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COST Action E 15
ADVANCES IN DRYING OF WOOD (1999-2003)
st
1 Workshop “State of the art for Kiln drying” in Edingurgh 13./14th, Oct 1999
How to Get a Relevant Material Model for
Wood Drying Simulation ?
Patrick Perré
ENGREF/INRA, Forest Products Team,
14 rue Girardet
F-54 042 Nancy Cedex - France.
Email : [email protected]
ABSTRACT
What is a relevant material model for wood drying simulation? The answer, which obviously
depends on the objective of the simulation code, should balance the possibilities and the
requirements of each solution.
The first part of this paper is devoted to the transfer phenomena involved in wood drying. The
important physical mechanisms and related coupling are explained. The main physical
formulations proposed in the literature to model these phenomena are presented and compared.
Some simulation examples have been chosen to highlight the possibilities and limitations offered
by these different formulations.
The second part of the paper presents techniques that can be used to deal with stresses and
deformations induced during drying by shrinkage. Starting with very simple configurations used
to explain shrinkage, shrinkage induced stress and the memory behaviour of wood, the whole
development of drying stress can be reconstructed from the evolution of the moisture content
profile. For interested readers, the mathematical formulation of these mechanisms is presented
thereafter.
The main objective of this paper is to help the reader with the often difficult decision of:
- the number of independent variables to be used in the physical formulation,
- the number of space dimensions to be treated in the numerical solution,
- the most consistent way to deal with drying stresses and deformations.
Obviously, no single answer exists: each solution is only a compromise between requirements and
possibilities that must account for the refinement expected for the process description, the
knowledge of the product properties and the numerical expertise of the team in charge of the
project.
1. INTRODUCTION
The simulation of wood drying has been the subject of numerous works, including the physical and
mechanical formulation, analytical and numerical solutions, physical and mechanical characterisation
and process experiments carried out in both the laboratory and industry.
However, in spite of these numerous works, the wood drying process still remains more or less based
on empirical knowledge. Numerical codes have only recently been used in the improvement of the
COST Action E 15
ADVANCES IN DRYING OF WOOD (1999-2003)
st
1 Workshop “State of the art for Kiln drying” in Edingurgh 13./14th, Oct 1999
industrial process and as yet still are not used effectively for process control. The main reasons for this
situation lie in the dramatic complexity of wood drying:
- Coupled heat, mass and momentum transfer occur during drying,
- Stresses and deformation due to shrinkage involve a complex mechanical behaviour, with an
important memory effect,
- The product to be dried is not only complex, but presents a huge variability (between and
within species, within trees -sapwood/heartwood, flatsawn/quatersawn-, within boards knots, annual rings, fibre angle, reaction wood…),
- In addition, the control of the industrial process must account for the kiln complexity (the air
flow around the load and within the stack, with important variations in space and time).
Nowadays, thanks to the increasing power of computers, the numerical simulation becomes a very
interesting tool that is able to cope with this tricky situation. However, the complexity of a numerical
model must remain consistent with a global project of wood drying improvement or kiln control. In
particular, the final choice must account for:
- the aim of a numerical model : What is the refinement expected for the process description ?
- the knowledge of the product properties : Are all required values available ? Are you able to
perform or propose measurements for the unknown values?
- the possibilities of software development : What is the numerical expertise of the team in
charge of the project ?
Furthermore, numerous other possibilities and options exist in the development of a computer code for
wood drying simulation, including:
- the number of independent variables to be used in the physical formulation,
- the number of space dimensions to be treated in the numerical solution,
- the most consistent way to deal with drying stresses and deformations.
This paper proposes some guidelines to help the reader with these thought processes.
The first part of this paper is devoted to the coupled heat and mass transfer phenomena that occur
during drying. Simple diagrams are used to outline the important physical mechanisms and related
couplings. Then, the most common formulations encountered in the literature are presented and
compared using the results of several numerical simulations. The conclusions of this part are
summarised in a table that highlights the possibilities offered by a code as a function of the number of
independent variables used in the physical formulation and the number of space dimensions computed
in the simulation.
The second part of the paper is devoted to the development of drying stresses and drying
deformations. Firstly, in a similar fashion to the presentation of the heat and mass transfer processes,
each single basic phenomenon is explained. Once put together, they allow the complete stress
evolution to be reconstructed as drying progresses. Then, the general mechanical formulation is
presented. The set of equations to be used is quite well defined for this mechanical part and no detailed
discussion has to be undertaken in this case.
COST Action E 15
ADVANCES IN DRYING OF WOOD (1999-2003)
st
1 Workshop “State of the art for Kiln drying” in Edingurgh 13./14th, Oct 1999
Finally, the reader should be aware that this text is not exhaustive. It is only focused on numerical
solutions of the drying process. In particular, neither analytical solutions (Crank 1975, Perré et al.
1999), nor global approaches (i.e. dimensionless drying curves proposed by Van Meel, 1958) are
presented, even if both can be useful in addressing specific problems.
Most of the diagrams and explanations used to explain the physical and mechanical mechanisms
(paragraphs 2.1, 2.2, 3.1 and 3.2) come from a tutorial presented at the 5th IUFRO Wood Drying
conference (Perré 1996).
2. HEAT AND MASS TRANSFER DURING DRYING
2.1.
Low temperature convective drying
Low temperature convective drying is the most widespread industrial process, i.e. conventional kiln
drying. In this case, the role of internal gaseous pressure is almost negligible and transfer occurs mainly
in the direction of the board thickness. Concerning drying mechanisms, we have to distinguish two
periods:
a) The constant drying rate period (Fig. 1)
This stage is very common for some porous media, especially for example with aerated concrete, but
is more difficult to observe for wood. However, it exists almost always for fresh boards consisting of
sapwood that are dried in moderate conditions (Perré et al. 1993, Perré and Martin 1994). During
this period, the exchange surface of the board is still above the fibre saturation point (FSP). As a
result, the vapour pressure at the surface is equal to the saturated vapour pressure, and is a function of
the surface temperature only.
Crossed heat and vapour transfer occurs in the boundary layer. The heat flux supplied by the air flow
is used solely for transforming the liquid water into vapour. During this stage, the drying rate is constant
and depends only on the external conditions (temperature, relative humidity, velocity and flow
configuration). The temperature at the surface is equal to the wet bulb temperature. Moreover,
because no energy transfer occurs within the medium during this period, the whole temperature of the
board remains at the wet bulb temperature.
