This is author version of article published as:

Perre, Patrcik and Turner, Ian W. (2001) Determination of the Material Property Variations Across the Growth Ring of Softwood for Use in a Heterogeneous Drying

Model. Part 2 Use of Homogenisation to Predict Bound Liquid Diffusivity and Thermal Conductivity. Holzforschung 55(4):pp. 417-425.

Copyright 2001 Walter de Gruyter

Determination of the material property variations across the growth

ring of softwood for use in a heterogeneous drying model Part II - Use of homogenisation to predict bound liquid diffusivity and

thermal conductivity.

Patrick Perré LERMAB (Laboratory of Wood Science) UMR INRA n° 1093 E.N.G.R.E.F., 14 rue Girardet, F-54042 Nancy Cedex, France

and

Ian Turner

School of Mathematical Sciences, Queensland University of Technology,

GPO Box 2434, Brisbane, Q4001, Australia

Key Words : Diffusivity, Homogenisation, Thermal conductivity.

ABSTRACT

In this work, the extensive knowledge of wood gained at the ENGREF by using a combination of microscopic observation and experimental work is being used to postulate material property correlations for a macroscopic heterogeneous “Cerne” growth ring model of softwood. In this second part, the method of homogenisation is used to capture the contrast of property between the gaseous phase and the solid phase. Homogenisation problems are computed using a Control Volume formulation. The tracheid model developed in part I has been used to define several elementary representative volumes. Consequently, bound water diffusivity and thermal conductivity can be predicted for each material direction as a function of local wood density. For practical applications, analytical models fitted from the computed values allow these property variations to be easily and accurately determined.

1. INTRODUCTION Wood is a material of plant origin, produced in tree by the secondary thickening. Consequently, appearance and properties of this material depend strongly on the

species, but also on biological diversity and growth conditions (Wilson and White 1984). Indeed, wood properties depend on the species, on the tree, on the position within the tree… In wood sciences, biometry is often used to address this problem of variability. However, derived from applied mathematics and widely used in the field of porous media (Sanchez-Hubert and Sanchez-Palencia 1992, Adler 1992, Hornung 1996) homogenisation techniques propose a deterministic approach that allows the macroscopic properties to be calculated from the microscopic properties and the morphology of the porous structure. Although very powerful, these tools just start to be used in wood sciences. Some examples can be found in wood mechanics (Farruggia 1998, Harrington et al. 1998). Concerning mass and thermal diffusion, only simple deterministic models, mainly parallel/series models, can be found (Siau 1995). In the present work, wood is considered as a natural composite material. The properties and the spatial organisation of its components are parameters that must be involved in the understanding of the macroscopic physical properties. As proposed in recent works (Perré 1998, Perré and Badel 2000), the geometrical description of the pore structure can come from image analysis performed on anatomical views. However, the purpose of the present work is to catch the average effect of structure variations within the annual ring of softwood, rather than the actual values calculated for one specific tree. This is why the tracheid model developed in part I will be used as a geometrical description of the microscopic structure along the annual ring, from earlywood to latewood.

2. DERIVATION OF THE MACROSCOPIC WOOD PROPERTIES

2.1 Basic variable definitions The following variables will be referred to and used often throughout this paper:

-3s

0

P orosity

Density of the cell-wall substance ( 1530 kg m )

Specific wood density (oven dry mass/ green volume)s

f

r r

r

º

º =

º

-3

o

Water density ( 1000 kg m )

Moisture content (kg water/ kg of dry wood)

Free water moisture content

Bound water moisture content

Fibre Saturat ion Point

TemperatureFSP

l l

l

b

X

X

X

X

T C

r rº =

º

º

º

º

º

Because wood is highly hygroscopic, bound water has to be separated from free water to give the expression:

b lX X X= + [1] where min( , )b fspX X X= and 0.325 0.001fspX T= -

Remark : at full saturation, maxsat fsp lX X X= + )

The saturation variable is calculated according to the free water content only:

max

ll

l

XS

X= [2]

Assuming that the volume of the pores is constant and that the density of bound water and free water are equal, the following expression has been derived for the volume fraction of the solid phase:

