Wood Material Science, advanced course
Dick Sandberg and Ove Söderström (Ed.)
Växjö University, TD Report No. 38, 2007
ISSN: 1651-0038
ISBN: 978-91-7636-561-8
PREFACE
In the Wood Mechanics Program the Department of Building Materials at
KTH and the Department of Technology and Design at Växjö University
have collaborated to give Wood Material Science, advanced course during
the spring term 2007. The course gives 5 academic points (7.5 ECTS) and
has created some articles written by the students and these articles are here
presented. The articles shall be considered as a part of the education and
not as fully processed scientific papers. All the authors themselves are
responsible for the content in their articles.
The following persons have been taking an active part in the realization of
this course and we would like to express our greatest gratitude to all of
them:
Prof. Johan Claesson, Chalmers University of Technology
Doc. Jarl-Gunnar Salin, SP Trätek
Prof. Hans Petersson, Växjö University
Prof. Sigurdur Ormarsson, Technical University of Denmark
Dr. Åsa Blom, Växjö University
Doc. Parviz Navi, EPFL, Swiss Federal Institute of Technology, Lausanne
Prof. Thomas Thörnqvist, , Växjö University
Doc. Andreas Steffen, Siempelkamp Maschinen- und Anlagenbau GmbH
& Co. KG, KREFELD, Germany
Dr. Mats Westin, SP Trätek
Dr. Magnus Wålinder, SP Trätek
Dr. Håkan Bard, Växjö University
Prof. Erik Serrano, Växjö University
Ing. Bertil Enqvist, Växjö University
Development manager, Bo Nilsson, Swedspan AB
Finally we would like to thank the financial support from the Woodtech
program and the Swedish Foundation of Strategic Research.
Stockholm and Växjö, April 2007
Ove Söderström
Professor
Dick Sandberg
Docent
I
II
Content
Capillarity phenomena and application on wood.
Karin Sandberg and Carmen Cristescu
1
Capillary suction in a closed tube -a dynamic study.
Martin Hägglund and Eva Frühwald
29
Kapillär vätskestigning i korta rör.
Lars-Elof Bryne och Jimmy Johansson,
47
Simulering av fukttransportvägar vid virkestorkning samt
vätskeinträngning i ved vid massakokning – med tillämpning
av perkolationsteori.
Johan Sjödin och Lars Eliasson
61
Analysis of deformations and section forces due to moisture
content variation in wooden structures.
Kirsi Salmela and Johan Vessby
91
Finite element study of moisture related stresses in
wooden structures.
Kristoffer Segerholm, and Janne Manninen,
105
III
Capillarity phenomena and application on wood
Karin Sandberg, SP Trätek, Skellefteå
Carmen Cristescu, Luleå Technical University
Problem
The model below was conceived to predict the capillary motion of water
in an open glass tube.
∂  ∂z 
8 ∂z
2σ cos ϕ w
z + gz −
=0
z +
2
∂t  ∂t  ρR ∂t
ρR
Inertia+viscous flow+gravity+capillary flow
For a more detailed approach of this equation, see appendix 1.
Tasks
1.
Analyse the model below in detail and discuss its physical relevance.
2.
“What is wrong with model?” Discuss the approximation and
assumption made.
3.
Formulate a criterion for the oscillation found.
4.
Could the model approximate the capillary rise of water in solid
wood?
1
1. Introduction
In order to be able to understand the capillary phenomenon and to model
it, some terms involved in this subject are briefly defined.
1.1. Definitions
Capillarity is a manifestation of surface tension by which the portion of
the liquid coming in contact with a solid is elevated or depressed,
depending on the adhesive or cohesive properties of the liquid”
As soon as the liquid comes in contact with the walls of the tube, a contact
angle is established. In the case of water and glass this angle is small
resulting in a concave meniscus. The surface tension acting on the fluid at
the point of contact of air-water-glass will create an upward force that will
make water rise in the tube until the weight of the water column balances
the upward surface tension component, see figure 1, (Sepulveda, 2007).
Figure 1.
Capillary rise.
Surface tension is an effect within the surface layer of a liquid that causes
that layer to behave as an elastic sheet, tending to minimize the area of the
surface. Surface forces, or more generally, interfacial forces, govern such
phenomena as the wetting or not wetting of solids by liquids, the capillary
rise of liquids in fine tubes and wicks, and the curvature of free-liquid
surfaces.
Contact angle is the angle at which a liquid/vapor interface meets the
solid surface. It plays the role of a boundary condition.
2
2. State of art on capillary rise
The literature in the capillarity field was found to be vast and wide,
therefore we decided to divide this state of art into three, according to the
information used by us to analyse the model.
2.1 Equations attempting to describe the capillary phenomena i.e.
capillary rise in cylindrical and square-shaped tubes.
Capillary penetrations are frequently interpreted by using the "Washburn
equation" to describe the movement of the liquid meniscus in the porous
solid (e.g., powder bed). The Washburn equation was derived from the
Hagen-Poiseuille equation, which relates the volumetric flow rate of a
fluid, through a straight circular tube, to the pressure drop imposed.
According to Szekley et al. 1971 The Hagen-Poiseuille equation, which
applies to laminar flow and to steady-state conditions, may be written as
dQ π∆ Pr 2
=
dt
8ηh
(Eq. 1)
where dQ/dt is the volumetric flow rate, r is the radius of the tube, η is the
viscosity of the fluid, AP is the pressure drop, and h is the liquid height in
the tube.
The Washburn equation consists of applying Eq. (1) to the motion of a
liquid meniscus in a capillary of radius r. In the special ease of capillary
rise in the vertical direction, h represents the distance of rise, from a given
datum line, dQ≡πr2 dh, and the net pressure driving force is made up as
the difference between the capillary pressure, which drives the fluid
upwards, and the static pressure exerted by the fluid. The capillary
pressure ∆Pc may be expressed as:
∆Pc =
2σ cos θ
r
(Eq. 2)
Where σ is the interfacial tension and θ is the contact angle.
Combining Eq. (1) and (2) and noting that the static pressure exerted by
the liquid is given by ρgh, one arrives at the Washburn equation:
3
1
dh
(∆P − ρgh )r 2
=
dt 8ηh
(Eq. 3)
here ρ is the fluid density and g is the acceleration due to gravity. Equation
(3) relates the rate of rise of the meniscus, at a given vertical height, to the
other property values of the system, according to Szekely et al. (1971).
In order to characterize the wettability of powders by this technique, only
the very early stages of the penetration can be utilized as the presence of
the liquid will rapidly alter the initial arrangement of the solid particles. In
light of these instabilities in the packing, caused by the penetrating liquid,
ideally one would wish to use the capillary rise data, extrapolated to time
=0 i.e. .h→0
Inspection of Eq. (3) shows, however, that:
dh

