# Flow in Timber

The term flow is synonymous with the passage of liquids through a porous medium such as timber, but the term is also applicable to the passage of gases, thermal energy and electrical energy; it is this wider interpretation of the term that is applied in this article, albeit that the bulk of the article is devoted to the passage of both liquids and gases (i.e. fluids).

The passage of fluids through timber can occur in one of two ways, either as bulk flow through the interconnected cell lumens or other voids, or by diffusion. The latter term embraces both the transfer of water vapour through air in the lumens and the movement of bound water within the cell wall (Fig. 1).

The magnitude of the bulk flow of a fluid through timber is determined by its permeability. Looking at the phenomenon of flow of moisture in wood from the point of view of the type of moisture, rather than the physical processes involved as described above, it is possible to identify the involvement of three types of moisture:

- Free water in the cell cavities giving rise to bulk flow above the fibre saturation point.
- Bound water within the cell walls, which moves by diffusion below the fibre saturation point.
- Water vapour, which moves by diffusion in the lumens both above and below the fibre saturation point.

It is convenient when discussing flow of any type to think of it in terms of being either constant or variable with respect to either time or location within the specimen; flow under the former conditions is referred to as steady‑state flow, whereas when flow is time and space dependent it is referred to as unsteady‑state flow. Because of the complexity of the latter, only the former is covered in this text; for more information you should consult Siau (1984).

One of the most interesting features of steady-state flow in timber in common with many other materials is that the same basic relationship holds irrespective of whether one is concerned with liquid or gas flow, diffusion of moisture, or thermal and electrical conductivity. The basic relationship is that the flux or rate of flow is proportional to the pressure gradient:

Flux/Gradient = k

where flux is the rate of flow per unit cross-sectional area, gradient is the pressure difference per unit length causing flow, and k is a constant, dependent on form of flow, e.g. permeability, diffusion or conductivity.

## Bulk Folw and Permeability

Permeability is simply the quantitative expression flow and permeability of the bulk flow of fluids through a porous material. Flow in the steady-state condition is best described in terms of Darcy’s law. Thus:

Permeability = Flux/Gradient

and for the flow of liquids, this becomes:

k = QL ÷ AΔP

where k is the permeability (cm^{2}/atm s), Q is the volume rate of flow (cm^{3}/s), ∆P is the pressure differential (atm), A is the cross-sectional area of the specimen (cm^{2} ), and L is the length of the specimen in the direction of flow (cm).

Because of the change of pressure of a gas and hence its volumetric flow rate as it moves through a porous medium, Darcy’s law for the flow of gases has to be modified as follows:

k_{g }= QLP ÷ A∆Pp̂

where k_{g} is the superficial gas permeability and Q, L, A and ∆P are as stated before, P is the pressure at which Q is measured, and p̂ is the mean gas pressure in the sample (Siau, 1984).

Of all the numerous physical and mechanical properties of timber, permeability is by far the most variable; when differences between timbers and differences between the principal directions within a timber are taken into consideration, the range is of the order of 107 . Not only is permeability important in the impregnation of timber with artificial preservatives, fire retardants and stabilising chemicals, but it is also significant in the chemical removal of lignin in the manufacture of wood pulp and in the removal of free water during drying.

**Flow of Fluids**

The bulk of flow occurs as viscous (or laminar) flow in capillaries where the rate of flow is relatively low and when the viscous forces of the fluid are overcome in shear, thereby producing an even and smooth flow pattern. In viscous flow Darcy’s law is directly applicable, but a more specific relationship for flow in capillaries is given by the Poiseuille equation, which for liquids is:

where N is the number of uniform circular capillaries in parallel, Q is the volume rate of flow, r is the capillary radius, ∆P is the pressure drop across the capillary, L is the capillary length and ɳ is the viscosity. For gas flow, the above equation has to be modified slightly to take into account the expansion of the gas along the pressure gradient. The amended equation is:

In both cases:

Q ∝ ∆P/L

or flow is proportional to the pressure gradient, which conforms with the basic relationship for flow. Other types of flow can occur, e.g. turbulent, non-linear and molecular diffusion, the last mentioned being of relevance only to gases. Information on all these types can be found in Siau, 1984.