0

1

( )

s b

ls

s sb fsp

l

X

X X

rr

er rr r

+= æ ö÷ç ÷ç + - ÷ç ÷÷çè ø

[3]

The porosity is defined as (1 )sf e= - and the volume fractions of the liquid and gaseous phases are given respectively by

( ) and 1gl l lS Se f e f= = - [4]

In order to clarify the homogenisation formulation, superscripts are used in the property notations: ̂ denotes a microscopic value and %denotes the homogenised value of the arrangement of cells (tracheids or ray cells). The macroscopic value (arrangement of tracheids with ray cells) has no superscript. 2.2 Tracheid Model The purpose of the present work is to catch the average effect of structure variations within the annual ring of softwood. Consequently, the tracheid model developed in part I will be used as a geometrical description of the microscopic structure along the annual ring, from earlywood to latewood. Although this model is quite simple, it enables the major features of softwood anatomy to be accounted for. In particular, the evolutions of cell wall thickness and radial radius along the ring are in good agreement with anatomical observations. Furthermore, in order to be consistent with the elaboration of wood cells in the cambium, the cells are aligned in the radial direction and in staggered rows in the tangential direction. This detail will be significant when performing homogenisation to predict the macroscopic mass and thermal diffusivity. From this tracheid model, the unit cell can be obtained by using a two cell representative volume. Figure 1 depicts this representative elementary volume in the case of low and high solid fraction (earlywood and latewood respectively).

2.3 Bound Liquid Diffusivity To derive the macroscopic bound liquid diffusivity it is necessary to analyse the bound liquid transport mechanisms that evolve at the microscopic scale. Microscopic bound liquid migration arises due to two distinct mechanisms – diffusion of the hygroscopically bound water through the cell walls and vapor diffusion due to Fick’s law. At the microscopic scale, these two fluxes can be expressed:

( )

( )sb b b

v g v v

D X Bound Liquid Flux

D V apour Flux

r

r w

= - Ñ

- Ñ

F

F [5]

In equation [5], and vbD D represent the microscopic bound liquid and vapor

diffusivities having units 2 1m s- and vw is the mass fraction of vapour in the gaseous

phase. Assuming isothermal conditions and constant total pressure, it is possible to simplify the microscopic vapour flux [5] to depend upon the gradient of bound liquid:

v vv v b

b

M PD XRT X

æ ö¶ ÷ç ÷ç= - Ñ÷ç ÷÷ç ¶è øF [6]

Equations [5] and [6] allow the microscopic pseudo-diffusivities (kg.m-1.s-1) to be defined accordingly:

ˆ ,

ˆ

sb b

v vv v

b

D D

M PD DRT X

r=

¶=¶

[7]

Using the bound liquid diffusivity data of Stamm (1959) it is possible to obtain the following exponential correlation to best fit, in the least squares sense, ˆ

bD :

4300ˆ exp 12.8183993 10.8951601273.15sb bD X

Tr

æ ö÷ç= - + - ÷ç ÷çè ø+ [8]

The expressions for the macroscopic bound liquid diffusivity in the radial and tangential directions are calculated using homogenisation techniques according to these microscopic properties, together with the pore morphology defined in the previous section (Figure 1). The method and numerical tool used to carry out the homogenisation is similar to that described in a previously published work (Perré 1997). However, the pore morphology used here, and hence the results, are slightly different. In the case of a medium hypothesized to be periodic, for which the micro and macro scales do not interfere, homogenisation reduces to that of solving a physical problem over the unit cell (Suquet 1985, Adler 1992, Sanchez-Hubert and Sanchez-Palencia 1992). Specific periodic boundary conditions are associated to this problem. A diffusive phenomenon, such as thermal diffusion or mass diffusion will be considered.

