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Theoretical Thermal Conductivity Equation for
Uniform Density Wood Cells John F. Hunt, P.E., Research Mechanical Engineer
Hongmei Gu, Post Doc. Research AssociatePatricia 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
RESEARCH GOALRESEARCH GOAL: :
Maintain a healthy and sustainable forest through an Maintain a healthy and sustainable forest through an understanding of wood and wood processing conditions that can understanding of wood and wood processing conditions that can be used to manipulate the properties of wood and wood be used to manipulate the properties of wood and wood composites for specific performance characteristics.composites for specific performance characteristics.
STUDY OBJECTIVESTUDY OBJECTIVE::
To understand and develop an easily definable heat transfer To understand and develop an easily definable heat transfer coefficient equation that can be used in heat and mass transfer coefficient equation that can be used in heat and mass transfer modeling. modeling.
Background
Understanding the flow and Understanding the flow and generation of heat energy generation of heat energy within wood is critical to wood within wood is critical to wood heating control to obtain the heating control to obtain the desired final product material desired final product material propertiesproperties..
Uncontrolled heating can Uncontrolled heating can cause serious defects in cause serious defects in the products if not the products if not appropriately applied, appropriately applied, either with conventional or either with conventional or alternative energy.alternative energy.
Why model transfer in wood?Background
Softwood is Anisotropic, axialSoftwood is Anisotropic, axial--symmetric, porous material.symmetric, porous material.
Each growth ring consists of Each growth ring consists of two significantly different two significantly different structures structures –– earlywood and earlywood and latewood.latewood.
Radial
Tangential
Background2D Cellular Heat Transfer Model
Input parameters:Input parameters:
@30C@30C
Cell porosityCell porosity
Cell alignmentCell alignment
CellwallCellwall propertiesproperties
Air propertiesAir properties
Vapor PropertiesVapor Properties
Water PropertiesWater Properties
Path-R Path-T
Line
Line
Radial
Tangential
Path-R
Line
Path-TRadial
Tangential
Line
Background2D Cellular Heat Transfer Model
Cellular ModelCellular Model
Evaluated Evaluated KKeffeff At:At:
4 MCs:4 MCs:
0% MC Oven Dry 0% MC Oven Dry
Fiber Saturation Fiber Saturation Point (FSP);Point (FSP);
50/50 water/vapor 50/50 water/vapor in the lumen;in the lumen;
Fully saturated;Fully saturated;
Free water in Cell Lumen
Air or Water Vapor
FS Cellwall
Background2D Cellular Heat Transfer Model
Background
Thermal conductivity Density Specific HeatSymbol (W/m.K) (Kg/m3) (J/Kg·K)
Cell wall substance (0%MC) 1k CW 0.410 1540 1260
Air in the lumen (0%MC) 2k air 0.026 1.161 1007
Bound water in cell wall 3k BW 0.680 1115 4658
Saturated cell wall (FSP) 4k f
7 0.489 1415 2256Water vapor in cell lumen 5
k V 0.018 0.734 2278Free water in cell lumen 6
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.
