capillary pressure: some static and dynamic views in ...

CAPILLARY PRESSURE: SOME STATIC AND DYNAMIC VIEWS IN SEMI-ORDERED MEDIA AND THEIR APPLICABILITY TO WOOD DRYING G. William Slade CSIRO Forestry and Forest Products Private Bag 10, Clayton South MDC, Victoria 3169 Australia Abstract

In most situations where vapor-liquid motion is simulated within porous media, one often sees the con icting notions of `continuous water column' used alongside the concept of `well-mixed liquid-gas system' within a control-volume. This assumption clearly breaks down when the computational control volume approaches the structural cell dimensions. The `grainyness' of the cell structure and the uid ow patterns cannot be neglected any longer, if accurate descriptions of water (or any other

uid) content are to be generated. This would be useful especially in predicting the onset of hygroscopicity in regions where the liquid saturation begins to vanish during drying. Using arguments based on a simple mechanical potential energy formulation, the multiplicity of equilibrium states for a uid/gas system in a semi-ordered structure (where capillary forces dominate, e.g. softwoods) will be explored. By these methods, it is hoped that insight into capillary motion on a mesoscopic scale can be gained.

INTRODUCTION The main motivation for this work is the desire to move away from dicultto-measure and often non-representative coecients like permeability and diffusion coecients which are used in the1 many continuum models that simulate

drying. It was desired to produce a model that would represent the interaction between the structure of the solid material and liquid distribution in intimate detail. It is hoped that by examining the microscopic (or mesoscopic) behavior of uids in a porous medium, such as wood, better understanding of the physics of water movement can be attained. In continuum models, smoothing of rapid changes in liquid distribution can possibly hide important processes. This work will propose a model of capillary force based on a nearest neighbor interaction energy (the tendency for the uid to want to clump together). Some simulations are presented to show, in a simple way, the multiplicity of pro les that the uid can assume and still have the same (or very nearly the same) surface energy. As a consequence of this multiplicity of states, it is proposed that the microscipic moisture distributions and their gradients are not uniquely de ned. In addition to an exploration of equilibrium states, a simple drying simulation is presented where uid `particles' are allowed to escape through the problem boundaries in a simple non-equilibrium open system. Regrettably, space does not permit an extensive discussion of solid-liquid interaction, however some notes are made on the eects of inhomogeneous pore size and wetting energy distributions. A further discussion on uid-structure interaction must be carried out later. ENERGY FORMULATION FOR WATER-WOOD-AIR SYSTEM The fundamental relation that this model has with wood structure is the simulation of cell-wall wetting and internal free surface energy (self-energy) and `communication' through the pits and permeable surfaces (mutual or interaction energy). The basic idea of this model is to minimize the amount of free surface area that might appear within and between cells while simultaneously maximizing the area of wettable solid-liquid interfaces. In order to formulate the simple model, the free energy is given in terms of the interface surface areas. Figure 1 shows a conceptualization of this idea. It shall be assumed that the air mixture-solid and air mixture-liquid interaction is negligible. The main contributions to the energy functional come from the energy of attraction between the solid material and the liquid as well as the surface tension energy of the free surfaces (air-liquid interfaces). It should be mentioned that the radii of the capillaries are assumed to be much smaller than the average meniscus radius of curvature. The eects of meniscus curvature are ignored in this work. Using these assumptions, one can write the energy of solid-liquid attraction

as

Uwet = Ssl;

(1)

Ufree = Sla :

(2)

and the free surface energy as The parameter de nes the nature of the solid surface anity for liquid. If < 0, the surface is wettable, if  0 the surface is non wettable and as increases, becomes repellant. Since the meniscus curvature is ignored in this case, the free surface area is taken as the cross-sectional area of the capillary tube or pit aperture. The solid-liquid interaction energy can be thought of as

Sla

Air

Air

Liquid

Liquid

Ssl

Liquid Bath

Figure 1: Simple capillary problem showing the surface de nitions. This illustration shows interaction with a wettable surface Ssl and the free surface Sla. Gravity is directed downwards. Some meniscus curvature is shown in the gure, but ignored in the present formulation. the energy of wetting. Essentially, dUwet gives information on how much energy it takes to push l
dSwet amount of water onto the solid surface. If the surface is water repellent, dUwet will be positive. If the surface is wettable, dUwet will be negative indicating that water will be drawn to the surface. Likewise, dUfree describes the energy needed to stretch the free surface by an amount dSfree. In order to nd the surfaces that yield a minimum of potential energy the