A linear relation exists between the flux vector q and the gradient of the driving force uÑ :

,i ij jq a u= [9]

In equation [9], the order two tensor a ij represents the property of the medium. In the case of a periodic medium, the homogenized tensor ijA is defined by:

,a a

Homogenized Average of the Correct ive+ termcoefficient microscopic coefficient

jpij ij ipA w= +

= [10]

Equation [10] indicates that the macroscopic value is the sum of two terms:

- The first term is the average, over the unit cell, of the microscopic values. In the case of diffusive phenomena, this term represents a fictitious medium for which all the components act in parallel.

- The second term is a corrective term whose calculation requires the solution of the physical problems defined on the unit cell (two problems for the diffusive phenomenon in two-dimensions):

( ), ,,

periodic

o Y

h

hij j ij ii

w Y

a w a ver

ìï -ïïïíï = -ïïïî

[11]

The periodic boundary conditions associated with these problems prove that the solution does not depend on the macroscopic configuration. These problems have to be solved once in order to obtain the property of the equivalent macroscopic medium. The numerical procedure developed to solve the system [11] uses a CV-FE (Control Volume - Finite Element) formulation. In addition to the accurate mass balance, this method permits the derivation of each flux in a homogeneous medium (the nature of the material does not change within an element). The linear system is solved via a pre-conditioned conjugate gradient strategy, which is particularly well adapted to dealing with periodic boundary conditions. In order to account for the staggered rows in the tangential direction, the unit cell, which must be able to reproduce the periodic porous medium, contains two tracheids (Figure 1). This unit cell is meshed using square four node elements ( 20.5 0.5 mm´ ). The calculations

are performed with dimensionless numbers, whereby both values of pseudo-diffusivity have been non-dimensionalised by ˆ

vD . The dimensionless bound water

diffusivity value used in the computation is ˆ

0.004ˆb

v

DD

= , which corresponds to a

bound water moisture content close to 12 % (Siau 1995).

Figure 2 depicts the microscopic mass flux calculated over the unit cell for each solid fraction (0.32 and 0.84). Two flow fields are depicted that correspond respectively to a unit macroscopic gradient in the radial and in the tangential directions. It is clear from this figure that the cell walls are not active parallel to the macroscopic moisture gradient. The other cell walls are active simply because they represent the shortest path to pass from one gaseous part to the other. Due to the staggered rows in the tangential direction, diffusion is slightly more difficult in the tangential direction for a squared cell. However, when the solid density increases, the cell becomes flattened in the R direction and the effect of the gaseous phase is much more efficient in the tangential direction than in the radial one (Figures 2 and 3). Consequently, because the conductive phase is not connected, the macroscopic coefficients are relatively close to the results obtained using a simple series model. However, using the present tracheid model, results depict a change in the anisotropic ratio with density - the R and T curves cross each other at around 0.3 for the solid fraction. The following correlations, valid in the range 200 to 1000 kg/m3, have been postulated to fit with good accuracy the value calculated by homogenisation (Figure 3):

( )( )1.81 1.6R

gseriesbD A e= +% [12]

1.8TseriesbD A= ´% [13]

with 1

ˆ ˆseries gs

vb

A

D Dee=

+

At this stage, we have to keep in mind that pseudo-diffusivity values have been used in the homogenisation procedure: this was necessary to have a consistent formulation of the mass fluxes within a non-homogeneous medium. In order to obtain the value of the standard diffusivity (m2.s-1), as it appears in the second Fick's law,

( )X D Xt

¶ = Ñ Ñ¶

[14]

the values must be divided by the wood density. In addition, although important for diffusion in the radial direction, the tracheid arrangement used for these calculations does not account for ray cells. A small percentage of ray cells has to be put in parallel to the cell arrangement. The final form of the radial macroscopic diffusivity becomes:

0 (1 )R R Rray ray rayb bD D Dr e e= - + ´% % [15]

Typical values for Rray rayand De % are .05 and ˆ0.1 vD´ respectively (values adapted

from Siau 1995). In order to be consistent, the same amount of ray cells should be put in series with tracheids in the tangential direction:

0

11

Tb

ray rayT T

rayb

D

D D

r e e= -+% %

[16]