COST Action E 15
ADVANCES IN DRYING OF WOOD (1999-2003)
st
1 Workshop “State of the art for Kiln drying” in Edingurgh 13./14th, Oct 1999
Boundary layers
T
Pv
EXTERNAL FLOW
VAPOUR
HEAT
Capillary migration
low MC
=
small radius
LIQUID FLOW
WOOD
high MC
=
large radius
Fig. 1 - Constant drying rate period : the moisture migrates inside the medium mostly by
capillary forces, evaporation occurs at the exchange surface with a dynamical
equilibrium within the boundary layers between the heat and the vapour flows.
The exchange surface is supplied with liquid water coming from the inside of the board by capillary
action : the liquid migrates from regions with high moisture content (liquid-gas interfaces within large
pores) towards regions with low moisture content (liquid-gas interfaces within small pores).
The constant drying rate period lasts as long as the surface is supplied with liquid. Its duration depends
strongly on the drying conditions (magnitude of the external flux) and on the medium properties. The
liquid flow inside the medium is expressed by Darcy's law (permeability ? gradient of liquid pressure).
When the moisture content decreases, the capillary forces usually increases but the permeability to the
liquid phase decreases dramatically. On the whole, the liquid flow tends to decrease for a same
gradient of moisture content. Consequently, the moisture content profile becomes steeper and steeper
as the drying progresses, and continues to do so until the liquid flow directed towards the surface
ceases. This is the end of the first drying period.
b) The decreasing drying rate period (Fig. 2)
Once the surface attained the hygroscopic range, the vapour pressure becomes smaller than the
saturated vapour pressure. Consequently, the external vapour flux is reduced and the heat flux supplied
to the medium is temporarily greater than what is necessary for liquid evaporation. The energy in
excess is used to heat the board, the surface at first, then the inner part by conduction. A new, more
subtle, dynamical equilibrium takes place. The surface vapour pressure, hence the external vapour
flow, depends on both temperature and moisture content. In order to respect the energy balance, the
COST Action E 15
ADVANCES IN DRYING OF WOOD (1999-2003)
st
1 Workshop “State of the art for Kiln drying” in Edingurgh 13./14th, Oct 1999
surface temperature increases as the surface moisture content decreases. This leads to a decreasing
drying rate (the heat supplied by the air flow becomes smaller and smaller).
Diffusion of vapour
and bound water
EXTERNAL FLOW
few vapour
molecules
VAPOUR
HEAT
many vapour
molecules
Capillary migration
BOUND WATER
AND VAPOUR
low MC
=
small radius
LIQUID FLOW
high MC
=
large radius
Fig. 2 - Second drying period : A region in the hygroscopic range develops from the exchange
surface. In that region, both vapour diffusion and bound water duffusion act.
Evaporation takes place partly inside the medium. Consequently, a heat flux has to be
driven towards the inner part of the board by conduction.
A two zones process develops inside the porous medium: an inner zone where liquid migration prevails
and a surface zone, less efficient, where both bound water and water vapour diffusion take place.
During this period, a conductive heat flux must exist inside the board to increase the temperature and
to evaporate the liquid driven by gaseous diffusion.
The region of liquid migration naturally reduces as the drying progresses and finally disappears. The
process is finished when the temperature and the moisture content attain, respectively, the outside air
temperature and the equilibrium moisture content.
2.2.
The effect of internal pressure on mass transfer
In order to reduce the drying time without decreasing the quality of the dried product, the drying
conditions must be so that the temperature of the product is above the boiling point of water. Such
conditions ensure that an overpressure exists within the material, which implies that a pressure gradient
drives the moisture (liquid and/or vapour) towards the exchange surfaces (Lowery 1979, Kamke and
Casey 1988).
COST Action E 15
ADVANCES IN DRYING OF WOOD (1999-2003)
st
1 Workshop “State of the art for Kiln drying” in Edingurgh 13./14th, Oct 1999
Saturated vapour
Atmospheric pressure
Pressure (kPa)
100
Boiling temperature
50
External pressure
0
0
20
40
60
80
100
120
Temperature (°C)
Fig. 3 - Vacuum drying seeks to reduce the boiling point of water in order to obtain a high
temperature configuration with moderate drying conditions.
At the atmospheric pressure, the boiling point of water equals 100°C. Consequently, in order to obtain
internal overpressure, the temperature of the porous medium must be above that level during at least
one part of the process. This is exactly the aim of convective drying at high temperature (moist air or
superheated steam) and a possible aim of contact drying or drying with electromagnetic field
(microwave or high frequency).
However, as shown in figure 3, it is possible to reduce the boiling point of water by decreasing the
external pressure and, consequently, to obtain a high temperature configuration with relatively
moderate drying conditions. This is the principle of vacuum drying, particularly interesting for materials
that would be damaged by high levels of temperature.
When overpressure exists inside a board, the high anisotropy ratios imply bi-dimensional transfers :
heat is often supplied in the thickness direction, while, in spite of the length, the effect of the pressure
gradient on gaseous (important for low MC) or liquid migration (important for high MC) takes place in
the longitudinal direction (Fig. 4). This is obviously a result of the anatomical features of wood. In the
case of very intensive internal transfer, the endpiece can be fully saturated, and sometimes, moisture
can leave the sample in the liquid state (this is quite easy to observe during microwave heating).
COST Action E 15
ADVANCES IN DRYING OF WOOD (1999-2003)
st
1 Workshop “State of the art for Kiln drying” in Edingurgh 13./14th, Oct 1999
HEAT
Vessel or
tracheid
VAPOUR
LIQUID
Pits
Endpiece
fully saturated
OVERPRESSURE
Liquid evacuation
possible in
microwave heating
Fig. 4 - Drying at high temperature (second drying period) : a high temperature configuration
means that an overpressure develops inside the medium. Depending on the MC, this
overpressure induces liquid and/or gaseous flow; In addition, wood being strongly
anisotropic, the most part of the flow occurs in the longitudinal direction (see the
magnifyed views).
2.3.
Physical formulation
Several sets of macroscopic equations are proposed in the literature for the simulation of the drying
process. The first fundamental difference between them lies in the number of state variables used to
describe the medium:
? one : moisture content (or an equivalent variable: saturation, water potential...),
? two : MC (or equ.) and temperature T (or an equivalent variable : enthalpy...),
? three : MC (or equ.), T (or equ.) and gaseous pressure Pg (or an equivalent variable : air
density, intrinsic air density...).