Z

following functional should be minimized (Finn 1986)

U = Sfree + Swet + dm + V :

(3)

The function  is the gravitational potential and dm represents a dierential unit of liquid mass. The domain is de nes the volume of liquid. The integral in the third term represents the contribution of gravitation in the potential energy. Typically, the eects of gravitation will be ignored in this work, but for completeness, was included in the previous expression. The fourth term V represents the eects of a constrained volume V ( is an undetermined multiplier). In some ways like a penalty method, this term constrains the volume to be xed while the surface con guration is allowed to change. It should be mentioned that the preceding is a static model. Dynamic situations add signi cantly to the complexity. (Schmittmann and Zia, 1995 contains some excellent references on the subject of phase interfaces.) Eects of viscosity and inertia can appear. However, one could assume that changes of state occur slowly and thereby augment the static model with a gas pressure term while ignoring kinetic energy terms. This essentially de nes the waterair-solid system as a relaxation problem where kinetic energy terms are small with respect to dissipation. This will not be discussed here in the interest of brevity. NUMERICAL PARTICULARS The minimization of (3) is in general very dicult to carry out. Unlike standard approaches on the continuum, it is desired to account for the discrete nature of the material structure as well as the discontinuous nature of capillary surfaces. In order to generate minimum-energy solutions, a cellularautomata (Vesely 1994 and Atsushi, et al. 1995) methodology is developed which attempts to minimize a simpli ed form of the energy equation in (3). The Metropolis Monte-Carlo method (Vesely 1994) introduces a statistical avor to the computations. The model is non-deterministic on a `microscopic' scale, but certain averaged `observables' are generally not random and follow de nite deterministic rules (as long as the statistical ensemble is large enough and is truly random). In the interest of simplicity, a reduced version of (3) can be written as U = Wici + Aij cicj : (4)

X

XX

i

i j (i)

The term ci contains the occupancy state of cell i. In this analysis, ci can be either `0' or `1'. A more complex approach could allow a range of values for ci.

The rst summation represents the energy associated with actually occupying the site (wetting energy) and the second term represents any free-surfaces that may appear between wood cells (pits, permeable surfaces, etc.). In the two-dimensional model, if we allow Wi and Aij to be constants over all lattice elements, we nd that

U = U0 +

X ci(4ci i

cE cN cW cS ):

(5)

The E; N; W; S subscripts denote the indices for the eastern, northern, western and southern cells adjacent to cell i. The energy of wetting can be ignored in the minimization process, since no preference is given to any particular cell being occupied over another based on self-energy. Only cell-cell interaction energies need to be considered. Figure 2 provides a geometric visualization of computing the interaction energy (intercell free surface area).

CN CW

Ci

CE

CS

= Empty

= Full

Figure 2: Illustration of a lattice cell and the interaction with its nearest neighbors. The contour drawn around the cell system represents the contour of integration for the `surface area' (surface length in 2-D) calculation. The undetermined multiplier constraint is not needed here, because it is a simple matter to x a ` uid' conservation law in the algorithm to keep the number of occupied cells constant during a simulation. (Later, changing values of cell occupation will be permitted.) The minimization of U in two dimensions consists of minimizing the length of the contour around connected and disconnected pockets of ` uid.' Similarities with the now classic Ising magnet models (Wannier 1966) and the