→ ∞ , as t 
→ 0
dt
(Eq. 4)
The apparent inadequacy of the Washburn equation for short times is due
to the fact that the quasi-steady state approximation implicit in the
derivation of Eq. (3) is inappropriate to conditions where rapid changes
occur in h and in dh/dt.
Szekely et al. 1971 offers a more rigorous formulation of the capillary
penetration process, with a view to resolving the above-mentioned
anomaly.
Close to the moment when the tube touches the liquid, Washburn's law
leads to an unphysical infinite velocity of imbibition. The problem was
deeply studied by Quere (1997) and Quere et al. (1999). He noticed that
inertia must be considered at that particular moment, and to stress its
effect, a similar experiment was performed with ethanol, a liquid of much
lower viscosity, while keeping the density and the surface tension roughly
constant.
Bico et al. (2002) discuss the rise of liquids and bubbles in angular
capillary tubes (see figure 2). By angular tubes the author means a
4
medium having corners. The fingers progressing along the corners can
easily be detected with a blotting paper placed at the top of the tube (much
higher than the position of the central meniscus).
By balancing the weight with the capillary force, they obtain good
correlation with data for equilibrium height of the central meniscus. A
much quicker argument for deriving the height would consist of reducing
the weight to the contribution of the central meniscus. The fingers
contribution is thus found to lower this quantity by about 6 %. Note finally
that the calculation supposes an infinite extent for the fingers, although the
tubes used for experiments have a length of the order of 10 cm. Varying
the total height of the tube, they observed that the central meniscus kept
the same height (taking a shorter tube did not make the meniscus rise),
which could be due to the possibility for the liquid to adjust its curvature
on the sharp edges at the extremity of the tube.
An interesting observation is that the shape of the meniscus is
approximately estimated as a curvature, but in reality the liquid surface is
plane in the middle and arched close to the walls. De Gennes in 1985
proposed that when the droplets or capillaries have a size radius~1 mm, all
curvature corrections should be negligible.
Figure 2.
Capillary rise in a square tube from Bico et al. (2002).
5
Zhmud et al. (2000) discuss the flow patterns effects in the front zone of
the liquid column and near the capillary entrance for hydrophobic tubes.
He assumes that the liquid present in the dipped part of the capillary, and
in the bulk reservoir the capillary is connected to, starts moving at the
same time.
2.2 Models of Capillarity in wood
The Comstock mode (1971) assumes that softwood is composed of a
homogeneous matrix of cells that overlap adjacent cells by a fraction ~ of
their total length; all bordered pits are assumed to be located on the
overlapped surfaces. A schematic of this structure is shown in figure 3,
where each lumen is assumed to have a square cross section of size W x
W and tapers only in one direction. It can be seen that the free liquid is
assumed to reside in the tip of each cell, the gas phase forming a centrally
located bubble. The meniscus that separates the two phases has principal
radii of curvature that depend on the size of the lumen. This is an
important assumption since previous explanations of capillary transport of
liquid during both penetration and drying in the longitudinal direction
assume that menisci formed in the bordered pits, which have much smaller
radii of curvature, control the capillary forces (Hawley 1931; Stature
1963; Siau 1971 from Spolek 1981)
Figure 3.
Unsaturated moisture distribution in Comstock structural
model.
6
Kouali et al. (1981) considers that free-water in wood is in liquid form in
the lumens or voids of the wood. The amount of free water which can be
held is limited by the porosity of the wood. As there is no hydrogen
bonding, free water is held only by weak capillary forces and cannot cause
normal swelling or shrinking, because the cell wall is already saturated by
the much more tightly bound hygroscopic water.
The capillary pressure in softwoods was modelled also by Spolek et al.
(1981). He stated that in any lumen partially filled with liquid, a meniscus
forms between the liquid and gas phases due to the surface tension forces
between the water molecules and the cell wall material. A balance of the
mechanical forces at this phase interface will show that the gas phase
pressure exceeds the pressure in the liquid. Therefore, the liquid phase
velocity can be expressed explicitly as a function of the wood saturation
gradient if the dependence of the capillary pressure on saturation can be
identified.
The theoretical model of Kowalski (2002) differentiates from those
presented in the above paragraphs with the mechanism of mass transport
in the pore space. Namely, one assumes here that the gradient of a
generalised chemical potential of the fluid (dependent on the
thermodynamical state) is the main thermodynamical force responsible for
the fluid transport in wood. The generalised chemical potential includes
the capillary and gravitational potentials. The chemical potential has to be
a function of state depending on the same parameters of state as the free
energy (temperature, strain tensor, fluid content). If one takes into account
that the pressure of the pore fluids differs from the pressure of free fluid
by capillary pressure, consequently the chemical potential can be split into
the standard potential and the capillary potential.
For Virta (2005), who studied the one-side soaking of a cladding-board,
the most critical part of the modelling appeared to be the neglected air
pressure function that was compensated by using an apparent surface
emission coefficient as a virtual boundary layer. The apparent surface
emission coefficient was used to compensate air pressure development
that occurs during short-term water soaking by fitting the modelling
results and the experimental observations together. On the basis of the
results, the apparent surface emission coefficient (S) was approximately
10-7 m/s for Norway spruce during short-term water soaking. Without the
7
apparent surface emission coefficient (S=1) the original water soaking
model gives too high moisture content values in short-term water soaking.
The main reason for this phenomenon may be the neglected air pressure,
because initial water soaking is rapid. Another reason may be the capillary
pressure function that is reported to be obscure in the case of empty or
partly filled voids.
Virta (2005) also states that the effect of capillary pressure on water flow
into wood is not very clear from the point of view of short-term free water
soaking. The capillary pressure function may be obscure in the case of
empty or partly filled voids. On the basis of the modelling, the free water
content of wood has a drastic effect on free water soaking in the capillary
structure of softwood. The permeability of wood depends significantly on
airflow within the capillary structure of wood. Air pressure increases
during free water soaking with the increase of free water content.
However, the pressure cannot be explained by air compression only. It
also results from dissolving of air in free water penetrating into the wood
structure.
2.3 Relevant influences of parameters for a wood capillary model.
- The properties of water, as viscosity (η) and surface tension (σ) change
with temperature see table 1. The surface tension of water is higher that of
all other liquids except mercury.
Table 1.
Properties of water (Fysikalia, Tabell – och formelsamling).
Temp
Kinematical
viscosity
Dynamic
viscosity
Surface tension, σ
Density, ρ
°C
10-6 (m2/s)
10-6 (Ns/m2)
10-3 (N/m)
(kg/m3)
0
1.792
1792
75.64
999.8
10
1.308
1519
74.23
999.7
20
1.004
1005
72.75
998.2
30
0.805
801
71.20
995.7
40
0.661
656
69.60
992.2
50
0.556
549
67.94
988.1
8
As seen from table 1 the viscosity of water makes has a big influence on
the surface tension.
One shouldn’t forget about the influence of impurities normally found in
water in wood, such as water soluble wood extractives and how they
influence the surface tension.
The contact angle measured on a wood surface is not directly applicable to
a wood substance surface, e.g. wood lumen. But Liptakova and Kudela
tried to solve this problem by a method which separated the effect of
permeation and the non-wood substance part of a wood surface. The
values are as follows: for air-filled wood the contact angle value is 55±9,
for water filled 19±6, for wood substance 21±6 (Wadsö, 1995).
Quere et al. (1999) emphasised the quantities of energy loss at entrance
of pipe, pressure loss and friction viscosity when calculating a theoretical
model.
Zhmud et al. (2000) considers that initially, the liquid sucked into the
capillary is accelerated by capillary force; however, soon thereafter the
capillary force is compensated for by the viscous drag so that a quasisteady state is achieved and eventually the rise is slowed down by gravity.
It should once again be emphasized that the viscous drag is the only factor
damping oscillations in the model. In practice, there is additional
dissipation of energy by liquid returned to the bulk reservoir, so the
damping will be faster than predicted.
The issue is that the moisture content of wood varies within the same
sample. Wood is not completely dried and there are always small
variations in MC within the wood even after condition for a long period.
Wood is extremely non homogenous, and its structure and chemical
variability is reflected in wide range in its physical properties as
permeability, capillary behaviour, thermal conductivity and diffusion of
bound water. Great uniformity is found between wood of conifers
(softwoods) and dicotyledonous angiosperms (hardwoods) that have
extreme structural difference. Because of great structural variation
especially in hardwoods there are a greater range in permeability and
capillarity behaviour. (Siau, 1984).
9
Softwood tracheids have a length of 2−4 mm in length with aspect ratio
(L/D) of about 100:1. Tracheids in softwood have a supporting and
conducting role. The small block-like cell some 200x30 µm in size, known
as parenchyma, are mostly located in the rays and are responsible for
storage of food material.
In hardwood four types of cells are presents but tracheids in a small
amount. Parenchyma cells are used for storage. Support is affected by long
thin cells with tapered ends, known as fibres, usually, about 1−2 mm
length and with an aspect ratio of 100:1. Conduction is carried out in cells
known as vessels or pores are usually (0.2 −1.2 mm) and are relatively
wide up to 0.5 mm (Dinwoodie, 1981).
Softwood has about 93 % tracheids, 6 % rays and 1 % resin channels.
Hardwood has about 50−55 % vessels, 26 %, tracheids and fibres, 3−5 %
parenchyma and 18 % rays.
This anatomy of wood is influenced by the circulation of sapwood during
growth. When seasonal growth commences, the dominant function
appears to be conduction, while the latter part of the year the dominant
factor is support. This changes in emphasis manifest itself in the
softwoods with the presences of thin-walled tracheids (about 2 µm) in the
early part of he season (earlywood) and thick walled (up to 10 µm) and
slightly longer (10%) in the latter part of the season (latewood)
(Dinwoodie, 1981). In some of the hardwoods, the earlywood is
characterised by presence of large-diameter vessels surrounded primarily
by parenchyma and tracheids; only a few fibres are present. In the late
wood, the vessel diameter is considerably smaller about 1/5 and the bulk
of tissue comprises fibres. Timbers with this characteristic two-phase
system are referred as having a ring-porous structure. Uniformity across
the growth ring occurs not only in cell size, but also in the distribution of
the different type of cells called diffuse-porous.
Table 2 gives a Summary of dimension of structural elements in normal
softwoods.
10
Table 2. Summary of dimension of structural elements in normal
softwoods (Siau, 1984).
Structural element
Softwoods (µm)
Tracheid length
3500
Tracheid diameter
35
Tracheid lumen diameter
20−30
Overall diameter of pith chambers of
boarded pits
6−30
Effective diameter of pit openings
0.02−4.0
Interconnection by means of pits occurs between cells to permit the
passage of mineral solutions and food in both longitudinal and horizontal
planes. There basic pits occur. Simple pits taking a form of straight- sided
holes with transverse membrane occur between parenchyma and
parenchyma, and between fibre and fibre. Between tracheids a complex
structure known as bordered pit occurs. Similar structures are to be found
interconnecting vessels in horizontal plane. Between parenchyma cells and
tracheids or vessels there occur semi–bordered pits, often referred to as ray
pits. These are characterised by the presence of dome on the tracheids or
vessel wall and the absence of such on the parenchyma wall: a pit
membrane is present, but the torus is absent (Dinwoodie, 1981).
In softwoods both tracheids and parenchyma cells have close ends end and
movement of liquids and gases must be way the pits (Dinwoodie, 1981).
Bordered pit is almost entirely restricted to the radial walls of the tracheids
towards the ends of the cells. The number of pits per tracheids varies from
50 to 300 in earlywood with fewer in latewood (Siau, 1984 and references
therein). Lumens are much larger and the cell wall layers much thinner in
earlywood than latewood (Siau, 1984).
It is in green timber the torus of the bordered pits is usually located in a
central poison and flow can be at a maximum. Since the earlywood cells
possess larger and more frequent bordered pits, the flow through the
earlywood is considered greater than that through latewood. However on
drying, the torus of the earlywood cells becomes aspirated, due to tension
11
stresses set up by the retreating water meniscus. In this process the margo
strands obviously undergo very considerable extension this could be as
high as 8 % in earlywood and the torus is rigidly held in a displacement
position by strong hydrogen bonding. This displacement of torus
effectively seals the pit and reduces the level of permeability of dry
earlywood to a value similar to that of green latewood. In the latewood,
only about half of the pits become aspirated and consequently the
percentage reduction in permeability with drying is much lower than in the
earlywood. It appears that aspiration is mainly irreversible. (Dinwoodie,
1981).
Permeability of heartwood is usually appreciably lower than of sapwood
due to the deposition of encrusting materials over the torus and margo
strands and also within the ray cells (Dinwoodie, 1981)
The longitudinally permeability is usually high in the sapwood of
hardwoods. This is because these timbers possess vessels, the ends of
which have been either completely or partially dissolved away. Radial
flow is again by way of the rays, while tangential flow is more
complicated, relaying on the presence of pits interconnecting adjacent
vessels, fibres and vertical parenchyma. Transverse flow rates are usually
much lower than in softwoods, but somewhat surprisingly a good
correlation exist between tangential and radial permeability; this is due in
part to very low permeability of the rays in hardwoods. Since the effect of
bordered pit aspiration, so dominant in controlling the permeability of
softwoods, are absent in hardwoods, and the influence of drying on the
level of permeability in hardwood is much less than is the case with
softwoods.
The final conclusion is that the flow will be different in green and dried
wood, in the longitudinal, radial and tangential direction, it will be very
different between wood species, within heartwood, juvenile wood,
sapwood, not to mention earlywood and latewood.
Since in capillarity phenomenon the impact between liquid (water) and
surface (wood) is crucial, the influence of the uneven texture of the
interior wall in the wood cell should be taken into account. In wood
vessel walls are not very smooth, they contain warts, ridges and other
irregularities especially due to the presence of pits. Therefore the
12
flowpattern is not ideally poraboloid but a good deal more complex in
living wood.
The problem of the moisture content of the cell wall at the moment of
liquid rise remains critical. If one considers the inner wood surface unsaturated and that the wood is filled with air, and the contact angle of 55º,
this situation will change for the following water molecules: they will
meet a water-filled wood, with a contact angle of 19 º. (Liptakova et al.
1994 from Wadsö, 1995). So the contact angle is not constant during the
capillary rise.
Zhmud et al. (2000) noticed how it has been neglected that in order to fill
a capillary, the capillary liquid has to displace another fluid, usually air or
vapor. Taking this into account would result in additional dissipation of
energy; the effect can be significant for low-viscous liquids.
The shape of the wood cells is more square or hexagonal rather than
circular therefore some other differences should be taken into account.
Concerning the movement of bubble air it was also found that a long
bubble does not remain trapped in a thin closed square tube as it would be
in a circular one, but rises because of gravity. Bico et al. (2002). These
results prove that the shape of the tube is a very important feature for
capillary rise.
3. Analyze of the given model
Calculation were performed with MathCAD to see the capacity of the
model to capillary rise in both softwood and hardwood cells. The values of
σ (surface energy), ρ (density) and η (viscosity) of water were chosen
considering that water is pure and has a temperature of 25°C. Lumen size
(radius) were chosen from the data in table 2.
The plots represent the evolution in time of the height of the meniscus in a
capillary rise according to the model developed by Prof. Ove Söderström
(2007). For the derivation of the theoretical model and plots see appendix
1 and appendix 2.
13
3.1 Results and discussion
The analyse of the plots we noticed that:
•
•
•
•
Oscillations are dependent on the value of the radius and of the
contact angle in the same time
When radius increases, the number of oscillations increases
When contact increases, the oscillation are influenced in a lower
manner
When both radius and angles are increased the frequency of
oscillations
According to the equations 6, the oscillations will occur when the radius is
larger than 477 µm : (Söderström, 2007)
rcrit = 5
32η 2σ cos θ
ρ 3g 2
(Eq. 5)
But in softwood species, the lumen radius varies from 10 to 18 µm,
therefore according to the calculations above (Eq.5), oscillation would
never occur at such a low value of the radius.
If thinking of a porous hardwood species like oak, with a lumen radius in
early wood reaching up to 200 µm, then, again, oscillation can not occur.
The conclusion is that this model looks at details which are not relevant
for wood. In reality oscillations are not an issue in wood capillarity,
nobody has ever proved their existence under any circumstances.
One lack of the model is that the length of capillary is not considered. As
seen from the literature, (Zhmud et al. 2000) this length related to the
radius gives a criterion for the oscillations in the plotted results. The
amplitude of oscillation decreases with increasing the capillary length.
The plots show a very rapid rise of capillary height (less then 0.5 s in all
situations). In reality the time of capillary movement in wood is slower
due to the roughness of the inside cell wall and most of all due to the
14
presence of pits. Pits are directing the water towards other neighbouring
cells. The movement is not completely vertical, especially in softwood.
If one considers wood cells as a capillars, they exist as open and closed
and they communicate through pits. Instead of an open tube model or a
closed tube model, a solution would be an interacting capillary bundle
model capable also to take into account the different behaviour in the 3
axes. We propose the use of an angular capillary tube and the presence of
air in lumen to be taken into account.
References
Bico J.,Quere D., 2002. Rise of Liquids and Bubbles in Angular Capillary
Tubes, Journal of Colloid and Interface Science, Vol. 247, p. 162-166.
de Gennes P.G. (1985). Wetting: statics and dynamics, Reviews of
Modern Physics, Vol. 57, No. 3, Part 1, p.827-863.
de Gennes P.G. (1996). Mechanics of soft interfaces, Faraday Discuss.,
Vol. 104, p.1-8.
Dinwoodie J.M., 1981. Timber its nature and behaviour, International
Student Edition, Van Nostrand Reinhold, Austria.
Kouali M. El., Vergnaud J. M., 1991. Wood Sci. Technol. 25:327-339
(1991).
Kowalski J.S.,Musielak G.,Kyziol L., 2002, Non-linear Model for Wood
saturation, Transport in Porous Media, 46, p.77-89
Quéré D. (1997). Inertial capillarity. Europhys. Lett., Vol. 39, No. 5, p.
533-538.
Quéré D., Raphaël É. and Ollitrault J.-Y. (1999). Rebounds in a Capillary
Tube. Langmuir, Vol. 15, p. 3679-3682.
Siau, J. F. 1984. Transport Processes in Wood. Berlin: Springer-Verlag.
15
Spolek G. A. & Plumb O. A, 1981, Capillary Pressure in Softwoods,
Wood Sci. Technol. 15:189-199 (1981) Wood Science and Technology9
by Springer-Verlag.
Szekely J., Neumann A.W. and Chuang Y.K. (1971). The Rate of
Capillary Penetration and the Applicability of the Washburn Equation.
Journal of Colloid and Interface Science, Vol. 35, No. 2, p. 273-278.
Söderström O. 2007. Private communication
Virta J., Koponen S., Absetz I (2005) Modelling moisture distribution in
wooden cladding board as a result of short-term single-sided water
soaking, Building and Environment, 41, 1593-1599
Wadsö, L. 1995. Capillarity in wood and related concepts; a critical
review. University of Lund. Report TVBM-3069.
Zmud B.V., Tiberg F. and Hallstensson K. (2000). Dynamics of Capillary
Rise. Journal of Colloid and Interface Science, Vol. 228, p. 263-269.
Sepulveda. 2007
http://citt.ufl.edu/Marcela/Sepulveda/html/en_capilaridad.htm
16
Appendix 1. Capillary dynamics
θ+dθ
θ
z
φ
R
r
dr
r
∂z
)
∂t
∂  ∂z 
d ( mv )
= 2πρrdr  z 
F=
∂t  ∂t 
dt
d ( dm ⋅ v ) = d ( 2πr ⋅ dr ⋅ z ⋅ ρ
 ∂v 
− 2πrσ sin θ + 2π ( r + dr )σ sin( θ + dθ ) − η   2πrz
 ∂r  r
∂  ∂z 
 ∂v 
+ η   2π ( r + dr ) z − 2πrdrzρg = 2πρrdr  z 
∂t  ∂t 
 ∂r  r + dr
 ∂ 2v 
∂  ∂z 
 ∂v 
rσ cos θdθ + σdr sin θ + ηrz  2 dr + η   ⋅ zdz − rzρgdr = ρrdr  z 
∂t  ∂t 
 ∂r  r
 ∂r 
ϕ=
σ
π
2
−θ
∂
∂ 2v
∂  ∂z 
( r sin ϕ ) + η 2 zr − ρgzr = rρ  z 
∂r
∂r
∂t  ∂t 
17
Laminar – turbulent flow
For a tube Reynolds number is defined as:
Re =
v⋅D
υ
,
where v =velocity, D =diameter and υ = kinematic fluid
viscosity. Turbulent flow if Re > 2300.
For wood we consider capillary tubes with diameter in the order of
magnitude of 100 µm water and the liquid is water
Re=
v ⋅10 −4
≈ 5v
18 ⋅10 −6
5 v < 2300
v < 500 m/s
We always have laminar flows in capillarity tubes if the flow mainly is
governed by surface tension.
Laminar flow
R 2 − r 2  dp 
v(r) =
− 
4η  dz 
R
R
1
q
v=
2πrv(r )dr = 2 ∫ v(r )rdr
2 ∫
πR 0
R 0
Def :
q
v= 2
R
v=
R
R 2 − r 2  dp 
∫0 4η  − dz rdr
R 2  dp 
− 
8η  dz 
18
 r2 
v(r ) = 2v1 − 2 
 R 
v(0) = 2v
4r
∂v
=− 2 v
R
∂r
4
 ∂v 
  =− 2 v
R
 ∂r  r = R
4v
∂ 2v
=− 2
2
∂r
R
Integration over r above with the assumption z(r) independent of r.
At r = 0, the φ = 0 (symmetry)
and φ = φw (the wetting angle) at r = R
R
R
R
R
R
∂
∂v
∂ 2v
∂ ∂z
r
dr
z
dr
z
(
cos
)
+
+
σ
ϕ
η
η
∫0 ∂r
∫0 ∂r
∫0 ∂r 2 rdr − ρgz ∫0 rdr = ∫0 rρ ∂t ( z ∂t )dr
σR cos ϕ w − ηz 2v − ηz 2v − ρgz
R2
R 2 ∂  ∂z 
=ρ
z 
q ∂t  ∂t 
2
∂z
∂t
2σ cos ϕ w
8 ∂z
∂  ∂z 
z − gz =  z 
−
2
ρR
∂t  ∂t 
ρR ∂t
v=
2σ cos ϕ w
8 ∂z
∂  ∂z 
=0
z + gz −
z +
2
∂t  ∂t  ρR ∂t
ρR
Inertia+viscous flow+gravity+capillary flow
19
∂4
=0
Stationary conditions: ∂t 4
zstat = H
2 σ cos ϕ w
= 0
ρR
2 σ cos ϕ w
H =
ρ gR
gH
−
Time constants: τ σ =
ρR2
τη =
8η
H
surface tension
g
viscosity
Dimensional units:
ξ=
v=
z
H
t
τσ
∂ξ
∂  ∂ξ 
+ ξ −1 = 0
 + Ωξ
ξ
∂v
∂v  ∂v 
whereΩ =
τσ
τη
20
Numerical solution
 ∂ 2ξ   ∂ξ 
∂ξ
 ξ 2  +   + Ωξ
+ ξ −1 = 0
∂v
 ∂v   ∂v 
2
ξ −ξ
+
 + Ωξ n n⋅ n−1⋅ + ξ n − 1 = 0
h
h
h


