3.4 Constitutive relations
3.4.3 Free water
The free water constitutive relation is usually assumed to follow the generalized Darcy’s law by which the mass average velocity of the free water is given by
vw=−KwKrw
µw
∇(pw+ρgz) (3.28)
or in terms of mass flux
jw=ρwvw=−ρw
KwKrw
µw
∇(pw+ρgz) (3.29)
where z is the upward directed coordinate opposite to which gravity acts, Kw is the absolute permeability,Krwthe relative permeability andµwthe dynamic viscosity of water.
The water pressurepwhas two different interpretations depending on whether the medium is partially or fully saturated. This is discussed below.
Partially saturated conditions
Under partially saturated conditions the water pressure is equal to the sum of the capillary pressure and the gas pressure
pw=pc+pv+pa (3.30)
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General mathematical modeling of moisture transport 3.4 Constitutive relations
100 8 6 4 2 0
0.2 0.4 0.6 0.8 1 1.2
(× 10 Pa)4 Capillary pressure Free water content Wf (kg/kg)
Figure 3.5Capillary pressure as function of free water content for softwood at 20◦C, according to [48].
where the capillarypcpressure is a function of the degree of saturation and temperature.
This is often taken as varying principally as
pc=σ(T)p(Wf) (3.31)
where σ(T) is the surface tension of water which depends only on the temperature and p(Wf) is a function which depends only on the degree of saturation, or alternatively, on the free water contentWf given by
Wf=ρw
ρ0
ϕS (3.32)
whereϕis the porosity andSthe degree of saturation.
In this way, the capillary pressure–saturation curve needs only determination at one tem-perature whereas the general temtem-perature dependence is correlated via temtem-perature de-pendence of the surface tension of water which is a well–described quantity.
If gas pressures and gravity are temporarily ignored, (3.29) can be expanded in terms of Wfas
jw = −ρw
KwKrw µw
∇
· σ(T)
µ ∂p
∂Wf
¶
∇Wf+p(Wf) µ∂σ
∂T
¶
∇T
¸
= KW∇Wf+KT∇T
(3.33)
Thus, a general non–isothermal constitutive relation is obtained through only one ex-perimentally determined curve, a typical example of which is shown in Figure 3.5. In comparison, the total moisture diffusion model would require separate determination of the coefficientsKWandKT at different moisture contents and temperatures, which, need-less to say, is a formidable task.
As for the relative permeabilitiesKrw, these should equal unity for fully saturated condi-tions and then decrease as the saturation decreases. This decrease expresses the gradual
3.4 Constitutive relations General mathematical modeling of moisture transport
T
L
(a)
T
L
(b)
Figure 3.6Free water distribution in wood at high (a) and at low (b) degrees of saturation.
loss of continuity of the water phase as the saturation decreases. The orthotropy of wood plays an important part in this regard. Looking in more detail at the mechanism of water transfer in softwood this can be expected to take place via the pits located primarily at the ends of the tapered section of the cells, see Figures 3.6 (a)–(b). As the resistance to flow is generally much smaller in the longitudinal direction than in the transverse direction it can be expected that the effective conductivity is also much larger in the longitudinal direction. However, because of the particular structure of wood this is only the case at relatively high degrees of saturation. This situation is depicted in Figure 3.6 (a). The water phase still has a high degree of continuity and longitudinal transfer will be domi-nant. At lower degrees of saturation the water collects at the tapered ends and the degree of continuity of the free water phase decreases. This means that longitudinal transfer is severely hindered while the transverse transfer is much less affected. Thus, in [48] the relative permeabilities in the three principal directions of a non–specified softwood were taken as
KrwL =S8 , KrwT =KrwR =S3 (3.34) whereas the absolute permeabilities were reported to be
KwL= 10−12m2, KwT = 12KrwR= 10−16m2 (3.35) The ratio between the longitudinal and tangential effective permeabilities are thus
KwLKrwL KwTKrwT = KL
KT = 104S5 (3.36)
which means that belowS ' 0.15 the transfer of free water is faster in the tangential direction than in the longitudinal direction. Indeed, as the saturation decreases below some value, the longitudinal free water flow becomes so small that it is practically zero.
This flow of free water at low degrees of saturation in wood and other porous materials has recently been the subject of some controversy. In some fields, e.g. those dealing with the flow of water in soils, the so–called irreducible saturation point is an integral part of almost all constitutive relations, e.g. the very commonly applied parameterization of van Genuchten [65]. That is, below some degree of saturation Sirr the flow of free water is
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General mathematical modeling of moisture transport 3.4 Constitutive relations
0 0.2 0.4 0.6 0.8 1
0 0.2 0.4 0.6 0.8 1
1.2 Total water content (kg/kg) Relative effective free water transfer coefficient
WFSP
S8 S3 S2
Figure 3.7Relative effective free water transfer coefficients as function water content.
assumed not to take place at all. For wood the irreducible saturation concept has also been used, withSirrtypically being taken asSirr'0.1 [58, 44]. Whereas the irreducible saturation concept may provide a good approximation to the conditions in soil, it has some rather undesirable consequences when it comes to hygroscopic materials such as wood. Thus, all experimental evidence, see e.g. [62], points to a rather smooth transition between the hygroscopic and partially saturated states and the effective transfer coefficient should thus be equally smooth. On the other hand, if the irreducible saturation concept is adopted, there exists a range above the fiber saturation point and below the irreducible saturation point where all water transfer takes place solely as vapour diffusion/convection.
