' • : ( ' ' / U if Q.25 | ,=2 ° «E 1- 10 r * 0 r=5.0cm, z=30.0cm, experimental — 6r = ±1.5 . . I . . . i Time (hrs) ft t- i 60 35 r " 2 5 , , 20 F 50 ^ 4 5 <-> „„ £ : : Time (hrs) 55 I S 30 • 10 • j- 0 60 //^ •-16 10 5 * /yy', , # ' • hull, 2.25 *"" 15 ^ ^?f^t 9.40 \ \ £ 35 15 30 : i < i n_ .. .. 10 " . Q f,_„ r=9.0cm, z=30.0cm, experimental 8r = ±1.5 + U f 1 r=13.0cm, z=30.0cm, experimental 5r = ±1.5 i 5 10 15 20 °0 5 10 Time (hrs) Time (hrs) (c) (d) 15 20 60 r 55 '- , . +. . 50 ^ 4 5 - * ' • i « i " t " f , ! ' " ' .. .. T ' ; * O „„ -' 2 35 r 3 • g 30 • : Q.25 r | 20 r ^ 15 : - 10 °0 ... r=15.0cm, z=30.0cm, experimental 5 10 15 20 Time (hrs) (e) Figure B.5: June 2004, BK Pine, —30cm, i2=15.0cm, T0=21.0°C, T00=51.9°C, M0=0.37 APPENDIX B. MODEL TO EXPERIMENTAL RESULTS OVERLAYS 60 60 55 7 ! ! i . | • I "l 55 50 7 I. t 'Vj-~—^—'—' 50 " ,^45 r ^ 4 5 T °_. 40 o .„ . o 40 \ .Jf E 35 • i i 1-M i i»ijJ^^-^—L—"r .•i^S^r* ,.-j^5r 2 35 1 3 , L /#" ' ro 30 §.25 152 a.25 =• ^ i r t 1 20 \T\ r~j 1-15 ""IS 10 • 5 7 0 _ n n ., .__, r=0.0cm, z=128.0cm, experimental 10 5r = ±1.5 0\ i 5 ) 10 15 20 5 (a) (b) i : 20 60 ! : ! . l . j - f"J" f M 55 _ ^ 4 5 *J>^^J •-"!' ^ i T () , ? 35 Y/ 3 IS 30 _ if/ / $25 it/ ^}^i^^^ : i.|-|-|-|-!i"l't -"""" ' | 50 i_!L-t—-fc:—=• lf A^' ^ 0 5 <-> , „ , - • " ' T" £ 35 ro 3 ° i-j a.25 20 1 t ^ « 20 >-1B on 10 . : 1 15 Time (hrs) 55 ! • *-15 10 Time (hrs) 60 r 50 E /I m t k a n r a t ! r=5.0cm, z=128.0cm, experimental 5r = ±1.5 , , i 5 «,„,•*• i • . . i . . i i ... 10 r=9.0cm, z=128.0cm, experimental 5r = ±1.5 — n — i - i 10 , , i .. ., . r=13.0cm, z=12B.0cm, experimental Sr = ±1.S 1 °D 5 10 Time (hrs) Time (hrs) (c) (d) 15 20 60 i !.!.). I i.i.t.M.i ! ' 55 50 r . • | • tT ^ 4 6 7" () , P 35 . 3 ro 30 ?! 25 7 W 2 0 I- p5 ... 10 • 1 ,. ... r=15.0cm, z=128.0cm, experimental . i . r 1 i . . , 1 1 , i , 1 Time (hrs) (e) Figure B.6: June 2004, BK Pine, z=128cm, #=15.0cm, T0=21.9°C, Toc=51.9°C, M0=0.37 APPENDIX B. MODEL TO EXPERIMENTAL RESULTS OVERLAYS 153 50 45 i^HJJ-U+H-1^ ^ ) j M ^ ^ 40 O 35 d) 30 3 15 25 a a> . 20 E £> 15 '- ^\y^ **/ s v " 10 5 - : 0 . ... r=4.5cm, theoretical r=4.5cm, z=10.0cm, experimental 8r = ±1.5cm n r=0.0cm, z=10.0cm, experimental 5r = ± 1.5cm • 5 10 15 20 10 Time (hrs) Time (hrs) (a) (b) 15 r 45 o 25 i_ • 15 , 15 •_ - _0_ , 4I (. , 10 « r=8.5cm, z=128.0cm, experimental 5r = ± 1.5cm 0 5 10 15 0 20 5 r=12.5cm, z=128.0cm, experimental 6r = ± 1.5cm • . . . . • 10 15 20 Time (hrs) Time (hrs) (c) (d) <» 30 2: <» .Q.20F- E ^ L 15 10 5 r=14.5cm, theoretical r=14.5cm, z=128.0cm, experimental Time (hrs) (e) Figure B.9: July 2004, BK Pine, z=128cm, #=14.5cm, T0=19.0°C, Too=45.20C, M0=0.50 APPENDIX B. MODEL TO EXPERIMENTAL RESULTS OVERLAYS 50 O 1. 