The Streeter-Phelps model can be used to compute the dissolved oxygen concentrat

Question

The Streeter-Phelps model can be used to compute the dissolved oxygen concentration in a river below point discharge (Fid.1) o = o_s - k_d L_o/k_d + k_s - k_e (e^-k_a t - e^-(k_d + k_s)t) - S_h/k_a (1 - e^-k_a t) Eq(1) where o = dissolved oxygen concentration (mg/L), o_s = oxygen saturation concentration (mg/L), t = travel time (d), L_o = biochemical oxygen demand (BOD) concentration at the mixing point (mg/L), k_d = rate of decomposition of BOD (d^-1), k_s = rate of settling of BOD (d^-1), k_d = reacration rate (d^-1), and S_b = sediment oxygen demand (mg/L/d), As indicated in Fig. 1, Eq. (1) produces an oxygen "sag" that reaches a critical minimum a level o_e some time t_e below the point discharge. This point is called "critical" because it represents the location where biota that depend on oxygen (like fish) would be the most stressed. Given that o_s = 10 mg/L kd = 0.2 d^-1 k_a = 0.8 d^-1 k_s = 0.06 d^-1 L_e = 50 mg/L S_b = 1 mg/Ld a) Write a VBA function (Do) for Eq (1) and determine the critical time and concentration using solver b) Using Newton's method, determine the critical travel time and concentration by writing a VBA function for the first derivative (DOprime) and second derivative (DOdoubleprime) of Eq (1). Use an initial time of t = 0 day and perform iterations until epsilon_a

Explanation / Answer

0=10-(0.2*50)/(0.2d-1 +0.06d-1 -0.8d-1)(e-(0.8/d)*(0)-e-(0.2/d+0.06/d)*(0))-1/0.8d-1(1-e-(0.8/d)*(0))

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