The three corresponding sets of equations are presented hereafter, starting with the most
comprehensive macroscopic formulation (Model 1, three state variables). At present, researchers
using three state variables agree with the macroscopic formulation to be used. The set of equations, as
proposed below, originates for the most part from Whitaker's works (Whitaker 1977, 1980) with
minor changes required to account for wood properties and drying with internal overpressure (Perré
and Degiovanni 1990). In particular, the reader must be aware that all variables are averaged over the
REV (Representative Elementary Volume), hence the expression "macroscopic". This assumes the
existence of such a representative volume, large enough for the averaged quantities to be defined and
small enough to avoid variations due to macroscopic gradients and non-equilibrium configurations at
COST Action E 15
ADVANCES IN DRYING OF WOOD (1999-2003)
st
1 Workshop “State of the art for Kiln drying” in Edingurgh 13./14th, Oct 1999
the microscopic level. This is why, although very powerful for a lot of different configurations, this set
of equations has some limitations (Perré 1998).
Model 1 (comprehensive)
Water Conservation
(1.1)
Energy Conservation
(1.2)
Air Conservation
(1.3)
where the gas and liquid phase velocities are given by the Generalised Darcy’s Law:
(1.4)
the quantities ? ?are known as the phase potentials and ? is the depth scalar. All other symbols have
their usual meaning.
Boundary conditions
For the external drying surfaces of the sample, the boundary conditions are assumed to be of the
following form :
(1.5)
COST Action E 15
ADVANCES IN DRYING OF WOOD (1999-2003)
st
1 Workshop “State of the art for Kiln drying” in Edingurgh 13./14th, Oct 1999
where J w and J e represent the fluxes of total moisture and total enthalpy at the boundary
respectively, x denotes the position from the boundary along the external unit normal.
COST Action E 15
ADVANCES IN DRYING OF WOOD (1999-2003)
st
1 Workshop “State of the art for Kiln drying” in Edingurgh 13./14th, Oct 1999
A more detailed description of these equations and related assumptions can be found elsewhere (Perré
1996, Turner and Perré 1996, Perré and Turner 1999). Because this formulation takes care of the
internal pressure through the air balance (equation 1.3), the set of equations proved to be very
powerful, able to deal with numerous configurations involving intense transfers: high temperature
convective drying, vacuum drying, micro-wave drying,…
Using Model 2, the porous medium is described with only two independent variables (temperature or
equivalent and moisture content or equivalent). This model can be analysed as a simplification of model
1. No account is taken of the internal gaseous pressure. Equation 1.5 vanishes. Darcy's law (equation
1.4) subsists only for the liquid phase, in which only capillary forces act. Assuming the capillary
pressure to be function on moisture content only (instead of moisture and temperature), both vapour
and liquid migration can be expressed as diffusion mechanisms, with global "pseudo-" diffusion
coefficients K v and K ?. This set is obviously unable to deal with the effect of internal pressure on
liquid and gaseous migration, hence not suitable for drying configurations involving intense migration.
However, through the difference of enthalpy between vapour and liquid, it accounts for the latent heat
of vaporisation. By this way, it is able to deal with the most important feature of drying : the
coupling between heat and mass transfer (this is due to the latent heat of vaporisation and the relation
between temperature and saturated vapour pressure).
Model 2 (heat and mass transfer)
Water Conservation
(2.1)
Energy Conservation
(2.2)
where
Boundary conditions
(2.3)
From model 2, it is still possible to imagine something simpler: just forget about temperature, and
equation 2.2 is not longer necessary. By doing this, you assume in fact that the product temperature
COST Action E 15
ADVANCES IN DRYING OF WOOD (1999-2003)
st
1 Workshop “State of the art for Kiln drying” in Edingurgh 13./14th, Oct 1999
immediately follows the air flow temperature. It is no more necessary to differentiate liquid flow from
vapour flow. The use of one global diffusion coefficient leads to Model 3.
COST Action E 15
ADVANCES IN DRYING OF WOOD (1999-2003)
st
1 Workshop “State of the art for Kiln drying” in Edingurgh 13./14th, Oct 1999
Model 3 (mass transfer only)
Water Conservation
(3.1)
Boundary conditions
(3.2)
If we remember that the objective of drying is to remove the moisture from the product, the concept of
a "one equation" model sounds relevant: it accounts for the moisture migration within the medium and
the moisture flux at the boundary. This simplistic arguing encouraged too many scientists to use model
3.
160
60
120
50
Temperature
80
40
T emperature (°C)
Moisture co ntent (kg/kg)
One equation (mass)
One equation (hm divided by 10)
Two equations (heat and mass)
Averaged moisture content
40
0
0
30
20
40
60
80
20
100
Time (hours)
Fig. 5 - Convective drying at low temperature (Dry bulb = 50°C, dew point = 30°C, board
thickness = 30 mm, h = 14 W.m-2.°C -1; hm = 0.014 m.s-1) : Model 2 versus model 3.
Figure 5 depicts 1-D computational results obtained for convective drying at low temperature (Dry
bulb = 50°C, dew point = 30°C, board thickness = 30 mm). Model 2 (two equations) exhibits a
COST Action E 15
ADVANCES IN DRYING OF WOOD (1999-2003)
st
1 Workshop “State of the art for Kiln drying” in Edingurgh 13./14th, Oct 1999
constant drying rate period, during which the board temperature equals the wet bulb temperature.
Model 3 (one equation) is not capable of capturing the reduction of vapour pressure at the surface,
which results from the surface cooling due to evaporation. Consequently, the drying process is much
faster than it should be when using Model 3 with reasonable physical values. This error can be
corrected, at least the evolution of the averaged moisture content, by fitting the mass exchange
coefficient hm (see figure 5, curve obtained with hm divided by a factor 10!). However, this trick
provides the mass transfer coefficient to keep its physical meaning, and, more crucial, a new value has
to be fitted each time the drying conditions change. For example, the factor 10, which allowed a good
drying curve to be calculated at 50°C, is no more valid for a dry bulb temperature of 70°C (Fig. 6). In
conclusion, because one variable prevents the strong coupling that always exists between heat and
mass transfer within the medium to be captured, Model 3 is a crude choice that must absolutely
be discarded, even for very simple configurations.
160
80
60
One equation (hm divided by 10)
Two equations (heat and mass)
80
40
40
20
0
0
10
20
30
40
50
T emperature (°C)
Moisture co ntent (%)
120
0
60
Time (hours)
Fig. 6 - Convective drying at low temperature (Dry bulb = 70°C, dew point = 30°C, board
thickness = 30 mm) : failure of model 3.
Model 2 (two equations) is much more reasonable and allows numerous configurations to be
simulated, including conventional kiln drying. This choice is the correct one to start with drying
simulation or to provide on-line help in kiln control. Nevertheless, the users of such a model must be
aware of its limitations. From figure 4, it is easy to understand that this formulation fails as soon as the
internal pressure has a significant effect on transfer. There is no way to simulate high temperature
drying, vacuum drying, microwave heating, or, simply, full saturation of a wood sample by cycles of
vacuum and pressure!