In this expression, the transverse ray cell pseudo-diffusivity TrayD% has to be used and

due to the ray cell morphology in this direction, its value can be obtained from equation [13] for a density value 0r of 500 kg/m3. The final expressions [15] and [16] use only diffusivities for the parallel and the series models, so that any value of ˆ

bD can be used, which allows the effect of moisture content or temperature to be considered easily through equations [7] and [8]. The only limitation is certainly due to the pit apertures that have been neglected in this work. Their effect becomes significant for very low moisture contents because ˆ

bD becomes very small in this domain. Consequently, it is better not to use these results for moisture contents less than 8-10%. Due to the considerable length of wood fibers, the longitudinal macroscopic diffusivity is simply derived using a parallel model of the form: 0

ˆ ˆLs g vb bD D Dr e e= + [17]

The results of the correlations [15-17] have been graphed in Figures (4-6) versus bound liquid moisture content for various densities. As expected there is little difference in the behavior of the bound liquid diffusivities in the radial (Figure 4) and tangential (Figure 6) directions. However, it is to be noted that the effect of ray cell becomes significant in radial direction at low moisture content (low value of ˆ

bD and

high value of ˆvD ). The longitudinal diffusivity (Figure 6) highlights completely

different trends in the curves. In this direction, the values are much higher than in the transverse plane and the curves exhibit maximum values in the range 0.05 to 0.10 of moisture content. Indeed, as a consequence of the pore structure, which leads to almost parallel diffusion in the longitudinal direction, the macroscopic mass diffusion is mainly driven by vapour diffusion in the gaseous phase. Note that all predicted values are in good agreement with data available in literature (Wadsö 1993, Siau 1995), provided the averaged density of solid wood is used. 2.4 Thermal Conductivity The thermal conductivity of wood can be computed using exactly the same homogenisation tool as discussed above. In this case, the air phase is the non-conducting phase while the solid phase, which is connected, has the highest conductivity. Figure 7 depicts the microscopic thermal flux calculated over the unit cell, for two solid fractions (0.32 and 0.84). Two flow fields are depicted that correspond respectively to a unit macroscopic gradient in the radial and in the tangential direction. In the radial direction, only the cell walls parallel to the heat flow (R walls) are active. The orthogonal cell wall (T walls) can just increase slightly the section of the conductive phase when present close to these active walls, that means

close to the intercept. This effect becomes significant when the double wall thickness is high compared to the radial dimension of the tracheid (see Figure 7 for 0.84se = ). In the tangential direction, the walls perpendicular to the macroscopic gradient (R walls) are most efficient, since they permit the heat flux to be continuous within the conducting phase from one T wall to the T wall of the next tracheid. As for mass diffusivity, we propose here two correlations able to give values very close to the calculated ones based on the conductivity values of the air and the cell wall substance. Because the conducting phase is continuous, these expressions involve a mixture law with a positive power or, the conductivity calculated with the parallel model: 0 0.46 0.54R

series parallell l l= ´ + ´% [18]

( )1

0ˆ with 0.6nT n n

g sair s nl e l e l ^= + =% [19]

where 1 ˆ

ˆs gseries s airparallelgs

airs

andl l e l e leell

^

^

= = ++

Properties with a subscript of 0 denote thermal conductivities at a moisture content equal to zero. Figure 8 depicts the calculated homogenized conductivity values and the proposed correlations. The values used for the gas and solid phase thermal conductivities in these calculations are given respectively as 1 10.023 W m Kairl - -=

and 1 1ˆ 0.5 W m Ksl - -^ = (Siau 1995, Ngoe-Ekam et al. 1994).

In the longitudinal direction, the wood anatomy allows a simple parallel model to be used:

0 / / / /ˆ ˆ ˆwith 2L

g sair ss sl e l e l l l ^= + » ´ [20]

As stated for mass diffusivity, the effect of ray cells must be taken into account in expressions [18] and [19]:

0 0 / /(1 )R R

ray ray rayl e l e l= - + ´% % [21]

