**5.2. Understanding Liquid Transport**

cant amount of interface pressure in the range from 10 to 50 mmHg, depending on severity of venous disease. The applied pressure by the compression bandage on the padding has two components: first, the pressure that is transferred through padding and finally appears on the skin; second, the pressure loss in the padding structure. This pressure loss can be related to

the energy loss *EL* as described in Eq. (6). So, the pressure results can be expressed as:

**Figure 6.** A typical curve of pressure-thickness data for a nonwoven

*f TlT* = -= - ´ = - ( ) *<sup>T</sup>* ( ) <sup>1</sup> ( ) *<sup>F</sup> P P P P f EL P f EL*

bandage which generates total pressure *PT* on the padding, *r* is the radius of the limb and *Pl*

*F* = ´ t e

the pressure loss that can be related to *EL*. Using a set of experimental results, one can easily obtain the function relating pressure loss and energy loss for a given padding sample. Once expressed, it would be easy to obtain the final pressure at any given applied force (*F*). The measurement of applied force is difficult to judge during wrapping but the applied extension *ε* is easy to measure. The applied force *F* is related to the extension applied in the compression

where, *Pf*

174 Non-woven Fabrics

bandage during wrapping:

*r*

is the final pressure on the surface of skin, *F* is the applied force to the compression

(8)

(7)

is

Liquid transport, that is, wicking, happens due to capillary action that occurs when the fibrous network is completely or partially immersed in a liquid or in contact with a limited amount of liquid from an infinite (unlimited) or limited (finite) reservoir [39]. During transport, the liquid in a nonwoven structure can transmit through the thickness of the sample, that is, transverse wicking, or it can move along the plane of the fabric, that is, longitudinal or in-plane wicking [39, 44-46]. The thickness of padding is small as compared to other dimensions; and therefore, the liquid transmission in the plane of padding is more relevant for the present case. The basic theory in the field of non-homogeneous flows was proposed by Young and Laplace, which is related to the equation of capillarity as [44, 47]:

$$
\Delta P = -\gamma \nabla \hat{n} \tag{9}
$$

where ∆*P* is the pressure difference across the fluid interface, *γ* is the surface tension (or wall tension) and *n* ∧ is the unit normal pointing out of the surface. This describes the capillary pressure difference sustained across a curved interface between two immiscible fluids, such as water and air, due to the phenomenon of surface or wall tension. Lucas and Washburn further extended their work on capillary-driven non-homogeneous flows, which has been frequently used in textile areas [45, 47-49]. The Lucas–Washburn theory relates the rate of fluid flow into a circular tube via capillary action. This theory is a special form of a laminar viscous flow of a Newtonian liquid in a cylindrical type as expressed by Hagen–Poiseuille law [50]:

$$\frac{dV}{dt} = \frac{\pi r^4 \left(p\_s - p\_b\right)}{8l\mu} \tag{10}$$

*dV dt* is the volume flow rate of the Newtonian fluid, *pa* <sup>−</sup> *pb* is the pressure difference between the tube ends, *r* and *L* are the radius and length of the tube, respectively, and *μ* is the viscosity of the fluid. The structure of a nonwoven resulted in complex network of pores in a threedimensional (3D) network. For simplification, a capillary or pore in the network is assumed as a cylindrical tube (radius *r*) in which the distance travelled by the liquid along the capillary axis is *l* (Figure 7a). The capillary pressure (*Pa*) and the hydrostatic pressure (*Pb*) can be expressed as:

$$p\_a = \frac{2\gamma\cos\theta}{r} \tag{11}$$

$$p\_b = l\sigma\varrho\cos\beta\tag{12}$$

where *θ* is the contact angle between the liquid surface and the capillary wall and *β* denotes the angle between the tube and the vertical axis (Figure 7a).

**Figure 7.** a) A tube (*L*) of a radius *r* is suspended in a liquid source. The distance travelled by the liquid along the capillary axis is *l*. (b) Liquid transport on the surface of a textile substrate

Substituting, Eqs. (11) and (12) in Eq. (10) and expressing the volume *V* as *πr* <sup>2</sup> *l*, we can obtain the Lucas–Washburn equation to express the flow rate as:

$$\frac{dl}{dt} = \frac{r\chi\cos\theta}{4\,\mu l} - \frac{r^2 \sigma g \cos\beta}{8\,\mu} \tag{13}$$

The parameters, *r, γ, θ, μ, σ, g and β* remain constant for a given system, so Eq. (5) can be simplified to:

$$\frac{dl}{dt} = \frac{a}{l} - b$$

where a and b are constants. When the penetration of liquid is horizontal (β = 90°), the effects of the gravitation field are negligible and the acceleration g vanishes and therefore the second term (*b*) can be neglected. Finally, the wicking length (*l*) can be solved as [47, 48]:

$$l = \left(\text{2at}\right)^{1/2} \tag{14}$$

The above equation can be used for our case where the liquid from a point source is poured at the centre of a textile substrate and the spreading occurs radially outward in horizontal plane (Figure 7b).