Material Properties in the cellular model
2D Cellular Heat Transfer Model
Effective thermal Effective thermal conductivity as a conductivity as a function of wood function of wood porosity;porosity;
No significant No significant difference between difference between radial and tangential radial and tangential thermal properties;thermal properties;
Comparing to an Comparing to an electrical circuit model electrical circuit model in a wood textbook;in a wood textbook;
k eff = -2.04E-01p 3 + 5.38E-01p 2 - 7.22E-01p + 4.09E-01R2 = 1.00E+00
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0% 20% 40% 60% 80% 100%
Porosity
Ther
mal
con
duct
ivity
(W/m
.K)
TangentialRadialEqua. 4Equa. 5Poly. (Radial or Tangential)
BackgroundFE Model vs Siau’s Model
BackgroundFE Model vs MacLean’s Model
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0 300 600 900 1200 1500 1800Oven Dry Density (Kg/m3)
Effe
ctiv
e th
erm
al c
ondu
ctiv
ity (W
/m.K
)
Model - 0% MC Model - Fiber Saturation PointModel - 50/50 water/vapor in lumen Model - Fully SaturateMacLean - 0% MC MacLean - Fiber Saturation PointMacLeans - 50/50 water/vapor in lumen MacLean - Fully Saturated
0%10%20%30%40%50%60%70%80%90%
Porosity
Theoretical Equation Development
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0 300 600 900 1200 1500 1800Oven Dry Density (Kg/m3)
Effe
ctiv
e th
erm
al c
ondu
ctiv
ity (W
/m.K
)
Model - 0% MC Model - Fiber Saturation PointModel - 50/50 water/vapor in lumen Model - Fully SaturateMacLean - 0% MC MacLean - Fiber Saturation PointMacLeans - 50/50 water/vapor in lumen MacLean - Fully Saturated
0%10%20%30%40%50%60%70%
80%90%
Porosity
Problem – Interpolation & Use
Theoretical Equation DevelopmentDevelopment of the Equations
The Equation needs to be easily useable for future The Equation needs to be easily useable for future applications;applications;
Development of one equation that could describe all Development of one equation that could describe all moisture content and density variations of wood cells;moisture content and density variations of wood cells;
Assumes no significant difference between radial and Assumes no significant difference between radial and tangential thermal properties;tangential thermal properties;
11stst Try: Polynomial Equation to fit the data. Nothing Try: Polynomial Equation to fit the data. Nothing worked.;worked.;
22ndnd Try: Resistive Equation. ….Try: Resistive Equation. …. effeff R
K 1=
Theoretical Equation DevelopmentResistive Electrical Equations
b. Series Circuita. Parallel Circuit
Theoretical Equation DevelopmentResistive Electrical Equations
b. Series Circuit
a. Parallel Circuit
323231312121321321
RRCCRRCCRRCCRRRCCCReffp ++
=
6564546546
3232514665544
XXXXXXXXXC
XXXXCXCRCRCRCReffs ++
++
+=++=
Theoretical Equation DevelopmentResistive Electrical Equations
b. Series Circuita. Parallel Circuit
)(11
aLKLYR
f −==
)()(322
baKa
baKaLYYR
fwf −+
−−=+=
)()()(6543
bKb
bKba
bKaLYYYR
vfwf
+−+−=++=
6564546546
3232514
XXXXXXXXXC
XXXXCXCReffs ++
++
+=323231312121321321
RRCCRRCCRRCCRRRCCCReffp ++
=
)(1
LKaLX
f
−=
)(2
aLKbaX
f −−=
)(3
aKbaX
fw
−=
)(4
aLKbX
f −=
)(5
baKbX
fw −=
)(6
bKbXv
=
665544 RCRCRCReffs ++=
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0 300 600 900 1200 1500 1800Oven Dry Density (Kg/m3)
Effe
ctive
ther
mal
con
duct
ivity
(W/m
.K)
Model - 0% MC Model - Fiber Saturation PointModel - lumen filled with 50% free water Model - Fully SaturateParallel model - 0% MC Parallel model - Fiber Saturation PointParallel model - lumen filled with 50% free water Parallel model - Fully SaturateSeries model - 0% Series model - FSPSeries model - lumen filled with 50% free water Series model - Fully Saturated
Theoretical Equation DevelopmentFE vs Parallel vs Series
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.
dairdcwOD PP ρρρ +−= )1( (21)
Pd is determined by rearranging Equ. 21, Equ. 22.
aircw
odcwdP
ρρρρ
−−= (22)
Dry porosity can also be described in geometrical terms, Equ. 23.
2
2
1aPd =
(23)
Theoretical Equation DevelopmentDevelopment 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.
)%1(%
bwcw
cwbwf V
VMC−
=ρ
ρ (24)
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
bwcwf
cwfbw MC
MCV
ρρρ+
≅% (25)
Theoretical Equation DevelopmentDevelopment of the Equations
With the addition of moisture into the cell, wet porosity, Pw, is introduced and can be
described using Equ. 26.
dbw
dbww PV
PVP%1
)%1(−−= (26)
Wet porosity can also be described in geometrical terms, Equ. 27.