**Flow Paths in Timber**

**Softwoods**: Because of their simpler structure and their greater economic significance, much more attention has been paid to flow in softwood timbers than in hardwood timbers. Both tracheids and parenchyma cells have closed ends and that movement of liquids and gases must be by way of the pits in the cell wall.

Three types of pit are present. The first is the bordered pit, which is almost entirely restricted to the radial walls of the tracheids, tending to be located towards the ends of the cells. The second type of pit is the ray or semi-bordered pit, which interconnects the vertical tracheid with the horizontal ray parenchyma cell, while the third type is the simple pit between adjacent parenchyma cells.

For very many years it was firmly believed that – since the diameter of the pit opening or of the openings between the margo strands is very much less than the diameter of the cell cavity, and since permeability is proportional to a power function of the capillary radius – the bordered pits would be the limiting factor controlling longitudinal flow.

However, it has been demonstrated that this concept is fallacious and that at least 40% of the total resistance to longitudinal flow in Abies grandis sapwood that had been specially dried to ensure that the torus remained in its natural position could be accounted for by the resistance of the cell cavity (Petty and Puritch, 1970).

Both longitudinal and tangential flow paths in softwoods are predominantly by way of the bordered pits, as illustrated in Fig. 2, while the horizontally aligned ray cells constitute the principal pathway for radial flow, though it has been suggested that very fine capillaries within the cell wall may contribute slightly to radial flow. The rates of radial flow are found to vary very widely between species.

It is not surprising to find that the different pathways to flow in the three principal axes result in anisotropy in permeability. Permeability values quoted in the literature illustrate that for most timbers longitudinal permeability is about 104 times the transverse permeability; mathematical modelling of longitudinal and tangential flow supports a degree of anisotropy of this order.

Since both longitudinal and tangential flow in softwoods are associated with bordered pits, a good correlation is to be expected between them; radial permeability is only poorly correlated with permeability in either of the other two directions, and is frequently found to be greater than tangential permeability. Not only is permeability directionally dependent, but it is also found to vary with moisture content, between earlywood and latewood, between sapwood and heartwood (Fig. 3) and between species. In the sapwood of green timber the torus of the bordered pit is usually located in a central position and flow can be at a maximum (Fig. 4a).

Since the earlywood cells possess larger and more frequent bordered pits, the flow through the earlywood is considerably greater than that through the latewood. However, on drying, the torus of the earlywood cells becomes aspirated (Fig. 4b), owing, it is thought, to tension stresses set up by the retreating water meniscus (Hart and Thomas, 1967).

In this process the margo strands obviously undergo very considerable extension, and the torus is rigidly held in a displaced position by strong hydrogen bonding. This displacement of the torus effectively seals the pit and markedly reduces the level of permeability of dry earlywood. In the latewood, the degree of pit aspiration on drying is very much lower than in the earlywood, a phenomenon that is related to the smaller diameter and thicker cell wall of the latewood pit.

Thus, in dry timber, in marked contrast to green timber, the permeability of the latewood is at least as high as that of the earlywood and may even exceed it (Fig. 3). Rewetting of the timber causes only a partial reduction in the number of aspirated pits, and it appears that aspiration is mainly irreversible.

Quite apart from the fact that many earlywood pits are aspirated in the heartwood of softwoods, the permeability of the heartwood is usually appreciably lower than that of the sapwood owing to the deposition of encrusting materials over the torus and margo strands and also within the ray cells (Fig. 3).

Permeability varies widely among different species of softwood. Thus, Comstock (1967) found that the ratio of longitudinal-to-tangential permeability varied between 500:1 and 80 000:1. Generally, the pines are much more permeable than the spruces, firs, or Douglas fir.

This can be attributed primarily, though not exclusively, to the markedly different type of semi-bordered pit present between the vertical tracheids and the ray parenchyma in the pines (fenestrate or pinoid type) compared with the spruces, firs, or Douglas fir (piceoid type).

**Hardwoods**: The longitudinal permeability is usually high in the sapwood of hardwoods. This is because these timbers possess vessel elements, 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, relying on the presence of pits interconnecting adjacent vessels, fibres and vertical parenchyma; however, intervascular pits in sycamore have been shown to provide considerable resistance to flow (Petty, 1981).

