## 2 Theory of Permeability  
The interrelationship between porosity, permeability, and particle surface area was first established by [PERSON] [9], who investigated the laminar flow of water through soil using a simple model in which a bundle of parallel circular channels is contained inside a solid block. He pointed out that owing to the tortuous character of the flow, the actual length of the equivalent channels (\(l^{\prime}\)) should be greater than the depth of the soil (\(l\)), as schematically demonstrated in Figure 1.  
The velocity (\(v\)) distribution of the viscous fluid in a steady axisymmetric flow in a single channel is shown schematically in Figure 2.  
The velocity profile can be determined by summing up the force resulting from the pressure drop between the entrance and the exit of the channel and the internal frictional force. Thus,  
\[\Delta\mathrm{p}\gamma\mathrm{\pi}^{2}+\tau 2\mathrm{\pi rl}=0 \tag{1}\]  
where \(\Delta\mathrm{p}\) is the pressure drop along the axis at a distance \(l\), and \(\tau\) is the shear stress in the flowing fluid. Figure 1: [PERSON]’s model of a permeable body. Figure 2: Schematic of a velocity distribution in a circular channel. In a Newtonian fluid, the viscous stress is related to the strain rate by [PERSON]'s law of viscosity, which for cylindrical coordinates becomes  
\[\tau=\mu\frac{\mathrm{dv}}{\mathrm{dr}} \tag{2}\]  
where \(\mu\) is the viscosity of the fluid. Combining Equations (1) and (2), and integrating with the assumption that the layer associated with the channel wall is immobile,  
\[\int_{0}^{\
u}\mathrm{dv}=-\frac{\Delta\mathrm{p}}{2\mu\mathrm{l}}\int_{\mathrm{ R}}^{\tau}\mathrm{rdr}\]  
gives the following velocity profile equation:  
\[\
u=-\frac{\Delta\mathrm{p}}{4\mu\mathrm{l}}\left(\mathrm{R}^{2}-\mathrm{r}^{ 2}\right) \tag{3}\]  
The volumetric rate of fluid flow through an annulus of radii \(r\) and \(r+dr\) (see Figure 2) is  
\[\mathrm{d}\mathrm{q}_{\
u}=\
u 2\pi\mathrm{rdr} \tag{4}\]  
The total volumetric flow rate through the channel is then obtained by inserting (3) into (4) and integrating Equation (4):  
\[\mathrm{q}_{\
u}=\frac{\pi\Delta\mathrm{p}}{2\mu\mathrm{l}}\int_{0}^{\mathrm{ R}}\mathrm{r}\left(\mathrm{R}^{2}-\mathrm{r}^{2}\right)\mathrm{dr}=\frac{\pi \Delta\mathrm{p}}{8\mu\mathrm{l}}\mathrm{R}^{4} \tag{5}\]  
Conversion of the total volumetric flow rate to the average fluid velocity (\(v_{av}\)) in the channel is accomplished by [PERSON]'s equation:  
Figure 2.0  
where \(D\) is the channel diameter. The volumetric flow rate through the permeable body (see Figure 1) can be directly related to the volumetric flow rate through all channels by the following equation:  
Figure 2.1  
where \(v_{b}\) is the linear velocity of fluid flow through the permeable body; \(A\) is the area of cross-section of the permeable body normal to the direction of flow (see Figure 1); and \(\theta\) is the fractional porosity of the permeable body. Assuming that their actual path length is through the permeable body \(l^{\prime}>l\), as shown in the bottom part of Figure 1, Equation (7) becomes  
Figure 2.2  
The apparent volume of any porous body is the sum of the pore volume (\(V_{p}\)) and the volume of the solid material (\(V_{s}\)). Since \(V_{s}=Al(1-\theta)\), and \(V_{p}=Al\theta\), therefore  
Figure 2.3  
where \(S_{V}\) is the specific surface area of the permeable body. Converting [PERSON]'s equation into the form  
Figure 2.4  
and adopting the values of \(v_{av}\) and \(D\) from Equations (8) and (11), respectively, Equation (6) may be re-written as follows:  
Figure 2.5  
Inserting \(K^{\prime\prime}=2l^{\prime}/l\), it takes the form of the Kozeny-Carman equation:  
Figure 2.6  
For permeable bodies composed of compressed fine particles, \(K^{\prime\prime}\) depends in a complex manner on porosity, particle shape, and other factors. Experimental data acquired from different packed powder beds indicate, however, that \(K^{\prime\prime}\) is most often close to 5 [10,11].