COST Action E 15
ADVANCES IN DRYING OF WOOD (1999-2003)
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1 Workshop “State of the art for Kiln drying” in Edingurgh 13./14th, Oct 1999
From the same figure 4, the need for the longitudinal direction to be computed in such cases becomes
obvious. Numerous configurations have been computed with success using Model 1 (comprehensive
set) with two space dimensions (length and thickness of the board). Several examples, compared to
the corresponding experimental results, including temperature and pressure measurements within the
board can be found in the literature (Stanish et al. 1986, Constant et al. 1996, Perré 1996, Perré and
Turner 1996, Johansson et al. 1997). One example of high temperature convective drying (Tdry =
140°C; Tdew = 85°C) has been chosen to describe the possibilities of Model 1 with two space
dimensions. This example depicts all the trends observed during drying of wood when the internal
pressure is important. At the end of the constant drying rate period (Fig. 7, around 3 hours), the
surface temperature increases first, then the centre temperature. Once the boiling point of water is
attained, an overpressure develops within the board. Because the longitudinal permeability of wood is
much higher than the transverse permeability (by a factor 1000 in this simulation), this overpressure
gives rise to an important longitudinal moisture migration towards the end piece. The latter is resaturated and remains fully saturated for a long period (3.5 to 7.5 hours). Typical 2-D transfers occur:
heat is supplied to the product mainly through the thickness and moisture partly moves along the
longitudinal direction (Fig. 8). As a consequence of this moisture migration, the endpiece temperature
remains close to the wet bulb temperature for a corresponding duration. This effect has often been
observed during the experiment using thermocouples (Perré et al. 1993). During the last stage of
drying, the overpressure remains important with less or no liquid water inside the product. However,
longitudinal migration of moisture is still important through the high percentage of vapour that exists in
the gaseous flux.
1m
end
30 mm
centre
surface
COST Action E 15
ADVANCES IN DRYING OF WOOD (1999-2003)
st
1 Workshop “State of the art for Kiln drying” in Edingurgh 13./14th, Oct 1999
160
2.0
Tsurf
120
Tend
T centre
80
1.5
MC average
MC surf
MC centre
Pcentre
40
0
0
4
8
12
16
Internal pressure (P/P atm )
Temperature (°C) or MC (kg/kg)
MCend
1.0
20
Time (hours)
Fig. 7 - Convective drying at high temperature (Dry bulb = 140°C, dew point = 85°C, board
thickness = 30 mm, borad length 1 m) : model 1, two space dimensions.
The simulation proposed in figures 7 and 8 concerns sapwood (high initial moisture content, high values
of permeability). When observing a stack of boards submitted to high temperature conditions, resaturation of the end piece occurs only for sapwood boards, or for sapwood parts of boards (Fig. 9).
Because we use a material model, based on a comprehensive physical formulation, the differences
between sapwood and heartwood can be simulated by changing material properties (Perré and Martin
1994). These differences lie in only two sets of parameters : the permeabilities and the initial moisture
content. The values of initial moisture content come measurements (150% to 200 % for sapwood,
40% to 80% for heartwood). Concerning the permeabilities, the values used to differentiate sapwood
from heartwood are based on considerations concerning pit aspiration.
COST Action E 15
ADVANCES IN DRYING OF WOOD (1999-2003)
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1 Workshop “State of the art for Kiln drying” in Edingurgh 13./14th, Oct 1999
MC
50
40
1.0
30
Le n
gth
(
cm )
0.5
20
10
0.0
2
1
0 0
W
h
idt
)
( cm
50
12 5
30
Temp
15 0
40
10 0
Le n
gth
(cm
)
20
75
10
50
1
2
0 0
W
h(
idt
)
cm
40
1.8
1.6
30
Le n
gth
(
Pres
50
1.4
cm )
20
1.2
10
1.0
2
1
0 0
W
th
id
)
( cm
Figure 8 - High temperature drying (140°C/85°C) of sapwood. Carpet plot at 5 hours of drying.
Model 1 (Comprehensive physical formulation), two space dimensions. Internal
overpressure, re-saturation of the endpiece, thermal conduction long the thickness and
endpiece close to the wet bulb temperature are evident on these plots.
COST Action E 15
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MC
50
40
1.0
30
Le n
gth
(
cm )
0.5
20
10
0.0
2
1
0 0
W
h
idt
)
( cm
50
12 5
30
Temp
15 0
40
10 0
Le n
gth
(cm
)
20
75
10
50
1
2
0 0
W
h(
idt
)
cm
40
1.8
1.6
30
Le n
gth
(
Pres
50
1.4
cm )
20
1.2
10
1.0
2
1
0 0
W
th
id
)
( cm
Figure 10 - High temperature drying (140°C/85°C) of heartwood. Carpet plot at 5 hours of
drying. Model 1 (Comprehensive physical formulation), two space dimensions. Note the
high value of internal pressure and the absence of endpiece re-saturation
Putting these new values in the model, all trends generally observed (Salin 1989, Pang et al. 1994,
Perré and Martin, 1994) can be computed. In particular, no re-saturation of the endpiece is observed
(Fig. 10). In addition, the constant drying rate period disappears. The overpressure develops from the
COST Action E 15
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1 Workshop “State of the art for Kiln drying” in Edingurgh 13./14th, Oct 1999
beginning of the process and remains high up to the end of drying. The maximum pressure is higher for
heartwood than for sapwood.
Sapwood
Heartwood
Fig.9 - A stack of boards after some hours of drying at high temperature.
Several configurations, always in good agreement with experimental data, have been computed using
this model (Model 1, two space dimensions ): high temperature convective drying (superheated
steam and moist air), microwave heating, vacuum drying… In this case of high temperature
configurations, the needs for two dimensions and for a comprehensive set of equations can be proved
by two additional simulations.
160
3.0
120
2.5
T centre
Pcentre
MC centre
80
2.0
MC average
MC surf
40
1.5
0
0
4
8
12
Time (hours)
16
1.0
20
Internal pressure (P/P atm )
Temperature (°C) or MC (kg/kg)
Tsurf
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Fig. 11 - Convective drying at high temperature (Dry bulb = 140°C, dew point = 85°C, board
thickness = 30 mm, borad length 1 m) : Model 1, one space dimension.
Using the same formulation (Model 1) in one space dimension (thickness only), provides the
longitudinal mass transfer to be computed (Fig. 11). The internal overpressure develops without
important effect on mass transfer (the transverse permeability is too small for a significant flux to be
obtained). Both internal temperature and internal pressure increases faster than for a 2-D simulation,
but the global drying curve is much slower.