Transverse flow rates are usually much lower than in the softwoods, but somewhat surprisingly a good correlation exists between tangential and radial permeability; this is owing, in part, to the very low permeability of the rays in hardwoods. Since the effects of bordered pit aspiration, so dominant in controlling the permeability of softwoods, are absent in hardwoods, the influence of drying on the level of permeability in hardwoods is very much less than is the case with softwoods.

Permeability is highest in the outer sapwood, decreasing inwards and reducing markedly with the onset of heartwood formation as the cells become blocked either by the deposition of gums or resins or, as happens in certain timbers, by the ingrowth into the vessels of cell-wall material of neighbouring cells, a process known as the formation of tyloses. Permeability varies widely among different species of hardwood. This variability is due in large measure to the wide variation in vessel diameter that occurs among hardwood species.

Thus, the ring-porous hardwoods, which are characterised as having earlywood vessels of large diameter, generally have much higher permeabilities than the diffuse-porous timbers, which have vessels of considerably smaller diameter; however, in those ring–porous timbers that develop tyloses (e.g. the white oaks) their heartwood permeability may be lower than that of the heartwood of diffuse-porous timbers. Inter-specific variability in permeability also reflects the different types of pitting on the end walls of the vessel elements.

**Timber and the Laws of Flow**

The application of Darcy’s law to the permeability of timber is based on a number of assumptions not all of which are upheld in practice. Among the more important are that timber is a homogeneous porous material and that flow is always viscous and linear; neither of these assumptions is strictly valid, but the Darcy law remains a useful tool with which to describe flow in timber.

By de-aeration and filtration of their liquid, many workers have been able to achieve steady-state flow, the rate of which is inversely related to the viscosity of the liquid, and to find that in very general terms Darcy’s law is upheld in timber (see, e.g., Comstock, 1967). Gas, because of its lower viscosity and the ease with which steady flow rates can be obtained, is a most attractive fluid for permeability studies.

However, at low mean gas pressures, owing to the presence of slip flow, deviations from Darcy’s law have been observed by a number of investigators. At higher mean gas pressures, however, an approximately linear relationship between conductivity and mean pressure is expected and this, too, has been observed experimentally.

However, at even higher mean gas pressures, flow rate is sometimes less than proportional to the applied pressure differential owing, it is thought, to the onset of non-linear flow. Darcy’s law may thus appear to be valid only in the middle range of mean gas pressures.

## Moisture Diffusion

Flow of water below the fibre saturation point embraces both the diffusion of water vapour through the void structure comprising the cell cavities and pit membrane pores and the diffusion of bound water through the cell walls (Fig. 1).

In passing, it should be noted that because of the capillary structure of timber, vapour pressures are set up and vapour can pass through the timber both above and below the fibre saturation point; however, the flow of vapour is usually regarded as being of secondary importance to that of both bound and free water.

Moisture diffusion is another manifestation of flow, conforming with the general relationship between flux and pressure. Thus, it is possible to express diffusion of moisture in timber at a fixed temperature in terms of Fick’s first law, which states that the flux of moisture diffusion is directly proportional to the gradient of moisture concentration.

As such, it is analogous to Darcy’s law on the flow of fluids through porous media. The total flux F of moisture diffusion through a plane surface under isothermal conditions is given by:

where dm/dt is the flux (rate of mass transfer per unit area), dc/dx is the gradient of moisture concentration (mass per unit volume) in the x direction, and D is the moisture diffusion coefficient, which is expressed in m^{2}/s (Siau, 1984; Skaar, 1988).

Under steady-state conditions the diffusion coefficent is given by:

where m is the mass of water transported in time t, A is the cross-sectional area, L is the length of the wood sample, and ∆M is the moisture content difference driving the diffusion.

The vapour component of the total flux is usually much less than that for the bound water. The rate of diffusion of water vapour through timber at moisture contents below the fibre saturation point has been shown to yield coefficients similar to those for the diffusion of carbon dioxide, provided that corrections are made for differences in molecular weight of the gases.

This means that water vapour must follow the same pathway through timber as does carbon dioxide, and implies that diffusion of water vapour through the cell walls is negligible in comparison to that through the cell cavities and pits (Tarkow and Stamm, 1960). Diffusion of bound water occurs when water molecules bound to their sorption sites by hydrogen bonding receive energy in excess of the bonding energy, thereby allowing them to move to new sites.