0

0

1 0.61T

ray rayT

ray

with nl e el l ^

= =-+% %

[22]

rayl ^ and / /ray

l are evaluated using equations [19] and [20] respectively, with 500

kg.m-3 as density. Contrary to bound water diffusion, which occurs when free water is not present in the medium, the thermal conductivity has to be predicted for all ranges of the moisture content. At first, one must note that the homogenisation software can be applied to a fully saturated medium, where it is sufficient to use l l , instead of airl . Because the contrast between l l and sl ^ is low, the homogenised conductivity calculated for a fully saturated medium is almost independent of the direction (R or T) and can be

evaluated with great accuracy using expression [18] with 1 10.6 . .W m Kl - -=l

instead of airl . Then, because the liquid conductivity is not significantly different from the conductivity of the solid, it is assumed that the presence of bound water affects only the solid fraction (equation [3]) and not the thermal conductivity of the solid fraction. When liquid water is present within the medium, one part of the pore is filled with water. A partially saturated medium is considered to be a mixture of dry wood and fully saturated wood, with proportions ( )1 lS- and lS respectively, lS being the saturation (fraction of pore volume filled with liquid). The conductivity of this mixture is calculated according to a mixture law, with the same value 0.6n = as for equation [18]. In the longitudinal direction, a parallel model has been used in the case of an oven-dry sample. Nevertheless, because free water is more likely located in the smallest possible radii, that means at both ends of the tracheids, we decided to use a mixture law between the dry medium and the fully saturated medium even in this direction. The final expressions read:

( )( )1

0ˆ( , ) (1 ) ( ) nnT T n n

s sl lX S Sl r l f l e l ^= - + +l [23]

( )( )1

0ˆ( , ) (1 ) ( ) nnR R n n

s sl lX S Sl r l f l e l ^= - + +l [24]

( )( )1

0 / /( , ) (1 ) ( )

nnL L nsl l s

X S Sl r l f l e l= - + +l% [25]

where 0.6n = for the three expressions. In these expressions, it should be noted that the phase volume fractions must be evaluated according to their definitions [equations 3-4]. The variation of thermal conductivity plotted versus moisture content for various densities is given in Figure 9 for the radial and Figure 10 for the tangential direction of wood. Figure 11 shows the variation in the longitudinal thermal conductivity with moisture content.

3. CONCLUSIONS A complete set of density dependent correlations, which include capillary pressure and absolute permeability (discussed in Part I) and bound liquid diffusivity and thermal conductivity (discussed in Part II) have been developed in this two part paper. These correlations will be used in future research together with a new version of an existing two-dimensional drying computational model known as TransPore to enable the drying of a heterogeneous sample of softwood to be simulated at the scale of the annual growth increment. The models presented throughout this paper have been based on innovative wood science arguments and solid mathematical frameworks. Throughout the paper, the local value of each parameter was determined with the aid of the local density only. This allows any microdensitometry profile within the annual ring to be accounted for. Moreover, in spite of numerical calculations, all results have been deliberately expressed as "open" expressions. The reader can use and modify these expressions in order to suit specific species (provided this is softwood), samples

or configurations (resistance factor of pits, amount of ray cells, presence of a fluid phase that is nor air neither liquid water…). Finally, some these parameters can be fitted to match experimental data.

4. ACKNOWLEDGEMENT

One of the Authors, Ian Turner, would like to thank the French organisation INRA for funding his visit to the ENGREF/INRA laboratory to enable this work to be realised. The travel assistance offered under the Personal Development Program at the QUT should be acknowledged also.

5. REFERENCES

Adler, P.M., 1992 "Porous Media : Geometry and Transports" Butterworth-Heinemann Series in Chemical Engineering (1992).

Farruggia F., 1998, Détermination du comportement élastique d'un ensemble de fibres de bois à partir de son organisation cellulaire et d'essais mécaniques sous microscope. Thèse de Doctorat-ENGREF.

Gibson L.J and Ashby M.F., 1988 "Cellular solids, structure and properties" Pergamon Press.

Harrington J.J., Astley R.J., Booker, R.E., 1998, Modelling the elastic properties of softwood, Part I, The cell wall lamellae. Holz als Roh -und Werkstoff, 56(1), 43-50.

Hornung, U., 1996 "Homogenization and porous media" Springer Verlag.