Since in this range the equilibrium vapour pressure is very close to the saturated vapour pressure, no significant gradients in vapour pressure exist and the transfer by this mode is not nearly enough to account for the smooth transition between the hygroscopic and partially saturated states observed experimentally.
Whereas the relative permeabilities (3.34) do not make explicit reference to an irreducible saturation point, the variation of the relative permeabilities with saturation is so extreme, that a de facto irreducible saturation point is created. This can be seen by the effective transfer coefficients
Keff=ρwKrKrw µw
µ∂pc
∂W
¶
(3.37) which have been plotted in Figure 3.7 for different expressions of the relative permeability (each curve has been normalized with respect to the value atW = 1.2). As can be seen the effective transfer coefficient in the case whereKr=S8practically disappears around W = 0.6, whereas the limit a which the transfer becomes negligeble gradually decreases as the polynomial exponent in the expression for the relative permeability decreases. Thus, a de facto irreducible saturation point is created, e.g. for Kr = S8 around W = 0.6.
This fact, which has also been pointed out by Couture et al. [12], is reflected in the corresponding simulations, e.g. those concerning drying, in a way which is regarded unre-alistic. Examples of this are shown in Figures 3.8–3.10 where the relative permeabilities
3.4 Constitutive relations General mathematical modeling of moisture transport
0 1 2 3 4 5 6 7 8 9 10
0 0.2 0.4 0.6 0.8 1 1.2
Distance (cm)
Total water content (kg/kg) "W "irr
WFSP
Figure 3.8Simulated one dimensional drying profiles with Krw = S8 corresponding to longitudinal transfer.
0 1 2 3 4 5 6 7 8 9 10
0 0.2 0.4 0.6 0.8 1 1.2
Distance (cm)
Total water content (kg/kg)
WFSP
Figure 3.9Simulated one dimensional drying profiles with Krw = S3 corresponding to tangential transfer.
corresponding to the effective transfer coefficients shown in Figure 3.7 have been used in a one–dimensional drying simulation.
The figures show the drying profiles at different times and as can be seen the results for Kr =S8 display a characteristic jump extending from the fiber saturation point and to aroundW = 0.6. For Kr = S3 the profiles look more realistic, whereas quite a smooth transition is achieved forKr=S2. For the last two simulations also the bound water and vapour diffusivities play a role in achieving this smooth transition, whereas the principal character of the profiles forKr cannot be altered unless grossly unrealistic values of the bound water and water vapour diffusivities are assumed.
To account in more detail for the mechanisms of water transfer at low degrees of satu-ration Goyeneche et al. [24, 25] have recently proposed a film–flow model for free water below a certain saturation. With this model they were able to compute the corresponding relative permeabilities such that the usual model could be applied directly. Furthermore, experiments on ideal capillaries seems to verify the model.
Thus, it seems that there is physical justification for assuming transport of free water in
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General mathematical modeling of moisture transport 3.4 Constitutive relations
0 1 2 3 4 5 6 7 8 9 10
0 0.2 0.4 0.6 0.8 1 1.2
Distance (cm)
Total water content (kg/kg)
WFSP
Figure 3.10Simulated one dimensional drying profiles withKrw=S2.
the entire range of the partially saturated conditions. As a minimum the relative perme-abilities should reflect this, and thus, the simple functional relationshipKr =Sα seems to be inappropriate for large values ofα.
Fully saturated conditions
Under fully saturated conditions the water pressure is a variable in its own right. In order to properly include the effects of this pressure, the water and porous medium compress-ibility should be considered. The general free water conservation equation without any sinks or sources can be written as
∂
∂t(ϕρwS) +∇ ·(ρwvw) = 0 (3.38) where ϕis the porosity and S the degree of saturation. Under partially saturated con-ditions the porosity and the density of water are usually assumed constant. However, compressibility can be included by the following linear relations [18]
ρw=ρ◦w(1 +cw(pw−p◦w)) (3.39) ϕ=ϕ◦(1 +cm(pw−p◦w)) (3.40) wherecw is the water compressibility,cmthe porous medium compressibility, andρ◦w and ϕ◦ reference values at the pressurep◦w. The accumulation term can now be expanded as
∂
∂t(ϕρwS) =A∂S
∂t +B∂pw
∂t (3.41)
where
A=ϕ◦ρ◦w£
1 +cmcw(pw−p◦w)2+cm(pw−p◦w) +cw(pw−p◦w)¤
(3.42) B=ϕ◦ρ◦wS[cm+cw+ 2cmcw(pw−p◦w)] (3.43)