OK, I wouldn’t do this if I wasn’t at my wit’s end, and I can already hear you all screaming about not doing other peoples homework but I really really really need a hand here so,
PLEASE help me with my homework!!
A sewage treatment plant has a concrete holding tank of one million litres capacity. Initially the tank contains 750 000 litres of sewage which consists of 60 000kg of organic material suspended in 690 000kg water.
Fresh sewage, with a concentration of suspended organic material of 2 wt%, runs into the tank at 30 000 litres per hour, and sewage leaves the tank at 20 000 litres per hour. The bottom of the tank is covered with a sludge of 40 000kg of precipitated organic material. The stirring caused by the liquid flow causes this precipitate to go into suspension at a rate proportional to the difference between the saturation concentration of 0.10 kg of suspended organic material per kg sewage and the actual
concentration of the suspended organic material in the tank. When there is no suspended organic material the precipitated sludge goes into suspension at a rate of 0.0005kg per minute per litre of solution.
(Hint: Use the second sentence in the previous paragraph to write an algebraic relationship between the rate of re-suspension and the concentration of suspended material. You will see that there is a missing parameter, but the last sentence in the paragraph enables you to evaluate this unknown parameter.)
Provide a material balance solution to this problem.
here’s that in pictures:
OK, so you can see there’s 6 unknowns: the masses of sludge, dissolved solids, and water in the tanks after t=0, and the flowrates out of the tank and (here’s the key) from the solid sludge to the dissolved sludge. I’ve solved everything except for that flowrate:
dMwt/dt = 29400 - (Mwt * 20000/(Mwt+Mot))
dMot/dt = (600 + Fst) - (Mot * 20000/(Mwt+Mot))
dMst/dt = 0 - Fst
Fst is proportional to 0.1 of the total mass of water and dissolved sludge in the tank, minus the mass fraction of dissolved sludge in the tank at any given time. Expanding out,
Fst = k (0.1*(Mot+Mwt) - Mot/(Mot+Mwt))
I need to solve for k in order to express Fst in a relationship involving mass rather than flows, so that I can solve all the equations simultaneously in MATLAB. I just can’t figure out how :?. I know it’s got something to do with Fst being 0.03 kg/hr (0.0005kg/min) when no sludge is dissolved in the tank, but the system is already past this point. So, anyone?
Oh God help me!!!