Appendix: Capillary Transport in a Tube

When two dissimilar materials, such as two immiscible fluids or a fluid and a solid, are brought into contact with each other, surface tension forces arise due to the energy needed to form an interface. At the point where a fluid-fluid interface meets a solid, a contact angle, θ, is defined by the planes tangent to the fluid-fluid interface and the fluid-solid interface (Figure 9). When the surface tension forces are balanced a static contact angle [16] is obtained such that Young's Equation is satisfied:

FIG. 9. A static contact angle θ, defined where the fluid/fluid interface meets a solid surface, is obtained as the results of the balance of surface tension forces.

γ_bs − γ_as = γ_ab^{cos θ} (3)

where γ_ab is the surface tension between the two fluids, γ _as is the surface tension between fluid a and solid, and γ_bs is the surface tension between the fluid b and solid. Surface tension forces can also create a pressure drop Δ P across the interface between two immiscible fluids. This pressure drop is described by Laplace's law:

Δ P = γ κ. (4)

where γ is the surface tension and κ is the average curvature of the interface. In order to pin an interface, which approximately forms a spherical cap, in a tube of radius r, it can be shown that the following pressure difference must be applied [16]

Δ P = 2γ cosθ / r. (5)

Otherwise, capillary transport results from the unbalanced surface tension forces between the fluid-fluid and fluid-solid interfaces.

In general, the average velocity, V, of the fluid, in a porous medium, may be obtained from Darcy's law

$$V = -\frac{K \Delta P}{l\mu}$$ (6)

where K is the permeability, µ is the viscosity, and l is the length over which the pressure drop ΔP is measured. For a capillary tube K = r² / 8 so that

$$V = \frac{r}{4 \mu l} \gamma cos\theta.$$ (7)

where we ignore gravity (assuming a horizontal tube) and effects due to the details of the contact line motion. Since V scales as the tube radius, porous media with smaller pores will absorb a liquid more slowly. Integrating Equation 5, where we take V = dl / dt, we obtain

$$l=\sqrt{\frac{r \gamma cos\theta t}{2 \mu}} \equiv St^{1/2}$$ (8)

where S is the sorptivity.

For the case of a vertical tube in a gravitational field the meniscus will reach an equilibrium height, h_∞, given by h_∞ = 2 γ _abcosθ / (r ρ g) where g is the gravitational constant. For a tube of radius 1 micrometer (a typical size of the larger capillary pores in many concretes), and γ_ab ≈ 7.2x10⁻² N/m for a water/air interface at room temperature [16], and assuming water is perfectly wetting ( cos θ = 0), h ≈ 14 meters. Also a simple analysis based on the the Washburn equation [16] shows, at long times, that

h ≈ h_∞ (1 − exp(− α t )) (9)

where α = r ²g / (8µ h_∞).

If one assumes a straight parallel tube model of porous media, then a similar approach may be applied to estimate the water sorption. Here, the permeability of the tubes are replaced with the permeability of the porous medium. Therefore, at early stages, the moisture absorbed should be described by

W / A = St^1/2 + S_o (10)

where W is the volume of water absorbed, A is the sample surface area exposed to water, S is the sorptivity and S_o is a correction term added to account for surface effects at the time the specimen is placed in contact with the water.

Of course, a random porous material like concrete cannot be accurately modeled as a collection of tubes. In addition, the above theory of capillary suction applies to the case where the porous medium is initially dry. Clearly, the rate of sorption will depend on the degree of saturation of the porous medium. A more general macroscopic description of capillary suction is given by the following equation [5]:

$$\frac{d\theta}{dt}=\nabla D \nabla \theta$$ (11)

where the water content, θ, is the volume fraction saturated at time t, and D is the capillary diffusivity. Note D must depend in a complicated way on the pore structure and capillary pressure, which in turn depends on the water content. A simple empirical form, commonly used to describe moisture transport in soils and construction materials is

D ( θ ) = D₀exp(B θ_r ) (12)

where θ_r = ( θ − θ₀) / ( θ₁ − θ₀), θ₀ is the initial saturation, θ₁ is the saturation at the fluid-solid interface, and B and D₀ are empirically determined constants.