156 50 4 0 • ' • ( , 4 0 • 0 -iz^' - • CD 3 ra 3t) CD J Q. t - 20 CD h- " •-- „ • — . — """" „...-- .^-"' • " " /' 1 \ ! i r=11.0cm, theoretical r=11.0cm, z=10.0cm, experimental Sr = ±1.5 5 . 9 40 ^ // - / // ) - 10 i i ". C 20 > 15 » 5 r-16.0cm, z=10.0cm, experimental 8r = ±1.5 i , , , , i , 10 15 20 Time (hrs) Time (hrs) (c) (d) 60 r 50 • to " • _ . . • ro Temperature 9 40 . • 10 • r=17.0cm, theoretical r=17.0cm, 2=10.0cm, experimental 0 C 5 10 15 20 Time (hrs) (e) Figure B.10: Dec. 2004, Spruce, z=10cm, R=17.0cm, T0=6.8°C, T00=43.3°C, M0=0.64 APPENDIX B. MODEL TO EXPERIMENTAL RESULTS OVERLAYS 40 40 9 30 9 30 157 I i4*s4 I ^^^^\ 1 o Temperature o /' \J/\ 0 _ " r=0.0cm, z=30.0cm, experimental 0 5 / JC/'' # * L ^T 8r = ±1.5 Jr. . . . . I , , , , I . Temperature ^r^^\ kJ 3 (D 2 O ° Q. • £ 0) H- 10 • • 4/ / / / / / ' / 'ill _IJ/ 'ij/ • 0 5 AA n „ t. , r=16.0cm, theoretical r=16.0cm, z=30.0cm, experimental r=11.0cm, z=30.0cm, experimental 5r = ±1.5 10 15 Sr = ±1.5 20 10 Time (hrs) Time (hrs) (c) (d) 15 50 : . I l-lj-l I I I ]- t . | . | . | rT^TTTf.| 40 I' o 0 J) 30 3 CO o Q.20 E 0J l- 10 A-l " t\ 1 J.L. 1 r=17.0cm, z=30.0cm, experimental 0 c 5 10 15 20 Time (hrs) (e) Figure B.ll: Dec. 2004, Spruce, *=30cm, i?=17.0cm, T0=2.0°C, T,00=43.3°C, M0=0.64 APPENDIX B. MODEL TO EXPERIMENTAL RESULTS OVERLAYS 158 • 40 : - ^ ^ r 9 30 : / ^ ^#4 • t ^ ^ ^ T " o 30 - ^ v_ to O 20 : fc 20 Q. E : ' - . r=0.0cm, z=128.0cm, experimental / X-' 0 ) 5 10 15 " r=6.0cm,z=128.0cm, experimental 6r = ±1.5 20 Time (hrs) Time (hrs) (a) (b) , . ..L_4 40 ' /^\ \* 9™ (D rtrpt^-p '7 f\1 //"" • / / 3 (0 .,, c 5 10 15 20 25 Time (hrs) (e) Figure B.13: Jan. 2005, Pine, z=10cm, fl=18.5cm, T0=4.5°C, r oo =53°C, Af0=0.49 APPENDIX B. MODEL TO EXPERIMENTAL RESULTS OVERLAYS r=0.0cm, z=30.0cm, experimental 8r = ±1.5 60 «tj. o °„, 40 0) 3 • • • - • ) 25 160 5 Time (hrs) i 10 15 20 , , , 25 Time (hrs) (b) r=10.0cm, theoretical r=10.0cm, z=30.0cm, experimental r=16.0cm, theoretical r=16.0cm, z=30.0cm, experimental 8r = ±1.5 10 6r = ±1.5 15 10 15 Time (hrs) Time (hrs) (c) (d) o 3 2 • > > . . 15 I . 20 I ^j 25 Time (hrs) Time (hrs) (a) (b) r=10.0cm, theoretical r=10.0cm, z=128.0cm, experimental 5r = ±1.5 O • / A O Temperature °C 60 • 10 A- ' 0 A 'A •k 1 ' 5 10 I . . 20 15 r=16.0cm, theoretical r=16.0cm, z=128.0cm, experimental 8r = ±1.S _1_ 10 15 20 25 L J I 25 Time (hrs) Time (hrs) (c) (d) 2 a) Q. ^ 20 r=18.5cm, theoretical r=18.5cm, z=128.0cm, experimental 10 15 Time (hrs) (e) Figure B.15: Jan. 2005, Pine, z=128cm, i?