so-called `Travelling Salesman Problem' (Press, et al. 1986) are apparent in this approach. The transition rules utilized in the Monte-Carlo method are devised such that the number of occupied sites are conserved. No `particle' is allowed to move on top of an adjacent occupied site or through an external boundary. If the transition energy
E = U + U  0 (U + is the total system energy after the transition and U is the system energy before the transition) and the desired neighboring site is unoccupied, the transition occurs always (probability = 1). If
E > 0, and the desired neighboring site is unoccupied and not on a boundary, the transition occurs with a probability exp(
E=kT ). The value of T is analogous to the thermodynamic temperature, k is like the Boltzmann constant. A random distribution of ` uid particles' can be `annealed' to a state close to the minimum energy by running a simulation where the temperature slowly decreases until no more transitions occur. Various `annealing' schedules can be used to produce dierent results. The preceding paragraph touches on a central point of this paper. Namely, that for a given initial distribution of uid many energy minima can exist which represent many dierent uid distributions. The distribution is not, in general, a unique function of the energy. There exists instead a density of liquid states that depends on energy (computing that density of states or partition function is generally a herculean task and is usually not done explicitly). On a macroscopic scale, the `average uid content' may look the same for several dierent samples while on a microscopic scale, the uid topology can be very dierent for each sample. The microscopic gradients take on a completely different character for each sample. It is proposed that this can have a signi cant eect in the onset of hygroscopicity and destructive drying stress in wood. To illustrate a couple of examples where capillary surface energy is at a local minimum somewhere near the global minimum, Figures 3 and 4 show two distributions of ` uid' in two dimensions for the simple regular lattice model. Figure 5 shows the initial (high-temperature) distribution of the uid. It should be noted that the global minimum energy state for this example is when the uid assumes a square-like shape (with some irregularity due to excess particles). Figure 6 shows the free energy reduction over the course of the simulation. In a semi-ordered media (such as wood) where capillary radii and wetting energy varied, the surface structure that the uid assumes in the global energy minimum will depend heavily on the solid structure and not as much on the initial distribution of uid in the high temperature state. The location of the

100

50

100 0 0

50

50

100

Figure 4: Another low energy pro le on the exact same grid as in Figure 3. All initial conditions were the same. The nal `surface energy' per particle is 0.432 for this distribution.

0 0

50

100

Figure 3: Illustration showing a low-energy local minimum `liquid' pro le computed using a MonteCarlo minimization of (5). The computations were done on a 100x100 grid with 49% occupancy. The nal `surface energy' per particle is 0.435 in this case.

100

50

0 0

50

100

Figure 5: The initial distribution used in the computation of pro les in Figures 3 and 4.

T_i = 400, linear decrease in T from 400-1 in final 15000 timestep interval 10 "Avg.out"

T=120 1 "Avg.out" 0.9

Avg. occupancy (Avg. MC)

Free surface energy/particle

0.8

1

0.7 0.6 0.5 0.4 0.3 0.2 0.1

0.1 2000

4000

6000

8000 10000 12000 Time (dimensionless)

14000

16000

18000

20000

Figure 6: The evolution of the `surface energy' per uid particle as a function of time. The temperature decreases linearly from 400 at timestep 5000 to near zero at time-step 20000.

0 20000

40000

60000

80000 100000 120000 140000 160000 180000 Time (dimensionless)

Figure 8: The average liquid content is plotted versus time in this illustration.

T=120 4 "Avg.out" 3.5

100 Surface energy per occupied cell

50

3

2.5

2

1.5

1

0 0

50

0.5

100 0 20000

Figure 7: An example capillary surface calculation halfway through a `drying' simulation. The `temperature' is xed to 120. Note the roughness of the surface of the contiguous `lump' of liquid.'

40000

60000

80000 100000 120000 140000 160000 180000 Time (dimensionless)

Figure 9: The surface energy per particle is depicted here. Notice the steep jump to a maximum value near the end of the simulation as the liquid mass nally breaks up.