At any one time the number of molecules with excess energy is proportional to the vapour pressure of the water in the timber at that moisture content and temperature. The rate of diffusion is proportional to the concentration gradient of the migrating molecules, which in turn is proportional to the vapour pressure gradient.

The most important factors affecting the diffusion coefficient of water in timber are temperature, moisture content and density of the timber. Thus, Stamm (1959) showed that the bound-water diffusion coefficient of the cell-wall substance increases with temperature approximately in proportion to the increase in the saturated vapour pressure of water, and increases exponentially with increasing moisture content at constant temperature.

The diffusion coefficient has also been shown to decrease with increasing density and to differ according to the method of determination at high moisture contents. It is also dependent on grain direction; the ratio of longitudinal to transverse coefficients is approximately 2.5:1.

Various alternative ways of expressing the potential that drives moisture through wood have been proposed. These include percentage moisture content, relative vapour pressure, osmotic pressure, chemical potential, capillary pressure and spreading pressure, the last mentioned being a surface phenomenon derivable from the surface sorption theory of Dent, which in turn is a modification of the Brunauer– Emmet–Teller (BET) sorption theory (Skaar & Babiak, 1982).

Although all this work has led to much debate on the correct flow potential, it has no effect on the calculation of flow; moisture flow is the same irrespective of the potential used, provided the mathematical conversions between transport coefficients, potentials and capacity factors are carried out correctly (Skaar, 1988).

As with the use of the Darcy equation for permeability, so with the application of Fick’s law for diffusion there appears to be a number of cases in which the law is not upheld and the model fails to describe the experimental data. Claesson (1997) in describing some of the failures of Fickian models claims that this is due to a complicated, but transient sorption in the cell wall; it certainly cannot be explained by high resistance to flow of surface moisture.

The diffusion of moisture through wood has considerable practical significance since it relates to the drying of wood below the fibre saturation point, the day-to-day movement of wood through diurnal and seasonal changes in climate, and in the quantification of the rate of transfer of vapour through a thin sheet such as the sheathing used in timber-frame construction.

What is popularly called ‘vapour permeability’ of a thin sheet, but is really vapour diffusion, is determined using the wet-cup test, and its reciprocal is now quantified in terms of the water vapour resistance factor (µ). Values of µ for timber range from 30 to 50, while for wood-based panels, µ ranges from 15 for particleboard to 130 for oriented strand board (see also Dinwoodie, 2000).

## Thermal Conductivity in Timber

The basic law for flow of thermal energy is ascribed to Fourier, and when described mathematically it is:

K_{h} = HL ÷ tAΔT

where K_{h} is the thermal conductivity for steady state flow of heat through a slab of material, H is the quantity of heat, t is time, A is the cross-sectional area, L is the length, and ΔT is the temperature differential. This equation is analogous to that of Darcy for fluid flow.

Compared with permeability, where the Darcy equation was shown to be only partially valid for timber, thermal flow is explained adequately by the Fourier equation, provided the boundary conditions are defined clearly. Thermal conductivity will increase slightly with increased moisture content, especially when calculated on a volume-fraction-of-cell-wall basis.

However, it appears that conductivity of the cell-wall substance is independent of moisture content (Siau, 1984); at 12% moisture content, the average thermal conductivity of softwood timber parallel to the grain is of the order of 0.38 W/mK.

Conductivity is influenced considerably by the density of the timber, i.e. by the volume-fraction-of-cell-wall substance, and various empirical and linear relations between conductivity and density have been established. Conductivity will also vary with timber orientation owing to its anisotropic structure; the longitudinal thermal conductivity is about 2.5 times the transverse conductivity. Values of thermal conductivity are given in Siau (1984) and Dinwoodie (2000).

Compared with metals, the thermal conductivity of timber is extremely low, although it is generally up to eight times higher than that of insulating materials. The average transverse value for softwood timber (0.15 W/mK) is about one seventh that for brick, thereby explaining the lower heating requirements of timber houses compared with the traditional brick house. Thermal insulation materials in the UK are usually rated by their U-value, where U is the conductance or the reciprocal of the thermal resistance. Thus:

U-value = K_{h}/L

where K_{h} is the thermal conductivity and L is the thickness of the material.