=18.5cm, T0=2.2°C, T00=53°C, M0=0.49 APPENDIX B. MODEL TO EXPERIMENTAL RESULTS OVERLAYS 200 200 150 150 Q100 o £ 50 Q100 fj 0 o , , , 1 5 10 15 20 25 30 Time (hrs) (e) Figure B.16: Feb. 2005, Pine, z=10cm, i2=18.5cm, T0=1.1°C, T00=45°C, A/o=0.49 APPENDIX B. MODEL TO EXPERIMENTAL RESULTS OVERLAYS r=0.0cm, z=30.0cm, experimental 5r = ±1.5 40 o r=5.0cm, theoretical r=5.0cm, z=30.0cm, experimental 8r = ±1.5 i - ls 40 • ^^, * 1*& y^T o 01 3 CD 2 20 2 20 3 / / E a. E 01 l- 1/1 // 0 , 0 r=16.0cm, theoretical n=16.0cm, z=30.0cm, experimental 8r = ±1.5 1 , 1 5 10 15 20 25 30 Time (hrs) Time (hrs) (c) (d) ^ 40 o 0 CD 3 CO I J I I |-|- " I ' l I ' 1 1 ! rTF 1-' ' t • , '' • . i v" ' CD 2 0 Q. E CD t- -.(*(- 1 C 5 11. 1 1 r=18.5cm, z=30.0cm, experimental 10 1 1 . . 1 . . 1 15 1 1 20 1 L . . 1 25 1 1 1 1 1 30 Time (hrs) (e) Figure B.17: Feb. 2005, Pine, z=30cm, R=l8.5cm, T0=1.4°C, T00=45°C, M0=0.49 APPENDIX B. MODEL TO EXPERIMENTAL RESULTS OVERLAYS * • r=0.0cm, z=128.0cm, experimental 8r = ±1.5 n p Temperature °C 40 r=5.0cm,z=128.0cm, experimental 8r = ±1.5 40 ^ssS*a**a*l o •'!*' 0 D ro 20 a. / E CD / ' " 0 1D 15 20 25 30 - ' • / ^ / / y /// //-"' <5 20 01 Q. E / / / • /// • 0 // / / , i • • ' 40 ' I I• I i /// CD H I 5 10 15 (b) "[ i ,-1 I ! • ' (a) -^fTp' 3 I i Time (hrs) ' C I T Time (hrs) Temperature °C O o , n t• I.I " T -f 0 r=10.0cm, z=128.0cm, experimental 5r = ±1.5 ^ ^ ^ f £ * *=esj 40 # I J"' /// • 5 // //' 1/ i******^ ^ 2! •• 164 20 25 30 " ^"^"" 'ZjPPf^^^^\"'\^ ~{0f i' 5 i • . . 10 ,,,,!,, 15 20 25 30 5 r=16.0cm, z=128.0cm, experimental 8r = ±1.5 i - i 10 15 Time (hrs) Time (hrs) (c) (d) 20 25 30 o o 20 r=18.5cm, theoretical r=18.5cm, z=128.0cm, experimental Time (hrs) (e) Figure B.18: Feb. 2005, Pine, z=128cm, #-18.5cm, T0=0.1°C, T^=45°C, M0=0.49 APPENDIX B. MODEL TO EXPERIMENTAL RESULTS OVERLAYS 165 o CD CD O. E CD _ - o , Temperature °C , I • o - o o 400 r c n,™ » h < » « i i M i • r=0.0cm, z=10.0cm, experimental 6r = ±1.5 1 1 . 1 1 1 1 1 1 1 1 4 6 8 10 r=5.0cm, z=10.0cm, experimenta 5r = ±1.5 -400 o a 4 6 8 10 Time (hrs) (a) (b) 400 r 12 14 16 400 r - 20D O 0 CD J3 o <5 o jgs --.- • • . . 1 6 . i 8 • • n »i-_ _f i r=5.0cm, z=30.0cm, experimental fir = ±1.5 -400 ' 10 0 2 4 6 8 10 Time (hrs) Time (hrs) (a) (b) 400 12 14 16 400 200 200 O I -400 =12.0cm, theoretical r=12.0cm, z=30.0cm, experimental 8r = ±1.5 « ) 2 4 6 8 10 12 14 o o Temperature O o Temperature - o Temperature 3 : ; - O 0 166 -400 0 16 2 4 r=21.0cm, z=30.0cm, experimental fir = ±1.5 6 8 10 Time (hrs) Time (hrs) (c) (d) 12 14 16 400 200 O O « • O O Temperature o —m -too * 2 4 _ c t. I U , r=23.5cm, z=30.0cm, experimenta 6 8 10 12 14 16 Time (hrs) (e) Figure B.20: Mar. 3, 2005, Spruce, z=30cm, i?.