When using Model 2 the internal pressure is supposed to have the same value as the external
pressure. Consequently, no mass transfer can be induced by internal vaporisation. Even if supplied by
capillary action, which explains the plateau near the wet bulb temperature, the endpiece moisture
content continuously decreases through the process (Fig. 12). For lack of important longitudinal mass
transfer, the drying curve is much slower than the one depicted in figure 7.
160
2.0
T surf
120
T end
80
1.5
MC centre
MC surf
MCaverage
40
Internal pressure (P/P atm )
Temperature (°C) or MC (kg/kg)
Tcentre
MC end
Pcentre
0
0
4
8
12
16
1.0
20
Time (hours)
Fig. 12 - Convective drying at high temperature (Dry bulb = 140°C, dew point = 85°C, board
thickness = 30 mm, borad length 1 m) : model 2, two space dimensions.
More recently, the same physical formulation has been computed in three dimensions (Model 1, three
space dimensions ). This allowed new mechanisms to be exhibited (Perré and Turner 1999). This
version leads to a comprehensive description of both physics and geometry. Such a model has
probably a great potential in the future, provided the computational time is dramatically reduced.
As a conclusion of this part devoted to transfer mechanisms involved in wood drying, table 1
summarised the main conclusions that can be drawn from the simulations proposed above and from the
literature. Boxes of this table are intercept between the number of independent variables (model 3 to
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1) and the number of space dimensions. The bold borders denote consistent choices, each adapted to
specific problems.
Finally, one must keep in mind that the computational effort dramatically changes according to these
choices. Table 2 gives an order of magnitude of computational time required for a complete drying
simulation on a classical Personal Computer. These values will decrease with the increase of computer
power. Nevertheless, because all equations are strongly coupled and highly non-linear, the use of nonefficient computational strategies can lead to computational times orders of magnitude (a factor 10,
possibly 100) higher than the values reported in table 2.
Table 1 - Heat and mass transfer during wood drying: simple guideline to help the choice
of a physical formulation and geometrical description.
Independent
variables
Space dimension
1-D
(Thickness)
Model 3
One variable
(MC)
Model 2
Two variables
(MC + T)
Model 1
Three variables
(MC + T + Pg)
Simple but very poor
physics (no coupling
between heat and mass
transfer) hence
misleading results (1)
Fast and easy
Comprehensive physics,
but unrealistic effect of
the internal pressure (no
longitudinal direction)
misleading results at
high temperature
Suitable for coarse
description of
convective drying at low
temperature
2-D
(Thickness + Width)
Correct description of
the section shape but
very poor physics (no
coupling between heat
and mass transfer)
hence misleading
results (1)
Correct description of
the section shape.
Suitable for convective
drying at low
temperature
Comprehensive physics,
but unrealistic effect of
the internal pressure (no
longitudinal direction)
misleading results at
high temperature
2-D
(Thickness + Length)
Useless additional work
compared to 1-D Very
poor physics (no
coupling between heat
and mass transfer) and
misleading results
Useless additional work
compared to 1-D
Comprehensive physics
and good geometrical
description. Suitable for
almost all drying
processes (effect of
internal pressure)
3-D
Silly choice : complex
numerical calculation
for very poor physics
and misleading results
Good geometrical
description. Can be
interesting to calculate
the global deformation
of entire boards
The best possible
description at the
macroscopic level
(geometry and physics)
(1) : Suitable only for diffusion in the hygroscopic range of very thick boards.
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Table 2 - Order of magnitude of the computer time required for a complete drying
simulation (Personal computer, Pentium II 400 MHz, up-to-date numerical
techniques)
Independent
variables
Model 3
One variable
(MC)
Model 2
Two variables
(MC + T)
Model 1
Three variables
(MC + T + Pg)
1-D
Less than one second
Around one second
Some seconds
2-D
Less than 30 seconds
Around one minute
Some minutes
3-D
Less than 30 minutes
Around one hour
Some hours
Space dimension
3. MECHANICAL PHENOMENA RELATED TO DRYING
3.1.
Mechanical behaviour of wood
Wood drying does not only consist in removing its moisture: the quality of the dried product is the main
requirement for the industrial process. Because wood shrinks during drying, deformations and stresses
develops than can lead to unusable products (Fig. 13). The simulation of these aspects must account
for the complex mechanical behaviour of wood, including its memory effect. Section 3.1 is devoted to
a simple presentation of this behaviour.
Figure 13 - Two exemples of mechanical degrade during wood drying : board deformation and
internal checking.
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Green
wood
FSP
Length
Ovendried
MC
0
FSP
Fig. 14 - Wood shrinkage. The FSP, usually close to 30% depends on the specie and the
temperature.
Firstly, we must refer to shrinkage which is the "driving" force for drying stress: without shrinkage, no
drying stress would develop ! Figure 14 exhibits the dimension variation of a sample free of load
versus the moisture content (the latter is assumed to be uniform). Under normal conditions, the
dimensions do not change until the moisture content attains the FSP. Then, the dimension variations are
almost proportional to the change of MC. This strain field is called free shrinkage.
A sample subjected to tensile or compressive stress (Fig. 15), exhibits at first the instantaneous
deformation (elastic part), which then increases in time (creep). When submitting the same specimen to
moisture content changes, the deformation increases much more quickly (mechano-sorptive effect).
MC
Time t=0 +
Constant
MC
LOAD
Time T
After cycles
of MC
LOAD
LOAD
Time
Load
Time
Length
variation
mechano-sorptive
viscoelastic
Time
Fig. 15 - Viscoelastic and mechano-sorptive behaviour of wood.
Applying the viscoelastic and mechano-sorptive creep as shown in figure 15, it is possible to predict
how the length of samples submitted to tensile or compressive load will evolve during drying (Fig. 16).
Indeed, a loaded sample undergoes viscoelastic creep because of the time required for drying, and
mechano-sorptive creep because of, obviously, the MC decreases from the initial value to the
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equilibrium. Therefore, a sample submitted to compressive stress (n°1) exhibits, at the end of drying, a
smaller length than a same sample free of load (n°2). This is the opposite for a sample under tension
(n°3).
Time t=0 +
High MC
Time t
Low MC
LOAD
1
2
3
Drying
1
2
3
LOAD
Fig. 16 - Dimension changes of a specimen loaded during drying.
