The fluid flow rate through a rigid porous medium is described by d'Arcy's Law, essentially a classical phenomenological relationship between force and flux. Because the medium is rigid it is dimensionally determinable and can be almost likened to a collection of capillary tubes of length t. Darcy's Law is generally expressed as

 (1)    F  =  kDP/tm

where m is the fluid viscosity and F, the flux term, is the volume flow rate across a rigid porous medium of thickness t, under a pressure head of DP, the force term. The phenomenological constant k is termed the permeability. Essentially, the flow rate depends basically on two medium characteristics, pore size and number, and on two fluid flow characteristics, fluid viscosity and pressure differential, all of which determines the nature of the flow across the fabric, whether laminar or turbulent flow. Ostensibly, any of these quantities should be readily measurable: pore size and number by photographic and weighing means; viscosity and pressure differential by several conventional viscosity determinators and transducers. Because a fabric is not a rigid porous medium however, the successful application of Darcy's Law to textiles becomes problematic.

In the case of woven and non-woven textile fabrics' fluid flow, it is the hydraulic pore size and the dynamic viscosity that is required, not their static values. As an additional complication, not only does yarn filaments penetrate into the pores, but the pore shapes themselves can alter under flow conditions, both significantly affecting the fluid flow streams. Worse, the thickness t is not only variable under flow conditions, but intrinsically the thickness of a fabric is difficult to express and almost impossible to rigorously delineate. Accordingly, an alternative phenomenological relationship can be expressed in which the thickness term is eliminated, yielding

(2)     F  =  P_LDP/m

The phenomenological constant P_L is termed the laminar permittivity. To measure textile permittivity several testing methods are relied upon. Figure 1.1 shows the simplest type now in service.  The fabric is classified by the water flow required to achieve a pressure difference of 50 mm.

Alternatively, the fabric to be tested is clamped to the base of the vertical hollow cylinder of the permittivity testing machine, as shown in Figure 1.2. As specified by ASTM D 4491-96, the fabric can be subject to either a Constant Head Test or a Falling Head Test.

According to the Constant Head Test, a fixed head of de-aerated fluid, at a fixed temperature, is maintained above the fabric, and the volume of fluid passing through the fabric is measured over a standard period with a stop-watch. The head and flow rate is documented.  The test is repeated, using higher constant heads until sufficient head:flow-rate data-point pairs are collected to permit a curve to be plotted.

According to the Falling Head Test, a diminishing head of de-aerated fluid, at a fixed temperature, is maintained above the fabric and the volume of fluid passing through the fabric is measured over a series of standard periods with a stop-watch. The head and flow rate is documented. The flow-rate must be slow enough to permit accurate time-measurements to be made. Accordingly, the pressure difference DP across the fabric can be calculated from the head and the flow rate F from the incremental volume-time measurements. The head pressure and the flow rates are then plotted, an example of which is shown in Figure 2.2, and the slope dF/dDP calculated.

The laminar-flow permittivity P_L is the slope of the linear portion of the curve. In practice, the laminar flow rate F of a newtonian fluid of viscosity m is proportional to the pressure difference DP imposed across the porous medium.  Accordingly

(3)     P_L =(dF/dDP)m

Turbulent flow is exhibited at a flow rate sufficiently high that the flow rate ceases to be linearly proportional to the pressure differential. A turbulent permittivity P_T must then be determined. In practice, for a rigid medium, the critical flow conditions at which streamline flow becomes turbulent is discernible: the latter following a second order relationship

(4)     P_T  = (dF²/dDP)m

For a textile fabric, however, the critical point is difficult to discern, if possible at all. In fact, the transition appears to occur over a range of flow rates, and the second order relationship may not be valid at all. All that can be observed is a gradual deviation from linear behavior. The major deficiencies in the ASTM D 4491-96 permittivity testing method is not only the slow tedious manual operation of the apparatus and its means of measurement, but, more importantly, the limited head and flow rates measurable and the minimal control over the testing operation.

The principal deficiency in all such  permittivity testing apparatus, however, is that they do not approximate the actual flow conditions in practice. Generally, in practice, fluid is in full contact with both sides of the fabric however, with the flow from the fluid at higher pressure to the fluid at lower pressure.