Ngoe-Ekam P.S., Lecomte D; and Menguy G., 1994 "Thermal conductivity of tropical wood : influence of moisture, cutting level and principal cutting plan" 9th International Drying Symposium, published in Drying’94, 751-758, Gold Coast, Autralia.

Perré, P., and Keller, R., 1994 "La prédiction des propriétés macroscopiques du matériau bois à partir de sa structure anatomique : besoin ou moyen de caractériser la paroi ?" Journal of Trace and Microprobe Techniques, 12(4), 277-288.

Perré P., 1998 "The Use of Homogeneisation to Simulate Heat and Mass Transfer in Wood: Towards a Double Porosity Approach" Plenary lecture, International Drying Symposium, published in Drying’98, 57-72, Thessaloniki, Grèce.

Perré P. and Badel E., 2000 "Properties of Oak Wood Predicted from X-ray Inspection: Representation, Homogenization and Localization" submitted.

Plumb, O.A., Brown, C.A., and Olmstead, B.A., 1984 "Experimental measurement of heat and mass transfer during convective drying of southern pine" Wood. Sci. Technol. 18, 187-204.

Siau, J.F., 1995 "Wood : influence of moisture on physical properties" Virginia Polytechnic Institute and State University.

Stamm A.J., 1959 "Bound-water diffusion into wood in the fiber direction" For. Prod. J. 9, 27-32.

Sanchez-Hubert, J. and Sanchez-Palencia E., 1992 "Introduction aux méthodes asymtotiques et à l'homogénéisation" Masson, Paris.

Wadsö, L., 1993 "Studies of Water Vapor Transport and Sorption in Wood" ,Building Materials, Lund University, Lund, 102 pages.

Wilson K. and White D.J.B., 1984 "The anatomy of wood : its density and variability" Storbart & Son Ltd, London.

FIGURES CAPTIONS

Figure 1 : The Unit Cell used for Homogenisation Calculations a) Earlywood, b) Latewood

Figure 2 : Microscopic mass flux calculated over the unit cell. For each solid fraction (0.32 and 0.84), two flow fields are depicted that correspond respectively to a unit macroscopic gradient in the radial and in the tangential direction

Figure 3 : Dimensionless diffusivity versus solid fraction for the radial and tangential directions of wood (markers come from homogenisation calculations).

Figure 4 : Variation of Wood Bound Liquid Diffusivity with Moisture Content in the Radial Direction for various Densities (markers come from homogenisation calculations).

Figure 5 : Variation of Wood Bound Liquid Diffusivity with Moisture Content in the Tangential Direction for various Densities

Figure 6 : Variation of Wood Bound Liquid Diffusivity with Moisture Content in the Longitudinal Direction for various Densities

Figure 7 : Microscopic thermal flux calculated over the unit cell. For each solid fraction (0.32 and 0.84), two flow fields are depicted that correspond respectively to a unit macroscopic gradient in the radial and in the tangential direction

Figure 8 : Conductivity versus solid fraction for the radial and tangential directions of wood

Figure 9 : Variation of Wood Thermal Conductivity with Moisture Content in the Radial Direction for various Densities

Figure 10 : Variation of Wood Thermal Conductivity with Moisture Content in the Tangential Direction for various Densities

Figure 11 : Variation of Wood Thermal Conductivity with Moisture Content in the Longitudinal Direction for various Densities

a) Earlywood

R

TThe unit cell used for homogenisation

b) Latewood

R

T

The unit cell used for homogenisation

Figure 1 : The Unit Cell used for Homogenisation Calculations a) Earlywood, b) Latewood

a) Eearlywood (solid fraction = 0.32)

0 0.5 1 1.5 2x

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

y

Radial flux

0 0.5 1 1.5 2x

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

y

Tangential flux

a) Latewood (solid fraction = 0.84)

0 0.5 1 1.5 2x

0

0.1

0.2

0.3

0.4

0.5

y

Radial flux

0 0.5 1 1.5 2x

0

0.1

0.2

0.3

0.4

0.5

y

Tangential flux

Figure 2 : Microscopic mass flux calculated over the unit cell. For each solid fraction (0.32 and 0.84), two flow fields are depicted that correspond respectively to a unit macroscopic gradient in the radial and in the tangential direction