− 4ξ n−1⋅ − ξ n− 2⋅ + hΩξ n−1⋅ − h 2
ξ 2 − h2
ξ n⋅ 2 −
ξ n + n −1
2 + hΩ
2 + hΩ
ξ n⋅ ⋅
ξ n⋅ − 2ξ n−1⋅ + ξ n−2⋅  ξ n⋅ − ξ n−1⋅ 
2
Succesive solution from n=q
ξ 0 andξ1 are needed
ξ0 = 0
ξ << 1
21
1 ∂  ∂ξ 2

2 ∂v  ∂v

1 ∂ξ 2
 + Ω ⋅
=1
2 ∂v

1 ∂ξ 2
γ =
2 ∂v
∂γ
+ Ωγ = 1
∂v
dγ
= dv
1 − Ωγ
∂ξ
γ = 0 where v = 0 as
finite at v = 0
∂v
 1
 −  ln(1 − Ωv) = v
 Ω
1
γ = (1 − e −Ωv )
Ω
ξ = 0 when v = 0
ξ=
ξ = ξ1 whereν = h
2
1 − e −Ωv
v−
Ω
Ω
2
1 − e−Ωh
h−
ξ1 =
Ω
Ω
22
Material data
0
σ := 73 ⋅10
−6
Ra := 800 ⋅10
−3
η := 1.01 ⋅10
4
Na := 10
gr := 9.81
ρ := 1000
2σ cos ( φw)
He :=
ρ ⋅gr⋅Ra
He = 0.019
He
gr
τσ :=
Time := 1 ⋅10
φw := 0
−3
τσ = 0.044
2
τη :=
Ω :=
ρ ⋅Ra
τσ
Ω = 1.00000 Ω
τη
Time
Tνmax :=
h :=
τη = 0.079
8 ⋅η
τσ
Tνmax
−3
h = 2.296 × 10
Na
ξ 0 := 0
ξ 1 :=
2
Ω
− Ω ⋅h
⋅ h−
1−e
−3
ξ 1 = 2.296 × 10
Ω
ξ1
n := 2 .. Na
2 ⋅h
= 0.025
Ω
2
ξ n :=
4 ⋅ξ n−1 − ξ n−2 + h⋅Ω ⋅ξ n−1 − h
2 ⋅( 2 + h⋅Ω )
2
 4 ξ n−1 − ξ n−2 + h⋅Ω ⋅ξ n−1 − h2 ( ξ n−1) 2 − h2
 −
+ 
2 ⋅( 2 + h⋅Ω )
2 + h⋅Ω


23
i := 0 .. Na
Hξ i := ξ i
Tν i := h⋅i
CHi := Hξ i⋅He
CHi := Hξ i
Timi := Tν i⋅τσ
24
max( ξ ) = 1.230 ξ Na = 1.001
Appendix 2. Plots
The following charts represent the evolution in time of the height of the
meniscus in a capillary rise according to the model developed by Ove
Söderström 2007. For the derivation of the theoretical model and plots see
appendix 1 and appendix 2. Calculation were performed with MathCAD
to see the capacity of the model to capillary rise in both softwood and
hardwood cells. The values of σ (surface energy), ρ (density) and η
(viscosity) of water were chosen considering that water is pure and has a
temperature of 25C. Lumen size (radius) were chosen from the data in
table 2.
1.230
0.03
1.5
0.023
0.02
1
CHi
Hξ i
0.01
0.5
0.000
0.000
0
0
Figure.
0
0.000
10
Tνi
20
22.963
0
0.000
0.5
Timi
1
1.000
Height of capillary rise in time: left side dimensionless
numerical solution; right side: capillary height in meters as
a function of time in seconds for a radius of 800*10-6 m.
Same equation but with different radius show that with a radius of 500 µm
and θ=0 the oscillations disappear. With a radius of an average tracheids
17.5 µm there are no oscillations.
25
0.029
0.04
0.03
0.025
0.02
Hξ i 0.02
CHi
0.01
0.000
0
0.000
0
0.000
0
2
3.396
Tνi
Radius of 17,5 µm and θ=0
1.004
0
0.000
0.5
Timi
1
1.000
Radius of 17,5 µm and θ=0
1.5
0.030
0.04
1
Hξ i
CHi 0.02
0.5
0.000
0
0.000
0
0.000
0
10
Tνi
18.154
Radius of 500 µm and θ=0
1.081
0
0.000
0.5
Timi
1
1.000
Radius of 500 µm and θ=0
1.5
0.027
0.04
1
Hξ i
CHi 0.02
0.5
0.000
0
0.000
0
0.000
0
10
Tνi
19.887
Radius of 600 µm and θ=0
0
0.000
0.5
Timi
1
1.000
Radius of 600 µm and θ=0
26
When σ, ρ and η is chosen for a temperature of 20°C and variation in
radius, end θ=20° the following oscillations are given.
0.06
0.046
0.016
0.02
0.04
Hξ i
CHi 0.01
0.02
0.000
0
0.000
0
0.000
2
0
4
5.321
Tνi
Radius of 17.5 µm and θ=20
1.000
0
0.000
0.5
Timi
1
1.000
Radius of 17.5 µm and θ=20
1.5
0.015
0.02
1
Hξ i
CHi 0.01
0.5
0.000
0
0.000
0
0.000
10
0
20
25.439
Tνi
Radius of 400 10 µm θ=20
1.177
0
0.000
0.5
Timi
1
1.000
Radius 400 µm and θ=20
1.5
0.015
0.012
1
0.01
Hξ i
CHi
0.5
0.005
0.000
0
0.000
0
0.000
10
20
Tνi
0
30
31.156
Radius 600 10 -6 mm θ=20
0
0.000
0.5
Timi
1
1.000
Radius 600 10 -6 mm θ=20
27
1.308
2
−3
0.01
9.914×10
Hξ i 1
CHi 0.005
0.000
0
0.000
0
0.000
0
20
Tνi
35.976
Radius 800 10 -6 mm θ=20
0
0.000
0.5
Timi
1
1.000
Radius 800 10 -6 mm θ=20
When σ, ρ and η is chosen for a temperature of 20C and variation in
radius, end θ=55 the following oscillations are given.
0.2
0.186
−3
3.499×10
Hξ i 0.1
CHi 0.002
0.000
0
0.000
0
0.000
10
Tνi
0
20
22.851
Radius 17.5 µm and θ=55
1.262
0.004
0
0.000
0.5
Timi
1
1.000
Radius 17.5 µm and θ=55
2
−3
0.0015
1.037×10
0.001
CHi
Hξ i 1
5 .10
0.000
0
4
0.000
0
0.000
50
Tνi
0
100
109.249
Radius of 400 µm and θ=55
0
0.000
0.5
Timi
1
1.000
Radius of 400 µm, and θ=55
28
Capillary suction in a closed tube - a dynamic study
Martin Hägglund, Lund University
Eva Frühwald, Lund University
Introduction
Capillary rising always involves two forces: the gravity force, which is
directed downwards, and the surface tension, which contributes with an
upward force. As long as the liquid is in movement, i.e. before a stationary
state is reached, inertia and viscous forces also affect the development of
suction height in time. In case the tubes are not open there will be an
additional downward force on the meniscus due to compressed air (or any
other gas). The size of this force may be determined from the ideal gas law
stating that the pressure times the volume of a certain amount of gas space
is constant under isothermal conditions - if the volume decreases, the
pressure must increase.
29
The Lucas-Washburn equation presents a quasi-steady state of the
capillary height, where the capillary force is compensated by viscous force
and gravity (Washburn 1921, Zhmud et al. 2000). However, this predicts
an infinitely high capillary velocity at zero time, which is inconsistent with
experimental results and even with theory (Szekely et al. 1971). The flow
assumed in the Lucas-Washburn equation is a laminar flow (Poiseuille
flow); however, infinitely high velocity would result in turbulent flow.
Quéré (1997) proposed another equation for inviscid fluids (fluids with
low or no viscosity), where the capillary height is only determined by the
balance of capillary force and gravity. At the beginning, the capillary
height is determined by the capillary force (z~t(1-Ct)), i.e. the velocity is
different from zero at t=0. Later on, the viscous force also influences the
capillary height, so that a quasi-steady state is achieved (z~t1/2 i.e.
Washburn equation). Finally, the capillary rise is influenced by gravity,
leading to a final capillary height (Stange et al. 2003). In this model, the
viscous drag leads to oscillations. If “turbulence” is accounted for (above a
barrier velocity), the results of this model fit experimental results very
well, especially at the beginning. Turbulence at high velocities results in
energy dissipation, which slows down the flow rate and thus the capillary
rise. One advantage with this equation is that is has no singularity (i.e.
acceleration is not infinite at t=0) at the beginning of the capillary rise
(zero time), which the Lucas-Washburn equation has. In a perfect model,
also the energy dissipation caused by the need of the rising liquid to
displace another fluid (air or water vapor) must be considered.
In the following text, a force balance equation encompassing inertia,
viscosity, gravity, surface tension and pressure caused by compressed air
in the case of a closed capillary tube is considered. It can be noted that the
governing differential equation resembles that of a spring-mass-damper
system. As for such a system, if the damper (dash-pot) is not of critical
magnitude or above, oscillation will occur around the steady-state level.
Analogously, oscillation in a capillary tube may also occur if the viscosity
(“equivalent” to the dash-pot) is low enough (Quéré and Raphaël 1999);
movements of a viscous fluid dissipate energy.
30
Theory
z
θ
θ+dθ
θ
gravity force g
r
dr
R
Figure 1.
R
Sketch of tube filled with capillary water.
A differential equation for capillary dynamics for open tubes (figure 1)
may be derived as:
 2 γ cos θ c 
∂  ∂z  8η  ∂z 
z  + gz
−
z  +
2 
{  ρR  = 0
∂
∂
∂
t
t
t
ρ
R




1424
3 14243 gravity 14243
inertia
viscosity
(Eq. 1)
surface tension
where z - suction height, R - tube radius, ρ - fluid density, η- fluid
viscosity, γ -surface tension, θc - contact angle (between fluid and tube
wall), g - gravity and t - time.
Note that for the stationary case, i.e. ∂z / ∂t = 0 , the equation simplifies to:
z static = H =
2 γ cos θ c
ρgR
(known as the capillary suction height)
31
ρR 2
H
and τη =
together with
g
gη
the dimensionless units ω = z / H and λ = t / τ γ , Eq. (1) is rewritten as:
After introduction of new constants τ γ =
∂  ∂ω  τ γ ∂ω
+ ω −1 = 0
ω
+ ω
∂λ  ∂λ  τ η ∂λ
1
∂ 2ω 1  ∂ω  τ γ ∂ω
+ 
+ 1 − = 0∑
 +
2
ω  ∂λ  τ η ∂λ
ω
∂λ
(Eq. 2)
2
⇔
In order to solve for ω, the second order differential equation is rewritten
once more by defining two new variables
ω1 = ω
and ω 2 =
∂ω
∂λ
and thus arrive at a system of two first order differential equations:
 ∂ω1
 ∂λ = ω 2

 ∂ω
τ
 2 = − 1 ω 22 − γ ω 2 − 1 + 1
ω1
τη
ω1
 ∂λ
(cf. a first order system:
 ω  
∂  ω1 
= F  1 , λ  )


∂λ ω 2 
 ω 2  
This ODE-system can now be solved in Matlab using e.g. the solver
ODE45.
Finding initial conditions (a t = 0):
Since suction start at height z = 0:
ω1init = ωinit = 0
For t close to 0 ⇒ ω << 1 (i.e. close to 0), Eq. (2) can be rewritten:
32
∂  ∂ω  τ γ ∂ω
=1
ω
+ ω
∂λ  ∂λ  τ η ∂λ
1 ∂  ∂ 2  1 τγ
 ω +
2 ∂λ  ∂λ  2 τ η
⇒
 ∂ 2
 ω  =1
 ∂λ 
(Eq. 3)
Solving for ω gives:
ω=
(
τ
2τ η 
 λ − η 1 − e −λτ γ / τη
τ γ 
τγ
) = Ω = ττ

for small λ

=
≈
x 
(series development of e )

(

=

η
2
1 − e − λΩ
 λ −
Ω
Ω
γ
)  =


(
2
1 − (1 − λΩ + (λΩ) 2 / 2 − (λΩ) 3 / 6
 λ −
Ω
Ω
(
=
λ Ω − (λ Ω) 2 / 2 + (λ Ω) 3 / 6
2
 λ −
Ω
Ω
=
λ2 Ω λ3 Ω 2
2
 λ − λ +
−
Ω
2
6
)  =


(
λ − λ2 Ω / 2 + λ3 Ω 2 / 6
2
 λ −
Ω
1

1  Ω λΩ 2
 = λ
 −
Ω  1
3


 = λ 1 − λΩ / 3

(Eq.
Consequently,
ω
init
2
 ∂ω 
=

 ∂λ 
init


λΩ
=  1 − λΩ / 3 −
= 1.

