Question

A wastewater treatment plant has a rupture in one of their flow tubes, thereby leading to the discharge of waste material into a local lake system. The rupture goes unnoticed for a long enough time that the concentration of the waste in the lake reaches a steady state. Assume that the lake can be modeled as a continuously stirred tank reactor, with inflow, outflow, and complete mixing, and that the lake flows so that the volume remains constant at 107 m³. Pollutant-free water enters the lake at a volumetric flowrate of 45 m³/s. The waste from the wastewater treatment plant (that flows into the lake) has a waste concentration of 120 mg/L and a volumetric flowrate of 6 m³/s. Thus, there are two inlets to the lake, one with unpolluted water and the other with polluted water. The waste naturally decays according to the rate relationship, k*C_waste , where k = 0.15 day^-1. Determine the steady state concentration of waste in the lake, in units of mg/L (milligrams per liter). Express your answer to one decimal place.

Answer 1

Rate of change in waste concentration in lake = Rate of inflow of waste(both the inputs) - Rate of consumption or decay of waste - Rate of outflow of waste

The equation is given by   

VdC/dt = Qin1*Cin1 + Qin2*Cin2 - V(k*C) - Qout*C

V= volume of the lake = 107 m³

dC/dt = Change of concentration with respect to time

Qin1 = Flow rate of pollutant free water m³/sec

Cin1 = Concentration of pollutant in pollutant free water = 0 mg/L = 0 g/m³

Qin2 = Flow rate of waste water = 6 m³/sec

Cin2 = Concentration of waste in waste water = 120 mg/L = 120 g/m³

Qout = Exit flow rate of water = Qin1 + Qin2 = 45 + 6 = 51 m³/sec

C = concentration of waste in lake at any time t.

k = rate constant = 0.015 day^-1 = 0.015/86400 sec^-1 = 1.736 * 10^-7 sec^-1 .

Now at steady state, dC/dt = 0. and we know that Cin1 = 0. Then above equation becomes,

0 = Qin2*Cin2 - V(k*C) - Qout*C

C = Qin2*Cin2 / ( V*k + Qout) = (6*120)/(107 * 1.736 * 10^-7 + 51 )

C = 14.117 g/m³ = 14.117 mg/L

In one decimal place,

Concentration of waste at steady state = 14.1 mg/L