Residence Times and Sewage Treatment

Consider a mathematical model for a more complex physical situation where the fluid streams contain solutes. This situation is common in industrial practice, for example in the treatment of residential sewage. 

Let s analyze the steady-state operation to determine the volumetric flow rates treated and recycle- We will assume that (1) the sewage density is independent of the bacteria concentration or organic matter concentration, (2) the process is at steady state, (3) the aeration tank is well mixed, (4) the flow rate of the inlet air is equal to the flow rate of the outlet air plus CO2, and (5) the growth rate of the bacteria is zero (this is controlled by the flow rate of air to the aeration tank). 

Given that there are two unknowns, treated and (2recyde, you should suspect that you need two independent equations. A mass balance on the entire system is trivial, namely that 5,000 gallons per minute enter and leave the treatment facility. We recognize, however, that in addition to the overall mass balance, we can apply the conservation of mass principle to the components that make up the fluid streams. Thus we can write a mass balance for the bacteria as well as for the total mass. 

Let s choose the system boundaries to be the aeration tank because we have the most information about this unit. We continue to use the symbol F for the rate of total mass input and Q for volumetric flow rates applying the conservation of mass to the aeration tank at steady state yields 

Because we assumed that the densities are equal and we are given that 2in = 5,000 gal/min, 

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