Let's solve the **first part**: **Construct a diffusion-reaction model for a spherical floc of radius \( R \), with oxygen diffusion coefficient \( D_{O_2} \) and nitrate diffusion coefficient \( D_{NO_3} \). Derive the critical floc radius \( R_c \) at which the denitrification rate is maximized, assuming zero-order kinetics for nitrate reduction and Monod kinetics for oxygen consumption.** --- ## Step 1: **Setting Up the Diffusion-Reaction Model** For a spherical floc, the steady-state diffusion-reaction equation for a substrate (e.g., oxygen or nitrate) is: \[ \frac{1}{r^2}\frac{d}{dr}\left( r^2 D \frac{dC}{dr} \right) = -r' \] where - \( D \) = diffusion coefficient - \( C \) = substrate concentration - \( r' \) = rate of substrate consumption (reaction rate) --- ## Step 2: **Kinetics Assumptions** - **Zero-order kinetics for nitrate:** \( r'_{NO_3} = k_{NO_3} \) (constant, as long as nitrate is present) - **Monod kinetics for oxygen:** \( r'_{O_2} = \frac{V_{max} C_{O_2}}{K_s + C_{O_2}} \) --- ## Step 3: **Critical Floc Radius \( R_c \)** The critical floc radius is where the **anoxic core** just forms—i.e., oxygen is depleted at the center (\( r = \)), enabling denitrification at maximum rate. Boundary conditions: - At \( r = R \): \( C_{O_2} = C_{O_2,bulk} \) - At \( r = \): \( \frac{dC_{O_2}}{dr} = \) (symmetry) For large flocs, the oxygen profile is such that \( C_{O_2}() = \) at the critical radius. --- ## Step 4: **Solving for \( R_c \) (Zero-Order Approximation)** Assume **zero-order** oxygen consumption (valid when \( C_{O_2} \gg K_s \)): \[ \frac{1}{r^2}\frac{d}{dr}\left( r^2 D_{O_2} \frac{dC_{O_2}}{dr} \right) = -k_{O_2} \] Integrate twice: First integration: \[ r^2 D_{O_2} \frac{dC}{dr} = -\frac{k_{O_2}}{3} r^3 + A \] At \( r = \), \( \frac{dC}{dr} = \Rightarrow A = \): \[ D_{O_2} \frac{dC}{dr} = -\frac{k_{O_2}}{3} r \] Second integration: \[ \frac{dC}{dr} = -\frac{k_{O_2}}{3 D_{O_2}} r \] \[ C(r) = -\frac{k_{O_2}}{6 D_{O_2}} r^2 + B \] Apply boundary condition at \( r = R \): \( C(R) = C_{O_2,bulk} \): \[ C_{O_2,bulk} = -\frac{k_{O_2}}{6 D_{O_2}} R^2 + B \] At \( r = \): \( C() = B \). For critical radius, \( C() = \): \[ = -\frac{k_{O_2}}{6 D_{O_2}} ()^2 + B \Rightarrow B = \] So, \[ C_{O_2,bulk} = -\frac{k_{O_2}}{6 D_{O_2}} R_c^2 \] \[ R_c^2 = \frac{6 D_{O_2} C_{O_2,bulk}}{k_{O_2}} \] \[ \boxed{R_c = \sqrt{\frac{6 D_{O_2} C_{O_2,bulk}}{k_{O_2}}}} \] --- ## **Final Answer** The **critical floc radius** at which denitrification rate is maximized (i.e., when oxygen is just depleted at the center) is: \[ \boxed{R_c = \sqrt{\frac{6 D_{O_2} C_{O_2,bulk}}{k_{O_2}}}} \] Here, \( D_{O_2} \) is the oxygen diffusion coefficient, \( C_{O_2,bulk} \) is the bulk oxygen concentration, and \( k_{O_2} \) is the zero-order oxygen consumption rate. This result assumes zero-order kinetics for oxygen and that nitrate reduction can proceed in the anoxic core.