6 1 − λΩ / 3  λ = 0

Adding a pressure force cause by closed tube
The size of this force is derived as
 H tube

− 1p atm
Ftube = πR 2 
 H tube − z 
Ftube
- pressure force cause by closed tube
33
(under isothermal conditions)
4)
)  =


)  =


ρatm
- atmospheric pressure
Htube
- tube height
The governing equation is consequently (cf. Eq.1):

 2γ cos θ c  1  H tube
∂  ∂z  8η  ∂z 




+
−
z
gz
+
−
1
z +


2
{  ρR  ρ  H − z  p atm = 0
∂
∂
∂
t
t
t
R
ρ






tube
1424
3 14243 gravity 14243 144
424443
inertia
viscosity
surface tension
closed tube pressure
(Eq. 5)
Remembering that ω = z / H , λ = t / τ γ , τ γ =
∂
∂λτ γ
ρR 2
H
and τη =
:
g
gη
2
2 



ωp atm
=0
 Hω ∂ωH  + 1 Hω ∂ωH + HωH − H + H 
2
2
2
2

 τ
 ρg(H − ωgτ ) 
∂
λτ
∂
λτ
τ
τ
τ
γ
η
γ
tube
γ
γ
γ
γ




After H 2 / τ 2γ is factored out and eliminated, the expression simplifies to
ωp atm
∂  ∂ω  τ γ ∂ω
+ ω −1+
=0
ω
+ ω
∂λ  ∂λ  τ η ∂λ
ρg ( H tube − ωgτ γ2 )
⇔
p atm
∂ 2ω 1  ∂ω  τ γ ∂ω
1
+ 
+1− +
=0
 +
2
ω  ∂λ  τ η ∂λ
ω ρg ( H tube − ωgτ γ2 )
∂λ
2
(Eq. 6)
Analogously to Eq. (2), Eq. (6) is rewritten as a system of differential
equations.
ω1 = ω and ω 2 =
34
∂ω
∂λ
 ∂ω1
 ∂λ = ω 2

 ∂ω
τ
p atm
 2 = − 1 ω 22 − γ ω 2 − 1 + 1 −
ω1
τη
ω1 ρg ( H tube − ω1 gτ γ2 )
 ∂λ
(Eq. 7)
(initial condition for this modified differential equation is not affected as
z=0 makes the closed tube pressure term vanish).
For the stationary case ( ∂z / ∂t = 0 ), Eq. (5) simplifies to:
gz −


2γ cos θ c 2  H tube
2  H tube
+ 
− 1 p atm = gz − Hg + 
− 1 p atm = 0
ρR
ρ  H tube − z 
ρ  H tube − z 
and solving for z gives:
closed tube
z static
= H closed tube =
(
1
H + H tube + µ − ( H + H tube + µ ) 2 − 4 H tube H
2
where µ =
p atm
.
ρg
35
)
Application
Experiments on capillary suction of small wood specimens (height 25
mm) were performed. The specimens, both spruce and larch, were
immersed into ink colored water and the time needed for the ink to reach
the upper surface was noted. It was found that the first visible spots
appeared after roughly 4 min and 30 sec; the colored surface continuously
increased and after about 30 min the entire surface was colored.
From literature it was found that the wetting angle for spruce is about 70
degrees and the diameter of the tracheid is estimated at 30 µm. The
viscosity for water is 0.89 × 10-3 Pa·s and the surface tension 73 ×10-3
N/m. In Figure 2 the capillary rising in time for an ideal capillary tube is
plotted demonstrating a much faster capillary rising than observed in the
small timber cubes. Timber tracheids have a defined length (2-4 mm) and
are connected to each other. The fluid has to flow through many tracheids
and bordered pits on its way to the top surface. This resistance within the
wood specimen due to the complex internal structure – the passage
between different tracheids –causes retardation. Furthermore, the air inside
the cell lumen must be displaced to leave room for the water. This causes
an additional downward force reaction. Furthermore, the rough surface in
the cell lumen (warty layer) might cause some retardation.
Capillary suction (H = 170 mm)
180
160
suction height, z [mm]
140
120
100
80
60
40
20
0
0
Figure 2.
100
200
300
400
500
time, t [s]
600
700
800
900
1000
Capillary rising for a tracheid assuming a diameter of 30
µm and a wetting angle of 70 degrees.
36
Examples of capillary rising
Short investigation on how different parameters influence the capillary
height. (ρ = 1000 kg/m3 , η = 1.01×10-3 Pa·s, θ = 0o, γ = 73 ×10-3 N/m and
a tube of glass unless otherwise stated).
Radius
25
suction height, z [mm]
20
R = 800 µm
15
2R
10
5
4R
0
0
0.1
0.2
0.3
0.4
0.5
time, t [s]
0.6
0.7
0.8
0.9
1
The frequency of the oscillations increase with the radius (cf. a stiffer
spring in a spring-mass system). It is also noted that the shape changes to
be more parabolic.
37
Contact angle
25
suction height, z [mm]
20
o
θc = 0
o
θc = 30
15
o
θc = 60
10
5
o
0
0
θc = 90
0.1
0.2
0.3
0.4
0.5
time, t [s]
0.6
0.7
0.8
0.9
1
Small change of the contact angle from 0o to e.g. 10o, 20o (30o) have little
effect on the static suction height as the vertical component of the surface
tension force will be “roughly” the same ( cosθc ≈ 1 for small angles). For
90o there will be no suction at all since no vertical component exists.
38
Different fluids
Water
Density
[kg/m3]
Viscosity
[Pa·s]
surface
[N/m]
1000
0.89 × 10-3
73 ×10-3
tension
-3
wetting angle
[o]
0
Ethanol
789
1.074 × 10-3
22 ×10
25*
Glycerol
1126
934 × 10-3
64 ×10-3
25*
* no exact values were found, 25o is an estimation
25
suction height, z [mm]
20
water
15
glycerol
10
ethanol
5
0
0
0.1
0.2
0.3
0.4
39
0.5
time, t [s]
0.6
0.7
0.8
0.9
1
Closed tubes
25
Htube = ∞ i.e. open tube
suction height, z [mm]
20
15
Htube = 10 m
10
5
H
tube
0
0
0.1
0.2
0.3
=1m
0.4
0.5
0.6
0.7
0.8
0.9
time, t [s]
As seen, closed tubes clearly reduce the suction height. Even a 10 m long
closed tube will reduced the height to about half that of open one.
Concluding remarks
Modeling capillary suction in wood is difficult. The nature of the tracheids
and their inter-connection to each other cannot be modeled by an open or
closed tube model. Because of e.g. friction and turbulence at the tube
boundaries there is a delay of the capillary rising (as compared to an
“ideal” tube). Perhaps one way to improve a model could be to use some
sort a retardation term (based on e.g. empirical data) in order to have a
more accurate time development.
40
1
References
Quéré D (1997). Intertial capillarity. Europhys. Lett. 39:5 pp. 533-538.
Quéré D, Raphaël É and Ollitrault JY (1999). Rebounds in a Capillary
Tube. Langmuir, 15, pp. 3679-3682.
Stange M, Dreyer ME and Rath HJ (2003). Capillary driven flow in
circular cylindrical tubes. Physics of fluids 15:9 pp. 2587-2601.
Szekely J, Neumann W and Chuang YK (1970). The rate of Capillary
Penetration and the Applicability of the Washburn Equation. Journal
fo Colloid and Interface Science, 35:2 pp. 273-278.
Washburn EW (1921). The Dynamics of Capillary Flow. The Physical
Review, vol. XVII:3 pp. 273-283.
Zhmud BV, Tiberg F and Hallstensson (2000). Dynamics of Capillary
Rise. Journal of Colloid and Interface Science 222 pp. 263-269.
41
42
Appendix - Matlab code
%---------------------------------------------------------function
wp=Capillary(lambda,w,tau_lambda,tau_ata,g,rho,p_atm,Htube)
;
%Closed tube
w1p = w(2);
w2p
=
-1/w(1)*w(2)^2-tau_lambda/tau_ata*w(2)-1+1/w(1)p_atm/(rho*g*(Htube-w(1)*g*tau_lambda^2));
%Not close tube
%w1p = w(2);
%w2p = -1/w(1)*w(2)^2-tau_lambda/tau_ata*w(2)-1+1/w(1);
wp=[w1p;w2p];
%---------------------------------------------------------%---------------------------------------------------------% Capillary dynamics
%close all;
clear all;
%---Parameters------------------------------------------%-------------------------------------------------------%Density
rho=1000;
%Viscosity
%ata=0.0728;
43
ata=1.01e-3;
%Surface tension
gamma=73e-3;
%Tube angle (degrees)
theta_C=0;
theta_C=theta_C*pi/180; %Tube angle (radians)
%Gravity
g=9.8;
%Radius
%R=800e-6;
R=0.0008;
%Atmospheric pressure
p_atm=101325; % Pa
1 atm = 101325 Pa
%Height of close tube
Htube=10; %
%-------------------------------------------------------%Stationary suction height
H=2*gamma*cos(theta_C)/(rho*g*R);
%Stationary suction height, closed tube
my=p_atm/(rho*g);
44
H_closed_tube=1/2*((Htube+H+my)-((Htube+H+my)^24*Htube*H)^(1/2));
%Introduced constants
tau_lambda=(H/g)^0.5;
tau_ata=rho*R^2/(8*ata);
%Beräkningspunkter
time_range=[0:0.001:1];
% s
lambda_range=time_range./tau_lambda;
%Startvärden
[w1_init,w2_init];
w1_init = 1e-20;
% ( =0
since z=0)
w2_init = 1;
w_init=[w1_init;w2_init];
[lambda,w]=ode45(@Capillary,lambda_range,w_init,[],tau_lamb
da,tau_ata,g,rho,p_atm,Htube);
% Time t and height z
t=lambda.*tau_lambda;
z=H*w(:,1)*1000;
% *1000 to get mm
%plotting
%close all;
plot([t(1)
tube
t(end)],[H
H]*1000,':g','linewidth',3);
%
Open
hold on;
plot([t(1)
t(end)],[H_closed_tube
H_closed_tube]*1000,':b','linewidth',3); % Close tube
plot(t,z,'k','linewidth',1.5);
45
xlabel('time, t
[s]');
ylabel('suction height, z
[mm]');
%title('Capillary suction - development in time');
title(['Capillary suction (H =',num2str(H*1000,3),' mm;
H_c_l_o_s_e_d _t_u_b_e =',num2str(H_closed_tube*1000,3),'
mm)']);
grid on;
%---------------------------------------------------------%---------------------------------------------------------%----------------------------------------------------------
46
Kapillär vätskestigning i korta rör
Lars-Elof Bryne, Kungl Tekniska Högskolan, KTH
Jimmy Johansson, Växjö universitet
Sammanfattning
En viktig egenskap hos trä är att det upptar och avger fukt i relation till
omgivningens relativa fuktighet och temperatur. Att förstå flödet i virket
är viktigt för att optimera t.ex. torkning, impregnering och ytbehandling.
Fuktinträngning i virke kan också leda till angrepp av bl.a. röta. Fukt i trä
kommer att röra sig olika snabbt i olika riktning i träet. Snabbast sker
transporten i träets longitudinella riktning dvs. längs med fibrerna. Detta
kallas kapillärt flöde.
I följande rapport studeras den kapillära transportmekanismen utifrån en
modell för kapillär transport i rör. Denna kopplas därefter till materialet
trä. Speciellt studeras vertikala rör som sätts i kontakt med vatten, och vad
som händer då dessa rör är kortare än den teoretiska kapillära stighöjden.
Modellerna studeras med Mathcad.
47
Inledning
Kapillär fukttransport i material som t.ex. trä är viktigt ur flera aspekter.
Den kapillära förmågan styr i det levande trädet hastigheten med vilken
trädets krona kan förses med vatten och när trädet väl fällts är
kapillariteten en styrande faktor för torkning av träet. Fenomenet är även
viktigt vid eventuell impregnering, ytbehandling, limning och behandling
med dimensionsstabiliserande kemikalier (se t.ex. Stamm 1964). Vidare är
kapillaritet viktigt för att förhindra eventuell vätskeinträngning i virkets
ändträ under användning, vilket kan leda till röta. Vid ytbehandling och
limning kan dessa kapillära fenomen med uppfuktning av trämaterialet
som följd så småningom leda till delaminering mellan träyta och
lim/ytbehandlingssystem.
Trä kan mycket förenklat ses som ett antal rör eller kapillärer av olika
storlek och längd som är sammanbundna med små öppningar i rörens
sidor. I en cylindrisk kapillär med ena änden i kontakt med en vätska
kommer vätskeytan att forma en menisk pga. ytspänning, figur 1, med en
kontaktvinkel θ mellan vätskan och kapillärrörets yta. Beroende på
storleken på krafterna mellan kapillärens yta och vätskan kommer ytan att
sjunka eller stiga i förhållande till vätskan utanför kapillären.
Vätskan kommer att lyfta om kontaktvinkeln, θ är mellan 0 och 90° och
sjunka om kontaktvinkeln är större än 90°. Att vätskeytan stiger benämns
med att vätskan "väter" materialet varför kontaktvinkeln även benämns
"vätningsvinkeln". För kontakten mellan vatten och trä antas att vatten
väter trä (Siau, 1995), dvs. 0<θ<90º.
I en vertikal kapillären kommer den kapillära stigningen uppnå ett
jämviktstillstånd (stationära tillståndet) då kontaktkraften mellan vätskan
och kapillärens yta (2πRγcosθ) balanseras av vätskans tyngd (πR²zgρ)
enligt ekvation 1.
48
Figur 1.
z=
Vätska som stiger respektive sjunker i en kapillär; (a)
Vätskan väter kapillärens yta och vätskan stiger (b)
Vätskan väter inte kapillären och vätskan sjunker (Siau,
1984).
2σ cos θ
r ρw g
(Ekv. 1)
där:
z = kapillär stigning (m)
σ = ytspänningen
(N/m)
θ = vätningvinkeln (°)
r = kapillärens radie (m)
ρw = densiteten för vätskan (vatten) (kg/m³)
g = tyngdacceleration (m/s²)
49
Kollmann och Côte (1968) anger att ytspänningen, σ för vatten kan
bestämmas enligt:
σ = 0, 0769(1 − 0, 00225 T )
(Ekv. 2)
där:
T = Temperaturen (°C)
För trä i kontakt med vatten kommer dock det icke stationära fallet att vara
speciellt viktigt för de applikationer som tidigare nämnts. I fallet trä
kommer även de kapillärer eller celler som bygger upp materialet att vara
av sådan längd och storlek att den kapillära stighöjden vid jämvikt
vanligen överstiger den enskilda cellängden. Detta gör det intressant att
studera vad som händer i det ickestationära fallet då vattenmenisken
närmar sig toppen på en kapillär. Eftersom trä även är ett material som
krymper och sväller i kontakt med vatten blir det också intressant att
studera vad som händer då radien på ett cellelement i kontakt med vatten
förändras i storlek.
Syfte
I följande studie ska en modell för den kapillära stigningen i det icke
stationära tillståndet skapas där kapillärernas längd är kortare än den
teoretiska kapillära stighöjden vid jämvikt. Modellen ska även kunna ta
hänsyn till att kapillärens radie varierar vid kontakt med vätskan.
Kapillär stigning i det icke stationära fallet
Det icke stationära tillståndet, dvs. då vätskan i en kapillär förflyttar sig
uppåt eller nedåt följer Newtons dynamiska ekvation för en viskös icke
kompressibel vätska enligt ekvation (3)
d
dz
8 ⋅η
dz
2 ⋅ σ ⋅ cos θ
⋅z⋅ + g⋅z −
=0
(z ⋅ ) +
2
dt
dt
dt
ρ ⋅r
ρw ⋅ r
där:
η = Vätskans dynamiska viskositet (Ns/m²)
50
(Ekv. 3)
Modellering av den kapillära stigningen
Kapillär stigning i långa rör
Från Ekvation 3 fås det stationära tillståndet (ekvation 1) då
dz
=0
dt
För att lösa ekvation 3 i det ickestationära tillståndet införs följande
konstanter:
τσ =
H
g
τη =
ρr ²
8η
ξ=
z
H
ν=
t
τσ
Ω=
τσ
τη
Ekvation 3 får därmed formen
ξ⋅
d
2
dν
2
2
 d ξ  + Ω ⋅ ξ⋅  d ξ  + ξ − 1


