Question

Consider a 20 * 10⁵ m³ lake fed by a polluted stream having a flow rate of 4.2 m³/s and pollutant concentration equal to 25 mg/L. There is also a sewage outfall that discharges 0.5 m³/s of wastewater having a pollutant concentration of 275 mg/L. The stream and sewage wastes have a second order decay rate of 0.32/day. Assuming the pollutant is completely mixed in the lake and assuming no evaporation or other water losses or gains, find the steady-state pollutant concentration in the lake. Derive the equation from the mass balance on C in the lake for full credit. Note that the quadratic equation will be useful for solving this problem.

Answer 1

Concentration of pollutant leaving the lake = C

Input rate of pollutant = Qs Cs + Qw Cw

= (4.2 m3/s x 25 mg/L + 0.5 m3/s x 275 mg/L) * 1000 L/m3

outlet rate of pollutant = (4.2+0.5) m3/s x C mg/L x 1000L/m3

decay rate = (0.32/day) x (1day/86400s) x (C mg/L) x (20 x 10^5 m3) x (10^3L/m3)

Input rate of pollutant = outlet rate of pollutant + decay rate

Qs Cs + Qw Cw = QmC + KCV

(4.2 m3/s x 25 mg/L + 0.5 m3/s x 275 mg/L) * 1000 L/m3 = [(4.2+0.5) m3/s x C mg/L x 1000L/m3] + [(0.32/day) x (1day/86400s) x (C mg/L) x (20 x 10^5 m3) x (10^3L/m3)]

242500 mg/s = 4700 C + 7407.407 C

C = 20.03 mg/L