`center-of- uid-mass' will be signi cantly shifted unlike the unbiased simple structure. This is because the wetting energy term does not vanish from
E during state transitions in the inhomogeneous case as it did for the simple model used to generate Figures 3-5. It is notable that the `center of mass' of the moisture distribution does not move signi cantly during the entire simulation indicating no net interaction with the structure. At this stage, it is worth showing the results of a simple drying model based on these methods. The annealing model is augmented such that particles are allowed to cross the external boundaries and vanish into an in nite `particle sink'. The temperature is xed such that the probability of particles `breaking o' from the uniform uid mass is signi cant over reasonable time scales. The only `forces' that are present in the model are the capillary tension and the particle concentration gradient. This is not exactly physically realistic, but it should give the reader a feel for the numerical process. Figure 7 shows the simulated capillary surface about halfway through the simulation. Note the presence of ` ngering' and surface roughness along the contour of the main ` uid' mass. This is a complex phenomenon which continuum models would not resolve but is likely to happen in physical situations (A nice picture in Frey, et al., 1994). Figure 8 shows the progression of the `drying' by tracking the average occupancy (moisture content) over time. Figure 9 shows the average free energy per particle which can give information on critical phenomena such as when the uid mass nally breaks up and disappears at the end of the simulation. CONCLUSIONS It should be realized that at this stage, this is a very simple model that does not take into account heat ow eects, uid kinetic energy eects, uidstructure interactions or uid-gas interactions. Based entirely on the interaction potential energy of a uid, realistic looking distributions of uid can be generated for a `spongy' medium using a discrete cellular-automata based Monte-Carlo method. Given that the solid structure pore sizes are uniformly distributed and are uniformly wettable, the assumption is made that the uid-structure interaction potential energy density (cell self-energy) is constant for all uid states. Only the `nearest neighbor' uid- uid interactions appear in the minimization and aect the distribution. However, in a material such as wood, the set of cells may have a range of wetting properties and pore sizes. There would exist in these cases a uid-solid interaction potential energy that would depend explicitly on the distribution of liquid. As an example, consider the dierence

in the anities of earlywood and latewood for lumen water. (Pore size is the main factor here.) It is not clear that a capillary pressure (potential energy) function based solely on local saturation and temperature (Hunter 1995a) contains enough information in cases where there is signi cant interaction with the structure. In order to address the question of moisture gradients (above ber saturation in wood), even the simple model presented here illustrates the ambiguity in moisture pro les that can occur. The discrete model illustrates the roughness that occurs in the capillary surface as well as the multiplicity of states that can exist within a small band of system energies. Given a small perturbation to the system, a shift to another (signi cantly dierent) distribution is possible in a short time. This is a problem of scale that cannot be readily addressed in a continuum model. Liquid gradients can, in general, have a microscopic scale which may not be de ned by smoothly varying contours. In the nal scheme of studying internal drying stresses in wood, it would seem that this microscopic activity would be a dominant factor in the onset of mechanical failures (checking) in timber. In addition to the inclusion of more energy terms (to account for other phenomena), realism could be enhanced by allowing partially lled cells. Moreover, the cell to cell neighbor interaction could be varied to simulate the pits that form uid paths between softwood cells. Also, the model could be generalized to three dimensions and allowing two interacting phases to exist within a cell. The water sorbed within the cell walls can be modelled using the idea of micro-capillaries (Hunter 1995b). Any, or all of these ideas could improve the simulation quality. Of course, the inclusion of more phenomena would increase CPU times. The results of the simulations look promising. Inclusion of the uid-solid and uid-gas interactions along with a realistic heat transfer model may yield realistic simulations for speci c wood samples (or any other porous medium) using a relatively simple and computationally ecient model. REFERENCES Atsushi, Y., Inamuro, T. and Adachi, T., 1995 Analysis of Shear Layers Based on the Lattice Gas Model, Int. J. Num. Meth. Fluids, (21), pp. 967-972. Finn, R., 1986, Equilibrium Capillary Surfaces, Grundlehren der mathematischen Wissenschaften 284, Springer-Verlag pp. 3-16. Frey, E., Tauber, U. C. and Schwabl, F., 1994, Crossover from Isotropic to Directed Percolation," Physical Review E, (49), pp. 5058-5072. Hunter, A. J., 1995a, Equilibrium Moisture Content and the Movement of Water through Wood above Fiber Saturation," Wood Science and Technology,

(29) pp. 129-135. Hunter, A. J., 1995b, A Complete Theoretical Isotherm for Wood Based on Capillary Condensation," Wood Science and Technology (in press). Press, W. H., Flannery, B. P., Teukolsky, S. A. and Vetterling, W. T., 1986, Numerical Recipes, Cambridge University Press, pp. 326-334. Schmittmann, B. and Zia, R. K. P., 1995, Statistical Mechanics of Driven Diusive Systems, Academic Press series on Phase Transitions and Critical Phenomena, (17), pp. 80-81. Vesely, F. J., 1995, Computational Physics, Plenum. Wannier, G., 1966, Statistical Physics, Dover, pp. 330-342.