 dν 
 dν 
(4)
ξ+
Denna ekvation löses numeriskt med t.ex. Mathcad. Figur 2 beskriver
lösningen till ekvation 4 för kapillärer med radier 200 och 800 µm. Som
synes genererar större kapillärer ett något märkligt fenomen med någon
slags instabilitet eller översläng som dämpas med tiden.
51
Kapillär stighöjd (m)
0.06
0.04
0.02
0
0
0.2
0.4
0.6
0.8
a)
Tid (s)
b)
Kapillär stighöjd (m)
0.03
0.02
CH i
0.01
0
0
0.2
0.4
0.6
0.8
timi
Tid (s)
Figur 2.
Kapillär stighöjd som funktion av tiden för en kapillär med
radien a) 200 µm, b) 800 µm.
52
Kapillär stigning i rör som är kortare än den kapillära
jämviktsstighöjden
Om en kapillär som är kortare än den kapillära stighöjden i
jämviktstillståndet sätts i kontakt med en vattenyta kommer kapillären att
fyllas. I vätskemenisken som når toppen på röret kommer vätningsvinkeln
att öka och den uppåtriktade kapillära kraften kommer att minska och då
vätningsvinkeln är 90° kommer kraften vara 0. När sedan vätningsvinkeln
fortsätter att öka kommer menisken att dras tillbaka av ytspänningen. Det
kan därmed antas att menisken kommer att brytas och kapillären svämmas
över om hastigheten är större än noll då vätningsvinkeln är θB.
Söderström (2005) beskriver att då den första termen (inertia) i ekvation 3
försummas, kan hastigheten på menisken när denna når toppen på röret
(höjden L) beräknas enligt följande:
v ( L) =
2 ⋅ σ ⋅ r ⋅ cos θ − g ⋅ L ⋅ ρ ⋅ r ²
8 ⋅η ⋅ L
(Ekv. 5)
Menisken kan enligt Söderström (2005) antas ha formen av en sfär. Den
totala volymen på vätskekolumnen när höjden L har nåtts är πr²(L-xp), där
xp definieras som att volymen πr²xp är samma volym som den sfäriska
koppen enligt figur 3. xp kan då bestämmas enligt:
x p = xw ⋅
3 ⋅ r ² + xw ²
6 ⋅ r²
(Ekv. 6)
där xw bestäms enligt:
xw = r ⋅
1 − sin θ
cos θ
(Ekv. 7)
53
fb
xB
xw
fw
Figur 3.
Geometri hos en vattenmenisk som närmar sig toppen på
ett rör vid kapillär stigning (från Söderström 2005).
Menisken kommer vidare att brytas när volymen är πr²(L-xπ), där xπ och
xB definieras på samma sätt som i ekv. 6 och 7 och figur 3. Därmed fås:
xπ = x B ⋅
3 ⋅ r ² + xB ²
6 ⋅ r²
(Ekv. 8)
x B = −r ⋅
1 − sin θ B
cos θ B
(Ekv. 9)
Den kinetiska energin för vätskekolumnen ges enligt Söderström (2005)
av den första termen i ekvation 3. Integration med avseende på stighöjden,
z görs från L-xp till L-xπ hastigheten vid respektive integrationsgräns är v1
för L-xp och 0 vid L-xπ i annat fall så bryts menisken. Efter reduktion fås:
v12 −
−g⋅
4 ⋅ η x w ⋅ (3 ⋅ r ² + x w )² + x B ⋅ (3 ⋅ r ² + x B )²
⋅
⋅ v1
3⋅ ρ
r4
x w ⋅ (3 ⋅ r ² + x w )² + x B ⋅ (3 ⋅ r ² + x B )² 2 ⋅ σ ⋅ cos θ x w2 − x B2
+
⋅
=0
ρ⋅L
3⋅ r2
r2
(Ekv. 10)
54
Därmed kommer menisken att brytas när v(L) (från ekvation 5) är större
än v1. För trä som antas vätas av vatten kan θ vara 0°. Detta gör att xw blir
lika med r. Det antas därefter att översvämning sker då θB = 180°.
Ekvation 1, 5 och 10 ger då ett villkor på längden av den kapillär för att
menisken skall brytas enligt följande:
1
λ( r) :=
128⋅ Ch( r)
1+
3