We have now sufficient knowledge to come back to our original concern: the development of drying
stresses. Let us assume that a board is exposed to an air flow. At the beginning of drying (constant
drying rate period), the entire board remains in the domain of free water. No shrinkage occurs, hence
stress build up is absent (the initial stress field, which can exist at the beginning, mainly growth stresses,
is supposed to be negligible throughout the following).
Second drying period
a
b
c
tensile
stress
PSF
compressive
stress
Fig. 17 - Appearance of drying stresses following upon shrinkage.
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End of drying
a
b
c
tensile
stress
PSF
Xeq
compressive
stress
Fig. 18 - Stress reversal due to the memory effect of wood.
At the beginning of the second drying period, shrinkage exists close to the exchange surfaces (Fig.
17a). At this moment, if we cut the section into slices, the external slices would have a shorter length
than the inner ones (Fig. 17b). This displacement field is not compatible and induces, in the actual
section, tensile stress in the surface layers and (because of equilibrium conditions) compressive stress
in the core layers (Fig. 17c). The section length is a weighted average of each slice length (the elastic
properties depend on moisture content). During this period, surface checking is possible. Note that,
from this point onwards, the wood layers dry under load.
As the drying proceeds, viscoelastic creep develops, together with mechano-sorptive creep. External
slices appear similar in configuration to that exhibited for slice n°3 in figure 16, while the internal slices
resemble slice n°1. Consequently, in spite of the flat moisture content profile, slicing the section at the
end of the drying would give picture b in figure 18: the core slices, dried under compression, are
smaller than the external ones, dried under tension. In the actual section, compressive stress exists in
the inner part (Fig. 18c). This phenomenon is known as the so-called stress reversal or casehardening.
The residual stress level depends on many parameters (drying conditions, species, sawing pattern,
thickness) which represents most of the problem of drying optimisation. In addition, we must keep in
mind that gradients of moisture content, strain and stress exist along the thickness. This explains the
curvature of the slices observed in prong test or cup method (Fig. 19). When the inner tensile stress is
too high, internal checking occurs (Fig. 13). An interesting simulation of this test can be found in
Dahlblom et al. (1994).
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Second drying period
End of drying
Prong
test
Cup
method
Fig. 19 - Two experimental methods used to determine drying stresses.
3.2.
Drying stress formulation
During drying, shrinkage appears in all parts of the board for which the moisture content X is within the
hygroscopic range. This phenomenon induces a deformation field noted ? sh defined in the material axes
by:
(1)
If this deformation field does not fulfil the geometrical compatibility, a strain tensor ? mec related to
stresses is generated. The constitutive equation, which represents the mechanical behaviour of the
material, relates this strain tensor ? mec and the stress tensor. Due to the memory effect of wood, this
tensor ? mec has to be divided into two parts : an elastic strain ? elas , connected to the actual stress tensor
and a memory strain ? mem which includes all the strain due to the history of that point (? mem can deal
with plasticity, creep, mechano-sorption... ):
(2)
The geometrical compatibility applies to the total strain field ? tot. When solving the mechanical problem
in terms of displacement, the total strain tensor is deduced from the displacement field and this
geometrical condition is automatically fulfilled within the domain. The stress field must satisfy the local
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mechanical equilibrium and the boundary conditions. Finally, the complete formulation of the problem
is given by :
(3)
Remarks:
• this static formulation is usable only for boundary and volumetric forces satisfying the global
equilibrium.
• in order to ensure the uniqueness of the solution, additional conditions are required on the
boundary conditions (
). Otherwise, the solution is defined within a rigid body
motion.
• wood being orthorhombic, each behaviour law involves nine independent terms. In fact, it is
more common to define the inverse of aijkl which, for the case of linear elasticity, leads to the
so-called generalised Hooke's law (Guitard 1984).
? How to express the memory effect?
In the memory strain field ? mem lies the entire problem of constitutive model for wood. Several works,
both theoretical and numerical proposed solutions. They can be very comprehensive, but at the same
time very complicated and, consequently, very difficult to characterised (Ranta-Maunus 1975). In fact,
wood exhibits so particular behaviour that each new experiment can lead to another expression. The
problem lies in the fact that the memory effect of wood depends not only on the temperature and
moisture content values, but also on their variations in time and on the history of their variations in time.
Nevertheless, in the case of drying, the moisture content only decreases and some simplifications
apply. Here, only the most common way to express creep and mechano-sorptive effect will be
presented. The general formulation of the time dependency of the creep property involves the whole
stress history:
(4)
Jijkl(t) is the creep compliance tensor and t the actual time. The experimental creep function is often
analysed as N Kelvin elements in series (Dahlblom 1987, Genevaux 1989, Martensson 1992,
Mohager and Toratti 1993). In the case of uniaxial load, this leads to:
(5)
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The temperature and moisture dependency of that function can be expressed using a material time or
changing the characteristic time ?n. The thermal activation, for example, is often expressed with the aid
of an Arrhenius law :
(6)
? Wn is the activation energy associated to element n.
The mechano-sorptive effect occurs as soon as the moisture content changes during load. The simpler
way to express this effect consists in assuming that the strain rate depends linearly on both the stress
field and the time derivative of the moisture content:
(7)
Remark : the mechano-sorptive strain rate is always in the direction of the stress field, hence
the factor sign(u).
A tri-dimensional resolution is very costly in terms of calculation time and memory space. The need for
such a cost is justified when the objective of the calculation lies in the global deformation of the board.
By this way, the effect of reaction wood, fibre angle, properties variations can be analysed. Nice
examples of these possibilities can be found in the literature (Dahlblom et al. 1994, 1996, Ormarsson
1999). Nevertheless, in order to study the stress development within a section far from the ends of the
board, a 2-D simulation is sufficient. A "plane displacement" formulation has to be used in this case
(Perré and Passard, 1995).
3.3.
Some simulation examples
The results presented in this section have been computed with a code able to solve the heat and mass
transfer and the mechanical phenomena occurring during drying. The mechanical part is described in
Perré and Passard (1995). In order to point out the possibilities offered by the complete simulation of
transfer and mechanical aspects of wood drying, two different configurations have been chosen :
conventional kiln drying and non-symmetrical convective drying.
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Fig. 19 – Board deformation during non-symmetrical drying. A complete code is used, that
accounts for both heat and mass transfer and mechanical behaviour following shrinkage.
For this configuration, a large displacement formulation is used.
In the case of non-symmetrical convective drying, one face of the board is impervious. This
configuration allows one part of the drying stresses to be transformed into a global deformation of the
sample section (Brandão and Perré 1996). Figure 19 depicts one example of deformed section during
non-symmetrical drying. In this case, the board section is described by a triangular finite element code,
which is used for mechanical calculations. The heat and mass transfer equations are solved using
control volumes built from the finite element mesh.
