Anaerobic digestion is used to reduce the secondary sludge volume. It is known t

Question

Anaerobic digestion is used to reduce the secondary sludge volume. It is known that for every 1 kg of secondary sludge decomposition in anaerobic condition, roughly another 0.1 kg of new biomass sludge, which is stable and more difficult to decompose, is formed. Calculate how many days, cultivation it would take to reduce the sludge volume by 50%. Please neglect the inoculum biomass for digester. Other information is listed as follows k for secondary sludge: 0.2 day-1 Initial VS in secondary sludge (So): 50 kg/m3 Biomass yield for sludge decomposition: 0.1 kg/kg First-order kinetics (e 2.718):C-So-e-

Explanation / Answer

S(t) the final sludge volume, that is the 50% of the initial volume, and per day 0.1 kg/kg new biomass sludge is generated. Then the sludge volume reduced by 50% means that 25 kg/m3, (So/2) which includes 2.5 kg (25 * 0.1 =2.5) of new sludge biomass. Then, the final sludge volume will be 25+2.5 = 27.5 kg/m3    

S(t) = S0 (1-e-kt)

27.5= 50 (1-2.718-0.2t)

0.55-1 = - e-kt

0.45 = e-kt

-kt= ln 0.45

-kt = -0.798

t = 0.798/0.2

t = 3.99 days required to reduce the sludge volume by 50%

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