Question 12 of 18: Oxygen Utilization in a Complete-Mix Reactor Without Recycle

Question 12: Oxygen Utilization in a Complete-Mix Reactor Without Recycle (8 marks)

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Given. Influent biodegradable soluble COD $S_0=500$ g/m³; effluent soluble COD $S=10$ g/m³; reactor/effluent VSS $X=200$ g/m³ (complete-mix, no recycle ⇒ effluent concentration equals reactor concentration); flow $Q=1{,}000$ m³/d; cell COD-equivalent $1.42$ g O₂/g VSS (from C₅H₄NO₂+5O₂→5CO₂+2H₂O+NH₃).

Find. (a) Observed yield, g VSS produced per g COD removed; (b) oxygen required, g/d.

Approach. In a complete-mix reactor without recycle, the effluent leaves the reactor once through (no return sludge, no separate wasting stream), so all VSS produced exits continuously at the reactor's own concentration $X$. Compute the daily mass of COD removed and VSS produced directly from $Q$, then use a COD mass balance — total COD removed splits between oxygen consumed by respiration and the COD retained in the new cell mass produced — to isolate the oxygen term.

1. Mass of COD removed per day. $$\Delta S = S_0-S = 500-10=490\ \text{g/m}^3$$ $$m_{COD,removed}=490\ \tfrac{\text{g}}{\text{m}^3}\times1{,}000\ \text{m}^3/\text{d} = 490{,}000\ \text{g/d} = 490\ \text{kg/d}$$
2. Mass of VSS (biomass) produced per day. $$m_{VSS}=X\times Q = 200\ \tfrac{\text{g}}{\text{m}^3}\times1{,}000\ \text{m}^3/\text{d}=200{,}000\ \text{g/d}=200\ \text{kg/d}$$
3. (a) Observed yield. $$\boxed{Y_{obs}=\frac{m_{VSS}}{m_{COD,removed}}=\frac{200{,}000}{490{,}000}=0.408\ \text{g VSS/g COD removed}}$$
4. (b) Oxygen required. A COD balance states $\text{COD removed}=\text{O}_2\text{ consumed}+1.42\times\text{VSS produced}$ (the cells themselves retain COD-equivalent oxygen demand): $$m_{O_2}=m_{COD,removed}-1.42\,m_{VSS}=490{,}000-1.42(200{,}000)=490{,}000-284{,}000$$ $$\boxed{m_{O_2}=206{,}000\ \text{g/d}=206\ \text{kg/d}}$$ equivalently, $206{,}000/490{,}000=0.420$ g O₂/g COD removed.