=23.5cm, T0=2.8oC, T00=44.7°C, Af0=1.49 APPENDIX B. MODEL TO EXPERIMENTAL RESULTS OVERLAYS 167 r 40 40 o o Temperature 0 j - =y?s 0 20 CD •. i"*" 'J- I ZX"~ 3 40 ^ -""'"r' £ 0) 20 1- 0 r=12.0cm, theoretical r=12.0cm, z=10.0cm, experimental 5r = ±1.5 2 I 2 r=23.5cm, theoretical r=23.5cm, z=30.0cm, experimental i 8 10 12 14 16 Time (hrs) (e) Figure B.23: Mar. 7, 2005, Spruce, z=30cm, #=23.5cm, T0-- J.8°C, T 0O =51.5 o C, Af0=1.49 APPENDIX B. MODEL TO EXPERIMENTAL RESULTS OVERLAYS so - 80 60 60 • " • • • • -20 „,.„ -40 -60 1X1 4 6 ,h„„,i„„l 9 3 20 - E 1— ro -60 • 0 • • • r=5.0cm, theoretical r=5.0cm, z=128.0cm, experimental = ±1.5 fir 2 4 6 8 10 Time (hrs) Time (hrs) (a) (b) 60 40 • T *7 • " • 20 • 12 14 16 — ' 2 4 6 - * -60 " ) • ' • • • 20 =12.0cm, theoretical r=12.0cm, z=128.0cm, experimental 8r = ±1.5 * -_ V, 0 ! J(0 L ; • -20 -60 • 80 : 40 0 T TT • 60 O. : -40 r=0 Ocrr ,z= 128. Ocm experiments 8r = ± 1 . 5 i i i • i . 4-Li. 8 10 12 80 s (U a . • • o 0 :• • Temperature r ro o 20 40 9 40 170 8 10 12 14 r=21.0cm,z=128.0cm, experimental 5r = ±1.5 • 16 Time (hrs) Time (hrs) (c) (d) 80 60 O 40 o - - 2 3 20 IS §. o E (D 1— -20 J|0 r=23.5cm, z=128.0cm, experimenta -60 , 2 1 1 , > , 4 i 6 8 10 12 14 16 Time (hrs) (e) Figure B.24: Mar. 7, 2005, Spruce, z=128cm, i?=23.5cm, T0=8.8°C, r oo =51.5°C, M0=1.49 APPENDIX B. MODEL TO EXPERIMENTAL RESULTS OVERLAYS 300 300 250 250 200 200 150 150 , • o o ^ 100 50 2 0 • • • • • • 171 ,„, • o 100 -ioo -150 -200 * -250 -300 5 =23.5cm, z=10.0cm, experimental 10 15 20 25 Time (hrs) (e) Figure B.25: Mar. 22, 2005, Spruce, M0=1.49 40cm, i?,=23.5cm, T0=11.2°C, Too=52.0°C, APPENDIX B. MODEL TO EXPERIMENTAL RESULTS OVERLAYS 800 600 800 - 600 400 400 O O o r = ±1.5 10 15 20 r=5.0cm, theoretical r=5.0cm, z=128.0cm, experimental Sr = ±1.5 25 10 _l_ 20 15 Time (hrs) Time (hrs) (a) (b) O 4t > j-i=t£ CD 1 . ^ - -"r< ^ '"I 1 1 ^••r' x t 5 - ' ' •ffi CD 20 s Q. E ^s "^ tffl s"^5" i ! 1"' CD tr=12.0cm, theoretical r=12.0cm, z=128.0cm, experimental 8r = ±1.5 r=12.0cm, theoretical r=12.0cm, z=128.0cm, experimental 5r = ±1.5 15 10 Time (hrs) Time (hrs) (c) (d) 60 O 25 •M-H-l-l. i 11 20 25 11 A . i « * * | ,« 40 0 CD 3 CD CD 20 Q. E CD V- 0 - „ . C +k a*V 1 r=23.5cm, z=128.0cm, experimental -20c 5 10 15 20 25 Time (hrs) (e) Figure B.27: Mar. 22, 2005, Spruce, z=128cm, /?=23.5cm, T 0 = 5.9°C, TOC=52.0°C, Af0=1.49