Theoretical Thermal Conductivity Equation for Uniform Density Wood Cells John F. Hunt, P.E., Research Mechanical Engineer Hongmei Gu, Post Doc. Research Associate Patricia K. Lebow, Mathematical Statistician USDA Forest Service, Forest Products Laboratory Madison, WI Presentation Overview Background • Research Goals and Study Objective • Why model heat transfer in wood • 2D Finite Element Heat Transfer Models • Comparisons with other Models Theoretical Equation Development • Problem – Interpolation & Use • Development of the Equations • Microwave Heating Rates Applications • Study of Heat Flux • Study of Ring Orientation & Density Background RESEARCH GOAL: Maintain a healthy and sustainable forest through an understanding of wood and wood processing conditions that can be used to manipulate the properties of wood and wood composites for specific performance characteristics. STUDY OBJECTIVE: To understand and develop an easily definable heat transfer coefficient equation that can be used in heat and mass transfer modeling. Background Why model transfer in wood? Uncontrolled heating can cause serious defects in the products if not appropriately applied, either with conventional or alternative energy. Understanding the flow and generation of heat energy within wood is critical to wood heating control to obtain the desired final product material properties.. Background 2D Cellular Heat Transfer Model Softwood is Anisotropic, axial- symmetric, porous material. Each growth ring consists of two significantly different Radial structures – earlywood and latewood. Tangential Background 2D Cellular Heat Transfer Model Path-R Radial Path-T Input parameters: Lin e @30C Line ¾Cell porosity ¾Cell alignment Tangential Path-R Path-T ¾Cellwall properties Radia ¾Air properties l ¾Vapor Properties Lin e Lin ¾Water Properties e Tangential Background 2D Cellular Heat Transfer Model Cellular Model Evaluated Keff At: FS Cellwall 4 MCs: ¾0% MC Oven Dry Free water in ¾Fiber Saturation ¾Fiber Saturation Cell Lumen Point (FSP); ¾50/50 water/vapor in the lumen; Air or Water Vapor ¾Fully saturated; Background 2D Cellular Heat Transfer Model Material Properties in the cellular model Thermal conductivity Density Specific Heat 3 Symbol (W/m.K) (Kg/m ) (J/Kg·K) 1 Cell wall substance (0%MC) k CW 0.410 1540 1260 2 Air in the lumen (0%MC) k air 0.026 1.161 1007 3 Bound water in cell wall k BW 0.680 1115 4658 4 7 Saturated cell wall (FSP) k f 0.489 1415 2256 5 Water vapor in cell lumen k V 0.018 0.734 2278 6 Free water in cell lumen k FW 0.610 1003 4176 Note: 1. Property values for cell wall substance at 0% MC was obtained from Siau's book (1995). 2. Air property values for air were obtained from Incropare (1981). 3. Density of bound water was obtained from Siau (1995). Thermal conductivity and specific heat of bound water was obtained based on water properties and assumption of the linear relationship with density. 4. Property of saturated cell wall was obtained by rule of mixture 5. Property values of water vapor was obtained from Ierardi (1999). 6. Property values of free water was obtained from Incropera (1981). 7. kf is constant when MC is over FSP, but changes with MC below FSP. Background FE Model vs Siau’s Model 0.7 Tangential ¾Effective thermal 0.6 Radial Equa. 4 conductivity as a Equa. 5 function of wood 0.5 Poly. (Radial or Tangential) 3 2 porosity; k eff = -2.04E-01p + 5.38E-01p - 7.22E-01p + 4.09E-01 R2 = 1.00E+00 0.4 ¾No significant difference between 0.3 radial and tangential thermal properties; 0.2 ¾Comparing to an (W/m.K) conductivity Thermal 0.1 electrical circuit model in a wood textbook; 0 0% 20% 40% 60% 80% 100% Porosity Background FE Model vs MacLean’s Model 0.8 Model - 0% MC Model - Fiber Saturation Point Model - 50/50 water/vapor in lumen Model - Fully Saturate 0.7 MacLean - 0% MC MacLean - Fiber Saturation Point MacLeans - 50/50 water/vapor in lumen MacLean - Fully Saturated 90% 80% 0.6 70% 60% 50% 40% 30% 20% 10% 0.5 0% Porosity 0.4 0.3 0.2 Effectivethermal conductivity (W/m.K) 0.1 0.0 0 300 600 900 1200 1500 1800 Oven Dry Density (Kg/m3) Theoretical Equation Development Problem – Interpolation & Use 0.8 Model - 0% MC Model - Fiber Saturation Point Model - 50/50 water/vapor in lumen Model - Fully Saturate 0.7 MacLean - 0% MC MacLean - Fiber Saturation Point MacLeans - 50/50 water/vapor in lumen MacLean - Fully Saturated 90% 80% 0.6 70% 60% 50% 40% 30% 20% 10% 0.5 0% Porosity 0.4 0.3 0.2 Effective thermal conductivity (W/m.K) 0.1 0.0 0 300 600 900 1200 1500 1800 Oven Dry Density (Kg/m3) Theoretical Equation Development Development of the Equations ¾The Equation needs to be easily useable for future applications; ¾Development of oneone equationequation thatthat couldcould describedescribe allall moisture content and density variations of wood cells; ¾Assumes no significant difference between radial and tangential thermal properties; ¾1st Try: Polynomial Equation to fit the data. Nothing worked.; = 1 Keff ¾2nd Try: Resistive Equation. …. Reff Theoretical Equation Development Resistive Electrical Equations a. Parallel Circuit b. Series Circuit Theoretical Equation Development Resistive Electrical Equations a. Parallel Circuit C1C2C3R1R2R3 R = effp C1C2R1R2 + C1C3R1R3 + C2C3R2R3 b. Series Circuit X2X3 X4X5X6 R =C4R4+C5R5+C6R6 =C4X1+C5 +C6 effs X2+ X3 X4X5+ X4X6+ X5X6 Theoretical Equation Development Resistive Electrical Equations a. Parallel Circuit b. Series Circuit = + + Reffs C4R4 C5R5 C6R6 C1C2C3R1R2R3 R = X 2X 3 X 4X 5X 6 effp C1C2R1R2 + C1C3R1R3 + C2C3R2R3 R = C4X1+ C5 + C6 effs X 2 + X 3 X 4X 5 + X 4X 6 + X 5X 6 L R1 = Y1 = L − a − X1 = K f (L a) K f (L) L − a a a − b a − b R2 = Y 2 +Y3 = + X 2 = X 3 = − − − K f (a b) K fw (a b) K f (L a) K fw (a) L − a a − b b b b b R3 = Y 4 + Y5 + Y6 = + + X 4 = X 5 = X 6 = − − K f (b) K fw (b) Kv (b) K f (L a) K fw (a b) Kv (b) Theoretical Equation Development FE vs Parallel vs Series 0.8 Model - 0% MC Model - Fiber Saturation Point Model - lumen filled with 50% free water Model - Fully Saturate Parallel model - 0% MC Parallel model - Fiber Saturation Point 0.7 Parallel model - lumen filled with 50% free water Parallel model - Fully Saturate ) Series model - 0% Series model - FSP Series model - lumen filled with 50% free water Series model - Fully Saturated 0.6 0.5 0.4 0.3 0.2 Effective thermal conductivity (W/m.K conductivity thermal Effective 0.1 0 0 300 600 900 1200 1500 1800 Oven Dry Density (Kg/m3) Theoretical Equation Development Development of the Equations The ODD or ρOD, of the cell is described by density of the cell wall, ρcw, density of the air, ρair, and the oven-dry porosity, Pd, in Equ. 21. ρ = ρ − + ρ OD cw (1 Pd ) air Pd (21) Pd is determined by rearranging Equ. 21, Equ. 22. ρ − ρ P = cw od (22) d ρ − ρ cw air Dry porosity can also be described in geometrical terms, Equ. 23. 2 = a Pd 2 1 (23) Theoretical Equation Development Development of the Equations With the addition of moisture into the cell, the volume percent of bound water, V%bw , in the cell wall needs to be described. By definition, fiber moisture content MCf is calculated by dividing bound water weight by the oven-dry cell weight, which is the cellulose material weight, when assuming UNIT volume for the wood cell. The MC in the fiber cell wall (MCf) can be calculated using Equ. 24. V % ρ MC = bw cw (24) f ρ − cw (1 V %bw ) By rearranging Equ. 24, V%bw can be determined for MC < 0.3. For MC > 0.3 V%bw is a constant at 0.293. For MC from 0 to 0.3 then MCf ≅ MC. For MC > 0.3 MCf = 0.3 MC ρ V % ≅ f cw bw MC ρ + ρ f cw bw (25) Theoretical Equation Development Development of the Equations With the addition of moisture into the cell, wet porosity, Pw, is introduced and can be described using Equ. 26. (1−V % )P P = bw d (26) w − 1 V %bw Pd Wet porosity can also be described in geometrical terms, Equ. 27. 2 = a Pw 2 L (27) The outer dimension, L, and area of the wet cell, L2, changes with increasing moisture up to the FSP and can be described using Equ. 28. ρ + MC ρ L2 = bw f cw (28) ρ + ρ bw MC f Pw cw Theoretical Equation Development Development of the Equations The specific equation for MC at all moisture conditions is described in Equ. 29. ρ 2 2 ρ − + vb Pw + ρ − b ( cwV %bw )(1 Pw ) 2 fw (1 2 )Pw MC = a a (29) ρ − − ( cw (1 V %bw ))(1 Pw ) The location and description of the free water is around the inner lumen that extents into the middle of the lumen having a dimension b. For MC < 0.3 there is assumed no free water then b = 0, Fig. 4. For MC > 0.3 it is assumed the cell wall is saturated and the remaining water goes into the lumen as free water which can be determined by rearranging Equ. 28 and solving for b, Equ. 30. a2{(1− P )[V %bwρ − (1−V % )MCρ ]+ P ρ } b = w bw bw cw w fw (30) ρ − ρ Pw ( fw v ) Theoretical Equation Development Development of the Equations Thermal conductivity for the fiber cell wall material, Kf, needs to also be determined before calculating the effective thermal conductivity.