3. Consider a porous medium composed of continuous, straight, parallel flow channels, whose cross-sectional area varies in a quasi-periodic manner (serial type non-uniformities). See Figure 2. Assume that the flow channels are all cylindrical in shape and that each channel is composed of segments of capillaries of two diameters, d, and d₂. Let N be the number of channels per unit cross-sectional area and L be the macroscopic length of the medium. Let L, and L2 be the total lengths of segments of sizes di and d2, respectively, in a channel (L, + L2 =L). (a) Develop a functional form of the intrinsic permeability of this medium. (b) Compare your result in part (a) to that predicted by assuming that the medium is composed of uniform parallel channels having the same mean hydraulic diameter as the system described above (Carman-Kozeny assumption). (c) To force the Carman-Kozeny model result to agree with that in part (a), what must you assume about the tortuousity of the medium depicted in Figure 2? Explain. (d) Extra Credit (10 points) - On page 259 of Dullien (1992), he presents an expression (eq. 3.3.10) which relates the permeabilities you computed in parts (a) and (b) above: k=kcx (1+y}{1+yx)/(1+yx)(1+y/x*), where x = d/d, and y=Ly/LI. Does his expression agree with your results? Show all work. L2 Li Figure 2 d₂

Fig: 1