20
1.2
Surface
5 mm
7.5 mm
Stress (MPa)
10
2.5 mm
1.0
Mid-thickness
5
0.8
0
0.6
Moisture content
-5
0.4
-10
-15
0
0.2
10
20
30
Time (hours)
40
0
50
Moisture content (kg/kg)
15
1.4
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Fig. 20 - Low temperature convective drying. Stress level at different positions from the surface
and moisture content versus time. Case of symmetrical drying conditions.
In order to simplify the graphs, one-dimensional results will be presented in the following (Fig. 20 and
21). Using these moderate drying conditions (dry bulb: 80°C, dew point: 50°C) and with a high initial
moisture content (100%), a constant drying rate period exists. The entire board remains in the domain
of free water during this period: no shrinkage, hence no stress appears during the first drying period.
Then, in the case of symmetrical drying, the section remains flat and the stress calculated at different
level is quite easy to understand: tension in the surface layers due to shrinkage and, because of
momentum balance, compressive stress in the inner part at the beginning of the second drying period.
As the process progresses, the stress reversal can be observed.
When only one face dries, the curve shapes are quite confuse and the understanding is almost
impossible without a detailed analysis of the calculated results. However, the section curvature depicts
an evolution similar to what can be measured (in particular, the negative curvature at the end of the
process).
Note that almost the same trend could be obtained using any memory effect: creep (Kawai 1984),
mechano-sorptive (Perré and Passard 1995) or both (Mauget 1996). As previously stated, wood
exhibits a very complex constitutive equation: several parameters can be taken into account, with
temperature and moisture dependency. The objective of a drying model is to achieve a good balance
between required knowledge and simulation possibilities. For example, in figures 22 and 23, the same
set of physical parameters have been used to simulate two different drying configurations: drying of a
loaded sample and non-symmetrical drying. The memory effect is assumed to be captured by two
constant parameters: one for the mechano-sorptive effect and the second for the viscoelastic creep.
For each single test, one parameter can be adjusted as a function of the other in order to obtain a good
agreement between experiment and simulation. However, by using both tests, only one set of
parameters works. Simulations depicted in figure 22 and 23 have been computed using this set
(Mauget 1996). This example proves that the problem of material characterisation is possible, but
certainly not simple. This constitutes the most difficult part of drying stresses prediction. In addition,
other phenomena exist that are even more complex and not really put in the existing codes: checking
and collapse.
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20
8
Stress (MPa)
10
4
2
0
0
Impervious face
-10
-4
2.5 mm
5 mm
-20
0
-2
7.5 mm
10
-1
Curvature
1/ radius of curvature (m )
6
Surface
-6
20
30
-8
50
40
Time (hours)
Fig. 21 - Low temperature convective drying. Stress level at different positions from the surface
and section curvature versus time. Case of non-symmetrical drying conditions.
80
70
Simulation
Deflection (mm)
60
Experiment
50
40
30
20
10
0
0
5
10
15
Time (hours)
Fig. 22 - Low temperature convective drying of a loaded sample. Evolution of the end
deflection versus time : experiment and simulation (Mauget 1996).
20
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0.15
0.1
Curvature (1/cm)
0.05
Simulation
0
0
2
4
6
-0.05
-0.1
8
10
Time (hours)
Experiment
-0.15
-0.2
Fig. 23 - Non-symmetrical low temperature convective drying. Evolution of the section
curvature versus time : experiment and simulation (Mauget 1996).
4. CONCLUSION AND PROSPECTS
In this paper, fundamentals of physical and mechanical aspects of wood drying have been simply
described. The comprehensive formulation able to account for these phenomena is proposed. A few
examples proved the ability of a numerical model to simulate the complete process (transfer and
mechanical aspects of wood drying). Simplified formulations and their limitations are also presented.
Concerning heat and mass transfer, three formulation are analysed :
- the simplest one (Model 3) has only one equation and one state variable. Not able to account for
the coupling between heat and mass transfers, it is misleading and must be discarded,
- the two equation model (Model 2) is suitable for most configurations of low temperature
convective drying. It can be used for drying optimisation or kiln control in one dimension (simple
and fast) or two dimensions (effect of section shape),
the comprehensive macroscopic formulation (Model 3) accounts for the internal pressure within the
product. Only this formulation can deal with processes involving internal vaporisation (high
temperature, vacuum, HF and microwave …). It can be used for 2-D models (thickness and length) or
3-D model (exhaustive geometrical description).
Note that the computation effort increases dramatically with the number of space dimensions, while it is
less sensitive to the physical formulation. Presently, in 1999, it is roughly a question of seconds in 1-D,
minutes in 2-D and hours in 3-D. This means that a 1-D model could be used for the simulation of the
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entire stack of boards put in a kiln with reasonable computer time. This is certainly a good way to
account for the biological variability of wood. In 2-D, only some boards can be computed at the same
time (flat- and quarter-sawn, sap- and heart-wood). Even if essential to model specific configurations
(microwave and HF drying, stresses within the section during drying at high temperature), the 3-D
computation will certainly stay in the laboratory for some more years.
The modelling of stress and deformation during drying is another big challenge for numerical simulation.
The formulation is well known and difficult to simplify: shrinkage, elasticity, creep and mechanosorption have to be taken into account. 1-D calculations can give an order of magnitude of stress level
during the process. 2-D (thickness and width) allows the section shape to be studied. Finally, 3-D
calculations can be justified for the analysis of global deformation of the board (reaction wood, fibre
angle, property variations).
Finally, one has to remember that any material model requires the knowledge of several properties.
The effort granted to numerical modelling must be consistent with the actual knowledge of the material
or the effort granted to experimental determination and experimental validation. In this field, discussions
on relative permeability curves or on constitutive equations are eloquent. New experimental techniques
(CT-scanning, NMR imaging…) will certainly allow significant improvements in validation and
characterisation (Davis et al. 1993, Lindgren 1992a, 1992b).
5. REFERENCES
Bird, R.B., Steward, W.E.and Lightfoot, E. N. "Transport phenomena" John Wiley and Sons, New York (1960).
Brandao A. and Perré P. "The "Flying Wood" - A quick test to characterise the drying behaviour of tropical woods"
5th International IUFRO Wood Drying Conference, 315-324, Québec (1996).
Comstock, G.L. and Cote, W.A."Factors affecting permeability and pits aspiration in coniferous sapwood" Wood
Sci. Techn.,Vol.2, 279-291 (1968).
Constant, T., Perré, P. and Moyne, C. "Drying with internal heat Generation : Theoretical Apects and Application to
Microwave Heating" AIChE Journal, 42(2), 359-368 (1996).