0

0.01

0.02

0.03

0.04

0.05

0.06

0 0.2 0.4 0.6 0.8 1.0

Correlation (Tangential)Correlation (Radial)Calculated (Radial)Calculated (Tangential)

Series model

Solid fraction

Dim

ensi

onle

ss d

iffus

ivity

Figure 3 : Dimensionless diffusivity versus solid fraction for the radial and tangential

directions of wood (markers come from homogenisation calculations).

10-11

10-10

10-9

0 0.05 0.10 0.15 0.20 0.25 0.30

900

800

300

400

500

600

700

1000

Density = 200 kg/m3

Moisture Content (kg/kg)

Diff

usiv

ity (m

2 /s)

Figure 4 : Variation of Wood Bound Liquid Diffusivity with Moisture Content in the Radial Direction for various Densities

10-11

10-10

10-9

0 0.05 0.10 0.15 0.20 0.25 0.30

300

400

500

600

800

1000

Density = 200 kg/m3

Moisture Content (kg/kg)

Diff

usiv

ity (m

2 /s)

Figure 5 : Variation of Wood Bound Liquid Diffusivity with Moisture Content in the Tangential Direction for various Densities

10-10

10-9

10-8

0 0.05 0.10 0.15 0.20 0.25 0.30

700

900

300

400

500

600

800

1000

Density = 200 kg/m3

Moisture Content (kg/kg)

Diff

usiv

ity (m

2 /s)

Figure 6 : Variation of Wood Bound Liquid Diffusivity with Moisture Content in the Longitudinal Direction for various Densities

a) Eearlywood (solid fraction = 0.32)

0 0.5 1 1.5 2x

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

y

Radial flux

0 0.5 1 1.5 2x

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

y

Tangential flux

a) Latewood (solid fraction = 0.84)

0 0.5 1 1.5 2x

0

0.1

0.2

0.3

0.4

0.5

y

Radial flux

0 0.5 1 1.5 2x

0

0.1

0.2

0.3

0.4

0.5

y

Tangential flux

Figure 7 : Microscopic thermal flux calculated over the unit cell. For each solid fraction (0.32 and 0.84), two flow fields are depicted that correspond respectively to a unit macroscopic

gradient in the radial and in the tangential direction

0

0.1

0.2

0.3

0.4

0.5

0 0.2 0.4 0.6 0.8 1.0

Correlation (Radial)Correlation (Tangential)Calculated (Radial)Calculated (Tangential)

Parallel model

Series model

Solid fraction

Con

duct

ivity

(W.m

-1.K

-1)

Figure 8 : Conductivity versus solid fraction for the radial and tangential directions of wood

(markers come from homogenisation calculations).

0

0.1

0.2

0.3

0.4

0.5

0.6

0 0.5 1.0 1.5 2.0

1000

900 800 700 600 500 400

300

Density = 200 kg/m3

Moisture content (kg/kg)

Ther

mal

con

duct

ivity

(W.K

-1.m

-1)

Figure 9 : Variation of Wood Thermal Conductivity with Moisture Content in the Radial Direction for various Densities

0

0.1

0.2

0.3

0.4

0.5

0.6

0 0.5 1.0 1.5 2.0

1000

900 800 700 600500 400

300

Density = 200 kg/m3

Moisture content (kg/kg)

Ther

mal

con

duct

ivity

(W.K

-1.m

-1)

Figure 10 : Variation of Wood Thermal Conductivity with Moisture Content in the Tangential Direction for various Densities

0.1

0.3

0.5

0.7

0.9

0 0.5 1.0 1.5 2.0

1000900

800700

600500

400

300

Density = 200 kg/m3

Moisture content (kg/kg)

Ther

mal

con

duct

ivity

(W.K

-1.m

-1)

Figure 11 : Variation of Wood Thermal Conductivity with Moisture Content in the Longitudinal Direction for various Densities