⋅ 1 +
1+
3

32⋅ Ch( r) 
(Ekv. 11)
där:
Ch( r) :=
η
2
2
3
ρ ⋅ gr⋅ r
0.04
Stighöjd (m)
0.03
0.02
0.01
1 .10
6
1 .10
5
1 .10
4
1 .10
3
Kapillär radie (m)
Villkor för brytning av vattenmenisken
Kapillär stighöjd
Figur 4.
Den kapillära stighöjden och villkoret för att menisken
skall brytas som funktion av kapillärradie.
55
0.01
Kapillär stigning i en kapillär som förändras vid vattenkontakt
Enligt Siau (1995) är det valigaste att vid studier av krympning och
svällning anta en konstant radie för cellumen. Detta antagande gäller dock
inte för alla träslag. Hos vissa träslag minskar lumens radie och hos andra
ökar densamma vid en ökad fuktkvot. En tänkt grund för en modell som
tar hänsyn till detta fenomen, dvs. med ökning alternativt minskning av
radien beroende på träslag diskuteras i avsnitt 4.
Diskussion
En förutsättning för ett kapillärt fuktflöde i trä är att materialet kan
betraktas som permeabelt vilket kräver att materialet är poröst men också
att hålrummen i materialet är sammanbundna. Detta kommer inte alltid att
vara sant för trä, då poröppningar kan vara aspirerade och extraktivämnen
och tyllbildning kan täppa till cellerna (Dinwoodie, 2000, Hart & Thomas,
1967, Stamm, 1964).
Trä har bäst permeabilitet i sin longitudinella riktning. Speciellt i
splintveden i lövträ kommer den longitudinella permeabiliteten vara hög. I
lövträ kommer enligt Dinwoodie (2000) den transversella permeabiliteten
i många fall vara mindre än i barrträ. I lövträ är den tangentiella och
radiella permeabiliteten lika eftersom märgstrålarna har relativt låg
genomsläpplighet. Detta skiljer sig åt från barrträ där permeabiliteten är
något högre i den radiella riktningen jämfört med den tangentiella
riktningen (Stamm 1964). Richter och Sell (1992) studerar silvergran och
visar att märgstrålarna för detta träslag jämfört med tall har liten betydelse
för vätskeflödet. Vätsketransportsegenskaper är således något som skiljer
sig mycket åt beroende på träslag. I barrträ sker den longitudinella och
tangentiella transporten enligt Dinwoodie (2000) via gårdade porer. Vid
torkning är det vanligt att dessa gårdade porer aspirerar dvs. stängs igen
vilket leder till att virkets permeabilitet blir lägre efter torkning. I lövträ
saknas dessa gårdade porer vilket leder till att torkningen inverkar lite på
permeabiliteten.
I lövträ är kärlen den viktigaste celltypen för transport. I denna celltyp har
ändarna i stort sett försvunnit och när cellerna placeras ovanför varandra
bildas ett effektivt rör med relativt stor diameter, upp till 0,5 mm i vilken
vätsketransport kan ske med kapillära krafter. Kärlen i lövträd går dock
56
inte perfekt longitudinellt utan avviker mer eller mindre från sin axiella
bana (Zimmermann, 1983). I barrträ är den dominerande celltyper
trakeider (90 %) och dessa står för merparten av den longitudinella
transporten. Trakeiderna har längden 2–4 mm och diametrarna 10–20 µm
(Dinwoodie 2000).
Cellernas storlek kommer att vara styrande för effektiviteten på
vätsketransporten och stora kärl ger en effektiv vätsketransport
(Dinwoodie, 2000; Zimmerman, 1983). Samtidigt tycks det enligt
Zimmermann (1983) finnas en övre begränsning på innerdiametrarna på
celler hos olika träslag på ca 0,5 mm. Detta antas bero på att stora kärl
också leder till minskad säkerhet eftersom effekten av en skadad cell, dvs.
en avbruten vätsketransport ökar med ökad cellstorlek.
Ringporiga träslag har vanligtvis stora kärl och en hög permeabilitet
medan ströporiga har mindre kärl och lägre permeabilitet. Ringporiga
träslag med stora kärl bildar dock ofta tyll i kärnveden vilket ger en
försämrad permeabilitet. Variationer i permeabilitet beror även på hur
poröppningarna mellan ändarna på kärlen ser ut (Dinwoodie, 2000).
Cellerna i trä är sammanbundna med porer. Torkning kan ofta leda till att
det tillbakadragande vattnet orsakar spänningar i cellerna vilket gör att de
gårdade porerna stänger sig eller aspirerar (Hart och Thomas 1967).
Morén (1993) beskriver att en färsk virkesyta är mycket reaktionsbenägen
och att poraspiration kan starta i ytskiktet på virke redan i den initiala
uppvärmningsfasen inför torkningen. Så länge porerna är öppna vilket de
oftast är i lövträ eftersom gårdade porer är sällsynta kommer det
longitudinella flödet inte begränsas av porerna (Dinwoodie, 2000), men
flödet kommer att bli trögare. Schulte och Gibson (1988) visar att porer
står för 12-70 % av det totala flödesmotståndet av vätska genom en cell till
nästa.
Det är även vanligt att det speciellt i kärnved finns extraktivämnen och i
lövträ som ek eventuell tyllbildning som förhindrar eller minskar
genomsläppligheten av vätska (Dinwoodie 2000, Siau 1995). För att få ett
mått på det kapillära flödesmotståndet använder t.ex. Choong och Tesoro
(1989) en effektiv kärlradie som i lövträ bestäms till 0,2–1 µm och i
barrträ en effektiv cellradie 0,1–0,2 µm. Med effektiv kärl- respektive
cellradie avses den radie som deltar i transport av vätska.
57
Extraktivämnena, som till viss mån, kan ses som ytaktiva medel kan
urlakas från cellväggen och sänka ytspänningen på vätskan, vid
kontakvinkelmätningar är detta ett känt fenomen (Wålinder, 2002). Vilken
effekt som detta skulle kunna ha på kapillära fenomen diskuteras av Zmud
(2000). En ökning av halten ytaktiva medel sänker överslängen som så
småningom försvinner helt. Detta skulle kunna innebära att om den
kapillära drivkraften inte påverkas lika mycket av den sänkta ytspänningen
så kommer menisken att brista vid vinklar enligt: 90 o < θ B < 180 o .
Vid uppfuktning av cellväggen sker antingen en ökning eller en minskning
av celldiametern (Siau, 1995). Enligt figur 2 borde då en ökning av
celldiametern resultera i ett förlopp för den kapillära stighöjden liknande
det i figur 2 b. Vid en minskning av celldiametern, som följd av
uppfuktning, skulle den kapillära stighöjden kunna anta det asymptotiska
förloppet som illustreras i figur 2 a. Instabiliteten som illustreras i figur 2 b
är dock nog med stor sannolikhet ej förekommande i trä, det krävs alldeles
för stora radier för att detta fenomen ska uppträda.
Slutsats och fortsatta studier
Studien indikerar att Newtons dynamiska ekvation är en bra utgångspunkt
för beskrivningen av ett kapillärt flöde. Modellens koppling till trä är dock
outvecklad med avseende på vissa faktorer hos trä t.ex. pormotståndet,
extraktivämnen och tyllbildning. Modellen för meniskens brytande i korta
kapillärer har ett relevant intresse vid vattnets transport genom
poröppningar. En intressant frågeställning är vad som händer om
ytspänningen för vatten skulle sänkas på grund av migrering av
extraktivämnen; kommer menisken att brista vid vinklar där θ B <180º?
Enligt modellen kommer smalare kapillärer antar ett asymptotiskt förlopp
för stighöjden och större kapillärer får en översläng eller oscillation.
I fortsatta studier bör översvämning av rören kopplas till materialet trä
genom studier av exempelvis vätning av ovansidan på träbitar som utsatts
för kapillärt vatten. Själva transportmekanismen bör också studeras på
cellnivå och speciellt hur vätska transporteras från en cell till nästa.
Problemet är här att hitta en lämplig testmetod. Även studier av vad som
händer vid varierande ytspänning är av stort intresse.
58
Referenser
Choong, E.T. och Tesoro, F.O. (1989) Relationship of capillary pressure
and water saturation in wood. Wood Science and Technology, 23:139–150
Dinwoodie, J.M. (2000) Timber: Its nature and behaviour. E & F N Spon,
London, ISBN: 0-419-25550-8
Hart, C.A. och Thomas, R.J. (1967) Mechanism of bordered pit aspiration
as caused by capillarity. Society of Wood Science and Technology, SWST
research paper, No. 19
Kollmann, F.F.P. och Côte W.A.Jr. (1968) Principles of wood science and
technology: I Solid Wood. Springer-Verlag
Morén, T. (1993) Creep, deformation and moisture redistribution during
air convective wood drying and conditioning. Luleå tekniska universitet,
Luleå
Richter, K och Sell, J. Untersuchung der kapillaren Transportwege im
Weisstannenholz (Studies of impregnation pathways in white fir (Abies
alba)). Holz als Roh- und Werkstoff, 50(9):329–336
Schulte, P.J. och Gibson, A.C. (1988) Hydraulic conductance and tracheid
anatomy in six species of extant seed plants. Canadian Journal of Botany,
66:1073-1079
Siau, J.F. (1995) Wood: Influence of moisture on physical properties.
Department of Wood Science and Forest Products, Virginia Polytechnic
Institute and State University. ISBN: 0-9622181-0-3
Stamm, A.J. (1964) Wood and cellulose science. The Ronald Press
Company, New York
Söderström, O. (2005) Dynamic capillary transport of water in porous
building materials. Nordic Symposium on Building Physics, Reykjavik
13-15 June 2005
59
Zhmud, B.V. (2000) Dynamics of Capillary Rise. Journal of Colloid and
Interface Science 228, 263-269
Zimmermann, M.H. (1983) Xylem structure and the ascent of sap.
Springer-Verlag, ISBN: 3-540-12268-0
60
Simulering av fukttransportvägar vid virkestorkning
samt vätskeinträngning i ved vid massakokning –
med tillämpning av perkolationsteori
Johan Sjödin, Växjö universitet
Lars Eliasson, Växjö universitet
Inledning
Perkolation betyder allmänt genomsilning som när ett lösningmedel får
passera genom en bädd av finfördelade ämnen. Exempel på sådan
användning är utvinning av vissa läkemedel med hjälp av alkohol ett annat
är kaffebryggning. Begreppet perkolation används oftast i samband med
vatten och hur detta silar genom marklagren ner mot grundvattennivån.
Perkolationsteori är ett sätt att modellera komplexa växelverkande
mångpartikel-system, som t ex ett bisamhälle, genom att förenkla
växelverkan mellan individerna och sedan låta systemet utvecklas genom
en simulering. Vanligen studerar man fenomen tvådimensionellt.
Perkolationsteori kan användas för att simulera skogsbränder,
oljeutvinning, kolonisation av kontinenter, torkning/uppfuktning av trä
(över fibermättnadspunkten) etc.
Simuleringsmodeller
Varje cell i en matris M är antingen tom (0) eller fylld (1) och placeringen
av nollor och ettor är slumpmässig oberoende av innehållet i granncellen,
figur 1. Sannolikheten för en fylld cell är p vilket innebär att i en matris,
med ett stort antal celler N, är antalet fyllda celler pN och antalet tomma är
(1-p)N.
Fyllda celler med grannar ”kant-i-kant” som också är fyllda bildar
tillsammans ett kluster. Dessa kluster eller öar i matrisen (nätverket)
61
representerar ett sammanhängande område. Cellerna är närmsta grannar
enbart om de har kantsidor och inte endast hörn som gränsar till varandra.
1
1
1
1
1
1
1
1
a)
Figur 1.
b)
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
c)
Defininition av perkolation och kluster, a) visar en del av ett
kvadratiskt nätverk, b) några av cellerna är fyllda, c)
sammanhängande celler (kluster) är markerade med olika
färg.
Det vanligaste mönstret man utgår från är kvadratiskt men andra tänkbara
mönster kan vara 6-kanter (bikakemönster) eller trianglar, figur 2. Dessa
mönster kan även genereras i 3D eller fler dimensioner.
Figur 2.
Exempel på olika utgångsmönster i 2D vid simulering.
62
1
Vid generering av ”fyllda” nätverk med olika värden på p kan 2Dmatriserna åskådliggöras enligt figur 3 som saknar klusterbildning från
kant till kant samt figur 4 och 5 med sammanhängande kluster från en kant
till motstående kant.
Figur 3.
Matrisstorlek 50*50, p=0,4948
63
Figur 4.
Matrisstorlek 50*50, p= 0,5968
Figur 5.
Matrisstorlek 50*50, p= 0.7140
64
Identifiering av kluster
Matrisens celler är slumpvis fyllda med ettor och nollor enligt ett bestämt
värde på p. Vid tillämpningar av mindre storlekar på nätverk blir det
faktiska värdet på p inte exakt det värde som sätts inledningsvis. På det
genererade nätverket identifieras sedan sammanhängande kluster och
ingående celler får ett för klustret unikt nummer. I detta fall har enbart 2Dnätverk tillämpats.
Klusteridentifikationen görs genom att varje cell i nätverket stegas igenom
för kontroll av status och motsvarande görs hos närmsta grannar. Första
gången en cell påträffas i ett nytt kluster får denna ett nummer som sedan
allteftersom tilldelas alla celler som tillhör detta kluster. Vid stora nätverk
är denna process tidskrävande. Beräkningstiden stiger snabbt med ökad
storlek på nätverket. Hoshen & Kopelman (1976) har beskrivit en algoritm
som gör identifikationen av kluster mindre tidskrävande.
Det finns ett kritiskt värde på p kallat den kritiska sannolikheten pc och i
ett stort nätverk bildas, under detta värde, inga kluster som når från en
kant till motstående kant. Vid p >= pc finns det en möjlighet för bildning
av ett klusterområde som är genomgående.
Staufer (2003) anger pc för en tvådimensionell infinit matris med
kvadratiskt mönster till 0,592746. Vid identifiering av kluster i ett finit
nätverk kan kluster bildas från kant till kant även vid värden på p som
ligger under det kritiska värdet och för mindre nätverk redan vid värden på
p som ligger långt under det kritiska värdet, figur 6.
Speciellt intressanta är de kluster som når från kant till kant.
Åskådliggörandet av bildade kluster kan göras med utskrift av cellnummer
och/eller färgläggning av klustrena i matrisen.
65
Simulations: 1004
Lattice: 50*50 square
pmin=0,5324
pmax=0,6368
Infinit lattice
pc = 0,592746
p
Figur 6.
Kluster från kant till kant kan uppstå även vid värden på p
under pc i ett finit nätverk
66
Perkolationsteori tillämpad på trä
Identifiering av kluster ”kant-till kant” i ett virkesstycke
Varje cell i nätverket antages representera en vedfiber i ett tvärsnitt av ett
trästycke. Den slumpmässiga fördelningen av fyllda celler utgör en tänkt
fördelning av celler med fritt vatten i lumen. Tomma celler representerar
fibrer som inte innehåller fritt vatten utan enbart vatten bundet till
fiberväggarna. Kopplingarna, eller övergångarna, mellan cellerna utgörs
av porerna i cellväggarna.
Vid genomförandet av denna simulering har inte mängden vatten i fibrerna
modellerats utan en fylld cell i nätverket motsvarar en vedfibrer som är
mer eller mindre fylld med fritt vatten. Mer eller mindre fyllda celler
representeras av 1 och tomma celler av 0. Fyllda celler som har närmsta
grannar som också är fyllda bildar kluster. Dessa kluster representerar
områden som har fibrer med fuktkvoter över fibermättnadsnivån. Speciellt
intressanta kluster är de som når från en kant till motstående kant. Dessa
genomgående kluster bildar ett sammanhängande vätskeledningssystem
(kapillärt nätverk) som effektivt leder vattnet till (eller från) virkesytorna.
Vid simulering av virkestorkning används ofta en diffusionsmodell för att
beskriva torkförloppet i splintved även vid fuktkvoter över
fibermättnadsnivån och Salin (2006a, 2006b och 2006c) använder istället
perkolationsmodellen för att modellera torkförloppet. Resultaten från
dessa simuleringar antyder att perkolationsmodellen stämmer bättre
överens med experimentella erfarenheter än den traditionellt använda
diffusionsmodellen.
Vätskeinträngning vid kokning av ved till pappersmassa
