Theoretical Thermal Conductivity Equation for … Thermal Conductivity Equation for Uniform Density Wood Cells John F. Hunt, P.E., Research Mechanical Engineer Hongmei Gu, Post Doc

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


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


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


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


b. Series Circuit

a. Parallel Circuit

323231312121321321

RRCCRRCCRRCCRRRCCCReffp ++

=

6564546546

3232514665544

XXXXXXXXXC

XXXXCXCRCRCRCReffs ++

++

+=++=


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


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)


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)


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)


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)


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.


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 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

Documents

Theoretical Thermal Conductivity Equation for … Thermal Conductivity Equation for Uniform Density Wood Cells John F. Hunt, P.E., Research Mechanical Engineer Hongmei Gu, Post Doc