Crank J. The Mathematics of Diffusion, Oxford University Press (1975).
Dahlblom O., Ormarsson S. and Petersson H. "Simulation of wood deformation processes in drying and other
environmental loading" Annals of Forest Science, 53, 857-866 (1996).
Dahlblom O., Petersson H. and Ormarsson S. "Numerical simulation of the development of deformation and stress in
wood during drying" 4th IUFRO Wood Drying Symp., 165-180 Rotorua (1994).
Davis, J.R., Ilic, J., and Wells, P., "Moisture content in drying wood using direct scanning Gamma-ray densitometry"
Wood and Fiber Sci., 25(2), 153-162 (1993).
Dullien, F.A.L., "Porous media - Fluid transport and pore structure" Second Edition, Academic Press (1992).
Genevaux, J.-M., 1989, "Le fluage à température linéairement croissante : Caractérisation des sources de
viscoélasticité anisotrope du bois" Thèse de Doctorat de l'Institut Nationnal Polytechnique de Lorraine, Nancy.
Guitard, D. "Mécanique du matériau bois et composites" Cepadues Editions, Toulouse (1987).
Johansson, A., C. Fyhr, A. Rasmuson, High temperature convective drying of wood chips with air and superheated
steam, International Journal of Heat and Mass Transfer 40(12), pp. 2843-2858 (1997).
Kamke, F.A. Casey, L.J. "Gas pressure and temperature in the mat during flakeboard manufacture", Forest Products
Journal, Vol.38, n°3, pp 41-43 (1988).
Kawai S. "Moisture movement and drying stresses in wood" Doctor Dissertation, Faculty of Agriculture, Kyoto
(1980).
COST Action E 15
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Lindgren, O., "Medical CT-scanners for non-destructive wood density and moisture content measurements"
Doctoral thesis, Luleå University, Sweden (1992a).
Lindgren, O., "Non-destructive wood density distribution measurements using computed tomography" Holz als
Roh- und Werkstoff, 50, 295-299 (1992b).
Lowery, D.P."Vapor pressure generated in wood during drying", Wood Science 5, pp.73-80, (1979).
Meyer, R.W. "Influence of pit aspiration on earlywood permeability of Douglas-Fir" Wood and Fiber, Vol. 2, p. 328339 (1971).
Mauget, B. "Simulation en grands déplacements du comportement mécanique du bois - Application au séchage
dissymétrique et séchage sous charge" thèse INPL, Nancy (1996).
Martensson, A., 1992, "Mechanical behaviour of wood exposed to humidity variations" Doctoral dissertation, Lund
Institute of Technology.
Mohager, S. and Toratti, T., 1993 "Long term bending creep of wood in cyclic relative humidity" Wood Sci.
Technology, vol. 27, 49 -59.
Pang, S. Keey, R.B. and Walker, J.C.F. "Modelling of the High-Temperature Drying of Mixed Sap and Heartwood
Boards" 4th Int. Wood Drying Conference, 430-439 (1994).
Perré P. et Degiovanni A. "Simulation par volumes finis des transferts couplés en milieux poreux anisotropes:
séchage du bois à basse et à haute température" Int. J. Heat and Mass Transfer, Vol.33(11), pp.2463-2478 (1990).
Perré, P., Moser, M., Martin M., "Advances in transport phenomena during convective drying with superheated
steam or moist air" Int. J. Heat and Mass Transfer, Vol.36(11), pp.2725-2746 (1993).
Perré, P. and Martin, M. "Drying at high temperature of heartwood and sapwood : theory, experiment and practical
consequence on kiln control" Drying Technology Journal, vol.12(8), 1915-1941 (1994).
Perré P. and Passard J. " A Control-Volume procedure compared with the Finite-Element method for calculating
Stress and Strain during Wood Drying ". Drying Technology Journal, Special issue Mathematical Modelling
and Numerical Techniques For The Solution of Drying Problems, 13(3), 635-660 (1995).
Perré P. "The Numerical Modelling of Physical and Mechanical Phenomena Involved in Wood Drying : an Excellent
Tool for Assisting with the Study of New Processes." Tutorial 5 th International IUFRO Wood Drying
Conference, 9-38, Québec (1996).
Perré P. and Turner I. "Using a set of macroscopic equations to simulate heat and mass transfer in porous media :
some possibilities illustrated by a wide range of configurations that emphasize the role of internal pressure "
Numerical methods and Mathematical modelling of the Drying Process, 83-156, edited by I.W.Turner and A.
Mujumdar, Marcel Dekker (1996).
Perré P. "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 (1998).
Perré P., Turner I. W. and Passard J. " 2-D solution for Drying with Internal Vaporization of Anisotropic Media "
AIChE. Journal, 45(1), 13-26 (1999).
Perré P. and Turner I. "A 3D version of TransPore : A Comprehensive Heat and Mass Transfer Computational
Model for Simulating the Drying of Porous Media" Int. J. Heat Mass Transfer, 42(24), 4501-4521 (1999).
Plumb, O.A., Brown, C.A., and Olmstead, B.A.,"Experimental measurement of heat and mass transfer during
convective drying of southern pine" Wood. Sci. Technol. 18, 187-204 (1984).
Ormarsson S. "Numerical Analysis o Moisture-Related Distrotions in Sawn Timber" PhD thesis, Chalmers
University of Technology, Sweden (1999).
Salin, J.G."Remarks on the influence of heartwood content in pine boards on final moisture content and degrade"
IUFRO Wood Drying Symp., 4-6 Seattle (1989).
Salin, J.G. "Simulation Models; from a Scientific Challenge to a Kiln Operator Tool" 6th International IUFRO Wood
Drying Conference, 177-185, Stellenbosch, South Africa (1999).
Siau, J.F., "Transport Processes in Wood" Springer-Verlag (1984).
Stanish M.A., Schajer G.S., Kayihan F. "A mathematical model of drying for hygroscopic porous media" AICHE
Journal, Vol. 32 (8), p. 1301-1311 (1986).
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Turner, I. W. and Perré, P. A synopsis of the strategies and efficient resolution techniques used for modelling and
numerically simulating the drying process, to appear in the Book on Numerical methods and Mathematical
modelling of the Drying Process, edited by I.W.Turner and A. Mujumdar (1996).
Van Meel D.A. “Adiabatic convection batch drying with recirculation of air” Chemical Engineering Science, 9, 36-44
(1958).
Wengert G. and Denig J. "Lumber drying - Today and Tomorrov" Forest Products J., 45(5), 22-30 (1995).