Equations 11.14 and 11.16 show that the frictional resistance for flow in porous media depends strongly on the porosity, ?. The porosity depends on the shape of the particles and on how they are arranged. Show the following:

(a) If we pack cubes of the same size, one above the other and side by side (rectangular array), the porosity = 0.

(b) If we stack spheres the same way (egg-crate packing), the porosity is 0.476.

(c) If we maintain the rectangular array in any one plane, but shift the planes so that the individual spheres in one plane fall into the holes between the spheres in the plane below (orthorhombic array), the porosity decreases to 0.3954.

(d) If we use the closest packing in each plane, and then place the planes so that they have the closest packing to the planes above and below (rhombohedral array), the porosity further decreases to 0.2595.

These arrays, and data on the porosities of other types of particles, are shown in Brown [13, p. 215].

F = 150 Vs(1 - 2) Ax D8 [laminar flow] (11.14) H 1.75 V Ax 1 - E D [turbulent flow] (11.16)