A biofilm with a thickness Lf (cm), grows on the surface of a solid (Fig. P24.14). After

Chapter 24, Problem 24.14

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A biofilm with a thickness Lf (cm), grows on the surface of a solid (Fig. P24.14). After traversing a diffusion layer of thickness L (cm), a chemical compound A diffuses into the biofilm where it is subject to an irreversible firstorder reaction that converts it to a product B. Steady-state mass balances can be used to derive the following ordinary differential equations for compound A: D d2ca dx2 = 0 0 x < L Df d2ca dx2 kca = 0 L x < L + L f where D = the diffusion coefficient in the diffusion layer = 0.8 cm2 /d, Df = the diffusion coefficient in the biofilm = 0.64 cm2 /d, and k = the first-order rate for the conversion of A to B = 0.1/d. The following boundary conditions hold: ca = ca0 at x = 0 dca dx = 0 at x = L + L f where ca0 = the concentration of A in the bulk liquid = 100 mol/L. Use the finite-difference method to compute the steady-state distribution of A from x = 0 to L + Lf , where L = 0.008 cm and Lf = 0.004 cm. Employ centered finite differences with x = 0.001 cm.