I det 2D-nätverket representerar varje cell en vedfiber och inledningsvis
finns inget fritt vatten i vedfibrerna, cellerna i nätverket är tomma. Mellan
vedfibrerna finns öppningar och i nätverket genereras storleken på dessa
slumpmässigt. Vid simulering av vätskeinträngning beaktas enbart den
största öppningen i varje cell, figur 7.
67
Figur 7.
Matris 50*50. Inträngning sker från samtliga sidor.
Delförstoringen till höger visar fyra celler med slumpvisa
storlekar på öppningarna i cellväggarna.
Vid simulering av vätskeinträngning kan öar bildas inne i nätverket som
inte blir fyllda. Dessa öar kan tänkas motsvara de delar av ett virkesstycke
som kan vara svåra att fukta upp (eller torka ut).
68
Resultat och diskussion
Identifiering av kluster i ett virkesstycke
Identifieringen av bildade kluster har gjorts med en algoritm som tagits
fram med hjälp av MatLab. Den algoritm som beskrivits av Hoshen &
Kopelman (1976) har inte tillämpats mer än till viss del men skulle
troligen vara nödvändig vid simuleringar på större nätverk. Den framtagna
algoritmen har i detta fall inte provats med matrisstorlekar över 200*200
eftersom beräkningstiden då blir alltför lång. Den kod som har tagits fram
finns i bilaga 1.
Beräkningsgång
1. En matris M (nr*nr) fylls med 0 eller 1 med hjälp av
slumptalsgenerering.
Sannolikheten för en fylld cell anges i kod med faktor.
2. Matrisen stegas sedan igenom där varje fylld cell kontrolleras om
den har någon närmsta fylld granne. Cell (M(1,1)), översta raden i
M (M(1,1:nr)) och den vänstra kolumnen (M(1:nr,1) behandlas
separat. Denna kontroll sker med ett antal if-satser.
3. Slutligen genereras en graf med unika nummer på cellerna inom
respektive kluster.
4. Med olika värden på p ges en uppfattning om storleken på pc i ett
finit nätverk
Efter ett antal upprepningar av klusteridentifikationen blir resultatet enligt
figur 8 och figur 9. Med värden på p som närmar sig pc är sannolikheten
större att det bildas kluster som når från en kant till motstående kant. För
ett infinit nätverk bildas det inga ”kant-till-kant” kluster vid värden på p
under pc.
69
Simulations: 1004
Lattice: 50*50 square
pmin=0,5324
pmax=0,6368
pc = 0,592746
Figur 8.
Klusterbildning från kant till kant i ett finit nätverk 50*50 vid
olika värden på p. Varje punkt i grafen anger andelen av
antalet simuleringar för respektive värde på p som gett ett
sammanhängande kluster från en kant till motstående
kant.
Simulations: 200
Lattice: 200*200 square
pmin=0,48897
pmax=0,66905
pc = 0,592746
Figur 9.
Klusterbildning från kant till kant i ett finit nätverk 200*200
vid olika värden på p.
70
Vid jämförelse av simuleringsresultaten ovan kan konstateras att redan en
matris med storlek 200*200 resulterar i en kurva som närmar sig värdet på
pc (som gäller för ett infinit nätverk).
I det eller de kluster som bildas från kant till kant bidrar vissa vägar mer
till vätsketransport från/till de inre delarna av nätverket. Några vägar är
återvändsgränder och tillför inget till vätsketransporten, figur 10 a. De inre
områden inom klustret med gemensam in och utfart har redigerats bort och
enbart huvudlederna är kvar i figur 10 b.
Totala antalet celler som ingår i det sammanhängande klustret är 712.
Efter redigering med borttagande av celler och områden som inte bidrar
till vätsketransport från/till de inre delarna är antalet aktiva celler 393,
eller 55 % av ursprunglig storlek.
a)
Figur 10.
b)
Matrisstorlek 50*50, p= 0.576.
a) visar klusterbildning mellan kanter, totalt ingår 712 celler
i klustret.
b) visar de vägar av betydelse för vätsketransport till/från
de inre delarna till ytterområdena där aktiva celler utgör
55 % av det från början sammanhängande området.
71
Vid genomräkning med olika värden på p blir resultaten att större eller
mindre kluster bildas, figur 11 och figur 12.
Figur 11.
Matrisstorlek 50*50, p= 0.51, saknar kluster från kant till
kant
72
Figur 12.
Matrisstorlek 50*50, p= 0.62, sammanhängande kluster
från kant till kant(er).
73
Inträngning av vätska i en massakokningsprocess
Simuleringen av kokvätskans inträngning har genomförts med en kod som
tagits fram med hjälp av Matlab. Den använda Matlab-koden återfinns i
bilaga 2.
Beräkningsgång
1. Ett fyrkantigt 2-dimensionellt nätverk (Figur 13) skapades. Varje
fiber får ett nummer tilldelat samt 4 stycken öppningar med
slumpvis valda storlekar.
2. Vätska tilldelas till randen (cellerna i nätverket fylls) längs alla
fyra sidorna.
3. I koden anges det antal steg (loopar) som programmet ska köras.
4. I huvudloopen finns det ett antal if-satser, dvs. villkorssatser.
Första villkors- satsen bygger på vilka steg som ska plottas.
Därefter kommer en sats som plockar vilka fibrer som är fyllda
med vätska. Dessa fibrer går sedan in i en ny villkorssats där
största
öppningen
tas
fram.
Är öppningen mot en tom fiber så blir den tomma fibern fylld. Om
öppningen istället är vänd mot en fylld eller mot någon kant så
händer inget utan programmet startar om på nytt, dvs. ett nytt steg
startas (Obs om detta inträffar så ändras storleken på den
öppningen till 0. Skulle detta inte göras så skulle programmet bara
köra på utan att någon annan fiber blir fylld). Programmet är i detta
utförande något oeffektivt då det måste starta om varje gång den
minsta öppningen är vänd mot en redan fylld fiber.
I figur 13 återfinns resultat ifrån en simulering. Det man kan se är att
fukten i de första stegen är ganska jämt utbredd, men att det sedan bildas
vägar där vätskan tränger igenom. Vid fyllning av en cell kan som mest tre
nya vägar öppnas där vätskan kan tränga in. Dessa nya öppningar där
storleken varierar slumpmässigt påverkar resultatet på så sätt att fukten
hittar en väg som den sedan följer.
74
Figur 13.
Steg 1
Steg 377
Steg 937
Steg 1791
Vätskeinträngning vid massakokning. Stegen anger antalet
loopar i programmet. Vid simuleringen bildas öar där
vätska inte tränger in. Vid aktuell simulering var merparten
av nätverket fyllt efter 1791 loopar.
75
Om man lägger till ett villkor som innebär att öppningen måste ha en
speciell storlek för att vätska ska kunna passera så inträffar en intressant
sak. Inträngningsförloppet stoppas upp om kravet är att öppningen måste
ha en viss storlek för att inträngning ska kunna ske. Detta fenomen kan
jämföras med tryckimpregnering av gran vilket svårligen låter sig göras.
Svårigheten att ”blöta upp” granveden inverkar positivt vid användning i
utomhusmiljö ovan mark.
76
Referenser
Salin J-G. 2006 a. Drying of sapwood analyzed as an invasion percolation
process. Maderas. Ciencia y technologia 8(3): 149-158, 2006
Salin J-G. 2006 b. Modelling of the behaviour of free water in sapwood
during drying. Part I. A new percolation approach. Wood Material
Science & engineering vol 1, No 1, 2006
Salin J-G. 2006 c. Modelling of the behaviour of free water in sapwood
during drying.Part II. Some simulation results. Wood Material Science &
engineering vol 1, No 1, 2006
Staufer D. Aharony A. 2003. Introduction to Percolation theory. Taylor &
Francis Group. London and New York.
Övrig litteratur
Pärt-Enander E. Sjöberg A. 2003. Användarhandledning för Matlab 6.5.
Uppsala universitet.
Hoshen J. Kopelman R. 1976. Phys. Rev. B, 14, 3428.
77
Appendix, Matlabkod
Identifiering av kluster
%---------------------------------------------------------------echo off
clear all;
clf;
home;
%--------Variabler-----------------------------------nr=50;
p=0.60
%----- System matrices -----------------------------------------Antal=1;
faktor=p-0.5;
% for o=1:nr
M=zeros(nr,nr);
V=zeros(nr,nr);
k=1;
for i=1:nr
for j=1:nr
M(i,j)=round(rand+(faktor));
Ex(k,:)=[j-1 j j j-1];
Ey(k,:)=[-(i-1) -(i-1) -i -i];
78
k=k+1;
end
end
k=1;
for i=1:nr
for j=1:nr
if i>1 & j>1
if M(i,j)>=1
V(i,j)=k;
k=k+1;
if (M(i-1,j)>=1) & (M(i,j-1)>=1)
aa=max([V(i-1,j) V(i,j-1)]);
bb=min([V(i-1,j) V(i,j-1)]);
V(i,j)=min([V(i-1,j) V(i,j-1)]);
for q=1:nr
for qq=1:nr
if V(q,qq)==aa
V(q,qq)=bb;
end
end
end
79
elseif M(i-1,j)>=1
V(i,j)=V(i-1,j);
elseif M(i,j-1)>=1
V(i,j)=V(i,j-1);
end
end
elseif i==1 & j>1 %--------------------------första raden utom (1,1)
if M(i,j)>=1
V(i,j)=k;
k=k+1;
if (M(1,j-1)>=1)
aa=max([V(1,j) V(1,j-1)]);
bb=min([V(1,j) V(1,j-1)]);
V(1,j)=min([V(1,j) V(1,j-1)]);
for q=1:nr
for qq=1:nr
if V(q,qq)==aa
V(q,qq)=bb;
end
end
end
elseif M(i,j-1)>=1
80
V(i,j)=V(i,j-1);
end
end
elseif i>1 & j==1 %--------------------------första kolumnenen utom M(1,1)
if M(i,j)>=1
V(i,j)=k;
k=k+1;
if M(i-1,j)>=1
V(i,j)=V(i-1,j);
end
end
elseif i==1 & j==1 %---------------------------M(1,1)
if M(i,j)>=1
V(i,j)=k;
k=k+1;
end
end
end
end
% --------------------------härslutar loop för j=1:nr
% --------------------------härslutar loop för i=1:nr
81
%----------------Utskrift av "nya" cellnummer
set(gcf,'DefaultTextColor','black')
title('\fontsize{20}\bf Exercise 0')
set(gcf,'DefaultTextColor','white')
k=1;
for i=1:nr
for j=1:nr
Exx=mean(Ex(k,:));
Eyy=mean(Ey(k,:));
if V(i,j)~=0
tal=int2str(V(i,j));
text(Exx-0.4,Eyy,tal,'FontSize',8);
end
k=k+1;
Exx;
Eyy;
end
end
T=0;
for i=1:nr
for j=1:nr
T=M(i,j)+T;
end
82
end
%--------------- beräkning av aktuellt p
p=T/(nr*nr)
%-------------- Färgläggning av bildade kluster
k=1;
for i=1:nr
for j=1:nr
if V(i,j)~=0
patch(Ex(k,:),Ey(k,:),sqrt(sqrt(V(i,j))))
end
k=k+1;
end
end
%-------------------------- end -----------------------------
83
Matlabkod
Vätskeinträngning vid massakokning
%---------------------------------------------------------------echo off
clear all;
%----- nätverket skapas -----------------------------------------nr=50; % nr*nr stort nätverk
M=zeros(nr,nr);
V=zeros(nr,nr);
k=1;
for i=1:nr
for j=1:nr
Ex(k,:)=[j-1 j j j-1];
Ey(k,:)=[-(i-1) -(i-1) -i -i];
k=k+1;
end
end
M(2,:)=ones(1,nr)*2;
M(nr-1,:)=ones(1,nr)*2;
M(:,2)=ones(nr,1)*2;
M(:,nr-1)=ones(nr,1)*2;
84
M(1,:)=ones(1,nr);
M(nr,:)=ones(1,nr);
M(:,1)=ones(nr,1);
M(:,nr)=ones(nr,1);
%----- slumpning av storlek på öppningar----------------------------------kk=1;
k=1;
for i=1:nr
for j=1:nr
v(k,:)=[rand rand rand rand];
xy(k,:)=[k,i,j];
k=k+1;
kk=kk+1;
end
end
kkk=1;
number=6000; %----Antalet loopar
verklig_nummer=1;
%----- Körningen startar----------------------------------for iii=1:number
if (iii==1)| (iii==number/5) | (iii==number/2) | (iii==number)
k=1;
subplot(2,2,kkk)
85
for i=1:nr
for j=1:nr
hold on
patch(Ex(k,:),Ey(k,:),-M(i,j))
k=k+1;
end
end
tal=int2str(verklig_nummer);
title(tal)
kkk=kkk+1;
end
kk=1;
k=1;
for i=1:nr
for j=1:nr
if M(i,j)==2
nr_pl(kk)=k;
vv(kk,:)=v(k,:);
kk=kk+1;
end
k=k+1;
end
end
86
a=vv(1,1);
for i=1:kk-1
for ii=1:4
if vv(i,ii)>a
a=vv(i,ii);
nr_pl_=nr_pl(i);
plats=ii;
end
end
end
if plats==1
ny_nr=nr_pl_-1;
x=xy(ny_nr,2);
y=xy(ny_nr,3);
if M(x,y)==0
M(x,y)=2;
verklig_nummer=verklig_nummer+1;
else
v(nr_pl_,1)=0;
end
end
87
if plats==2
ny_nr=nr_pl_-nr;
x=xy(ny_nr,2);
y=xy(ny_nr,3);
if M(x,y)==0
M(x,y)=2;
verklig_nummer=verklig_nummer+1;
else
v(nr_pl_,2)=0;
end
end
if plats==3
ny_nr=nr_pl_+1;
x=xy(ny_nr,2);
y=xy(ny_nr,3);
if M(x,y)==0
M(x,y)=2;
verklig_nummer=verklig_nummer+1;
else
v(nr_pl_,3)=0;
end
end
88
if plats==4
ny_nr=nr_pl_+nr;
x=xy(ny_nr,2);
y=xy(ny_nr,3);
if M(x,y)==0
M(x,y)=2;
verklig_nummer=verklig_nummer+1;
else
v(nr_pl_,4)=0;
end
end
end
89
90
Analysis of deformations and section forces due to
moisture content variation in wooden structures
Kirsi Salmela, SP Trätek
Johan Vessby, Växjö University
Introduction
Wooden structures are often subjected to moisture induced stresses or
deformations. The reason for this may be varying temperature and relative
humidity in different parts of the structure. One such case is shown in
figure 1, where the moisture content in the roof structure initially was 18
% in the whole structure and then changed to 12 % in the inner parts, on
the attic, and 5 and 8 % in the parts facing the outside and living areas
respectively.
The aim with this current project was to analyze the deformations and
section forces caused by the moisture variation in the structure. The
behaviour due to moisture variation may affect a structure adversely when
subjected to further loading by for instance snow or wind loading.
91
Figure 1.
Geometry and moisture content in the different parts of the
studied roof structure.
Implementation
The task was solved by using Matlab and the finite element toolbox
CALFEM. The structure was modelled with the two dimensional beam
element routines for linear static analysis. The two routines beam2e.m,
used to obtain the global stiffness matrix, and beam2s.m, used to obtain
the section forces T, N and M (shear force, normal force and moment) had
to be modified to take the moisture loading into account. They were
modified to be able to manage linear variation of the moisture content, w,
linear variation of the modulus of elasticity, E, and a constant shrinkage
coefficient, α, over the rectangular cross- sections of the members in the
frame A and the specific moment of inertia I.
The initial moisture content of the whole structure was assumed to be
18%. This reference moisture content was assumed to decrease so that a
linearly varying steady state moisture content over the member
cross-sections were obtained. The moisture content was changed to 12 %
92
in the inner parts, on the attic, and to 5 and 8 % in the parts facing the
outside and living areas respectively.
The member cross-section dimensions and the moisture content variations
over the members of the structure are shown in Figure 1. The linear
variation of the modulus of elasticity over the cross-section was assumed
to be the same for all five members. The variation were assumed to be
dependent on the pith distance r as 10+100*r GPa giving the modulus of
elasticity 10 GPa at the edge where the pith is located. In the calculations
the shear deformation of the beams where not considered. The shrinkage
coefficient α of the wood was set to 0.01.
The model assembly of the roof structure with 5 finite beam elements and
totally 12 degrees of freedom is shown in 2.
The system of equations that has to be solved in order to consider the
moisture variation may be formulated as follows;
K ⋅ a = f , f = f l + f 0u + f 0w
b
f = ∫ N eT qdx
e
l
a
b
0
f 0eu = ∫ B eT
u EAε dx
a
Figure 2.
Finite element assembly of the wooden roof structure. Five
finite elements are used.
93
b
f 0ew = ∫ BeT
w EIκ 0 αdx
a
where K is the global stiffness matrix of the assembled structure, a are the
unknown displacements and rotations and f is the total load vector. The
load vector can be divided into three different parts, where fl is the load
due to external loading of the structure as for instance point loads or
distributed loads, f0u is the load vector acting in the axial direction due to
the internal moisture state, and finally f0w is the load vector acting on the
torsional degrees of freedom due to the change in curvature by the
moisture variation.
The extra contribution to the load vector due to the normal strain and the
curvature respectively is based on the derivatives of the shape functions
and can be written in vector form as follows;
f 0eu
− E ⋅ A ⋅ α ⋅ ∆ω0 