2
2
LaPw =
(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.
cwwfbw
cwfbw
PMCMC
Lρρ
ρρ++
=2 (28)
Theoretical Equation DevelopmentDevelopment of the Equations
The specific equation for MC at all moisture conditions is described in Equ. 29.
)1))(%1((
)1()1)(%( 2
2
2
2
wbwcw
wfwwv
wbwcw
PV
Pab
aPbPV
MC−−
−++−=
ρ
ρρρ (29)
The location and description of the free water is around the inner lumen that extents into themiddle of the lumen having a dimension b. For MC < 0.3 there is assumed no free water thenb = 0, Fig. 4. For MC > 0.3 it is assumed the cell wall is saturated and the remaining watergoes into the lumen as free water which can be determined by rearranging Equ. 28 andsolving for b, Equ. 30.
)(}])%1(%)[1{(2
vfww
fwwcwbwbww
PPMCVbwVPa
bρρ
ρρρ−
+−−−= (30)
Theoretical Equation DevelopmentDevelopment of the Equations
Thermal conductivity for the fiber cell wall material, Kf, needs to also be determined beforecalculating the effective thermal conductivity. Using rules of mixtures for MC < 0.3 then, Kfcan be determine by Equ. 31.
bwbwbwcwf VKVKK %)%1( +−= (31) For MC > 0.3 the fiber cell wall is fully saturated with water and is assumed does not change.Using the rule of mixtures for MCf = 0.3 then the fiber cell wall thermal conductivity is aconstant at Kf = 0.4891, Table 1.
Theoretical Equation DevelopmentDevelopment of the Equations
The effective thermal conductivity, Keff, for the entire range of moisture contents and densities can be determined by only knowing the ODD and MC conditions for the sample by substituting Equ.s 15-20 are into Equ. 14 and simplifying the equation as Equ. 32.
)))()(
6)(
5)((4)1)(((
)(
vfwffwff
feff
bKKabKLabC
aKKLaCbaKC
LaLa
KaLK
−−+−+
−−−+−−
−= (32)
By solving for and substituting the appropriate values for variables a, b, L, Kf and substituting the appropriate C4, C5, and C6 parameters in Table 2, the entire range of thermal conductivities can be determined. Similarly, the equation can be used to determine estimated thermal conductivity values where FSP is other than 30%. By changing the volume percent of bound water, V%bw, (Equ. 25) and recalculating the values.
Theoretical Equation DevelopmentDevelopment of the Equations
Theoretical Equation DevelopmentEquation with Constants
Study of Heat FluxHeat flow in a wood board:Heat flow in a wood board:
Significant heat flux follows the latewood rings;Significant heat flux follows the latewood rings;
Higher energy is shown being transferred through the Higher energy is shown being transferred through the latewood rings, then across into the earlywood from one latewood rings, then across into the earlywood from one latewood ring to the next toward the center of the boardlatewood ring to the next toward the center of the board
Earlywood
Latewood
Application
Study of TemperatureApplication
Study of TemperatureApplication
20
30
40
50
60
70
80
90
100
0 2000 4000 6000 8000 10000 12000TIME (sec)
TEM
PER
ATU
RE
(C)
H_Ring_0.14"
H_Ring_0.28"
H_Ring_0.56"
Q_Ring_0.14"
Q_Ring_0.28"
Q_Ring_0.56"
20
30
40
50
60
70
80
90
100
0 2000 4000 6000 8000 10000 12000TIME (sec)
TE
MPE
RA
TU
RE
(C)
Loc_A1Loc_A5Loc_B1Loc_B4Loc_CLoc_DLoc_E
Study of RingOrientation & Density
Application
In summary: I would like to say In summary: I would like to say Thank You Thank You
Any QuestionsAny Questions
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