0




0
=
,
0 
⋅
⋅
α
⋅
∆
ω
E
A




0


0


f 0ew
0




0


− E ⋅ I ⋅ κ0 ⋅ α 
=

0




0


 E ⋅ I ⋅ κ 0 ⋅ α 
κ0 is the direction of the moisture variation and is defined as
κ0 =
∆ω2 − ∆ω1
h
where ∆ω1 and ∆ω2 are the difference in moisture content to the initial
reference moisture content of 18 %.
If the modulus of elasticity is varied over the cross-section the f0u and f0w
load vectors has to be complemented with factors that takes the
displacement of the neutral axis into account.
94
Model verification
The modified finite element beam routines were verified by modelling a
6 m long beam divided into two finite elements as shown in the top of
Table 1. The dimensions of the beam were set to 45 x 150 mm2. The
boundary conditions, moisture content ω and modulus of elasticity E were
varied to study different effects on the beam. The shrinkage coefficient
α = 0.01 was assumed to be constant in all of the calculations. The
displacements at the degrees of freedom and section forces N normal
force, T shear force and M moment are presented in vector and matrix
form in the left column together with a deformation figure.
In load case 1 the modulus of elasticity was assumed to be 10 GPa over
the cross-section. The moisture content change was assumed to be
uniform, ∆ω = -0.10, over the whole beam. The axial normal force
induced by the moisture change for a beam fixed at both ends is easily
checked by hand calculations by using the equation
F = E ⋅ A ⋅ α ⋅ ∆ω
In the verification model the beam is modelled with fixed support
conditions at both ends meaning that the translational degrees of freedom
1, 2, 7 and 8 are fixed. The calculated induced axial normal force by the
moisture content change is 67.5 kN for both the hand calculation and the
verification model. If the moisture content is changed to a nonuniform
variation over the beam with a moisture content of ∆ω = -0.10 on one side
and ∆ω = -0.05 on the other, the induced axial normal force at the neutral
layer, at the centre of the cross-section, will change to 50.6 kN and the
moment will be 422 Nm. The corresponding boundary condition for the
verification model is obtained by preventing rotations at both supports.
The calculated section forces are the same made by hand calculation and
by the finite element model as shown in load case 2 in Table 1.
In load cases 3, 4 and 5 the modulus of elasticity was still constant and the
moisture content ω over the cross-section still had a nonuniform variation
with ∆ω = -0.10 on one side and ∆ω = -0.05 on the other, but the
distribution was reversed in the two finite beam elements as shown in the
figure on the left in table 1.
95
In load case 3 the boundary conditions were chosen to give displacements
and zero section forces in the beam by assuming the beam to be simply
supported. This allows translations in the axial direction and rotations at
the supports to take place which means that no section forces are induced
by a moisture content variation. In load case 4 the beam was assumed to
be fixed in both ends, which should result in just normal forces in the
beam. In load case 5 all the rotational degrees of freedom were fixed,
which should give just moment forces in the beam. The results of the finite
element calculations corresponded to the expected results as seen in Table
1.
Table 1.
Load cases, displacements and section forces of
verification calculations.
Load Case /
Displacements /
Boundary conditions, BC
Section forces
3
2
1
1
6
L
5
6
2
4
9
Displacement = [ 1
8 9]
8
7
1
2
3
4
5
6
7
Section forces = [ Normal force Shear force Moment ]
(N)
0.2
0
-0.2
0
(N)
(Nm)
2
1
1
2
3
4
5
6
x
Displacement = [1
2
0
BC = [1 0 ; 2 0 ; 7 0 ; 8 0]
2
0
3
4
0
0
5
0
6 7 8 9]
0 0 0 0
Section forces = [ Normal force Shear force Moment ]
0.2
0
-0.2
0
1
67.5 kN
0N
0 Nm
2
67.5 kN
0N
0 Nm
2
1
1
2
3
4
5
6
x
Displacement = [1
BC = [1 0 ; 2 0 ; 3 0 ; 7 0; 8 0 ; 9 0
]
0
2
0
3
0
4
0
5
0
6 7 8 9]
0 0 0 0
Section forces = [ Normal force Shear force Moment ]
96
1
50.6 kN
0N
422 Nm
2
50.6 kN
0N
422 Nm
3
Displacement = [1 2
9 ]
3
4
5
6
7
8
0 0 -0.0050 -0.0023 0 0.0050 -0.0045 0
-0.0050
BC = [1 0 ; 2 0 ; 8 0]
Section forces = [ Normal force Shear force Moment ]
1
0N
0N
0 Nm
2
0N
0N
0 Nm
4
Displacement = [1 2
]
3
4 5
6
7 8
9
0 0 -0.0050 0 0 0.0050 0 0 -0.0050
Section forces = [ Normal force Shear force Moment ]
BC = [1 0 ; 2 0 ; 7 0 ; 8 0 ]
1
50.6 kN
0N
0 Nm
2
50.6 kN
0N
0 Nm
5
BC = [1 0 ; 2 0 ; 3 0 ; 6 0 ; 9 0 ]
Displacement = [ 1
8 9]
2
3
0
0
0 -0.0023 0
0
4
5
6
0 -0.0045
0
Section forces = [ Normal force Shear force Moment
6
E = 25 GPa
1
0N
0N
422 Nm
2
0N
0N
422 Nm
]
E = 25 GPa
Displacement = [ 1
8
9 ]
E = 10 GPa
7
E = 10 GPa
2
3
4 5 6
7
0.0030 -0.0058 0.0039 0 0 0 -0.0030 -
α = 0.01
0.0058 -0.0039
BC = [ 4 0 ; 5 0 ; 6 0 ]
Section forces = [ Normal force Shear force Moment
97
]
7
E = 25 GPa
1
0 kN
0N
0 Nm
2
0 kN
0N
0 Nm
E = 25 GPa
Displacement = [1 2
8
9 ]
E = 10 GPa
E = 10 GPa
α = 0.01
BC = [1 0 ; 2 0 ; 8 0]
3
4
5
6
7
0 0 0.0129 -0.0023 0.0194 0 -0.0045
0 -0.0129
Section forces = [ Normal force Shear force Moment
1
-10.5 kN
0N
1171 Nm
2
-10.5 kN
0N
1171 Nm
To study the effects of a modulus of elasticity that varies over the crosssection the moisture content change was set to be uniform ∆ω = -0.10 over
the whole beam and the modulus of elasticity at the top edge of the
cross-section was set to 25 GPa and to 10 GPa at the bottom edge. To
check that the beam may have displacements without any section forces
induced the translations in both axial and transverse direction as the
rotation were fixed at the middle of the beam. The set up and results are
presented in load case 6 in Table 1. In load case 7 the effects of both a
variation of moisture and modulus of elasticity is studied. The boundary
conditions are set to correspond to a simply supported beam and the
results are presented in the table. The resulting section forces are induced
due to the displacement of the neutral axis in the beam.
98
]
Results from the roof structure analysis
Depending on the boundary condition, the moisture content in the crosssection and on the modulus of elasticity in the five different wooden
members the deformation and stress distribution varies considerably.
Throughout the analysis the five elements are assumed to be fixed to each
other in the three degrees of freedom, which mean that there are no
additional degrees of freedom than the 12 displayed in Figure 2. In a real
roof structure of this kind the connections between the members would be
more complicated.
The shrinkage coefficient alpha is set to a constant value α = 0.01 in all
members throughout the analysis. The effects of boundary conditions, the
moisture content and the modulus of elasticity will be separately in the
following. The deformation figures, the distribution of moment and
normal forces will be presented for each load case.
Varying the boundary conditions
Two different cases of boundary conditions were studied. In both cases the
modulus of elasticity is assumed to be constant 10 GPa and the moisture
variation is assumed to be ∆ω = -0.10 in all elements.
The first case, Case bc A, corresponds to simply supported frame in which
translation in the degree of freedom 1, 2 and 8 respectively are prevented,
see Figure 2. In the second case, Case bc B, additional degrees of freedom,
5 and 7, are fixed.
In Case bc A there are no induced normal forces or moments due to the
uniform moisture content change in the members. On the other hand
horizontal displacements of 6 mm at the most and vertical translations of
1.5 mm occures. In Case bc B both deformations and forces are induced
due to the moisture change. The normal force at the bottom cord 94.5 kN
corresponds to a stress level of approximately 14 MPa.
99
Table 2. Results from analysis when the boundary conditions are varied.
Load case
Case bc A
Case bc B
BC
[1 0;2 0;8 0]
[1 0;2 0;5 0;7 0;8 0]
3
3
2.5
2.5
2
2
1.5
(maximum
displacement)
1
1
3
5
4
3
y
y
Deformed
figure
1.5
0.5
5
4
0.5
0
1
2
0
-0.5
1
2
-0.5
-1
-1
-1.5
-1.5
0
1
2
3
x
4
5
6
0
1
2
3
x
(24 mm @ dof 11)
(6 mm @ dof 7)
Normal
forces
(maximum
normal force)
(N = 0)
(94.5 kN @ el 5)
compression
Moment
(maximum
momentum)
(M = 0)
(141 Nm @ dof 3 and 9)
100
4
5
6
Varying moisture content
The initial value of the moisture content is assumed to be 18% in the
whole frame. The new moisture content values in the new changed climate
are presented in figure 3. If the modulus of elasticity is assumed to be
constant and the boundary conditions are kept the same as in the first case,
Case bc A, the deformations are increased and more complex due to the
moisture variation over the cross-section of the members. The variation
also introduces section forces in the members as shown in figure 4.
Figure 3.
The moisture content when the frame was subjected to a
new climate.
2.5
2
1.5
1
y
3
5
4
0.5
0
1
2
-0.5
-1
-1.5
0
1
2
3
x
4
5
6
Deformed figure
Normal force distribution
Moment distribution
(4.7 mm @ dof 7)
(31.8 kN @ el 5) tensile
(1.05 kNm @ el 5)
Figure 4.
Deformed figure and stresses respectively while the
structure is being subjected to varying moisture content.
101
Varying modulus of elasticity
The effect of a linear variation of the modulus of elasticity over the
cross-section of the wooden members and the same moisture variation as
in the previous was studied next. The boundary conditions were kept
simply supported as in the previous analysis. Two analyses are performed
with different distribution of the modulus of elasticity on the members. In
the first case, Case Emod A, the variation is only applied to the elements 1
and 2. In these elements the modulus of elasticity is assumed to be 25 GPa
at the top edge of the cross-section and 10 GPa at the bottom edge. In the
rest of the members (element 3, 4 and 5) the modulus of elasticity is
assumed to be the same ass as in the previous analysis, that is 10 GPa. In
the second case, Case Emod B, all the elements are assumed to have a
linearly varying modulus of elasticity according to E = 10000 + 100000·r
MPa, where r is the radius in meter, see Figure . The bottom edge of the
cross-section is assumed to have the lowest modulus of elasticity (r = 0).
MOE [MPa]
The results of the analysis are presented in Table 3. Comparing the
deformed figure in Figure 5 to the one with Case Emod A it is obvious
that the distribution of the modulus of elasticity counteracts the
deformations due to the distribution of the moisture content. In the case
Emod B very high compressive forces are obtained in element 5.
45000
40000
35000
30000
25000
20000
15000
10000
5000
0
0
0,05
0,1
0,15
0,2
0,25
0,3
Pith distance [m]
Figure 5.
Variation of the modulus of elasticity from the pith to the
bark.
102
Table 3. Results from analysis of varying modulus of elasticity.
Load case
Case Emod A
Case Emod B
3
3
2.5
2.5
2
2
1.5
1.5
(maximum
horizontal
displacement)
1
3
5
3
y
1
y
Deformed
figure
4
5
4
0.5
0.5
0
0
1
1
2
2
-0.5
-0.5
-1
-1
-1.5
-1.5
0
1
2
3
x
4
5
6
0
1
2
(4.8 mm @ dof 7)
(4.8 mm @ dof 7)
(49 kN @ el 3 and 4)
(221 kN @ el 5)
compression
tensile
3
x
4
Normal
forces
(maximum
normal force)
Moment
(maximum
momentum)
(1.31 kNm @ dof 3 and 9)
(1.53 kNm @ dof 6)
103
5
6
Conclusions and discussion
The modified finite element routines for beams were shown to bee
efficient to use in order to consider initial moisture variation in timber
members. The beam elements have been modified to handle linearly
varying moisture content as well as linearly varying modulus of elasticity
over a cross - section.
It has been shown that a moisture change in a wooden beam member of a
roof structure can lead to deformations and/or forces in the structure,
depending on which boundary condition is assumed. The deformations
and force distributions in the structure are dependent not only on the used
boundary conditions, but also on the variation of the shrinkage parameters
and the modulus of elasticity in the different members. In the performed
analyses of the roof structure moisture induced forces corresponding to
stresses higher than 10 MPa was found. Anyhow, the boundary conditions
and assumptions of connections between the members that are used to
obtain such stresses are not realistic and such stresses will most likely not
occur in practice.
The current way of modelling the influence of moisture induced stresses is
very effective since the analysis can be held computational cost efficient.
Further there are several ways to develop the model further, as for instance
to consider a nonlinear variation of the moisture content and the stiffness,
or to vary the shrinkage parameter over the cross-section.
References
1.
Austrell, P.-E., CALFEM : a finite element toolbox : version 3.4.
2004, Lund: Structural Mechanics LTH.
2.
Ormarsson, S., Numerical analysis of moisture-related distortions
in sawn timber. Publication / Department of Structural Mechanics,
Chalmers University of Technology, 99:7. 1999, Göteborg:
Chalmers tekniska högsk. [16], 213.
3.
Ottosen, N., Petersson, H., Introduction to the Finite Element
Method. 1992: Prentice Hall, London.
104
Finite element study of moisture related stresses in
wooden structures
Kristoffer Segerholm, Royal Institute of Technology, KTH
Janne Manninen, TKK, Wood Technology, Finland
3
2
4
1
Description of the problem
The aim of this project was to model how stresses in wooden beams are
affected by moisture changes. The project was solved by the finite element
program CALFEM. In analysis of moisture related stresses in timber
beams, the transient moisture content variation has to be estimated over
the cross sections of the beams. The moisture content can be calculated by
a two-dimensional moisture flow analysis. Here we simplified the analysis
by assuming a steady state moisture condition and uniaxial bending
condition with a linear moisture gradient over the cross section. Tasks 1
and 2 assumed also constant modulus of elasticity and shrinkage
properties over the cross section. On tasks 3 and 4 it was studied how
situation changes, if the longitudinal MOE and the longitudinal shrinkage
properties vary over the cross section. Tasks 1 and 2 were mandatory to
solve, whereas tasks 3-5 were voluntary. Only task 1 and task 2 were
solved. The specific tasks are defined below.
105
Task 1
Modify the beam routines beam2e.m and beam2s.m in CALFEM in order
to be able to analyse moisture related deformations and section forces
caused by initial normal strain ε0 and constant curvature κ0 over the cross
section.
Task 2
Use the modified routines to analyse the deformations, section forces and
surface stresses in the wood structure shown in the figure. The reference
moisture content (stress free condition) is ωref = 0.18, the MOE El = 10000
and
the
shrinkage
coefficient
N/mm2
αl = 0.01
Assumptions and simplifications
All the tasks were simplified by assuming a steady state moisture
condition and uniaxial bending condition with a linear moisture gradient
over the cross section.
Tasks 1 and 2
Tasks 1 and 2 assumed also constant MOE and shrinkage properties over
the cross section. Moisture gradient assumed not to have effect on the
MOE. The boundary conditions are simply supported at node 1 (x1,y1=0),
and y4 are also zero, node denotation can be seen in the figure on the first
page.
Calculations
Tasks 1 and 2
Modified beam2e, beam2s and main Matlab code are presented in
appendices. beam2e calculates the element stiffness matrix K e and the
element load vector f l e in global coordinates according to the following
expressions:
106
e
K e = GT K G
and
e
fle = G T f l
where:
 EA
 L

 0

 0
e

K =
EA
−
 L
 0


 0

0
0
12 EI
L3
6 EI
L2
6 EI
L2
4 EI
L
0
0
−
12 EI
L3
6 EI
L2
6 EI
L2
2 EI
L
−
−
EA
L
0
0
0
0
0
 qx L 


 2 
 qy L 
 2 
 q L2 

 y
e

f l =  12 
q L
 x 
 2 
 qy L 


 2 2
 qy L 
− 12 
107
0
12 EI
L3
6 EI
− 2
L
−
0
12 EI
L3
6 EI
− 2
L


6 EI 

L2 
2 EI 
L 

0 

6 EI 
− 2
L 
4 EI 

L 
0
and G is the transformation matrix (6x6) containing the direction cosines.
The new initial load case caused by the moisture loading was calculated
and added to the element load vector:
 ε EA
e
= 0
f moisture
0 κ 0 EI
 2
ε 0 EA

0 − κ 0 EI 

2
e
e
f l e = G T ( f l + f moisture
)
where;
κ =
0
∆ω 2 − ∆ω1
h
ε 0 = α∆ω 0
In beam 2s the axial strain caused by the moisture is recalculated to
represent a distributed load in the axial direction, and then added to q x :
q x ,new = q x + ε 0 ⋅
EA
L
The curvature ( κ 0 ) caused by the difference in moisture content is only
d 2v
. It is also
acting as a constant moment on the beam and is inserted in
dx 2
integrated two times and inserted in the equation for the deformations in
the v direction, the integration constants are zero due to boundary
conditions. The modified expressions in beam2s are:

q L2
M = C ⋅ (G ⋅ ed '− 0 0 0 − x
2 EA

−1
108
q y L4
24 EI
+
κ 0 L2
2

+ κ 0 L
6 EI

q y L3
[
v = x3
x2
 M (2)
 M (3)  x 4 q
x 2κ 0
y
+
x 1 ⋅
+
 M (5)  24 EI
2


 M (6) 
]
 M (2)
 M (3)  x 2 q
d v
y
+
+ κ0
= [6 x 2 0 0] ⋅ 
2
 M (5)  2 EI
dx


 M (6) 
2
109
Results
Tasks 1 and 2
Deformations and section forces has been solved for the whole structure.
The deformations, normal forces, shear forces and moment are plotted
below. Note that the scale of the deformation curve in the first figure is
magnified.
Figure:
Deformations of the structure
Figure:
Normal forces
110
Figure:
Shear forces
Figure:
Moment
111